Photonic quantum metrology

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 $m≫1$⁠), each interacting a single time with the system under study, the estimation error will scale at best as $∼m−1/2$⁠. 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 $∼m−1$⁠, 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) E_x;

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

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,

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,

in the case of single measurement result x_i. 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: $limm→∞Λ¯=λ ∀λ$⁠.

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

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

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

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

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

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,

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 protocols^195–198 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 distribution $P(λ)$ 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 probability $P(λ|x)$⁠, is updated by Bayes' rule,

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

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

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 $k∈ℤ$⁠. For such a parameter, a circular mean, calculated over the posterior distribution, can be defined,

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

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, $VunbiasH=|⟨eiΦ⟩|−2−1$⁠, and coincides with the standard variance for sufficiently sharply picked distribution. With biased estimators, the variance is $VbiasH=|⟨ cos (Φ−ϕ)⟩|−2−1$⁠. 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 

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 

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

where E_x 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 ρ_j 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

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

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 E_x. This defines the Quantum Fisher Information (⁠$FQ$⁠),⁶ 

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

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

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

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

If the evolution of the interaction is unitary, $ρλ=eiλHρ0e−iλH$⁠, 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 $ρ0=∑nbn|Φn⟩⟨Φn|, FQ$'s explicit expression is²¹¹ 

Furthermore, for pure initial states $ρ0=|Ψ0⟩⟨Ψ0|$⁠, evolving in unitary evolution, a simple expression for $FQ$ can be found,^{2,3,10,11,211}

where the variance $(ΔH)2≡⟨(H−⟨H⟩)2⟩$ 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 ρ_2, can be quantified by the normalized Bures distance: $D̃B(ρ1,ρ2)=1−F̃(ρ1,ρ2)$⁠, 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

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 

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 $Uλi=Uλ ∀ i$⁠.

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: $ρtot=ρ1⊗ρ2,···,⊗ρm$⁠. The value of $FQ$ for a separable state is

where for the first equality the additivity of $FQ$ has been exploited and $FQmax$ represents the maximum of $FQ$ over the states ρ_m. Then, in the presence of ν independent packets of m classical correlated probes, from Eq. (19), the minimum uncertainty $Δλ$ scales as^9,220

Since $FQmax$ is a constant factor, the error scaling with the number of the probes m is $Δλ∝1/m$⁠. 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 $e−iλH, FQ$ assumes the form of Eq. (25). The bound is saturated by states of the form

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 

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

The term $(hS−hs)$ is a constant so, here, the error scaling with the number m of probes is $Δλ∝1/m$ 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 $π/[m(hS−hs)]$⁠.^198,221,222

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

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 transfer^160–162 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,

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 $ℏωk→$⁠. Such operators obey the following bosonic commutation rules:

where $ki$ and $kj$ are two modes of the field. The number operator$nk$ along mode k is represented by: $nk=ak†ak$⁠, and the energy can be written as: $Hem=∑kℏωk→(nk+12)$⁠.

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

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

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,

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 $ak|0⟩=0 ∀k$⁠. We can then generate any Fock state from vacuum by iteratively applying creation operators on the modes,

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,

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

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

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

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:²⁴² $ak→∑j(ηkjaj+βkjaj†)+γk$⁠, 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 

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

The second evolution term $∑k>lgkl(2)ak†al+h.c.$ 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 $∑klgkl(3)ak†al†+h.c.$⁠, 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

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,

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

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

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

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: $bi†=∑kUik ak†$⁠. 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

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): $σx=a1†a2+a2†a1, σy=−i(a1†a2−a2†a1)$⁠, and $σz=a1†a1−a2†a2$⁠. The following relations hold:^265,266

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 photons^267–275 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.

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 $mℏ$⁠, where $m∈ℤ$ 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: $|L⟩|LGm⟩→QP|R⟩|LGm+2q⟩$ and $|R⟩|LGm⟩→QP|L⟩|LGm−2q⟩$⁠, where $|R⟩$ (⁠$|L⟩$⁠) is the right (left) circularly polarized state, and $q∈ℤ$ 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-bin^311–313 and time-frequency^314–317 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).

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 $k→p$ and frequency ω_p) passing through the crystal is annihilated, while a pair of photons is generated: idler (momentum $k→i$ and frequency ω_i) and signal (momentum $k→s$ and frequency ω_s). Given $|α⟩p$ the pump coherent state, the process in the Fock basis representation reads $|α⟩p|0⟩i|0⟩s→apai†as†|α⟩p|0⟩i|0⟩s≈|α⟩p|1⟩i|1⟩s$⁠, where a_x 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) $k→p=k→i+k→s$⁠. 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 $|α⟩p1$ and $|α⟩p2$⁠, the first order process generating a single pair is: $|α⟩p1|α⟩p2|0⟩i|0⟩s→ap1ap2ai†as†|α⟩p1|α⟩p2|0⟩i|0⟩s≈|α⟩p1|α⟩p2|1⟩i|1⟩s$⁠. The energy and momentum conservation relations are: $ωi+ωs=ωp1+ωp2$ and $k→p1+k→p2=k→i+k→s$⁠.

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 

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 technologies^353,354 such as Silica-on-Silicon^353–355 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) technique^365–367 exploits laser pulses to write waveguides inside glass (but also crystalline or polymeric) material substrates. Such a technique guarantees low cost, low loss (⁠$∼0.1−0.3$ 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 semiconductors^376–380 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.

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

where $α∈ℂ$ 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

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: $α=(⟨x⟩+i⟨p⟩)/2$⁠. 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

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

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

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

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

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.

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

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 $|Ψ⟩N00N$ is: $(Δn)2=N2/4$⁠; hence, from Eq. (25), the sensitivity achievable by N00N states is

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 $(|N,0⟩+|0,N⟩)/2→Uϕ(|N,0⟩+eiNϕ|0,N⟩)/2$⁠, 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

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 path^418–420 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 

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

The goal is to avoid all the non-N00N contributions $|k,N−k⟩$ with $k≠0,N$⁠. 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 path^434,435,439 degrees of freedom, in four⁴³⁹ and two^434,435 mode bulk interferometers. Some of such works^432,434,435 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 process^260,443 (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

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 

The reported experimental violation was $ηNV2N≈1.23$⁠, thus showing genuine quantum enhancement. In order to reach this violation, high efficiency (>0.95) detectors, low losses circuits, and high visibility fringes (⁠$V≈0.98$⁠) 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 written^458,459 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 $L=N−M$ photons along the other,

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

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:¹² $FQN00N=ηNN2$⁠, 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.

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:

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: $⟨X1|X2⟩=δ(X1−X2)$ and $⟨P1|P2⟩=δ(P1−P2)$⁠. 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-distributions^471,472 one of the most used is the Wigner function^{12,407,468,471,473–475} that, for a d-mode state ρ, is defined as

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: $∫−∞+∞ddX ddP Wρ(X,P)=1$⁠. From $Wρ(X,P)$⁠, one can recover the real marginal quadrature distributions,

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 states^{242,244,468,480–483} 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): $Δx Δp=1$⁠. 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,

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