Quantum metrology is one of the most promising applications of quantum technologies. The aim of this research field is the estimation of unknown parameters exploiting quantum resources, whose application can lead to enhanced performances with respect to classical strategies. Several physical quantum systems can be employed to develop quantum sensors, and photonic systems represent ideal probes for a large number of metrological tasks. Here, the authors review the basic concepts behind quantum metrology and then focus on the application of photonic technology for this task, with particular attention to phase estimation. The authors describe the current state of the art in the field in terms of platforms and quantum resources. Furthermore, the authors present the research area of multiparameter quantum metrology, where multiple parameters have to be estimated at the same time. The authors conclude by discussing the current experimental and theoretical challenges and the open questions toward implementation of photonic quantum sensors with quantum-enhanced performances in the presence of noise.

One of the basic pillars of physical science is the measurement process. The goal of a measure is to associate a value with a physical quantity, giving an estimate of it. Together with each experimental estimate, an uncertainty has to be provided, that is, the “parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measured quantity” [definition from ISO (Ref. 1)]. The statistical error, which affects the measurement result, can have different natures: technical or fundamental. The technical one is mostly represented by the accidental error, due to out-of-control imperfections in the measurement process. Conversely, there are fundamental limits on uncertainty, such as those due to Heisenberg relations, that are imposed by physical laws. Quantum mechanics is the most successful, predictive, and fundamental theory describing small scale phenomena. For this reason, a study of the measurement process and the ultimate achievable precision bounds has to be done under the light of such theory.2,3 On the one hand, quantum mechanics imposes fundamental limits on the estimate precision. On the other hand, in order to achieve such limits, quantum resources have to be employed.

Remarkably, the exploitation of quantum systems to estimate unknown parameters overcomes the precision limits that can be, in principle, obtained by using only classical resources. This idea is at the basis of the continuously growing research area of quantum metrology that aims at reaching the ultimate fundamental bounds on estimation precision by exploiting quantum probes.4–12 This field has attracted a large research effort in the last few years, leading to notable progress both theoretically and experimentally as reported in previous review papers concerning theoretical aspects of quantum metrology and phase estimation problems,10,11,13–17 multiparameter estimation scenario,18–20 optical metrology,12,21–30 and metrological tasks performed by different physical systems.15,31–33

Given an unknown parameter to be estimated and m classical probes (with m1), each interacting a single time with the system under study, the estimation error will scale at best as m1/2. This classical limit is a consequence of central limit theorem and is called Standard Quantum Limit (SQL). The term “classical” stands for probes that are at most classically correlated. If quantum probes are allowed, the SQL can be surpassed so that the uncertainty of the estimator reaches the more fundamental scaling m1, improving the precision by a factor m with respect to the SQL. Such new scaling represents the ultimate limit on estimation precision and is called the Heisenberg Limit (HL).

A key role of quantum enhancement in estimation theory is played by entanglement. This quantum phenomena is one of the most striking aspects of quantum mechanics and radically differs from the classical view of the world, as recognized by Einstein et al.34 The relevance of entanglement in the fundamental divergence between classical and quantum predictions was theoretically demonstrated by Bell.35 Realizing his famous test, recent experiments showed, in a nearly unambiguous way, that the observed quantum correlations arising from distant entangled systems cannot be explained by any classical theory obeying local causality.36–39 Quantum correlations also represent an extraordinary resource allowing to enhance performances in informational tasks.40–43 Quantum advantage with respect to the best classical strategies lies at the basis of the research area of quantum technologies, comprising branches such as quantum computation, communication, simulation, and precision metrology.41 

Besides the fundamental interest about ultimate precision limits, quantum metrology presents different applications. Indeed, quantum-enhanced sensitivity can benefit from different research branches, such as measurement on biological systems,28,32,44,45 gravitational waves detection,46 atomic clocks,47–51 interferometry with atomic and molecular matter waves,52–55 plasmonic sensing,56–58 magnetometry,59–79 spectroscopy and frequency measurements,80–88 lithography,89–94 microscopy and imaging,21,25,27,95–134 localization of incoherent point sources,117,135–137 Hamiltonian estimation,138–144 fundamental physics effects,145–159 coordinate transfer, synchronization, and navigation,160–168 absorption measurements,169–173 thermometry,174–183 and general sensing technologies.31,184

Different physical systems can be employed to realize quantum metrology tasks.15,31 The most appropriate quantum systems for several scenarios are photons, due to their properties such as high mobility and low interaction with the environment, together with the available technology for their generation, manipulation, and detection.24,185,186 Hence, the development of platforms and techniques to generate suitable quantum photonic states is of significant importance, able to provide quantum-enhancement in different metrology tasks. In particular, since many physical problems can be mapped into phase estimation processes, interferometry represents one of the most relevant scenarios.8–11 

Furthermore, the unknown parameters to be estimated in a physical problem can be, in general, more than one. Also in this case, quantum resources can enhance the simultaneous estimation of all parameters.18,20 This represents a relatively new research area with different experimental and theoretical open questions, such as the capability of reaching the quantum ultimate bounds in the simultaneous estimation of all parameters.

In this review, we present an overview of quantum metrology through photonic platforms by analyzing recent advances and discussing open problems. In Sec. II, we will briefly review the definitions of the basic quantities, such as Fisher Information and Cramér-Rao bound, that characterize a general estimation process, together with the most relevant theorems. In particular, the treatment is focused on single parameter estimation. In Sec. III we describe photonic platforms for the generation of different quantum states with the corresponding applications in quantum metrology and, in particular, on phase estimation problems. Section IV is devoted to adaptive protocols able to enhance the estimation processes. In Sec. V, we study the generalization of single parameter to multiparameter quantum metrology. We introduce the generalized theoretical framework and then describe the state-of-the-art of experimental photonic realizations of simultaneous multiparameter estimation. In Sec. VI, we conclude by describing the challenges toward genuine quantum enhanced metrology and provide some perspectives on future developments.

In this section, we describe the fundamentals of single parameter estimation, while the problem of estimating more than one parameter is described in Sec. V. In particular, some results of the former scenario do not apply for the latter one.

An estimation process aims at measuring an unknown parameter λ embedded in a physical system. Through the interaction between a probe and the system, information about the parameter is encoded in the probe state. The goal is to extract the information so that the estimation converges to the real value of the parameter. The general scheme can be described through four steps (Fig. 1):

  • preparation of a probe state ρ0, such that it is sensitive to variations of the unknown parameter λ;

  • interaction of the probe with the system through a unitary evolution Uλ depending on λ (for simplicity, we consider only unitary evolution but this can be extended to nonunitary maps). From such interaction, the evolved state ρλ encodes the information on the unknown parameter: ρλ=Uλρ0Uλ;

  • the information is extracted by means of a suitable positive operator valued measure (POVM) Ex;

  • finally, a suitable estimator, based on measurement results x, provides an estimate Λ(x) of the unknown parameter.

Fig. 1.

Conceptual scheme of a parameter estimation. An initial probe is prepared (red box) in a state ρ0 (eventually, from an initial state ρin). Then, it interacts with the unknown parameter λ through an evolution Uλ (green box). The state ρλ encoding the information on λ is measured by a POVM Ex (blue box) generating outcome x. Based on the outcomes x, a suitable estimator provides an estimate Λ(x) of the parameter λ.

Fig. 1.

Conceptual scheme of a parameter estimation. An initial probe is prepared (red box) in a state ρ0 (eventually, from an initial state ρin). Then, it interacts with the unknown parameter λ through an evolution Uλ (green box). The state ρλ encoding the information on λ is measured by a POVM Ex (blue box) generating outcome x. Based on the outcomes x, a suitable estimator provides an estimate Λ(x) of the parameter λ.

Close modal

Repeating this process ν independent times, the final estimator Λ(x) in general depends on the complete sequence of measurement results x=(x1,,xν).

A consistent estimator asymptotically converges to the real value of the parameter. An estimator is said to be unbiased if its mean value coincides with the unknown parameter,

Λ¯=xP(x|λ)Λ(x)=λλ,
(1)

where P(x|λ) represents the conditional output probability of obtaining a sequence of measurement result x, given a certain value of the parameter λ. Such probability is also called likelihood and is given by the Born rule,

P(xi|λ)=Tr(Exiρλ),
(2)

in the case of single measurement result xi. For ν independent measurements, P(x|λ)=i=1νP(xi|λ). Furthermore, a locally unbiased estimator is an unbiased estimator only for certain range of parameter's values, so that it satisfies the relation: Λ¯/λ=1. Finally, an estimator is asymptotically unbiased when it converges to the real value in the limit of infinite number of probes: limmΛ¯=λλ.

In order to quantify the accuracy of an estimation process, the Mean Square Error (MSE) can be defined,

MSE(λ)=x(Λ(x)λ)2P(x|λ).
(3)

For unbiased estimators, the MSE is equal to the variance on the estimate,

Δλ2x(Λ(x)Λ¯)2P(x|λ).
(4)

In general, an estimator Λ(x) can be nondeterministic. In this case, it is possible to take into account the probability Pexp(Λ|x) to generate an estimate Λ, given the experimental outcomes x. More specifically, the following probability distribution has to be considered:

Pest(Λ|λ)=xPexp(Λ|x)P(x|λ).
(5)

This expression provides the probability to obtain an estimate Λ given the parameter λ. In this case, the MSE of the estimation becomes

MSE(λ)=ΛPest(Λ|λ)(Λλ)2.
(6)

Quantum resources can be used both for the estimation of continuous unknown parameters and for parameters assuming discrete values, where the goal is to distinguish between them. This latter case corresponds to the quantum channel discrimination problem.24,187–193

1. Estimators

Different approaches exist to post-process experimental data and provide optimal estimation of the unknown parameter.14 

One of the most widely adopted estimators is the maximum likelihood estimator (MLE).194 It is the value of the parameter that, given a list of experimental results x, maximizes the likelihood probability,

ΛMLE(x)=arg[max{λ}P(x|λ)].
(7)

In the asymptotic limit, the MLE is unbiased, consistent and saturates the Cramer Rao bound (see Sec. II B). Other estimators are Bayesian estimator or the Method of Moments, the latter not requiring full knowledge of the likelihood function.16 

While MLE with the relative estimation bounds (see Sec. II B) is based on a frequentist interpretation of the probability, in the Bayesian approach, the conceptual meaning of probability is that of a degree of belief. In this sense, the Bayesian approach can be exploited as a framework to devise estimation protocols195–198 In this approach, the unknown parameter λ and the experimental result x are treated as random variables. Here, the relevant quantity is the degree of ignorance (or knowledge, equivalently) about the parameter. During a Bayesian estimation, such knowledge, that can be regarded as subjective (degree of belief), is updated according to the measurement results.

The starting point of the process is the prior distributionP(λ) that quantifies the initial ignorance on the unknown parameter. The experimental setup probing the system is described by the likelihood function P(x|λ) [Eq. (2)]. Once a measurement result x is obtained, the degree of knowledge, described by the posterior probabilityP(λ|x), is updated by Bayes' rule,

P(λ|x)=P(λ)P(x|λ)dλP(λ)P(x|λ),
(8)

where the integral in the normalization term has to be replaced by a sum when the unknown parameter λ assumes discrete values. The posterior in Eq. (8) contains the updated information from which interesting quantities can be calculated.

For instance, the mean square error (MSE) [Eq. (3)], of an estimator Λ(x), averaged over the parameter λ is obtained as

MSE(λ)=dλdxP(λ)P(x|λ)(Λ(x)λ)2.
(9)

By minimizing Eq. (9), the optimal Bayesian estimator Λopt(x) is calculated,

Λopt(x)=dλλP(λ|x),
(10)

that corresponds to the mean value of the parameter over the posterior distribution. Also other moments, such as the third moment, of such distribution can be informative on the estimation, especially to detect possible biases.199 

The phase shift ϕ estimated inside an interferometer is a circular parameter, where ϕ=ϕ+2kπ with k. For such a parameter, a circular mean, calculated over the posterior distribution, can be defined,

ϕcirc=arg[dϕeiϕP(ϕ|x)].
(11)

The standard variance with circular variables is no more adequate and the Holevo varianceVH can be defined, as a function of a quantity S called Sharpness200 

VH=S21S=|eiϕ|,
(12)

where the mean · is calculated over the probability distribution of the estimation process under study. The Holevo variance can describe the variance of unbiased phase estimators, VunbiasH=|eiΦ|21, and coincides with the standard variance for sufficiently sharply picked distribution. With biased estimators, the variance is VbiasH=|cos(Φϕ)|21. A Bayesian analysis of the sensitivity of coherent states in an optical interferometer for the estimation of a phase shift has been carried in Ref. 201.

A fundamental feature of a Bayesian approach is its direct application to adaptive protocols, described in Sec. IV. Note that, since a Bayesian approach allows exploiting prior knowledge on the parameters, the sensitivity bounds can be different from those relative to the frequentist approach:202,203 for MLE (frequentist approach), the error is defined by the mean square fluctuations, while in a Bayesian approach the uncertainty is quantified by the variance of the posterior, which is a different concept with respect to MSE.

Let us consider the estimation scenario where probes and measurements are fixed. A fundamental tool allowing to study the achievable bounds on estimation uncertainties is the Fisher Information (F). It is a quantity able to catch the amount of information encoded in output probabilities of the estimation process and is defined as194 

F(λ)=xP(x|λ)(log(P(x|λ))λ)2=x1P(x|λ)(P(x|λ)λ)2.
(13)

Note that all the previous definitions, written for the case of discrete values of measurement outcomes x, can be extended to the continuous case in which the sums are replaced by integrals over x.

We introduce a useful quantity called symmetric logarithmic derivative (SLD), Lλ, defined as the self-adjoint operator satisfying2 

ρλλ=(ρλLλ+Lλρλ)2.
(14)

The Fisher Information is related to the SLD operator through the following relation:

F(λ)=xRe[Tr[ρλExLλ]]2Tr[Exρλ],
(15)

where Ex is a chosen POVM measurement. F has the two following properties:16,204,205

  • Convexity: given a general mixed state ρ=jcjρj with jcj=1 and a fixed measurement, then: F(λ)jcjFj(λ), where F(λ) is the Fisher Information of the state, while Fj(λ) is the Fisher Information calculated for the single state ρj of the mixture;

  • Additivity: given ν independent probes measured independently, the F of the total ensemble will be: Ftot(λ)=iFi(λ), where Fi(λ) is the Fisher Information relative to the ith probe together with its measurement.

Intuitively, F being proportional to the derivative with respect to the parameter of the output probabilities, it allows quantifying the sensitivity of the system to a change of λ. More specifically, a larger amount of information is associated with larger variations of the output probabilities. This intuition was formalized with a fundamental result, called Cramér Rao bound (CRB). It links F to the ultimate bound achievable by the variance of any arbitrary estimator, with fixed ν identical and independent probes and measurements,206,207

Δλ2=x(Λ(x)Λ¯)2P(x|λ)(Λ¯/λ)2νF(λ).
(16)

In the presence of a locally unbiased estimator (Λ¯/λ=1), the CRB becomes

Δλ21νFλ.
(17)

An estimator that is able to saturate the inequality (17) is said to be efficient.

In the limit of a large number of measurements, the maximum likelihood estimator is efficient, since its distribution normally converges to the real value with a variance that saturates the CRB.208 Also, the Bayesian estimator is asymptotically efficient in the limit of large number of probes.16,202,208,209 Nevertheless, in the case of limited measurements and data, the saturation of the bound is no more guaranteed.210 

In Sec. II B, we have considered the scenario in which both probes and measurements are fixed, and we have discussed the ultimate limits optimizing for the best possible estimator. In this section, we review the ultimate precision limits obtained by optimizing over all possible measurements.

Given a state ρλ encoding the information on the parameter λ, one can maximize the Fisher Information F over all possible POVMs Ex. This defines the Quantum Fisher Information (FQ),6 

FQ(ρλ)=max{Ex}F(λ).
(18)

By definition, FQ(λ)F(λ) and the CRB can be extended to the Quantum-CRB (QCRB). For asymptotically locally unbiased estimators, such bound reads

Δλ21νF(λ)1νFQ(λ).
(19)

In other words, having a fixed probe state, the right-hand side of (19) represents the ultimate achievable precision bound regardless of the measurement. The only dependence is on the probe state ρλ.

FQ is related to the symmetric logarithmic derivative operator Lλ. Indeed, it can be demonstrated that the following relation holds:2,3,6

FQ(ρλ)=(ΔLλ)2=Tr[ρλLλ2],
(20)

where (ΔLλ)2 is the variance of Lλ over the state ρλ. If we write the probe state on the basis of its eigenstates ρλ=nan|ΨnΨn|, then FQ can be explicitly written as6 

FQ(ρλ)=n(λan)2an+2ij(aiaj)2ai+aj|Ψi|λΨj|2.
(21)

Analogously to F, FQ satisfies the properties of convexity and additivity,

FQ(jcjρλj)jcjFQ(ρλj),
(22)
FQ(iρλi)=iFQ(ρλi).
(23)

If the evolution of the interaction is unitary, ρλ=eiλHρ0eiλH, or equivalently λρλ=i[ρλ,H], where H is an Hermitian operator, FQ does not depend on the unknown parameter. Then in this case, FQ is function only of the initial state ρ0 and of the generator H. Given ρ0=nbn|ΦnΦn|,FQ's explicit expression is211 

FQ=2ij(bibj)2bi+bj|Φi|H|Φj|2.
(24)

Furthermore, for pure initial states ρ0=|Ψ0Ψ0|, evolving in unitary evolution, a simple expression for FQ can be found,2,3,10,11,211

FQ(ρλ)=4(ΔH)2(hmaxhmin)2,
(25)

where the variance (ΔH)2(HH)2 is calculated on the initial state ρ0 and hmin are the maximum and minimum eigenvalues of H, respectively. For the more general case of mixed probe states, the relation becomes: FQ(ρλ)4(ΔH)2, then a mixed probe state cannot perform better than a pure one (see Sec. II D for the discussion on optimal probes).

The maximum sensitivity of a quantum state for a parameter estimation is intimately related to the metric of the state.6,13,212,213 In particular, the distinguishability of the probe state for small variation of the parameters is directly linked to FQ. The distinguishability between two states, ρ1 and ρ2, can be quantified by the normalized Bures distance: D̃B(ρ1,ρ2)=1F̃(ρ1,ρ2), where F̃(ρ1,ρ2)=Tr[ρ1ρ2ρ1]2 is the standard fidelity. Given the state ρλ and an infinitesimal change δλ of the parameter, the normalized distance squared between ρλ and is proportional to FQ(ρλ),10,214

D̃B(ρλ,ρλ+δλ)2=18FQ(ρλ)(δλ)2.
(26)

From this expression, it is clear that the more ρλ and ρλ+δλ are “distant” (distinguishable) the greater is FQ(ρλ) and thus, the sensibility of the state to λ.

One of the goals of quantum metrology is to find measurements that are able, given a probe state, to reach the ultimate precision and then to saturate the QCRB in Eq. (19). This task is equivalent to find the POVM such that the Fisher Information F associated with the process becomes equal to the corresponding FQ associated with the probe state. The aim is then to find the measurement such that F=FQ. Indeed, if a large number of probes are available, the estimators to asymptotically saturate QCRB are known, such as maximum likelihood and Bayesian estimators. In the single parameter case, it is always possible to saturate the QCRB through suitable measurements.6 Because of FQ's additivity property [Eq. (23)], it is possible to saturate QCRB using local adaptive measurements for each probe without entangling measurements.9,215–219 Then quantum resources in the measurement stage do not enhance the estimation process.8,9 In particular, an optimal choice of POVMs is realized by the projectors over the eigenstates of the symmetric logarithmic derivative Lλ. Furthermore, an explicit expression for the optimal estimator Oλ is10 

Oλ=λ1+LλFQ(λ).
(27)

Since in general the optimal POVMs can depend on its value, it may be necessary to have a priori knowledge on the parameter. This difficulty can be overcome through adaptive estimation protocols (see Sec. IV).

In Sec. II C, we described the optimization of the estimation precision over all possible quantum measurements. The last step, in order to find the ultimate fundamental bounds, is the optimization over all possible input states. This task can be done by optimizing FQ over the initial probes.

For this analysis, we consider the parallel strategy depicted in Fig. 2. Here, m probes interact with the system, independently, with a separable linear unitary Utot=i=1mUλi, with Uλi acting only on the ith probe and such that Uλi=Uλi.

Fig. 2.

Conceptual scheme of a parallel parameter estimation. The measurements here considered are separable. Indeed, empolying entanglement in the measurement process does not allow to obtain better performances than the optimal separable strategy. Conversely, state preparation can lead to quantum enhancement by exploiting entanglement between probes. Described in Ref. 9.

Fig. 2.

Conceptual scheme of a parallel parameter estimation. The measurements here considered are separable. Indeed, empolying entanglement in the measurement process does not allow to obtain better performances than the optimal separable strategy. Conversely, state preparation can lead to quantum enhancement by exploiting entanglement between probes. Described in Ref. 9.

Close modal

In order to describe the relation between the optimal sensitivity and quantum states, we follow the approach presented in Ref. 220, which clarified and distinguished, using the general formalism of mixed states, the role of classical and quantum correlations.

The first property of optimal probes can be derived from the convexity (22) of FQ: the maximum of FQ is always achieved by pure states.

We initially focus on m probes that are classically correlated, that is, nonentangled. The total state can be then written as a convex combination of separable states, each one of the following forms: ρtot=ρ1ρ2,···,ρm. The value of FQ for a separable state is

FQ(ρ1ρ2,···,ρm)=imFQ(ρi)mFQmax,
(28)

where for the first equality the additivity of FQ has been exploited and FQmax represents the maximum of FQ over the states ρm. Then, in the presence of ν independent packets of m classical correlated probes, from Eq. (19), the minimum uncertainty Δλ scales as9,220

Δλ1νmFQmax.
(29)

Since FQmax is a constant factor, the error scaling with the number of the probes m is Δλ1/m. Namely this statistical bound is called Standard Quantum Limit. Such bound corresponds to the QCRB optimized over any arbitrary classically correlated probe state and can be seen as a consequence of the central limit theorem.

In Sec. II C, we have seen that quantum resources in the measurement stage are not necessary to reach the QCRB. Conversely, quantum resources employed for the preparation of probe states can enhance the sensitivity with respect to classical approaches, beating the SQL.4,5,7–9,11,82

In the case of pure states and unitary evolution eiλH,FQ assumes the form of Eq. (25). The bound is saturated by states of the form

|hmax+eiγ|hmin2,
(30)

where |hmax and |hmin are the eigenvectors corresponding to the maximum and minimum eigenvalues hmax and hmin, respectively. Let us define |hS and |hs to be the single probe eigenstate of the generator H relative to the maximum and minimum eigenvalues hS and hs, respectively. Then, if we have m probes, the optimal state in Eq. (30) is realized by9 

|hSm+eiγ|hsm2,
(31)

which is a maximally entangled state. For such state, (ΔH)2=m2(hShs)2/4. Hence, by using Eq. (25), we find that F=m2(hShs)2. Then, for ν independent packets of states in Eq. (31), the QCRB becomes9 

Δλ1νm(hShs).
(32)

The term (hShs) is a constant so, here, the error scaling with the number m of probes is Δλ1/m that corresponds to an improvement of the precision by a factor m with respect to SQL. This enhanced scaling is the ultimate limit on estimation precision and is called Heisenberg Limit (HL). The Heisenberg limit in Eq. (32) is saturable in the limit of large independent repetitions ν of m entangled states (Table I). For the case of a single measurement ν = 1, when a finite amount of a priori information (independent of m) is available, the ultimate limit becomes π/[m(hShs)].198,221,222

Table I.

Table of the relevant metrology quantities, indicating which step of the single parameter estimation protocol is optimized.

QuantityProbe ρ0POVM ExEstimator Λ(x)
MSE(λ) Fixed Fixed Fixed 
F(λ) Fixed Fixed Optimized 
FQ(λ) Fixed Optimized Optimized 
SQL Classically optimized Optimized Optimized 
HL Quantum optimized Optimized Optimized 
QuantityProbe ρ0POVM ExEstimator Λ(x)
MSE(λ) Fixed Fixed Fixed 
F(λ) Fixed Fixed Optimized 
FQ(λ) Fixed Optimized Optimized 
SQL Classically optimized Optimized Optimized 
HL Quantum optimized Optimized Optimized 

A key role to obtain quantum enhancement is played by entanglement. In particular, assuming unitary evolution, the relation

FQ(ρ0,H)>m(hShs)
(33)

is a sufficient condition for the presence of entanglement in the probe state ρ0.220 It turns out that entanglement, in the considered estimation scheme, is necessary in order to have an enhancement in estimation. Equation (33) can be exploited to detect entanglement:17,223–226 if the Fisher Information, that can be extracted for instance from an experimental characterization of the state, satisfies Eq. (33), then the state is entangled. Furthermore, this condition is necessary and sufficient to achieve estimation precision beyond SQL.15,16 This captures the fundamental relation between entanglement and quantum enhanced metrology. However, not all entangled states are able to satisfy inequality (33), then this condition is sufficient but not necessary for the presence of entanglement. Then such a relation defines also the concept of useful entanglement for quantum metrology.220 In particular, in Refs. 12 and 227, the authors investigate the role of mode- and particle-entanglement for quantum-enhanced performances in parameter estimation.

Until now, we have defined the HL scaling for the case of parallel estimation strategies and linear unitary evolutions, in which the Hamiltonian does not generate correlation between different probes. If we consider schemes with nonlinear interactions between probes and system, the scaling can be different.228–231 Furthermore, if one exploits resources other than the number of particles, the SQL can also be beaten with nonentangled probes.9 This is obtained for instance through multiround protocols,232–237 in which the additional employed resource is the running time of the estimation process.

Quantum metrology and in general quantum information tasks can be realized through different physical systems. Among them, photons possess different properties, which make them fundamental in different scenarios.21,24,29,185,186,238–240 Indeed, photons present high mobility and, at the same time, possess very low interaction with the environment (then low decoherence). This makes them the optimal choice for tasks like quantum communication241 or for long distance metrological problems such as coordinate transfer160–162 or large interferometers.46 Furthermore, different degrees of freedom of light can be exploited to encode and extract information. Photonics accomplishes suitable technologies for the generation, manipulation, and detection of quantum states encoded in the various degrees of freedom of light.185,186,238

The quantized electromagnetic field has an associated Hamiltonian of a quantum harmonic oscillator, with total energy,

Hem=kωk(akak+12),
(34)

where k is the wave vector, while k represents the electromagnetic mode that comprises the wave vector, polarization, frequency, time bin, and in general any degree of freedom of the field. The operators ak and ak represent, respectively, the annihilation and creation operators of photons with energy ωk. Such operators obey the following bosonic commutation rules:

[aki,akj]=[aki,akj]=0,[aki,akj]=δij,
(35)

where ki and kj are two modes of the field. The number operatornk along mode k is represented by: nk=akak, and the energy can be written as: Hem=kωk(nk+12).

The eigenstates of the Hamiltonian along mode k are the Fock states, |Nk, having fixed photon-number Nk, and corresponding energy ENk=ωk(Nk+12). The action of annihilation (creation) operators on Fock states is to destroy (create) a photon along mode k, according to the relations

ak|Nk=Nk|Nk1,ak|Nk=Nk+1|Nk+1.
(36)

The number of photons excited in a particular mode is given by the photon-number operator nk,

nk|Nk=akak|Nk=Nk|Nk.
(37)

Since the photon-number operators corresponding to different modes are commuting observables [see relations (35)], and each acts only on the corresponding mode, it is possible to completely describe the whole radiation field, at a fixed number of photons along d modes, by taking the tensor product of the individual states,

|{Nk}=k|Nk=|Nk1|Nk2,,|Nkd|Nk1Nk2,,Nkd.
(38)

Note that, since in this notation a mode comprises all degrees of freedom, the photons along each single mode ki (i=1,...,d) in Eq. (38) are indistinguishable.

The state in which the occupation numbers of all modes are 0 is called vacuum state |{0}|0, defined as the state such that ak|0=0k. We can then generate any Fock state from vacuum by iteratively applying creation operators on the modes,

|Nk=akNkNk!|0.
(39)

In second quantization, the dimensionless position-like and momentum-like operators xk and pk, also called quadratures, can be defined and expressed as a function of annihilation and creation operators,

xk=ak+akpk=i(akak).
(40)

Note that xk and pk are Hermitian operators, and, therefore, represent field observables. The commutation relations follow from (40) and (35),

[xkl,xkj]=[pkl,pkj]=0,[xkl,pkj]=2iδlj.
(41)

Equivalently to Eq. (34), we can write the energy as a function of xk and pk,

Hem=kωk4(pk2+xk2).
(42)

Quadratures are useful to describe photonic states in the phase space; in particular, this formalism provides insights for the study of continuous variable (CV) states (see Sec. III C 4). General quadratures rotated by an angle θ are defined as

xk(θ)=eiθak+eiθak,pk(θ)=i(eiθakeiθak),xk(θ)=cosθxk+sinθpk.
(43)

The electromagnetic field can undergo different evolutions. A common and important class of transformations is represented by the linear and bilinear ones that allow mode operators to evolve through an arbitrary Bogoliubov transformation:242akj(ηkjaj+βkjaj)+γk, where the matrices ηj,βj are related by the so-called Bloch-Messiah reduction for bosons.243 The Hamiltonian evolution of such processes, in a space of d different modes, has the following form:244 

H=k=1dgk(1)ak+k>l=1dgkl(2)akal+kl=1dgkl(3)akal+h.c.,
(44)

where g(i) represent the coefficients of the corresponding evolution terms.

The second evolution term k>lgkl(2)akal+h.c. conserves the photon number and describes linear mixing between modes. Such operations can be implemented by passive optical elements such as beam splitters and phase shifters (PSs) [Eq. (51)]. Conversely, the first and third terms, kgk(1)ak+h.c. and klgkl(3)akal+h.c., describe transformations that do not conserve the total number of photons and are associated with displacement and squeezing operators, respectively (see Sec. III C 4).

Photons represent a fundamental probe for quantum metrology. In particular, a paradigmatic scenario is phase estimation, in which the unknown parameter is a phase shift between different optical modes.

One of the most relevant scenarios for quantum metrology is phase estimation.8,16,245 The problem consists of estimating an unknown phase shift ϕ between two different modes, such as polarization, orbital angular momentum (OAM) or different paths. A lot of physical problems can be cast in a general phase shift estimation, and different physical probes can be employed. Tasks such as measurements of atomic properties,246,247 atomic clocks,48,248 and measurements of forces,249–251 require the use of atomic probes.15 Conversely, for tasks like the estimation of phase shifts produced by gravitational waves,46 lithography,89–91,93 imaging,21,25,27,95–98,100,103,104,107,110,113,119,123–125,132,134,172,252,253 sensing on biological systems,28 quantum key distribution,254 measurements of velocity, displacements, and lengths,11 photons are the most suitable systems. Besides the practical applications, phase estimation represents also a standard benchmark for general metrological protocols.

Consider an estimation of a phase shift between two paths. The transition of a system through a phase shift along a mode, say mode 1, is described by the unitary evolution

Ups=eiϕHps=eiϕa1a1,
(45)

where a1 is the particle annihilation operator along mode 1. The generator and conjugated operator255 of the phase shift is the number operator n1 along the corresponding mode,

Hps=a1a1=n1.
(46)

For the number operator n1, the difference of possible eigenvalues, with a single probe, is hShs=1. Then the SQL, Eq. (29), for phase estimation reads

ΔϕSQL1νm,
(47)

corresponding to the standard quantum limit. Conversely, the HL then reads

ΔϕHL1νm.
(48)

Since a general definition of a standard self-adjoint operator associated with phase shift measurement is problematic, its direct sharp measurement is not possible.256–258 Nevertheless, a phase shift can be treated as an evolution parameter and estimated from other observables whose values depend on it. In particular in optical phase estimation, the phase shift is a difference between optical paths that can be estimated through interferometers. One of the most common and simple two-mode optical interferometers, suitable for phase estimation, is the Mach–Zehnder interferometer (MZI).259,260

The two key elements of a MZI are the phase shifter (PS) and the beam splitter (BS). The former adds a phase shift ϕ between two modes whose annihilation operators are a1 and a2. The beam splitter (BS) represents a basic optical element that allows mixing between two input electromagnetic modes. It can be realized with a partially reflective mirror that transmits or reflects the incoming light. In particular, we consider here the balanced BS whose transmission and reflection probabilities are equal to 0.5.

The action of these elements is described by

PS(ϕ)=(100eiϕ),BS±=12(1±i±i1),
(49)

where BS+ and BS differ of an irrelevant, for our purpose, phase shift. The mode operator bi, generated by a unitary evolution U on modes ak, will be: bi=kUikak. The cascaded combination of these two elements can realize any unitary linear operation in arbitrary dimension.261–263 Such decompositions represent the basis for the realization of universal linear optics circuits.264 

A MZI interferometer is composed of cascaded two BS interspersed with a PS (Fig. 3). In the lossless scenario, up to a global phase, it is described by

MZI(ϕ)=BS+PS(ϕ)BS=(cos(ϕ2)sin(ϕ2)sin(ϕ2)cos(ϕ2)).
(50)
Fig. 3.

Scheme of a Mach–Zehnder interferometer. A MZI is composed of two beam splitters (BS) and a phase shift ϕ between the modes, 1 and 2, of the interferometer.

Fig. 3.

Scheme of a Mach–Zehnder interferometer. A MZI is composed of two beam splitters (BS) and a phase shift ϕ between the modes, 1 and 2, of the interferometer.

Close modal

The first BS can be seen as a preparation step of the estimation process, while the last one as part of the measurement step. In general, the output probabilities of photons exiting from a MZI depend on the phase ϕ. Since the Fisher Information depends on the derivatives of the output probabilities, the probe is more sensitive to a phase shift change for a larger variation of the fringe pattern.

A convenient way to express the MZI operation on electromagnetic modes is based on Pauli matrices expressed through the annihilation operators for modes 1 and 2 (a1 and a2): σx=a1a2+a2a1,σy=i(a1a2a2a1), and σz=a1a1a2a2. The following relations hold:265,266

PS(ϕ)=eιϕσz/2,BS±=e±ιπσx/4,MZI(ϕ)=BS+PS(ϕ)BS=eιϕσy/2.
(51)

Two cascaded independent PSs interspersed by a MZI can realize any unitary belonging to Lie SU(2) group. MZI transformation is used also as an interferometer in other degrees of freedom like polarization, for which the BSs are replaced by half-waveplates (HWPs) rotated by 22.5°.

Besides the applications to quantum metrology and in general to quantum information tasks, a MZI can be also the testbed for foundational tests, like those exploring wave-particle duality of photons267–275 or even quantum gravity phenomena when the probes are massive systems.276–278 

Photons represent the ideal probes for several interferometric tasks and quantum sensing.23,24 Here, we briefly review some of the basic photonic platforms exploited for quantum information and, in particular, metrology tasks. Note that recent and more in-depth reviews on photonic technologies for quantum information can be found in Refs. 185 and 186.

1. Photonic degrees of freedom

Quantum information can be encoded in photonic states by exploiting different degrees of freedom. A first possibility is provided by the polarization, or spin angular momentum (SAM), that can encode two-dimensional quantum states spanned by the orthonormal basis of horizontal and vertical polarizations {|H,|V}.279 The interaction between polarized light and matter allows the manipulation of the polarization through linear optical elements like waveplates. These are birefringent materials that introduce a phase delay between the two orthogonal polarizations. Any unitary transformation in the two-dimensional Hilbert space of polarization can be realized by a suitable sequence of waveplates. Projection in polarization space can be realized through a polarizing beam splitter, that spatially separates orthogonal polarizations.279 Polarization is then a largely used encoding for quantum information protocols and is also often coupled with other degrees of freedom.185 

Besides SAM, angular momentum of light is also composed of another contribution carrying orbital angular momentum (OAM) that is related to light spatial distribution.280,281 SAM and OAM can be considered separately in the paraxial limit and are exploited as independent carriers of quantum states. OAM can be described conveniently by Laguerre–Gauss (LG) modes, carrying, in a single photon regime, a quantized amount of angular momentum m, where m indicates the azimuthal phase structure of the beam. Thanks to the unbounded Hilbert space in which OAM lives, which is spanned by the basis of optical vortexes |LGm, states of arbitrary discrete dimensions (qudits) can be realized.282–287 Generation and manipulation of OAM modes can be performed by means of different devices. Two of the most important techniques are: q-plate (QP) and spatial light modulator (SLM). A QP is an inhomogeneous anisotropic material that, based on light polarization, changes the OAM state: |L|LGmQP|R|LGm+2q and |R|LGmQP|L|LGm2q, where |R (|L) is the right (left) circularly polarized state, and q represents the topological charge of the QP.288–291 Such devices, naturally entangle OAM and polarization degrees of freedom, thus are capable of generating vector-vortex beams.292–296 The latter are a class of states with several applications in quantum information, and very recently, the possibility of fiber propagation has been also demonstrated.297 Conversely, a SLM can induce directly phase and intensity changes on the optical beam in correspondence of each pixel.298,299 An SLM, followed by coupling in single mode fibers, can be exploited to measure different OAM modes.283,300–303 Other methods different from QP and SLM can be exploited to manipulate and measure OAM states, as shown in Refs. 304–310.

Time-bin311–313 and time-frequency314–317 degrees of freedom can be also exploited as quantum resources. For such encodings, photon manipulation can be done through interferometric schemes.185 In parallel, field quadratures, amplitude and phase-squeezing, can suitably encode continuous variable states.33,318,319

In the context of quantum metrology, one of the most widely adopted photonic degrees of freedom for quantum metrology is path encoding. The latter corresponds to a set of spatial modes occupied by the photons.7,8,279 In this framework, the two elements that allow a complete manipulation of single photons in spatial modes are the beam splitter and the phase shifter.279 Through these devices, interferometric setups, that are at the basis of most quantum metrology tasks, can be realized (see Sec. III A).

2. Generation and detection of photons

Different optical platforms are used to generate photonic quantum states, in particular, single or entangled photons. Photon sources based on the spontaneous parametric down conversion (SPDC) process in nonlinear χ2 materials are those more commonly employed.320,321 During this process, a pump photon (momentum kp and frequency ωp) passing through the crystal is annihilated, while a pair of photons is generated: idler (momentum ki and frequency ωi) and signal (momentum ks and frequency ωs). Given |αp the pump coherent state, the process in the Fock basis representation reads |αp|0i|0sapaias|αp|0i|0s|αp|1i|1s, where ax is the particle annihilation operator of photon x with x=p,s,i. SPDC processes must satisfy the following conditions: (i) energy conservation ωp=ωi+ωs and (ii) momentum conservation (phase matching condition) kp=ki+ks. SPDC sources permit generating both heralded single photons and entangled states in the different degrees of freedom.300,321–326

Another process that can be exploited to generate single and entangled photons is spontaneous four-wave-mixing (SFWM) inside χ(3) nonlinear waveguides.327–329 Inside the waveguide, two pump photons (with frequencies ωp1 and ωp2) interact and generate two photons, signal and idler. Given two coherent pumps |αp1 and |αp2, the first order process generating a single pair is: |αp1|αp2|0i|0sap1ap2aias|αp1|αp2|0i|0s|αp1|αp2|1i|1s. The energy and momentum conservation relations are: ωi+ωs=ωp1+ωp2 and kp1+kp2=ki+ks.

Both SPDC and SFWM are probabilistic processes, and the probabilities to generate photon pairs are typically low. In order to achieve a deterministic on-demand generation of single photons, other kinds of sources have to be employed, such as quantum dots,330–334 color centers,335 and others.321 

At the end of a process, photons have to be measured in order to extract the encoded information. A lot of technologies exist to detect single photons.321,336,337 Depending on the photon wavelength, different platforms are employed for detector technology. In the visible range, Si-based avalanche photodiode detectors are commonly used,321 while for the telecom range (around 1310 and 1550 nm), superconductive nanowire single photon detectors reach efficiencies above 0.95.338–340 Transition-edge sensor detectors are suitable for high efficiency photon number resolving detection, in both visible and telecom ranges.341,342 Other detectors are those based on quantum dots343 and up-conversion.344 In order to measure the spatial profile of the photons and realize imaging studies, single-photon sensitive cameras can be used.345–348 

3. Integrated photonic circuits

The realization of linear optical platforms for quantum applications requires a large number of components, especially when dealing with operations on the path degree of freedom. Bulk optical platforms possess limitations on the scalability on the experimental platforms. Complex schemes can require hundreds of optical elements, and thus, the size of the bulk apparatus would be far greater than a standard optical table. In particular, the most severe limitation concerns the stability of the apparatus: without a strict control over temperature, vibrations, and other environmental noise effects, it can be impossible to reach a sufficient accuracy in the phase control.

Then, while simple interferometers involving a small number of modes can be implemented with bulk optics, other platforms are necessary for more complex interferometers. A possible solution to these issues is provided by integrated photonic circuits.185,186,238,264,349–351 Such circuits provide stability, scalability, miniaturization, flexibility, cost reduction, standardization, greater efficiency, and precision for quantum applications with respect to a bulk approach.

In integrated circuits, light is confined in waveguides fabricated inside or on the device materials. Interactions between optical modes are realized through directional couplers. The latter elements are composed of two waveguides brought close together, so that the evanescent fields inside the waveguides overlap and tunneling between the two modes can happen. Also, tunable phase shifts between optical paths can be implemented by appropriate technologies depending on the integrated platform. Indeed, different platforms can be used for the integration of optical circuits each with its own advantages and drawbacks.185,186,349,352

Common and powerful platforms for integrated components are Si-based technologies353,354 such as Silica-on-Silicon353–355 and Silicon-on-Insulator platforms.353,356–358 These devices allows high density circuits thanks to the strong difference of refraction indexes between the substrate and surface, even if polarization qubits are not supported. Furthermore, several additional components, such as electronics elements and fast optical modulators, are compatible with this technology and can be exploited also for high rate quantum key distribution.359–362 Silicon waveguides are naturally suitable also for the generation of single and entangled photons through the SFWM process.363,364

The femtosecond laser waveguide writing (FLW) technique365–367 exploits laser pulses to write waveguides inside glass (but also crystalline or polymeric) material substrates. Such a technique guarantees low cost, low loss (0.10.3 dB/cm), and high speed fabrication and allows for 3D circuit geometries.368,369 Furthermore, thanks to the low birefringence of glass, polarization insensitive waveguides can be realized370 and polarization qubits can be supported and manipulated.371–375 

Among other integrated platforms, there are those based on III–V semiconductors376–380 and UV writing.381,382

The final goal of quantum integrated photonics is the complete simultaneous integration of all steps of a quantum information protocols in a single chip: generation, manipulation, and detection of quantum photonic states. Furthermore, integrated circuits should possibly be able to support all the degrees of freedom of light. During the last few years, much progress, toward this goal, has been made.

Integrated sources of single and entangled photons have been developed for different circuits and different generation processes.363,364,383–390 Efforts for integration of single photon detectors have also been performed.391–394 Regarding integration of different degrees of freedom, path and polarization are within current state of the art as previously discussed. Furthermore, progress has been made toward integration of time395 and OAM.396–398 Finally, it has been demonstrated that quantum states can be converted between different degrees of freedom on a chip.399 

The final target in quantum phase estimation is to reach a genuine enhancement with respect to the classical limit (SQL) when all the employed resources are carefully taken into account. More specifically, the relevant parameter is the total number of photons effectively employed throughout the experiment. In this way, we have to consider in the resource count also those photons that are lost and are not detected by the final measurement stage. Crucial parameters of the experimental setups to reach unconditional violation of SQL are thus the total transmission, the detection efficiency, and the visibility of quantum interference.400 Such parameters depend on the technical details of the particular experimental implementation and of the employed optical elements. Furthermore, photons can be lost during the post-selection process eventually needed for the implementation of the employed scheme. In this case, even with lossless optical elements, an enhanced sensitivity, with respect to classical resources, could be impossible to reach.401 Additionally, we note that attention has to be paid to the accounting of the external phase reference in the study of sensitivity in optical interferometry.402 A complete review on the limits achievable in quantum optical interferometry can be found in Ref. 403.

A less stringent requirement for quantum phase estimation is super-resolution: the achievement of fringes of interference that oscillates faster than any classical state.404 The simple super-resolution of a phase is less demanding than the unconditional violation of SQL, and can be obtained also starting from classical coherent states by applying suitable filtering protocols.234,405 Several photonic realizations of quantum phase estimation experiments demonstrated a super-resolution with schemes that, in principle, allow unconditional super-sensitivity, while experimental imperfections prevented to effectively reach it.

We describe now some important states of light used for phase estimation, and the corresponding schemes for their experimental implementation.

1. Coherent states

Among the most relevant classical states are coherent states. A coherent state |α of a single radiation field mode is the eigenstate of the annihilation operator a

a|α=α|α,
(52)

where α is called displacement. If we expand the state in terms of the complete basis of number states and substitute the expression into the eigenvalues equation, we find the general expression

|α=e(|α|2/2)N=0αNN!|N.
(53)

The coherent state is not an eigenstate of the Hamiltonian in Eq. (34). Hence, the continuous eigenvalues α of a are time-dependent, and their values change following the time evolution of the system |α(t)=eiωt|α. Mean values of quadratures are related to α by: α=(x+ip)/2. Since |α is a linear superposition of states with different numbers of photons, it does not possess a definite number of photons. However, the expectation value and variance of the photon number operator n in such a state can be calculated as

n=α|aa|α=|α|2,
(54)
Δn2=n2n2=α|aaaa|α|α|4=|α|2.
(55)

Hence, the root-mean-squared deviation is Δn=n. This is a typical property of the Poisson distribution. Indeed, it can be shown from Eq. (53) that the photon number probability distribution is

P(N)=|N|α|2=enN=0(|α|2)NN!=enN=0nNN!,
(56)

with mean value and variance equal to n=Δn2=|α|2. The relative value then decreases as Δn/n=1/n. Analogously to the vacuum state, coherent states symmetrically saturate the Heisenberg relation for quadratures. Hence, they are states of minimum uncertainties with Δx=Δp=1.

Coherent states can be also expressed through the displacement operator D(α). Such operator is associated with the first term of the evolution in Eq. (44), being defined by the relation

D(α)=eαaα*a.
(57)

The annihilation and quadrature operators, along mode k and under displacement evolution, are transformed in the following way:

akD(α)ak+α,akD(α)ak+α*,xkD(α)xk+Re[α],pkD(α)pk+Im[α].
(58)

Then, D(α) produces a translation in phase space without changing the uncertainties. Coherent states can be seen as the result of a displacement operation applied to the vacuum state

|α=D(α)|0.
(59)

An experimental approximation of the displacement operation on a target state can be obtained by injecting a coherent state |α, used as ancilla, in an unbalanced beam splitter together with the target.406 

Furthermore, these states describe with good approximation the light emitted from lasers and said to be “semi-classical” for their statistical properties.407 Therefore, in many contests, coherent states represent a suitable classical benchmark to be surpassed in order to certify quantum performances of other states. In this spirit, the statistics of a state is called nonclassical if it cannot be simulated by proper mixtures of coherent ones.408 

When applied to phase estimation problem, coherent states can reach the SQL but not the HL. The scaling of the error on a phase estimation with coherent states is 1/N=1/|α|, consistent with the SQL.

2. N00N states

One of the most important and paradigmatic classes of quantum states with a fixed number of particles, which enables quantum enhanced phase estimation, is represented by the so-called N00N states that are maximally entangled multipartite states distributed along two modes,7,49

|ΨN00N|N,0+eiγ|0,N2,
(60)

where N is the number of particles of the state and γ is the relative phase between the two components of the balanced superposition. The variance of the number operator calculated on |ΨN00N is: (Δn)2=N2/4; hence, from Eq. (25), the sensitivity achievable by N00N states is

ΔϕN00N1N,
(61)

that is, the Heisenberg limit in Eq. (48). For this reason, N00N states play a key role in quantum metrology and in particular in phase estimation processes.7,49,409 In an ideal interferometer, with a relative phase shift ϕ, a N00N state evolves as (|N,0+|0,N)/2Uϕ(|N,0+eiNϕ|0,N)/2, thus acquiring an amplified shift equal to Nϕ. This faster change in the phase shift, proportional to the number of photons, is at the basis of the improved metrological performances of such class of states. N00N states are optimal for local estimation when the unknown phase shift is small. However, such states cannot distinguish phase shifts that differ of π/N without a prior knowledge on the unknown phase shift.7,49,410

For the particular case of N00N states with N = 2, there exists a deterministic generation recipe. This is possible by exploiting indistinguishability of photons which, entering along the inputs of a beam splitter, interfere through the Hong-Ou-Mandel effect (HOM)411 (Fig. 4). Consider two monochromatic photons which share the same degrees of freedom (frequency, polarization, and so on…) apart from spatial modes. Each photon is injected along each of two inputs of a beam splitter, respectively. If the beam splitter is symmetric, the probability to find the photons in different output modes is zero. This is a bunching effect due to purely quantum interference. The final state at the output of symmetric BS will be

|Ψout=i2(|2,0+|0,2),
(62)

that is, a N00N state with N = 2. This kind of effect is due to bosonic quantum interference happening inside a BS. Indistinguishability of photons then leads to entanglement and a BS acts as an entangling gate, even if the two photons are independent and not properly interacting. In general, indistinguishability is considered a quantum resource.412–417 Generation of N = 2 N00N states via Hong-Ou-Mandel interference has been demonstrated with bulk optics configurations in path418–420 and polarization421 degrees of freedom. Polarization two-photon N00N states were also generated using the nonlocal correlation of entangled photons,422 and were exploited for imaging of samples that surpasses SQL, even if without unconditional violation, because of losses and technical imperfections.110 Two-photon polarization N00N states were also used to probe, in a nondestructive way, an atomic spin ensemble exploiting Faraday rotation.423 

Fig. 4.

Scheme of HOM effect with the symmetric beam splitter. Two indistinguishable photons are injected along the two input of a symmetric beam splitter. The final state is a balanced superposition of states in which the two photons are along the same output mode.

Fig. 4.

Scheme of HOM effect with the symmetric beam splitter. Two indistinguishable photons are injected along the two input of a symmetric beam splitter. The final state is a balanced superposition of states in which the two photons are along the same output mode.

Close modal

For numbers of photons N > 2, the generation of N00N states is not straightforward. Indeed, besides the case of N = 2, no scheme is able to generate in a deterministic way a N00N state with arbitrary N. Consider a general state, with fixed photon-number, along two modes

k=0Nck|k,Nk.
(63)

The goal is to avoid all the non-N00N contributions |k,Nk with k0,N. Such undesired terms can be in general discarded through post-selection schemes, thus introducing losses in the generation and/or measurement stages. Hence, most of the protocols, realizing higher photon-number N00N states, rely upon post-selection schemes,424–428 in the same spirit of linear optic quantum computation.262,429 In this way, the desired states are successfully generated, conditioned on the occurrence of specific probabilistic events. This capability has the cost of a nonunitary generation probability.

Conversely, another approach is to employ, as probes, states that are much easier to generate compared to N00N states. For this purpose, even classical states can be employed, given their greater robustness against losses. From such states, one can finally recover N00N typical super-resolution, through suitable state projections that discard all the undesired terms of Eq. (63), so that only a super-sensitive N00N contribution is observed.234,430–435

The first N00N state with N > 2 was realized in the polarization degree of freedom of three photons (N = 3) along a single spatial mode,436,437 in a post-selected configuration. States with N = 3 were also generated through the photon subtraction technique applied on two pairs of photons.438 Another demonstration of N00N super-resolution at N = 3 was obtained through state-projection measurements.431 

The case of N = 4 super-resolving photon states was demonstrated in polarization432 and path434,435,439 degrees of freedom, in four439 and two434,435 mode bulk interferometers. Some of such works432,434,435 studied interference fringes after state-projection measurements on contributions showing super-resolution. Super-resolution with N00N states of up to four photons was also obtained for quantum lithography application by using optical centroid measurement440 that allows reaching higher efficiencies. Indeed, such a technique does not require photons to be detected all at the same point.441,442

A clever way to generate N00N-like states is to let a coherent state interfere with the state generated by a SPDC process260,443 (Fig. 5). Such a scheme was experimentally realized, in the polarization degree of freedom, to perform imaging with N = 2, 3110 and allowed to generate (in a post-selected configuration) N00N states with up to N = 5 photons.440,444,445

Fig. 5.

High photon-number N00N state generation. Conceptual scheme for the generation of N00N state in the path degree of freedom, with high photon-number N. A coherent state |α and the output state |ξ of a SPDC state interfere in a beam splitter (BS) at the input of a Mach–Zehnder interferometer. Finally, after evolution, the photon number of the output ports of the final BS is measured by detectors D.

Fig. 5.

High photon-number N00N state generation. Conceptual scheme for the generation of N00N state in the path degree of freedom, with high photon-number N. A coherent state |α and the output state |ξ of a SPDC state interfere in a beam splitter (BS) at the input of a Mach–Zehnder interferometer. Finally, after evolution, the photon number of the output ports of the final BS is measured by detectors D.

Close modal

States showing super-resolution with photon number N = 6 were demonstrated through a coherent probe state and suitable state-projection measurements along N00N states in a time-reversal configuration.234 The probe state interacting with the system and embedding the phase is a classical one. Hence, even with ideal optical elements, SQL cannot be unconditionally beaten. Such a scheme was also used to perform imaging through a polarization state of N = 2, 3 photons.446 

A class of N00N-like states, where photons are distributed along different modes (for example N photons along N spatial modes maximally entangled in the polarization degree of freedom), are called Greenberger–Horne–Zeilinger (GHZ) states.447 Such states were realized with up to 10 spatially separated photons entangled in the polarization degree of freedom (with an experimentally observed super-resolution up to N = 8),448 with up to 10 qubits encoded in two degrees of freedom (polarization and path) of five spatially separated photons (with a super-resolution shown for N = 8 qubits),449 and with up to 18 qubits encoded in three degrees of freedom (polarization, path and OAM) of six spatially separated photons.450 GHZ states were also exploited for noisy phase estimations. In Ref. 451, each qubit of a four-photon polarization GHZ state is locally encoded in the diagonal basis, in order to improve the robustness of the phase estimation under dephasing noise along the computational basis.452 Similarly, Ref. 453 demonstrated the robustness to transversal noise using up to six-photon GHZ polarization states.

The first unconditional quantum violation of SQL, taking into account all the employed resources, was obtained in 2017 with a N00N state (N = 2) in polarization degree of freedom evolving through a bulk Mach–Zehnder interferometer.400 The SQL, corrected by the efficiency η and the visibility V, has the following form:400 

ηNV2N<1.
(64)

The reported experimental violation was ηNV2N1.23, thus showing genuine quantum enhancement. In order to reach this violation, high efficiency (>0.95) detectors, low losses circuits, and high visibility fringes (V0.98) were obtained.

The generation of N00N-like states and their use for quantum phase estimation can be realized inside integrated circuits that allow for high stability and fine tunability. Super-resolution with generated N00N states was achieved for N = 2 in silica-on-silicon,454–456 UV-written,381 silicon-on-insulator,363,457 and femtosecond laser written458,459 circuits. The case N = 4 was obtained with state projections in silica-on-silicon circuits,454 also in a heralded configuration.455 In these circuits, also states that resemble N00N ones and are ideal against symmetric photon-losses403 were generated, by adopting a heralded configuration.455 Such states contain M photons along one mode and L=NM photons along the other,

|M::L12(|M,L+eıϕ|L,M).
(65)

Equation (65) corresponds to N00N states when L = 0. They are entangled when ML, and show interference fringes whose frequency is enhanced by a factor |ML|.

Note that photons can be generated outside and then injected inside the circuits, where they interfere to generated N00N states, or they can be directly generated on chip through nonlinear processes.387,460 The SFWM generation process is naturally exploited in Si-based circuits.363,364,461,462 Also the SPDC process can be realized in integrated waveguides to generate N00N states.388,463–465 Finally, quantum dots can be exploited to generate N00N states.466,467

In conclusion, the use of N00N states for practical quantum metrology is limited by two factors: (i) it is hard to generate high photon-number states without relying on post-selection or filtering, and (ii) in the noisy regime, N00N states maintain their optimality only for small N. Indeed, the quantum Fisher Information, in the presence of symmetric losses η, is:12FQN00N=ηNN2, and thus, it decreases exponentially in N. For these reasons, for most of the practical applications where a high number of photons are required, other states such as squeezed ones are employed.

3. Continuous variable states

Fock states are expressed in the discrete variable (DV) formalism of mode operators ak. However, in order to describe some states and processes, it is useful to employ an equivalent formalism, based on continuous variables (CV).244,318,468–470 A convenient choice of CV is represented by quadrature operators xk and pk in Eq. (40), defining the phase space. Such observable operators do not commute [Eq. (41)] and then have to satisfy the Heisenberg relation. The latter imposes the following constraint on uncertainties:

ΔxkΔpk1.
(66)

Contrary to the photon number operator, the spectrum of the quadrature operators is continuous and their eigenstates form a complete orthogonal basis: {|X} of x with eigenvalues {X} and {|P} of p with eigenvalues {P}. Such states satisfy the following conditions: X1|X2=δ(X1X2) and P1|P2=δ(P1P2). Since x and p are conjugate observables, their eigenstates |X and |P are related by a Fourier transformation.

In order to describe a general state ρ in quantum phase space, a useful tool is provided by quadratures quasi-probability distributions P(X,P). Among the possible quasi-distributions471,472 one of the most used is the Wigner function12,407,468,471,473–475 that, for a d-mode state ρ, is defined as

Wρ(X,P)==1(2π2)d+ddXddPTr[ρei[P(xX)X(pP)]],
(67)

where X=(X1,,Xd) and P=(P1,,Pd) are the values assumed by quadrature operators x=(x1,,xd) and p=(p1,,pd), respectively, along the d modes. The Wigner function is normalized: +ddXddPWρ(X,P)=1. From Wρ(X,P), one can recover the real marginal quadrature distributions,

P(P)=+ddXWρ(X,P),P(X)=+ddPWρ(X,P).
(68)

However, the global Wigner function Wρ(X,P) is not a proper distribution and can also assume negative values. Indeed, the joint probability distribution P(X,P) for two no-commuting quantum operators x and p cannot be properly defined because of Heisenberg uncertainty relations (66). In particular, the negativity of the Wigner function can be used to certify the nonclassical nature of quantum states. Different experimental techniques are available to measure and reconstruct Wigner functions of photonic states.471,476–479

An important class of CV states useful also for quantum metrology is represented by Gaussian states242,244,468,480–483 that are described by a Wigner function corresponding to a multidimensional Gaussian distribution. Then, in order to characterize a Gaussian state, it is sufficient to acquire knowledge of the two moments of the associated distribution. The simplest Gaussian state is the vacuum state |0, with zero valued mean quadratures, x=p=0 and minimum symmetric uncertainties saturating relation (66): ΔxΔp=1. Conversely, thermal states are Gaussian states with x=p=0 but their uncertainties do not saturate Eq. (66). Restricting on a single mode, they can be expressed in the photon number basis as an incoherent mixture,

ρth=N=0nN(1+n)1+N|NN|,
(69)

where n is the mean photon number. The photon-number probability P(N) follows the distribution describing black-body radiation: P(N)=nN(1+n)1+N in which n=1/(e/KBT1), where KB is the Boltzmann constant and T is the temperature. The quadrature fluctuations of such states are:242 Δx2=Δp2=1+2n.

An intense research activity is devoted to study the interferometric properties of Gaussian states.481,482,484–491 In Sec. III C 4, we provide a description of some of the most used Gaussian states for quantum metrology tasks. Different reviews providing a more detailed description of Gaussian states are available, such as those in Refs. 242, 244, 318, and 468. In particular, several papers investigated the form of the Quantum Fisher Information for general Gaussian states.480,481,483,487,492,493

4. Squeezed states

Squeezed light states are among the most used Gaussian states to enhance quantum metrology tasks.22,26,244,468,494–498 A quantum state is said to be squeezed when an observable on this state presents a fluctuation (second moment of the Wigner distribution) that is lower than that of vacuum state. Generally, the continuous squeezed variables for quantum metrology tasks are quadratures. Furthermore, squeezed states saturate the Heisenberg relation and are minimum uncertainty states. However, such saturation is asymmetric due to squeezing of one quadrature and the corresponding anti-squeezing of the conjugate one,

{Δx<1andΔp>1orΔx>1andΔp<1ΔxΔp=1.
(70)
a. Single mode squeezed states

All single mode states saturating Heisenberg inequality are called single mode squeezed states.499 Hence, coherent and vacuum states are particular cases of such class of states. The operation of quadrature uncertainty squeezing is related to the third term in Eq. (44) and can be described through the squeezing single mode operator S(r),

S(r)=e12ra212r*a2,
(71)

where r=|r|eiθ, with |r| called squeezing factor and θ being the squeezing angle. The annihilation and quadrature operators, under squeezing evolution along mode k, are transformed following the relations:

akS(r)cosh|r|ak+eiθsinh|r|ak,akS(r)cosh|r|ak+eiθsinh|r|ak,xkS(r)cosh|r|xk+cosθsinh|r|xk+sinθsinh|r|pk,pkS(r)cosh|r|pk+sinθsinh|r|xkcosθsinh|r|pk.
(72)

In the case of θ = 0, the uncertainty on p is squeezed by a factor e|r|: Δp2S(|r|,θ=0)Δp2/e|r| while that of x is anti-squeezed by the same factor: Δx2S(|r|,θ=0)e|r|Δx2. Therefore, the squeezing factor can be estimated as the ratio between the uncertainty before and after squeezing, reducing exponentially one quadrature uncertainty at the expense of the other. For this reason, it represents the most relevant parameter in squeezing generation and it is commonly expressed in dB [(Δpsqueezed/Δpnotsqueezed)dB|r|].

The simplest single mode squeezed states are the squeezed vacuum states generated by applying the squeezing operator in Eq. (71) to the vacuum. Such class of states can also be written in Fock basis,

|rS(r)|0=1cosh|r|N=0(eiθtanh|r|)N(2N)!2NN!|2N.
(73)

Such states have zero mean quadratures, are composed of a linear combination of only even photon number states, and have a mean photon number n=sinh2|r|.

A general single mode squeezed state, with mean quadratures different from zero, can be generated by the application of the squeezing operator [Eq. (71)] and of the displacement one [Eq. (57)] on a vacuum state,

|ΨsqueezedsmD(α)S(r)|0.
(74)
b. Two mode squeezed states

Squeezing along two modes is a fundamental resource for quantum metrology.26,244,468,494,496,500 The evolution operator associated with two-mode squeezing is the contribution of the third term in Eq. (44) that, in contrast to the single mode operator in Eq. (71), involves two different modes k1 and k2,

S2(r)=er*ak1ak2rak1ak2,
(75)

where again r=|r|eiθ. The mode operators evolve under S2(r) according to the following relations:

ak1S2(r)cosh|r|ak1+eiθsinh|r|ak1,ak2S2(r)cosh|r|ak2eiθsinh|r|ak2.
(76)

The simplest states, generated by the two-mode squeezing operator, are the two mode squeezed vacuum states or twin-beam states, which present nonclassical correlation involving two modes (mode-entanglement).12 They are obtained by the action of S2 on the vacuum state, leading to the following output state in the Fock basis:

|r2S2(r)|0,0=1cosh|r|N=0(eiθtanh|r|)N|N,N.
(77)

For this state, the total number of photons along the modes is even for each superposition component, and the mean values of quadratures are zero. The mean total photon number n=nk1+nk2=2sinh2|r|. These states, in photon number basis, clearly show entanglement between the two modes. In particular, when the squeezing parameter |r| (large squeezing regime), this state corresponds to the EPR state.501 

Since the variance of the number operator is 2(n2+n), we find that its application to phase estimation in a MZI leads to

Δϕsqueezed1221n2+n,
(78)

which for large n shows Heisenberg-limited scaling.

Recently, a more general concept of nonlinear squeezing, applicable to non-Gaussian states, has been introduced.502 

c. Homodyne measurements

Since squeezed states are well described by CV quadratures, their detection mostly relies on measuring such variables. In order to measure quadratures, a suitable technique is represented by homodyne detection.26,244,468,477,503 To experimentally implement a homodyne apparatus, the idea is to interfere in a beam-splitter the target optical signal together with an additional coherent state of the same frequency that acts as an additional phase reference. Such a reference beam is called a local oscillator.242,244,468

A first scheme employs a balanced beam splitter and is called balanced homodyne detection. Consider a target state, with annihilation operator atgt, whose quadratures have to be measured. Such state is injected in one input mode of a balanced beam splitter. At the same time, a coherent state |αLO, with αLO=|αLO|eıθ and associated photon annihilation operator aLO, is injected along the other input of beam splitter. The output annihilation operators are then composed by the superposition of the operators relative to the two input beams: aout1=12(atgt+aLO) and aout2=12(atgt+aLO). The light exiting from BS outputs is then detected by photodiodes that reveal a current with intensity proportional to the number of photons: Iout1nout1=aout1aout1 and Iout2nout2=aout2aout2. Then, the difference of the two output intensities for the above input state in the limit of |αLO|1 (aLOαLO) is

ΔI=Iout1Iout2|αLO|(eıθatgt+eıθatgt).
(79)

This expression corresponds to the rotated quadrature xtgt(θ) [see Eq. (43)] for the target state. Note that, by tuning the local oscillator phase θ, one can measure all the quadrature space. For instance with θ=π/2, quadrature ptgt is measured since xtgt(θ=π/2)=ptgt(θ=0) [Eq. (43)]. Through this kind of measurement, it is then possible to perform quantum tomographies in the phase space.503–506 Furthermore, homodyne detection can be performed with high quantum efficiency.

When the beam splitter is unbalanced, the previous scheme is called unbalanced homodyne detection.244 In this case, only one BS output is measured and, tracing out the local oscillator, quadrature statistics can be measured in the limit |αLO|1.

Homodyne detection allows also to perform entangled measurements, such as Bell measurements, in quadrature space through multihomodyne detector schemes.244,468 Alternatively, such measurements can be realized by heterodyne detectors,507 in which the local oscillator has a different frequency with respect to the target state.244 

d. Phase estimation experiments with squeezed states

Several proof-of-principle experiments have demonstrated that squeezed states of light are fundamentals to beating shot-noise96,508 and improve interferometer sensibility.26,33,509–514 Squeezed states and homodyne detection have been recently exploited to deterministically achieve simultaneously the super-sensitivity and the super-resolution conditions.515 First generation of squeezed light was realized by exploiting nonlinear processes as four-wave mixing (FWM) and parametric down conversion inside sodium atoms and nonlinear materials, together with optical cavities and fibers.516–518 The three main technologies that have been adopted to generate squeezed states of light are: atoms, nonlinear crystals, and optomechanical systems.519 Squeezing by atoms exploits the third-order nonlinear susceptibility, through the FWM process.516,520,521 Photonic single mode squeezing can be experimentally generated exploiting parametric down conversion-based setups.522 Here, a second order nonlinear crystal is pumped by a pump of frequency 2ν with a phase able to create amplification (optical parametric amplification, OPA). In the absence of other excitation field, a squeezed vacuum state at frequency ν is generated. The pumped cavity is maintained slightly below the oscillation threshold so that no bright light is generated. The squeezing factor is higher as the working point is closer to the oscillation threshold. In order to enhance the amplification efficiency of the OPA, the nonlinear crystal can be placed inside an optical cavity, realizing an optical parametric oscillator (OPO).26,509,523–530 Nonlinear processes can be also employed to generate two mode squeezed states. Indeed, this class can be realized combining in a beam splitter two squeezed states generated through a type I parametric down conversion process with an opportune phase shift, or also by dividing in a polarizing beam splitter a squeezed state generated by a type II OPA.26 Finally, coupling optical fields with mechanical modes of given structures, such as crystalline resonators and membranes,531–536 allow squeezing generation in optomechanical systems.

Currently, gravitational wave detection represents the most direct and relevant metrological application.537 For this purpose, squeezing factors above 10 dB have been experimental achieved: 10 dB,526 12.7 dB,527 12.3 dB,528 11.6 dB,529 and 15 dB.530 All these realizations exploit cavity-enhanced optical-parametric amplification, working below its oscillation threshold and using a pumped type-I nonlinear crystal.

To date, photonic implementation using atoms, optomechanics, and nonlinear crystals has produced the largest squeezing factor amount relatively of 14.9 dB,519 25 dB,531 and 19 dB,527 respectively. Squeezed light has been generated at different wavelengths, such as 795 nm,521,531 860 nm,538 946 nm,539 1064 nm,527,529 1540 nm,532 and 1550 nm.528,540 Squeezed vacuum states have been demonstrated only exploiting atoms519 and nonlinear crystals.527 Although optomechanics demonstrated largest squeezing,531 in gravitational wave frequencies (audio-band regime), nonlinear crystals hold the record both in terms of generated and measured squeezing.527 Two-mode squeezed states can potentially enhance interferometry150,541 and are fundamental for practical quantum metrology. Two mode squeezing was achieved on platforms able to reach squeezing factors even greater than 10 dB.542–546 Furthermore, while most of the largest squeezing values were achieved in bulk optics, squeezed light generation and measurement have been also investigated in integrated platforms.532,547–556 Even thermal mixtures of squeezed states can be exploited for quantum metrology486,557 (also combined with other states558) as experimentally analyzed using nondegenerate OPO to realize thermal squeezed states with different purity and balanced homodyne detection.559 Finally, realization of squeezed states in the polarization degree of freedom has been reported509,560–562

As discussed above, one of the most direct applications of squeezed states relies on the enhancement of estimation of gravitational waves.30,513,563–568 Starting from the recent observations,569–571 investigation of gravitational waves is a very challenging research area and represents the first actual application of quantum metrology.4,509,510,516,572–575 The small amplitude (1022) of gravitational waves needs a very long interferometer to be measured, together with very low overall noise.30,568 Thermal contributions and radiation pressure576 over free-falling interferometer mirrors represent fundamental problems in reaching the sensitivity sufficient to detect such amplitudes. Furthermore, detectors are affected by photon-counting noise (or shot-noise), which follows the Poisson statistics. Therefore, improving signal-to-shot noise ratio (SNR) requires increasing the number of input photons. In contrast, thermal mirror displacement is proportional to the input laser power. Quantum metrological techniques, such as those adopting squeezing resource as the input state in the interferometer,577 seem to be the only possible short-term solution, able to reduce the quantum noise contributes without increasing the laser power. In particular, an interferometer-based detection depends on the optical path difference between internal arms. Entanglement in the probe state can correlate both noise and signal between internal arms in a way that noise is cancel out, thus enhancing the SNR. For example, if a coherent state is overlapped with a vacuum squeezed state into the input beam splitter of a Michelson interferometer, correlation after this interference permits squeezing in the output state and the improvement of the SNR.30 In particular, the higher the squeezing factor, the greater the SNR enhancement. Indeed, it has been demonstrated that strong squeezing could lead to approximately tenfold improvement in gravitational wave detection.30 Furthermore, the capability of reducing the quantum noise in gravitational wave interferometers depends on the frequency which has to be detected.578 Frequency-dependent squeezing schemes can be used to reduce quantum noise in the audio-band spectrum, and an EPR entangled squeezing-based setup has been proposed to achieve a broadband solution.579–581 One of these proposals has been recently realized in Refs. 582 and 583. Different studies have investigated the fundamental limits achievable in these interferometers and schemes to evade radiation pressure.584–590 

Starting from 2007, the GEO600 gravitational wave detector has successfully adopted squeezed light for its detection.46,591 Recently, after preliminary tests,592 squeezed vacuum states have been used in the Advanced Laser Interferometer Gravitational Wave Observatory (LIGO) detectors.593,594 Other gravitational wave detectors have almost achieved best technological performances in their several components571,595–597 and seem to find further improvements only in squeezing enhancement.

The conceptual scheme of the LIGO Michelson interferometer, seeded by squeezed states, is sketched in Fig. 6. The 4 km-long arms of the interferometer contain Fabry–Perot cavities composed of two mirrors that reflect the light forcing it to travel across the arms multiple times, so enlarging the effective length of the arms and, consequently, the sensitivity of the interferometer. Recycling power mirrors are used to increase the optical intensity inside the interferometer. When single squeezed states are used to enhance the sensitivity, they are injected in the input of the interferometer [Fig. 6(a)]. Also a two-mode squeezed state can be exploited to enhance the detection. In this case, the two squeezed fields, with a fixed relative angle, interfere in a beam splitter generating a two-mode entangled state. One mode is injected in the interferometer, while the other interferes with the output of the interferometer in a second beam splitter. Finally, a homodyne measurement of the quadratures is performed [Fig. 6(b)]. Thanks to the simultaneous measurement of the quadratures, the possible disturbance signals can be recognized with respect to the signal to be analyzed.598 Note that squeezed states can enhance the performances of gravitational wave detection, but they only represent a useful complement to classical resources. Indeed, the overwhelming majority contribution in the employed number of photons comes from coherent light, and then, even if an enhancement over SQL can be obtained, the sensitivity is still far from Heisenberg limit.

Fig. 6.

Conceptual scheme of LIGO interferometer seeded by squeezed states. Squeezed states are injected in the interferometer by means of Faraday isolators, allowing light to pass along only specific directions. (a) A single mode squeezed state |ξ, generated by an OPO, is injected in the Michelson interferometer. (b) A two-mode squeezed state |ξ2 is generated by two squeezed states interfering in a BS. Then, one mode of |ξ2 is injected in the interferometer, while the other mode interferes with the output of the interferometer in a beam splitter (BS). Described in Ref. 26.

Fig. 6.

Conceptual scheme of LIGO interferometer seeded by squeezed states. Squeezed states are injected in the interferometer by means of Faraday isolators, allowing light to pass along only specific directions. (a) A single mode squeezed state |ξ, generated by an OPO, is injected in the Michelson interferometer. (b) A two-mode squeezed state |ξ2 is generated by two squeezed states interfering in a BS. Then, one mode of |ξ2 is injected in the interferometer, while the other mode interferes with the output of the interferometer in a beam splitter (BS). Described in Ref. 26.

Close modal

For a more in depth recent analysis on squeezed states, we refer to the review in Ref. 26.

5. Other states

Holland and Burnett599 (HB) have introduced the adoption of twin-Fock states600 for phase estimation. Such states are interesting states with a fixed number of indistinguishable photons that are equally distributed along two different modes, so that their state can be written as: |N/2,N/2, with N even. Such states can be used as the input of interferometric setups to provide some advantages in nonideal conditions with respect to N00N states.15,17,405,601 Furthermore, twin-Fock states with a high number of systems N can be realized with physical platforms like Bose–Einstein condensate-based systems.223,602

Let us consider the scenario where noises, losses, and low detection efficiency are present,405,601 or when detection is restricted to projections onto states containing definite photon numbers along each output of interferometer (e.g., with a four-photon state along two modes, the detection could be restricted to |3,1 and |1,3 terms). In this configuration, the twin-Fock states can be a resource to outperform N00N states for N > 4 (see Ref. 603). The idea is to generalize the HOM effect, injecting two beams of indistinguishable N/2 photons (with N even) in the two input ports of a beam splitter. The output state |ΨHB, that we label as the HB state, after a relative phase shift ϕ between the two output modes, can be written in the following form:

|ΨHB=n=0N/2Cn|2n,N2n,Cn=ei2nϕ(2n)!(N2n)!2N/2n!(N/2n)!.
(80)

When N = 2, Eq. (80) coincides with a N00N state. Such states reach Heisenberg scaling up to a constant factor 2. The two beams injected along the input modes of the beam splitter can be generated with a SPDC process. HB states with four photons604 and six photons603,605 were realized in polarization degree of freedom, through the SPDC process.

Twin-Fock states are strictly related to a more general class of states with a fixed number of subsystems, which are called Dicke states.606–608 Such class has also been defined when dealing with distinguishable qubits. In this framework, a symmetric Dicke state |DNk, composed of N distinguishable qubits (in the basis {|0,|1}) with k excitations, has the following form:

|DNk=(Nk)12lPl(|1k|0(Nk)),
(81)

where the balanced superposition involves all the permutations Pl of k qubits in the excited state |1 and the other Nk in the state |0 (note that in this case the notation |0 and |1 stands for logical values of the qubits and not for the number of photons). For instance, the Dicke state with N = 4 different qubits and k = 2 is: |D42=(|0,0,1,1+|0,1,0,1+|1,0,1,0+|1,1,0,0+|1,0,0,1+|0,1,1,0)/6. Photonic symmetric Dicke states [Eq. (81)] were generated in the polarization degree of freedom through bulk optics schemes with up to N = 6 photons.608–610 Furthermore, such states were generated by exploiting path-polarization hyper-entanglement, with two photons carrying N = 4 qubits.611 

A more general class of states, robust against noises, are Dicke squeezed states.612 It has been recently demonstrated that the multimode photon states emitted by the phenomenon of collective Dicke superradiance can be a resource for quantum metrology.613 

Other classes of states can be deterministically generated by exploiting the generalized HOM effect, implemented by a quantum Fourier transformation (QFT) acting on N indistinguishable photons, one along each input mode of a N-mode interferometer. The QFT transformation is described by a unitary matrix Uj,k=1/Nexp[ı(j1)(k1)/N]. QFT states with N = 2 coincide with the N00N state, while they are different for N > 2. For instance, states at the output of a QFT with N = 3 have the following expression in Fock basis: 2/3(|3,0,0+|0,3,0+|0,0,3)+1/3|1,1,1. A typical interferometer in this framework is composed of a QFT that entangles the photons through the generalized HOM effect, and a QFT transformation that disentangles the photons after their phase-dependent evolution.614–616 In this way, phase super-sensitivity, beating SQL, can be observed if the unknown phase is put along one of the interferometer modes. However, this is true only for N < 7.615,616 An experimental realization of such a scheme was performed with N = 2, 3, 4, exploiting both path and polarization degrees of freedom in a bulk optical multimode interferometer.616 

Other states, suitable for experimental generation and robust with respect to noises, are entangled coherent states (ECS)617–621 along two modes that in Fock basis read as

|ECSα=12(1+e|α|2)(|α1|02+|01|α2),
(82)

where |0i and |αi represent, respectively, a vacuum and a coherent state along mode i. Such a state can be realized by injecting a coherent state |α along an input of a beam splitter and the state (|α+|α)/(2+2e2|α|2), also called Schrödinger cat state, along the other input.620,622 Approximated entangled coherent states [Eq. (82)] were generated by mixing squeezed vacuum states and coherent light inside a beam splitter in Ref. 623. The measurement was realized by performing photon-number resolving detection.

Different classes of states able to achieve quantum enhanced sensitivities in phase estimation can be realized through the generalization of the scheme used for N00N states generation in Fig. 5. Such generalization involves different kinds of states as inputs of the Mach–Zehnder interferometer. Given a coherent state injected along an input port of the first beam splitter, the optimality of a squeezed vacuum state, injected along the other input port, was demonstrated in Ref. 624. If we consider a Fock state in a single mode of the interferometer, and an arbitrary state with the same average number of particles in the other port, one can reach the Heisenberg limit for phase sensitivity, as demonstrated in Ref. 495 and experimentally realized through motional states of a single ion.625 This kind of scheme can be further generalized to a MZI in which an arbitrary state is injected along one input port and an odd or even state along the other port. The conditions to achieve the best sensitivities for this case were studied in Ref. 626. Furthermore, optimal product states along the input modes of the setup were found, both with a fixed photon number and fixed mean photon number, in Ref. 627.

Finally, also cluster states (employed as a flexible resource for efficiently generating different probe states)628 and random symmetric states629 are useful for quantum metrology tasks. In Table II we report some of the photonic platforms recently employed to perform phase estimation with fixed photon-number quantum states.

Table II.

Table of some platforms used to measure phase shifts with fixed photon-number quantum probes, realized during the last 10 years.

ProbeReferencesPlatformNumber of photons
N00N 110, 446  Bulk polarization two-mode MZI microscope N = 2, N = 2, 3 
N00N 445  Bulk polarization two-mode MZI N = 5 
N00N 400  Bulk polarization two-mode MZI unconditional enhancement N = 2 
N00N 455  Integrated silica-on-silicon heralded two-mode-MZI N = 2, 4 
N00N 454  Integrated silica-on-silicon two-mode-MZI N = 2, 4 
N00N 457  Integrated silicon-on-insulator two-mode-MZI N = 2 
N00N 363  Integrated silicon-on-insulator two-sources two-mode-MZI N = 2 
N00N 423  Bulk polarization two-mode-MZI N = 2 
N00N 440  Bulk polarization to path two-mode-MZI with optical centroid measurement N = 2, 3, 4 
N00N 458,459  Integrated FLW two-mode-MZI N = 2 
GHZ 448  Bulk polarization N = 8 
GHZ 451, 453  Bulk polarization noisy estimation N = 4, N=1,2,3,4,6 
GHZ 449  Bulk polarization-path hyper-entanglement N = 4 (8 qubits) 
GHZ 450  Bulk polarization-path-OAM hyper-entanglement N = 6 (18 qubits) 
QFT 616  Bulk path/polarization multi-mode MZI N = 3, N = 4 
HB 603–605  Bulk polarization two-mode-MZI N = 2, 4, N = 2, 4, 6 
Dicke 608,610  Bulk polarization N = 6 
Dicke 611  Bulk path-polarization hyper-entanglement N = 2 (4 qubits) 
ProbeReferencesPlatformNumber of photons
N00N 110, 446  Bulk polarization two-mode MZI microscope N = 2, N = 2, 3 
N00N 445  Bulk polarization two-mode MZI N = 5 
N00N 400  Bulk polarization two-mode MZI unconditional enhancement N = 2 
N00N 455  Integrated silica-on-silicon heralded two-mode-MZI N = 2, 4 
N00N 454  Integrated silica-on-silicon two-mode-MZI N = 2, 4 
N00N 457  Integrated silicon-on-insulator two-mode-MZI N = 2 
N00N 363  Integrated silicon-on-insulator two-sources two-mode-MZI N = 2 
N00N 423  Bulk polarization two-mode-MZI N = 2 
N00N 440  Bulk polarization to path two-mode-MZI with optical centroid measurement N = 2, 3, 4 
N00N 458,459  Integrated FLW two-mode-MZI N = 2 
GHZ 448  Bulk polarization N = 8 
GHZ 451, 453  Bulk polarization noisy estimation N = 4, N=1,2,3,4,6 
GHZ 449  Bulk polarization-path hyper-entanglement N = 4 (8 qubits) 
GHZ 450  Bulk polarization-path-OAM hyper-entanglement N = 6 (18 qubits) 
QFT 616  Bulk path/polarization multi-mode MZI N = 3, N = 4 
HB 603–605  Bulk polarization two-mode-MZI N = 2, 4, N = 2, 4, 6 
Dicke 608,610  Bulk polarization N = 6 
Dicke 611  Bulk path-polarization hyper-entanglement N = 2 (4 qubits) 

OAM states can be exploited to perform ultra-sensitive measurements of rotations.630–635 In this scenario, the value of angle of rotations can be embedded in relative phase shifts between OAM components. For instance, let us consider the state of a photon in a superposition of opposite OAM modes, with modulus of OAM number |m| as (|m+|m)/2. If a rotation of an angle θ is performed, whose sign is based on the value of the ancillary mode (say +θ for mode 1 and θ for mode 2), the state will evolve to (eımθ|m+eımθ|m)/2. Such state shows a rotation amplified by a factor m. Hence, the two angular momentum orientations ±m will show N00N-like interference fringes and are able to reach a sensitivity with an improved factor of m as Δθ=1/(mν), where ν is the number of single photon probes. Such enhanced sensitivity arises from the superposition of m-quanta of OAM. Since single photon probes are exploited, advantages can be obtained both for generation, detection, and robustness to losses with respect to N00N states.

Other classes of useful sensitive high dimensional states are the so-called Kings of Quantumness states.636 Those classes of states have been experimentally realized up to dimension 21 through OAM states.637 

If entangled photons carrying OAM are employed in interferometric setups, an amplification of the sensitivity due to both the carried OAM and the entanglement between probes can be obtained. This has been experimentally demonstrated with photonic platforms in Refs. 632–634.

Quantum enhanced sensitivities can be also obtained in SU(1,1) interferometers, in which the linear optical beam splitters (BS) are replaced by nonlinear active optical interactions.265,638–642 These include parametric amplifiers based on FWM processes643 or parametric down conversion644 that generate useful entanglement inside the interferometer. These platforms are robust against noises, thus guaranteeing quantum enhanced performances also in the presence of losses.645–649 General nonlinear effects in estimation strategies can lead to scalings beyond the Heisenberg limit.229,650–652

Different photonic techniques exploiting quantum states for imaging have been reported.21 For instance, spatial correlations between photons can be exploited for ghost imaging.27,653–659 In this case, two correlated photons, such as those belonging to pairs generated by a SPDC process, are employed to probe a sample with the goal of reconstructing its image. One of the photons propagates through the sample, and is measured by a photodetector without spatial resolution. Conversely, the other photon does not interact with the sample and is measured by the spatial resolving detector. The image of the object is reconstructed by the combined information from the two correlated photons: the spatial information comes from detecting the photon that did not interact with the sample, and is triggered by detection of the photon that interacted with it. Hence, by exploiting the spatial correlation of the photons, one can recover spatial information on the object. This is obtained by spatially resolving only the photon that did not interact with the object and discarding the spatial information obtained from detection of the interacting photon. Quantum ghost imaging has been realized exploiting different techniques for the spatial resolving measurement stage.346,347,660–662 The key ingredient of ghost imaging is the correlation between the two photons (or beams). Hence, also classical correlated light can be used for this purpose.663–666 However, in the low probe intensity regime, quantum probes can provide better performances.104,654,667,668 Quantum ghost imaging has also been realized to perform 3D tomography,669 and in a configuration exploiting entanglement swapping.670 

Imaging on samples with a pair of photons can be performed even without detecting the photons interacting with the sample.348,671 This technique exploits two identical nonlinear crystals pumped by two laser beams coming from the same split laser. The pumped crystals emit pairs of nondegenerate photons, signal and idler. The idler photon from one crystal, once separated from the signal one, interacts with the sample and is subsequently sent along the second nonlinear crystal. The latter element is also pumped by the laser and then can emit pairs of photons as well. In this way, there are two different paths for the signal photons, which are then sent to interfere in a beam splitter. Information is acquired from quantum interference between the two signal beams that did not interact with the object. Hence, one can perform phase and intensity imaging without measuring those photons, which have interacted with the object. An experimental demonstration of such a protocol has been demonstrated in Ref. 348. Finally, quantum mechanics allows acquiring information about an object without any system directly interacting with it. This is possible through interaction-free measurements.672–676 For a recent more detailed review on quantum imaging techniques, we refer to Ref. 21.

Another approach which can be used for quantum phase estimation exploits weak measurements. In particular, this approach leads to interesting quantum phenomena including weak values677,678 that can be employed for practical tasks in quantum metrology.401,679–686

The contrast of an object image can be enhanced by entangled photons through a scheme named quantum illumination.687–690 In this protocol, two entangled beams are generated, and one of the two beams is sent through a partially reflective object (sample). The reflected beam is finally jointly measured with the other beam who did not interact with the sample. Hence, exploiting the correlation between the beams one can recover the reflected light from the object, discriminating it from background noise. For this purpose, a two-mode squeezed vacuum state is nearly optimal for different performance criteria.691 Experimental realizations of quantum illumination were performed using parametric down conversion beams detected by the CCD camera692 and using microwave frequency beams.693 The quantum enhancement of such scheme resists also in strongly noisy environments.134,687,688,694,695

Finally, quantum interferometry can be realized also in the time-energy domain of photons, exploiting Franson interferometers.696–699 Furthermore, estimation of frequency as well as temporal separations between incoherent signals can be performed through mode-selective photon measurements.700 

Different estimation protocols have been defined9,701 and can be included in a few fundamental categories. A first example is provided by parallel protocols (Fig. 2) in which all the probes, entangled or not, interact in parallel with the system.2,211 A second class is composed of sequential (or multiround) protocols233,702,703 where single probes interact multiple times with the system. Finally, ancilla-assisted schemes232,704–708 are those where a part of the probe, generally entangled with the other part, does not interact with the system and is directly measured. All these protocols can be nonadaptive11 or adaptive.198,709 Here, we focus on adaptive techniques that represent a powerful tool to enhance the performances of estimation processes.198,233,709–713 In nonadaptive estimation protocols, the available m probes are sent through a fixed apparatus and, after collecting the full data set, a final estimate of the unknown parameter ϕ is obtained. Conversely, adaptive techniques make use of suitable controls on the experimental setup, namely, some physical parameters θ, such as additional feedback phase shifts, that can be adjusted during the estimation. Adaptive and entangled protocols can enhance metrology tasks, especially in the presence of noise.701,714 Adaptive protocols do not give advantages with respect to nonadaptive schemes when the estimation involves quantum channels that are (jointly) covariant with teleportation, such as the Pauli or erasure channels. In this case, Ref. 191 showed that the optimal performance is limited to the SQL, by adapting techniques previously developed for quantum communication.715 A discrete-time class of adaptive protocols can be schematically represented through the repetition, for each probe, of the four-step cycle as shown in Fig. 7.

Fig. 7.

Conceptual scheme of an adaptive estimation protocol. The cycle of a general adaptive estimation protocol starts from an initial state ρin that is prepared (blue box) in a state ρ0(θ) through the action of Uθ. Such state interacts with the unknown parameter ϕ (brown box), and then the output state ρfin(θ,ϕ) undergoes an appropriately chosen measurement Πx (gray box). After such measurement, the results X are exploited to define a suitable action Uθ(X) (orange box), employed to prepare the initial state of the next probe. In this way, the cycle is repeated for all the probes. At the end of the process, an estimator provides the final estimate of θ.

Fig. 7.

Conceptual scheme of an adaptive estimation protocol. The cycle of a general adaptive estimation protocol starts from an initial state ρin that is prepared (blue box) in a state ρ0(θ) through the action of Uθ. Such state interacts with the unknown parameter ϕ (brown box), and then the output state ρfin(θ,ϕ) undergoes an appropriately chosen measurement Πx (gray box). After such measurement, the results X are exploited to define a suitable action Uθ(X) (orange box), employed to prepare the initial state of the next probe. In this way, the cycle is repeated for all the probes. At the end of the process, an estimator provides the final estimate of θ.

Close modal
  • The first step is dedicated to the preparation of an initial probe ρin, through a process Uθ(x) that depends on certain parameters θ and, if available, on the results x of previous measurements.

  • At a second stage, the prepared probe ρ0(θ) interacts with the studied system and evolves under a unitary Uϕ (for simplicity we are assuming unitary evolution) in ρfin(θ,ϕ).

  • Then, a measurement Πx is performed and its outcome x is recorded.

  • The final step of the cycle is post-processing of the measurement results. This step includes the choice of the parameters θ determining the action Uθ(x) to apply to the initial probe of the successive cycle.

This cycle is repeated for all the probes. Finally, an estimator Φ(x) based on all measurement results x provides an estimation of the unknown parameter ϕ.

Exploiting adaptive protocols for quantum metrology was proposed in 1995 by Wiseman.709 Such protocols are necessary in order to overcome different issues. For instance, they can be used for the realization of the optimal POVM to saturate the QCRB. In certain scenarios, such POVMs can be hard or impossible to implement. In this case, approximation of such measurements can be achieved by adaptive techniques.215,711 In particular, to approach the QCRB one has to maximize the Fisher Information of a given setup. However, the latter quantity depends in general on the unknown parameter. More specifically, given an initial probe, the QCRB is attainable only when the unknown parameter takes a value, which maximizes the Fisher Information. Nevertheless, it can be demonstrated that, even with no prior knowledge on the unknown parameter, the QCRB can be asymptotically saturated by exploiting adaptive techniques.716 

A second scenario where an adaptive approach represents a useful resource is found for those systems where the output probabilities, calculated at different values of unknown parameter, take the same value. For instance, in a Mach–Zehnder interferometer seeded by single photons, the output probability (P1=1P0=cos2ϕ/2) is such that, in the range ϕ[0,2π], two different values of the phases lead to the same probability P10. Indeed, the latter is not a monotonic function of the phase. Hence, without changing the relative phase shift during the experiment, it is impossible to discern the two equivalent phases leading to the same probability. Conversely, by changing the total phase shift during the estimation process, for instance through another known control phase, it is possible to solve such issues. In any case, when the output probabilities are periodic with a period less than 2π, it is impossible to distinguish some phases. In such cases, one could employ an adaptive protocol where the probe state can change at each iteration, thus changing the likelihood function and its periodicity. For instance, during the first steps one can employ probes whose likelihood has no periodicity, in order to restrict the range of possible unknown phase values. When the range is sufficiently small, more sensible states with smaller periodicity can be used.409 However, the validity of such recipe depends on the problem symmetries.

Furthermore, an important task where adaptive protocols can be helpful is the convergence to the ultimate precision bounds in the limited data scenario.210,717 The latter regime characterizes different realistic conditions where the amount of resources that can be employed is restricted. In the single-parameter case, theorems guarantee that it is always possible to define suitable measurements and estimators, allowing to reach the minimum error achievable with a given probe state (see Sec. II C). However, this capability of reaching the ultimate bounds is guaranteed only in the asymptotic regime. Conversely, when only a limited number of probes is available, identifying the optimal strategies is a difficult task. To this end, one can employ adaptive protocols, leading to a boost in the convergence to the asymptotic limits.

Finally, adaptive protocols have to be taken into account to achieve the true quantum limits.221 Importantly, feedback and error-correction schemes can be exploited to face noises and/or time-varying parameters.714,718–731

There exist two prominent approaches to feedback-based phase estimation:

  • Online schemes: At each step of the estimation protocol, the feedback is calculated according to the previous measurement result and a heuristic. An important class of these schemes is represented by Bayesian adaptive protocols. Here, at each step of the protocol, the posterior probability evolves based on measurement results. In this way, the posterior is used to calculate the optimal feedback action to be applied at next step. Note that optimality is defined depending on the particular problem and heuristic.

  • Offline schemes: The feedback values used during the experiments are computed before the estimation process. The goal is then the optimization of such sequence of feedback values. Different optimization techniques based on trial and error approaches can be exploited, such as those based on Particle Swarm Optimization (PSO)732,733 and Differential Evolution (DE).734–736 

Finally, adaptive protocols can also be exploited to enhance state discrimination and more general in quantum tomography.737–740 This has been experimentally demonstrated in the estimation of the photon polarization.741–745 A detailed review on implementations of feedback controls in quantum systems can be found in Ref. 713.

Bayesian estimation (Sec. II A 1) naturally fits the requirements for adaptive protocols. In this framework, the posterior distribution is updated at each repetition of the estimation cycle [Eq. (8)]. The information encoded in this distribution can be exploited to choose the optimal feedback action according to the protocol heuristic.

One of the first adaptive phase estimations, providing an experimental demonstration of the proposal in Ref. 709, was realized exploiting adaptive homodyne phase measurements on coherent states.746 Coherent states with homodyne measurements were also employed for adaptive estimation of a continuously varying phase, beating the nonadaptive filtering limit.747 HL scaling, in this kind of schemes, cannot be achieved by employing coherent states as probes. However, an enhancement of a constant factor with respect to SQL can be obtained. Also coherent state discrimination can be performed through adaptive schemes.748 

When employing quantum states, one can reach improved scaling in the estimation process. In this regime, when the phase to be estimated is completely unknown (flat prior distribution), adaptive techniques can be employed.198,749 This is the goal of an ab initio quantum phase estimation experiment that was experimentally realized by Ref. 604 using HB states. In such realization, a sequence of different states is used. In particular, single-, two-, and four-photon HB states in the polarization degree of freedom were generated through a type-I SPDC process followed by interference in a polarizing beam splitter. These output states were detected by a probabilistic photon-number resolving detection. The employed Bayesian protocol is composed of a first step with random feedback. Subsequently, after the measurement of a group of single-photon events, the posterior probability is updated and the next feedback is calculated by optimizing the expected sharpness function [Eq. (12)] over the possible results of the next measurements. Using suitable sequences of states (with photon numbers N = 1, 2, and 4), the SQL was surpassed.604 

The SQL can be overcome by employing other classes of states, such as Gaussian squeezed states with squeezing parameter r that reach a value for the variance482,750 equal to V=1/[2Nsinh(2r)]. Since for this resource state the optimal Fisher Information depends on the unknown phase, an adaptive protocol has to be employed, and Bayesian estimation can be exploited for this purpose. Given this class of input states, a Bayesian protocol for ab initio phase estimation has been experimentally realized using squeezed states and homodyne detection, together with real-time feedback.751 The phase of a squeezed state is measured with respect to a local oscillator through homodyne detection. More specifically, a first set of data is exploited to perform a rough estimation of the phase. Then, the local oscillator phase is adjusted to the value that leads to the minimum error in the estimation process.751 Finally, also two mode squeezed states can be exploited in adaptive protocols.752 

Bayesian adaptive estimation can be used to reach the HL with single photons in a multipass configuration without the need for entanglement as demonstrated in Ref. 233. In this case, single photons are employed for a multipass polarization interferometer estimating phases through a generalized Kitaev's algorithm.753 An adaptive hybrid approach, exploiting simultaneously polarization entangled two-photon states and a multipass configuration (with N = 3 passes per state, two for one photon and one for the other), achieved within 4% the exact value of HL at a finite number of resources.754 This implementation demonstrated the theoretical proposal of Ref. 711. The optimal state for this protocol is:711,754|ψopt=c0|Φ++c1|Ψ+, with |Φ+=(|0,0+|1,1)/2,|Ψ+=(|1,0+|0,1)/2 and cj=sin[(j+1)π/5]/k=01sin[(k+1)π/5]2 and was realized through a probabilistic control-Z gate755 between two SPDC photons.

An efficient and robust adaptive Bayesian phase estimation protocol, called rejection filtering,756 was realized exploiting the evolution of pairs of photons in a silicon circuit. The latter implemented adaptive unitaries that depend on single events, extracted from collections of photon statistics.757 

An adaptive estimation experiment based on single-photon inputs was realized in a bulk Mach–Zehnder interferometer in the path degree of freedom, implementing two different Bayesian techniques:758 (i) particle guess heuristic, in which at each step the feedback phase is randomly drawn from the posterior distribution756 and (ii) an optimal heuristic, which is derived analytically by optimizing the Bayesian mean square error of the future events over the feedback, under the assumption of narrow Gaussian prior.758 In particular, the last optimized technique shows better performances than the PSO (discussed in details below) and the particle guess heuristics. Furthermore, such an optimized technique has been experimentally demonstrated to be robust against different classes of noise.

Offline machine learning techniques can be exploited to enhance quantum phase estimations. Machine learning techniques759,760 applied to physical problems represent a new, rich, and continuously growing research area in which learning tools are used to enhance quantum information tasks.274,761–806 Such techniques can also be used to calibrate quantum sensors.807 Remarkably, machine learning-based protocols have been developed also for adaptive quantum metrology142,732–736,756–758,801,802,808,809 and entanglement-assisted supervised learning in an entangled sensor networks can be exploited for sensing tasks.810 

Two significant machine learning techniques employed for quantum metrology with an offline approach are PSO732,733 and DE.735,736 Such techniques are able to self-learn the optimal feedback strategy to reach the ultimate limits on the scaling of the phase estimation uncertainty, with limited number of measurements. They are both based on reinforcement learning that is model-free, since it does not necessarily rely on the explicit model of the problem, but mainly on experience acquired from data. Even if a mathematical model is available, reinforcement learning techniques can surpass gradient-based greedy algorithms for nonconvex optimizations in high-dimensional problems. In particular, PSO and DE are evolutionary algorithms.811,812 Such algorithms often resemble biological evolution mechanisms and are characterized by the following features: the presence of a population of points in the search space, the existence of a figure of merit called fitness to be maximized and, finally, stochastic evolution of the solutions. One of the biggest advantages of evolutionary computation is the low probability of getting stuck at local optima of the function, since the space is explored by many candidate solutions and the optimization of the searching process happens in a quasi-random way.

For phase estimation tasks, such approaches are applied to calculate, prior to the experiment, the sequence of optimal feedback phases shifts to be used during the adaptive experiments with N probes. Considering a Mach–Zehnder interferometer, at each step k of the experiment, the optimal feedback phase Φk can be updated according to the following Markovian rule with a logarithmic-search heuristic,

Φk=Φk1(1)xk1ΔΦk,
(83)

where Φk1 is the feedback phase at previous step, and xk1={0,1} is the result of the measurement at step k – 1. The list of optimal phase shifts {ΔΦk} for k=1,N is called policy. The final estimate for the unknown phase ϕ coincides with the last value ΦN of the adaptive feedback phase at the end of the process according to Φest=ΦN.

PSO is part of a class of unsupervised reinforcement learning algorithms for optimization problems,813,814 and can be exploited to compute the list of phase shifts {ΔΦk} discussed above. The goodness of a policy is quantified by the sharpness of Eq. (12) relative to the estimation errors. Hence, the average of the sharpness is calculated over P(θ|ρ), that is, the probability distribution of the error θ on the estimate given a policy ρ. In this way, the sharpness in Eq. (12) is the objective function that is maximized by PSO over the policies and is related to the Holevo variance. When the sharpness is maximized, the Holevo variance is minimized. Given the number N of employed photons in the estimation process, the goal of the PSO algorithm is to find the optimal policy by maximizing the associated sharpness. At each iterative step of the algorithm, every policy is mapped to a vector and compares its fitness with those relative to its neighborhood and to its past history. Then, the policies are updated according to a stochastic evolution rule depending on global and local optima. After a certain number of iterations, the last global optimum represents the solution of PSO. In Ref. 758, an adaptive scheme using PSO policies was realized using single photons in a path Mach–Zehnder interferometer, and SQL was approached after few photons (20).

However, it has been observed that the PSO algorithm converges to optimal solutions only when the number of probes is small, and this limitation can be overcome by other techniques like Differential Evolution.734,735 DE is an evolutionary algorithm that performs a global optimization in the policies space by selecting and rejecting candidate policies according to their sharpness value. In particular, after a random initialization of candidate policies, at each iteration of the algorithm new polices are generated by combining randomly chosen policies. The policies with highest fitness values are then selected for the next step. This procedure is iterated until a halting condition for the fitness of the best policy is reached. These techniques are also resilient to different models of noise.736 

In general, a physical process can involve more than one parameter. Analogously to the single parameter case, the estimation of multiple parameters can be enhanced by using quantum resources, giving rise to the emergent field of multiparameter quantum metrology.18,20 A large effort has been made to such generalization of the single parameter case.2,10,18–20,103,124,218,227,441,442,598,815–932 In this scenario, while it may be possible to estimate separately the single parameters, in most of the cases simultaneous approach has to be adopted. Two main motivations can be identified in this direction: (i) simultaneous estimation of the parameters can be more efficient, in terms of employed resources, with respect to the separate estimation; (ii) in certain conditions, even if the parameter of interest is a single one, the estimation process unavoidably involves other parameters, such as noises, which have to be estimated simultaneously.

A large class of problems involves the estimation of multiple parameters and can then benefit of a quantum enhancement. Different examples are parameter estimation for gravitational wave detection,30,933 multiple phases,227,876–892 phases and noises,893–899,901–904 estimation of mixed qubit states and quantum tomography,218,905–911 multidimensional fields,846 force sensing,823,843 spin rotations,854 rotations about unknown axes,855 general functions of unknown parameters,930,931 displacements in phase space,598,832,912–916 squeezing and displacement of radiation,829 imaging,11,441,442 localization of incoherent point sources,103,124,917–922,924–926 spatial deformation of sources,849,850 local beam tracking,934 sensing on biological systems,28 range and velocity measurements,840 atomic clocks networks,857 quantum sensing networks,835,836,935 magnetic field imaging,852,927,936–940 Hamiltonian parameters,141,839 and general sensing technologies.848 

Despite the large number of applications, multiparameter quantum metrology is characterized by several open questions with respect to the single parameter scenario. For instance, the possibility of saturating the ultimate quantum bound is not always guaranteed.2,815,824,858,859,885 In parallel to investigations on the theoretical framework, there is growing interest in experimental implementations where vast unexplored areas still remain.18,20

In Sec. V A, we will briefly introduce the theoretical framework of multiparameter quantum metrology. Then, we will describe some specific multiparameter problems that have been studied with photonic platforms. Note that recent in-depth reviews on multiparameter quantum metrology can be found in Refs. 18 and 20.

The general scheme of a multiparameter estimation follows the same steps of the single parameter case (Fig. 1): preparation of the probe state, interaction and parameter encoding, probe measurement, and the estimator function.

Consider a multiparameter estimation task where d unknown parameters λ=(λ1,λ2,,λd) are obtained through a set of estimators Λ(x)=(Λ1(x),Λ2(x),,Λd(x)), after x measurement results. Each parameter λi, with i=1,,d, can represent a physical quantity. When more than one parameter is involved in the process, the Fisher Information is generalized to the real-valued symmetric Fisher Information matrix (F)

F(λ)ij=x[1P(x|λ)P(x|λ)λiP(x|λ)λj].
(84)

The sensitivity of an estimator is quantified by its covariance matrix, which is defined as

C(λ)ij=x[Λ(x)λ]i[Λ(x)λ]jP(x|λ),
(85)

with i,j=1,,d. The covariance matrix provides a measure of the sensitivity relative to each parameter, while taking also into account the possible correlations between them.

In analogy with the single parameter case, a vector of estimators Λ(x) are said to be unbiased if the following relation holds:

x[Λ(x)λ]P(x|λ)=0.
(86)

A locally unbiased estimator is an unbiased estimator that satisfies the following constraint:

xΛk(x)P(x|λ)λi=δik.
(87)

Note that, in the case of continuous-valued measurement outcomes x, an integral over x will replace the sums in all these expressions.

For an unbiased estimator, the Cramer-Rao bound (CRB)208,824 in the multiparameter case is generalized to the following matrix inequality:

C(λ)F1(λ)/ν,
(88)

where ν is the number of independent measured probes. The CRB is well defined only when F is strictly positive, and thus invertible. In this case, the inequality in Eq. (88) can always be saturated by the maximum likelihood estimator824 in the limit of large ν. Conversely, local unbiased estimators can reach the CRB for any number of measurements ν.

In complete analogy to the single parameter scenario, it is possible to define a Quantum Fisher Information matrix,866,941FQ, that only depends on the initial probe state ρ0 and on the transformation Uλ. It is defined as

FQ(λ)ij=Tr[ρλLiLj+LjLi2],
(89)

where Li is the symmetric logarithmic derivative of ρλ with respect to the parameter λi, defined as λiρλ=(Liρλ+ρλLi)/2. FQ has the following properties:866 

  • Semi-definite positivity: FQ0;

  • Convexity: FQ(pρ1+(1p)ρ2)pFQ(ρ1)+(1p)FQ(ρ2) for any ρ1,ρ2 and p[0,1];

  • Additivity: given ν independent probes ρi (i=1,,ν), the Quantum Fisher Information matrix of the total product state ρtot=i=1νρi is: FQ(ρtot)=i=1νF(ρi);

  • FQ(ρ)=FQ(UρU), for any unitary U independent of the unknown parameters λ.

A review on the properties and applications and calculation techniques of Quantum Fisher Information matrix can be found in Ref. 866, including a discussion on the infinitesimal generators of the parameters.

The Quantum Cramer-Rao bound (QCRB) in the multiparameter case is the following matrix inequality:

C(λ)F1(λ)νFQ1(λ)ν.
(90)

In particular, by summing over the diagonal elements of the matrix inequality (90), one can estimate the precision of a multiparameter estimator as the trace of the covariance matrix in Eq. (85) that obeys the scalar bound,

i=1d(Δλi)2Tr[F1(λ))]νTr[FQ1(λ))]ν.
(91)

The QCRB is saturated when the equality in the second part of Eq. (90) is reached. Note that it is also possible to define other quantities and bounds, and classify different multiparameter problems.18,874 For instance, in the multiparameter scenario, different perspectives can be provided by other metrics such as the right logarithmic derivative Ri relative to a state ρλ815,824 defined as: iρλ=ρλRi. In this case, a matrix I(λ)ij can be defined as I(λ)ij=Tr[RiρλRj]. The following bound can be demonstrated:13,C(λ)I1(λ). In some multiparameter cases, this bound can be tighter, that is more accurate, than the QCRB in Eq. (90). Also different other methods and techniques, beyond Quantum Fisher Information matrix, can be more suitable to capture the incompatibility of measurements and other problems in quantum multiparameter estimation: a complete review on this topic can be found in Ref. 19. However, hereafter we will consider only the QCRB with the metric defined by symmetric logarithmic derivative.

Considering pure probe states (ρλ|Ψλ), FQ can be expressed according to the following relation:

FQ(λ)ij=4Re[λiΨλ|λjΨλ]+4λiΨλ|ΨλλjΨλ|Ψλ,
(92)

where |λiΨλ|Ψλ/λi.

In order to find the best possible accuracy on the estimation, it is fundamental to find necessary and sufficient conditions to saturate the QCRB. As previously anticipated, the possibility of achieving the ultimate quantum bounds in multiparameter estimations is not guaranteed,2,815,824,858,859,885 at variance with the single parameter case.10 Indeed, when different parameters have to be estimated, the corresponding optimal measurements may not commute, thus making impossible their implementation in a single experiment.716 In this way, the capability of achieving the ultimate bounds is forbidden.

A necessary condition for the attainability of the multiparameter QCRB inequality is provided by the following constraint:825,858

Tr[ρλ[Li,Lj]]=0.
(93)

The latter equality corresponds to requiring that the optimal measurements for the estimation of the single parameters are compatible observables, which in general may not be satisfied. Importantly, for pure states there exists a necessary and sufficient condition for the saturation of the QCRB. If FQ corresponding to state |Ψλ is invertible, the QCRB can be saturated if and only if858 

Im[Ψλ|LiLj|Ψλ]=0i,j.
(94)

Here, Li(λ) has the following expression for pure states: Li(λ)=2(|λiΨλΨλ|+|ΨλλiΨλ|). In Ref. 885, the authors generalize such results. In particular, in the case of pure states, necessary and sufficient conditions on projective measurements are derived such that the Fisher Information matrix F is equal to FQ even if FQ is not invertible. If FQ is invertible, such conditions are necessary and sufficient also for the saturation of QCRB. When the generators of the parameters commute and the probe state is pure, the QCRB can be saturated.825,885

In parallel to the single parameter case, FQ is related to the geometric distance between states, generalizing relation (26). Let us consider an infinitesimal variation δλi of the parameter vector λi. The following equality holds:10 

D̃B(ρλ,ρλ+δλ)2=18ijFQij(ρλ)δλiδλj,
(95)

where D̃B is the Bures distance. A technique able to optimize Bayesian multiparameter estimation in the presence of limited data has been proposed in Ref. 864.

Several studies, such as Refs. 227, 825, 836, 877, and 881, have investigated, in different scenarios, the potential advantages of performing multiparameter estimation with respect to sequential single-parameter strategies. Despite the broad range of applications, the number of experimental implementation of quantum multiparameter estimation tasks is surprisingly few. In this scenario, photons can be employed with different schemes and approaches.18,20,847,869 In the next sections, we list some of the problems that have been approached through photonic platforms.

An important task in quantum multiparameter estimation is provided by those problems where the physical quantities to be estimated are multiple phases. This scenario has been intensively studied in the last few years.227,836,847,861,876–883,885–892 More specifically, the unknown parameters are relative phases corresponding to different paths in an interferometer with respect to a common reference. Besides direct mapping of this problem to quantum imaging, multiphase estimation can represent a benchmark suitable for tests of quantum multiparameter protocols. Its importance and generality derive also from the fact that unitary evolutions generally introduce a phase in the evolved states.

Let us now consider multiphase estimation in a multiarm interferometer. Here, the unknown parameters are a set of phases (relative to a reference) along d arms of an interferometer: ϕ=(ϕ1,ϕ2,,ϕd). The general scheme of a multiphase estimation is sketched in Fig. 8. Preparation of the probe along the (d+1) paths is realized by an operation UA, considered to be unitary for simplicity. After the evolution Uϕ, that depends on the unknown phases ϕ1,,ϕd, the state is measured through a second unitary UB and projective measurements performed on the output paths. Finally, a suitable estimator Φ(x)=[Φ1(x),Φ2(x),,Φd(x)] provides an estimate of the phases by exploiting the m measurement outcomes x=(x1,,xm).

Fig. 8.

Multiphase estimation scheme. An initial probe ρin, living in the space of the (d + 1) paths, is prepared in a state ρ0 through a unitary evolution UA. Then, the probe interacts with the phases ϕ1,,ϕd according to an evolution Uϕ. The state is measured by means of a unitary UB followed by a projective measurement, giving outcome x. Finally, an estimate of the unknown phases is given by a suitable estimator Φ(x).

Fig. 8.

Multiphase estimation scheme. An initial probe ρin, living in the space of the (d + 1) paths, is prepared in a state ρ0 through a unitary evolution UA. Then, the probe interacts with the phases ϕ1,,ϕd according to an evolution Uϕ. The state is measured by means of a unitary UB followed by a projective measurement, giving outcome x. Finally, an estimate of the unknown phases is given by a suitable estimator Φ(x).

Close modal

For pure input probes, prepared in |Ψ0, the state after the phase unitary evolution Uϕ reads |Ψϕ=Uϕ|Ψ0, where Uϕ=e(ii=1dOiϕi). In this expression, each Oi represents the generator of the phase shift ϕi along the mode i. When the operators Oi mutually commute, and hence [Oi,Oj]=0i,j, the Quantum Fisher Information matrix FQ takes the following form:

FQ(ϕ)ij=4[OiOjOiOj],
(96)

where the average · is calculated with respect to state |Ψϕ. When the phases are those corresponding to independent modes, the generators are Oi = ni (see Sec. III A), where ni is the photon number operator for mode i. Since [ni,nj]=0i,j, from (96) we find that FQ(ϕ)ij=4[ninjninj]. Hence, the quantum Fisher Information FQϕi of a single phase ϕi corresponds to

FQϕi=FQii=4(Δni)2,
(97)

where (Δni)2 is the variance of the photon number operator ni.

One of the first studies on simultaneous quantum enhanced estimation of multiple independent phases was performed in Ref. 877. The authors considered probe states with a fixed number of photons, and a number d of independent phase differences to be estimated for d modes of an interferometer with respect to an additional reference mode. The simultaneous estimation of the phases can provide an advantage in the variance that scales as O(d), with respect to the best quantum strategy that estimates such phases individually.877 In particular, this result is demonstrated using suitable optimized projective measurements on the optimal quantum probe states of the form

|Ψopt=1d+d[|0,N,,0,0++|0,0,,N,0+|0,0,0,N]+dd+d|N,0,0,0,
(98)

where N is the number of photons contained in the probe state. The state is distributed along d + 1 modes and the last term of the superposition indicates N photons occupying the reference arm. Such optimal states lead to a total variance equal to

i=1d(Δϕi)opt2TrFQopt1=(1+d)2d4N2.
(99)

This leads to an advantage (in the variance) of a factor O(d) with respect to the optimal separate quantum single-phase estimation leading to Tr[FQsep1]d3/N2. This enhancement achieved by performing simultaneous estimation can be found also with noncommuting unitary parameter generators846 and in the presence of a small amount of losses.815 A simultaneous multiphase estimation can even provide a higher advantage by using entangled coherent states.861,879 In particular, Ref. 861, generalizes the result of Ref. 877 studying generalized multimode N00N-like states with arbitrary states along the nonvacuum mode.

Multiphase estimation in multimode interferometers has been theoretically studied in Refs. 882 and 942. A bound on the achievable sensitivity using separable probe states has been obtained,882 providing conditions of useful entanglement for the simultaneous estimation. A multimode interferometer is composed of two cascaded (d+1)-mode balanced multiport splitters [the (d+1)-mode extension of beam splitters], resembling the structure of a Mach–Zehnder interferometer. The internal modes include d independent phase shifts between the different internal paths with respect to one of the modes acting as a reference. In Ref. 882, the authors study input multimode Fock states |11|12··|1d+1|11··1, where |1i represent a single photon along the mode i. The benchmark for the sensitivity in Eq. (91) is given by the lower estimator variance, achievable by using m separable photons to jointly estimate the d phases,220,882

i=1dΔϕi2Tr[F1(ϕ))]mdm.
(100)

This limit is valid for each separable state transformed by the action of the phase generators, and for all possible POVMs. Hence, it represents the classical limit in this scenario. Useful entanglement is then present in the state when the variance of the estimator is lower than the bound (100). Such bound can be surpassed by injecting indistinguishable photons into the multimode interferometer.882 To reach optimal and symmetric bounds for each value of the jointly estimated phases, an adaptive estimation protocol can be in principle exploited. This is obtained by employing additional control phases along the mode of the interferometer to perform adaptive measurements.882 

A deeper insight into multiphase estimation is obtained by using the CRB/QCRB inequality in its matrix formulation of Eq. (90): C(ϕ)F1(ϕ)/νFQ1(ϕ)/ν, ν being the number of repeated independent measurements. The relevance of considering the covariance matrix C(ϕ) to study the sensitivity bounds is highlighted by the possibility to compare any target scenario, with corresponding Fisher Information matrix Ftarget, with a benchmark state associated with a Fisher Information Fbench. As shown by Ref. 227, such comparison can be studied through the matrix FtargetFbench. Indeed, the number of positive eigenvalues of this matrix corresponds to the number of independent combinations of the unknown parameters227 for which the target state provides an enhancement compared to the benchmark state.

1. Photonic platforms for multiphase estimation problems

Photonic systems represent the most natural platform for multiphase estimation problems. Surprisingly, not many experimental realizations of quantum multiphase estimation have been reported.

As previously discussed, a relevant benchmark problem is represented by the estimation of different optical phases along different spatial paths, with direct application in the vast area of imaging. Integrated circuits represent an ideal and scalable platform to investigate experimentally such scenarios. Besides the quality of spatial mode interactions, integrated photonics provides the stability that is necessary to estimate relative phases along different paths, which is almost impossible to achieve in bulk optics platforms because of thermal fluctuations and mechanical vibrations.

In Ref. 891, the authors realized the first experimental implementation of multiphase estimation enhanced by quantum states. The employed platform is an integrated three-mode interferometer realized through the femtosecond laser writing technique. Such a device is composed of two cascaded tritters (the three-mode analog of beam splitters)942,943 and includes six reconfigurable thermo-optic phase shifters [see Fig. 9(a)]. The first tritter, described by a unitary UA, prepares the input probe state starting from indistinguishable photons, through an HOM interference effect. The final tritter, described by UB, is part of the measurement process together with single-photon detectors. After calibrating the device, the capability to achieve quantum advantage in multiphase estimation was experimentally demonstrated by performing two-photon measurements.891 In particular, the Fisher Information of the device Fexp was estimated from experimental data and compared with that relative to the optimal simultaneous strategy with separable probes (Fcl). For some values of the unknown phases, the matrix Fexp - Fcl has two positive eigenvalues demonstrating a quantum advantage reached by the circuit. Such advantage can be in principle extended to all pairs of phases through adaptive protocols. The sensitivity enhancement was achieved experimentally with respect to classical strategies, considering as resources the number of effectively detected coincidences.891 The same setup has also been exploited in Ref. 892 for the implementation of a Bayesian adaptive multiphase estimation140 using single photon inputs.

Fig. 9.

Photonic platforms for multiparameter problems. (a) Integrated platform for the simultaneous estimation of two phases Δϕ1=φ1φref and Δϕ2=φ2φref. The probe states are two indistinguishable photons. The unitaries UA/B represent the 2-D decomposition of tritters, while Ri are the resistors used to tune the phases. Described in Ref. 891. (b) Scheme of the apparatus for the distributed sensing of the average of four phases φ1,,φ4. In the entangled estimation, the probes are squeezed coherent states generated in the optical parametric oscillator (OPO) and distributed along the four nodes through 50:50 beam splitters (BS). The measurement of the phase quadrature pi, along each node i, is performed by the homodine detection HDi. Finally, the average estimation is performed. Described in Ref. 888. (c) Scheme of the Mach–Zehnder interferometer in polarization, realized to perform measurements of phase and visibility of different samples. The probes are two-photon N00N state in polarization. Described in Refs. 903, 904, and 960. (d) Experimental apparatus for the estimation of the parameters describing unknown processes in the polarization space. The probes are four-photon HB states in polarization and the final measurement of the coincidence events (CC) is realized by a probabilistic photo-number detection. Described in Ref. 910.

Fig. 9.

Photonic platforms for multiparameter problems. (a) Integrated platform for the simultaneous estimation of two phases Δϕ1=φ1φref and Δϕ2=φ2φref. The probe states are two indistinguishable photons. The unitaries UA/B represent the 2-D decomposition of tritters, while Ri are the resistors used to tune the phases. Described in Ref. 891. (b) Scheme of the apparatus for the distributed sensing of the average of four phases φ1,,φ4. In the entangled estimation, the probes are squeezed coherent states generated in the optical parametric oscillator (OPO) and distributed along the four nodes through 50:50 beam splitters (BS). The measurement of the phase quadrature pi, along each node i, is performed by the homodine detection HDi. Finally, the average estimation is performed. Described in Ref. 888. (c) Scheme of the Mach–Zehnder interferometer in polarization, realized to perform measurements of phase and visibility of different samples. The probes are two-photon N00N state in polarization. Described in Refs. 903, 904, and 960. (d) Experimental apparatus for the estimation of the parameters describing unknown processes in the polarization space. The probes are four-photon HB states in polarization and the final measurement of the coincidence events (CC) is realized by a probabilistic photo-number detection. Described in Ref. 910.

Close modal

Recently, distributed quantum sensing of the linear combination (arithmetic average) of multiple small phases along four distant nodes was performed.888 The scenario914 is a network of M nodes along which independent relative phase shifts ϕi, with i=1,,M, one for each node, are experienced by the probes. The final goal is to estimate the arithmetic average of the phases: ϕ¯=i=1Mϕi/M. The employed probe state is a squeezed coherent state of the form D(α)S(r)|0, where D(α) is the displacement operator [Eq. (57)] with amplitude α, and S(r) is the squeezing single mode operator [Eq. (71)] with squeezing parameter r. The output state is detected through homodyne detectors along each node, thus measuring the phase quadratures Pi (i=1,,M) representing the estimators for the phases. Given such kind of state, two classes of estimation experiments are possible: (i) separable estimation in which M independent and identical squeezed coherent probes are sent each along a single node, thus separately estimating the associated phases, and (ii) entangled estimation in which a single initial squeezed coherent state is equally divided along the M nodes by initial beam splitters, that generate mode entanglement in the probe state [Fig. 9(b)]. The authors in Ref. 888 showed that, in the ideal case of unitary transmission, the optimal sensitivity for the entangled estimation shows a Heisenberg scaling 1/(MN) in both the number of modes M and mean number of photon N. This is obtained by optimizing over the initial probe state. Conversely, a separable estimation leads to a SQL scaling in M and Heisenberg scaling in N: 1/(MN). The authors experimentally demonstrated this entangled advantage in a network of M = 4 nodes and a probe state generated by an OPO at wavelength 1550 nm. In particular, using optimal probes containing N2.5 photons per mode, the measured standard deviation of ϕ¯ estimated was found equal to σent=0.099±0.003 for the entangled estimation strategy, while being equal to σsep=0.118±0.002 for the separable estimation one.888 The optimality of the demonstrated setup for estimating the average phase has been proved in a general framework by Ref. 889.

A similar implementation of an entangled sensor network, based on Ref. 914, was experimentally realized in Ref. 944 through a reconfigurable radiofrequency photonic platform. The probe is a phase squeezed state that is prepared through tunable beam splitters that allow to generate a CV multipartite entangled state along three separated sensors. Tuning the beam splitters, different states can be produced in order to maximize the sensitivity for different tasks, such as phase gradient and mean amplitude estimations.

Realistic scenarios involve the unavoidable presence of noisy channels. The effects of noisy parameters inside optical interferometers were considered in several theoretical and experimental investigations.31,403,405,452,896,945–956 In practical applications of phase estimation problems, the theoretical achievable quantum-enhanced precision is limited by photon losses403,405,947,952 and phase diffusion,896,945,946,948,957 and vanishes when significant noise occurs.895,957 In order to achieve effective quantum enhancement, optical quantum sensors require to take into account all these imperfections, often resulting in trade-off conditions on the achievable sensitivities. In this scenario, multiparameter estimation of both phase and noise represents a valid solution. A possible approach can be performing an a priori characterization of noise before the estimation process. However, in many cases time-varying systematical errors cannot be characterized in advance, such as phase oscillations due to thermal or mechanical fluctuation of optical systems.958,959 In these cases, simultaneous estimation of phase and noise is necessary.894 All these studies generally require calculation of multiparameter bounds in which noise is considered as a nonunitary evolution.

We describe below different classes of multiparameter phase and noise estimation implemented in photonic platforms: phase and phase diffusion, and phase and visibility.

1. Phase and phase diffusion estimation

Characterization of phase diffusion mechanisms can provide more complete information on the quantum sensor. The problem of estimating a single phase inside an interferometer in the presence of phase diffusion can be modeled by an out-of-control random phase shift, according to a Gaussian distribution of standard deviation Δ.893,894 Such width represents the noise strength, and the associated nonunitary evolution can be described by the following action in the Fock basis:894 

cΔ=eΔ2(mn)2|mn|,
(101)

which causes an exponential damping of the coherence terms. Hence, starting from a two-dimensional pure state |Ψ0=cosθ2|0+sinθ2|1, after the a phase shift evolution eiϕ along mode 1 and a dephasing process, the final mixed state will be

ρdep=(cos2(θ2)sin(θ2)cos(θ2)eiϕΔ2cos(θ2)sin(θ2)eiϕΔ2sin2(θ2)).
(102)

Multiparameter estimation of phase and dephasing has been investigated in various scenarios and platforms, from single qubit systems825,863 to a larger number of qubits,893 as well as considering both independent825 and collective dephasing.898 The quantum Fisher Information matrix FQ relative to the considered qubit |Ψ0 reads894 

FQ(θ,Δ)=(e2Δ2004Δ2e2Δ21).
(103)

In this case, the necessary and sufficient condition to saturate the QCRB [Eq. (93)] is satisfied. Hence, there exists an optimal measurement such that the errors on both the two parameters reach the ultimate limits predicted by the QCRB. It is then necessary to define the optimal measurement. Such task in this scenario highlights a fundamental difference between single and multiparameter estimation. In the former case, quantum resources in the measurement stage do not improve the achievable sensitivity.9 In the latter case, entangled measurements of probes can lead to an advantage in certain conditions.894,901 The estimation of phase and phase diffusion is one of these scenarios. To formalize the problem, let us consider the following quantity κ (for an estimation of d parameters):820,894,901

κ=1νi=1d1FQiiFii1,
(104)

ν being the number of independent probes. This figure of merit allows quantifying how much the chosen measurement is close to saturate the quantum limits relative to the employed probes and described by FQ. The question is when the QCRB, κd, is saturated.

In the qubit case (comprising N00N states), it has been demonstrated that κ1 for any estimation involving two parameters, if the separate probes are independently measured.215,894 Then, in this case the QCRB cannot be saturated. In contrast, for the estimation of phase and dephasing, we find κ1.5 when two qubit probes are collectively measured.894,901 Hence, in this case, even if the probes are separable, an advantage can be obtained with entangling measurements. Note that collective measurements do not always provide an enhancement for any multiparameter estimation. For instance, entangling measurements do not improve multiphase estimation tasks.901 

An experimental implementation of such improvement on simultaneous phase and dephasing estimation was demonstrated in Ref. 901. In this work, the probes are polarization qubits and the two-qubit entangled measurements are Bell measurements realized through a probabilistic controlled-sign gate. Given an arbitrary two-qubit state, such gate introduces a phase sign “−1” on all terms |V|V in which both qubits are vertically polarized. The controlled-sign gate can be realized in the polarization degree of freedom through partially polarizing the beam splitter whose transmission coefficients are appropriately chosen.961–963 In this way, the authors performed simultaneous estimation of phase and dephasing, and an optimal value κopt=1.18±0.02 was obtained.901 

In a different work,863 weak measurements were exploited to experimentally perform multiparameter estimation of a phase shift and its phase diffusion with classical probes.

2. Phase and visibility estimation

In certain scenarios, noise can be detrimental to the estimation of a parameter. In these cases, even if the noise value is not of interest, a multiparameter approach can be used to estimate the desired unknown parameter. A certain class of noise processes in interferometers can be modeled by the interference visibility v1. This parameter quantifies the quality of probes and apparatus, such as the visibility of HOM interference of two photons in a beam splitter.

In Ref. 903, an optical phase shift ϕ and noise over the probe state, measured in terms of visibility v of the interference fringes, have been simultaneously estimated. The output probability distributions of the system depend on both ϕ and v. and this implies the necessity of a multiparameter estimation. If the visibility is not properly estimated, the estimation of the phase shift would suffer of a bias. In the employed apparatus, that is a Mach–Zehnder, a N = 2 N00N state in the polarization degree of freedom interferes with the optical phase [Fig. 9(c)]. Coincidence measurements are performed to estimate the two unknown parameters. First, two-parameter estimation is made for the pre-calibration of the apparatus. Then, the scheme is exploited to study the optical activity of two different biological samples, fructose and sucrose aqueous solutions, through a Bayesian learning approach. The quality of the experimental estimation with respect to the CRB has been verified through a likelihood ratio test, defined as: l=m2Tr(F·C(ϕ,v))m(lndet(C(ϕ,v))+lndet(mF))2. More specifically, the null hypothesis corresponds to the covariance matrix C(ϕ,v) of the two parameters saturating the CRB. This quantity is distributed as a χ2 variable with three degrees of freedom. For the fructose solution, the authors obtained a sensitivity lf=2.63, while for sucrose, ls=0.10. Both these values are compatible with the null hypothesis in a 95% confidence interval.

The same scheme has been used also to perform tracking of a chemical process relative to the acid hydrolysis of sucrose.904 In this case, the change of the sample optical activity from dextrorotatory to levorotatory, due to the chemical reaction, turns in a phase variation between the polarizations and is measured with quantum super-resolution. More specifically, the real-time visibility v(t) and phase ϕ(t) are monitored as a function of the time t.

Finally, this setup was used for estimation of real-time invertase enzymatic activity, through a Bayesian adaptive technique.960 At each step, an additional known and controlled phase is chosen to maximize the Fisher Information depending on the current knowledge of the unknown phase.

Photonic sensors are exploited in other multiparameter scenarios. Here, we briefly list some examples.

A first scenario is the estimation of separations between incoherent point sources.103,111–114,121,124,129,135,184,252,700,917–922,924–926,965,966 In the case of two point sources, the two parameters are the difference of the two positions λ1=x1x2, and the corresponding centroid λ2=(x1+x2)/2. In Ref. 964, the authors proposed and realized experimentally the simultaneous estimation of the centroid and separation of two point sources, exploiting the HOM effect. In this experiment, photons generated by the two sources are sent to the input ports of a beam splitter. The coincidences between outputs of the beam splitter and two-photon events along the same output are measured with a spatial resolution technique.967 

A different multiparameter task is related to system characterization. A quantum-enhanced tomography of an unknown unitary process acting on the polarization degree of freedom was realized using multiphoton quantum states in Ref. 910. The task is also called Quantum Process Tomography.968–973 In a multiparameter approach, the aim is to simultaneously estimate the parameters characterizing a quantum evolution, by exploiting opportunely chosen probes and measurements. In Ref. 910, quantum process tomography was performed on unknown unitary evolutions in the two-dimensional space of photonic polarization. Given a polarization single photon state cH|H+cv|V, where H and V are polarization states and cH and cV are the corresponding amplitudes, a unitary evolution U acting on this state can be described by a matrix U=(a+ibc+idc+idaib), with a,b,c,d and a2+b2+c2+d2=1. Hence, the estimation regards three independent parameters and is performed by measuring three output probabilities pHV, pAD, and pRL along three different polarization bases. The authors employed HB four-photon polarization states split along the two polarization modes H and V: |2H|2V. The state is generated by a type-I noncollinear SPDC process in which the polarization of photons emitted along one mode is rotated by 45°. All photons along the two modes are recombined in a single spatial mode through a polarizing beam splitter. After its preparation, the four-photon HB state passes through the unitary evolution, performed by three cascaded waveplates, and is finally measured by a polarization selection stage and probabilistic photon-number resolving detection [Fig. 9(d)]. The probabilities are estimated from data through a maximum-likelihood technique and a quantum enhancement is observed in the estimation of random unitaries, considering the four-photon detected events as the employed resources.910 

In measurements of squeezed light, scattering of photons from the meter can cause parastic signal, called parasitic interference.974,975 This affects also measurements of gravitational waves, limiting the possible quantum advantages.571 The authors in Ref. 598 theoretically and experimentally demonstrated the concept of quantum dense metrology. In this scheme, a two-mode squeezed state is exploited in order to identify the parasitic noise and discard the corrupted data recovering quantum advantage. The setup is the same represented in Fig. 6(b). To this end, both the quadratures are simultaneously estimated beyond the SQL. The quantum dense metrology technique was applied also in modified protocols able to enhance the reduction of noises.913 In Table III we report some of the photonic platforms employed to perform multiparameter estimations with quantum strategies.

Table III.

Table of some photonic platforms exploiting quantum schemes for multiparameter estimation problems.

Estimation problemsReferencesPhotonic platformMean number of photonsProbe
Two phases 891  Integrated three-arm interferometer N = 2 Indistinguishable photons 
Average of four phases 888  Four-node network N4×2.5 Squeezed coherent states 
Phase and phase diffusion 901  Bulk two-mode polarization interferometer with entangling measurement N = 2 Separable two-qubit state 
Phase and visibility 903  Bulk two-mode polarization MZI interferometer measurements on fructose and sucrose N = 2 N00N state 
Phase and visibility 904  Bulk two-mode polarization MZI interferometer real-time measurement of sucrose acid hydrolysis N = 2 N00N state 
Phase and visibility 960  Bulk two-mode polarization MZI interferometer adaptive real-time measurement of invertase enzymatic activity N = 2 N00N state 
Centroid and separation of two incoherent sources 964  HOM interference in a BS and spatial resolution detection N = 2 Indistinguishable photons 
Unitary process in polarization space 910  Bulk polarization interferometer N = 4 HB states 
Estimation problemsReferencesPhotonic platformMean number of photonsProbe
Two phases 891  Integrated three-arm interferometer N = 2 Indistinguishable photons 
Average of four phases 888  Four-node network N4×2.5 Squeezed coherent states 
Phase and phase diffusion 901  Bulk two-mode polarization interferometer with entangling measurement N = 2 Separable two-qubit state 
Phase and visibility 903  Bulk two-mode polarization MZI interferometer measurements on fructose and sucrose N = 2 N00N state 
Phase and visibility 904  Bulk two-mode polarization MZI interferometer real-time measurement of sucrose acid hydrolysis N = 2 N00N state 
Phase and visibility 960  Bulk two-mode polarization MZI interferometer adaptive real-time measurement of invertase enzymatic activity N = 2 N00N state 
Centroid and separation of two incoherent sources 964  HOM interference in a BS and spatial resolution detection N = 2 Indistinguishable photons 
Unitary process in polarization space 910  Bulk polarization interferometer N = 4 HB states 

In this review, we provided an overview of the current state of the art in photonic technologies for quantum metrology applications. In particular, starting from the theoretical fundamental ingredients, we have discussed the most commonly adopted strategies and platforms for photonic systems, with particular attention toward application of adaptive strategies for efficient extraction of information. Finally, we have provided an overview of the recently expanding field of multiparameter estimation, which has potential applications in a large variety of fields where the process inherently involves multiple physical quantities at once.

Several open points, both from a theoretical and experimental point of view, still have to be addressed toward development of photonic quantum sensors capable of providing quantum enhancement in realistic noisy conditions.191,452,701,954,956,976–979 For photonic systems, the main challenges are represented by losses within the apparatus. Theoretical studies11,949,950,953,957 have indeed shown the detrimental effect of losses toward reaching sub-SQL performances, in particular, when the number of involved photons N in the prepared probes is large. Indeed, no quantum enhancement (in terms of scaling in N) can be achieved when losses are large enough, leaving space only for a constant improvement in this regime. All these results lead to a large effort toward development of appropriate metrological strategies capable of providing a more robust behavior in a noisy and lossy scenario.708,947,952,967,980–991 In parallel, experimental effort has been devoted to achieving technological advances in photonic systems. Such effort has enabled the first recent experimental demonstration of unconditional violation of the SQL in a two-photon experiment.400 In this direction, a significant amount of work still has to be done to improve the performances of photonic platforms to obtain quantum-enhancement in more complex scenarios. To this end, a significant intermediate step before reaching an unconditional violation of the SQL for larger sensors would be to obtain improved performances with respect to classical strategies, by comparing the achieved sensitivities in the presence of the same noise conditions. Note that, in the presence of noise, entangled estimation protocols employing external ancillas can lead to a higher sensitivity than the one achievable by using sequential unentangled estimations.701,705,706,992 In this sense, noisy cases are those where quantum advantage becomes evident.12 

A second promising research direction can be found in the multiparameter scenario, which has recently received growing attention for its wide range of applications.18,20 Additionally, recent studies have shown that a multiparameter approach can provide some advantages in the presence of noise.894,903 While the last few years have reported significant advances in the field, both in terms of theoretical background and of technological platforms, there are still several open points. Indeed, general recipes for the definition of optimal probe states for a general multiparameter scenario are still lacking. A similar issue is present for the definition of the optimal measurement strategies, in particular in the presence of noise which inherently requires considering mixed probe states. Finally, minimally invasive scenarios such as those involving biological systems993 require the development of quantum strategies tailored to obtain optimal performances when only limited data are available.717,994

E.P. and M.V. contributed equally to this work.

The authors thank Ilaria Gianani, Marco Barbieri, Stefano Pirandola, and Dominic Branford for useful discussions. This work was supported by the Amaldi Research Center funded by the Ministero dell'Istruzione dell'Università e della Ricerca (Ministry of Education, University and Research) program “Dipartimento di Eccellenza” (CUP:B81I18001170001), by MIUR via PRIN 2017 (Progetto di Ricerca di Interesse Nazionale): project QUSHIP (2017SRNBRK), by QUANTERA HiPhoP (High dimensional quantum Photonic Platform; Grant Agreement No. 731473), and by the ID No. 61466 grant from the John Templeton Foundation, as part of the “The Quantum Information Structure of Spacetime (QISS)” Project (qiss.fr) [the opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation] and by the Regione Lazio programme “Progetti di Gruppi di ricerca” legge Regionale n. 13/2008 (SINFONIA project, Prot. No. 85–2017-15200) via LazioInnova spa.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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