Extremal quantum states

Despite the heated foundational debate concerning its interpretation,¹ it seems indisputable that the notion of a quantum state is a keystone of quantum theory.² Quantum states are ultimately responsible for the various counterintuitive features that cannot be fathomed in classical scenarios: these include quantum superpositions, coherence, no-cloning, entanglement, and quantum correlations, to cite but a few.³ The hunt for useful quantum technologies in fields as diverse as computing, metrology, and information processing relies precisely on exploiting these remarkable resources.

A natural question arises: how quantum is a state? What makes one state more quantum than another? Is there a trait to which one can point and say “this embodies quantumness”? In this respect, we deem quantumness to be the most appropriate term to refer to the quantum nature of a state, rather than nonclassicality, which is quite often used and implicitly assumes a classical perspective.

In quantum optics, quantumness is largely linked to the particle aspects of photons, whereas photons' wave aspects can be well understood in classical terms. However, this is too heuristic to be of any quantitative use. The quantumness of a system can be measured through a number of indicators, the most popular being squeezing,^4–6 sub-Poissonian statistics,⁷ and negativity of the Wigner function.⁸ In terms of these indicators, numerous quantifiers have been discussed; relevant examples include Mandel's Q parameter,^9,10 distance-based measures,^11–17 nonclassical depth,^18–20 moment methods,^21–23 operator ordering sensitivity,²⁴ and resource-based indicators.²⁵ All these approaches have their own insights and merits in quantifying the quantumness of a state, but each highlights only one particular aspect of the problem. It seems impossible that there exists a unique and universal quantifier for quantumness suitable for all purposes.

These quantifiers also have some drawbacks. First, most of them are bespoke, tailored to continuous variables or other special cases. For example, the established negativity of the Wigner function is merely a proxy for the non-Gaussianity of the state expressed in canonical position-momentum coordinates; this has no significance for other dynamical variables, such as the ones associated with spins and orbital angular momentum, for which the concept of Gaussianity is equivocal. Second, many of them have no clear operational meaning and are hard to implement in the laboratory. This is the true, e.g., for the distance-based measures; they require a full tomography to be evaluated (which is challenging in many cases), and there are no physical arguments that favor any of the multiple metrics studied in the literature. Third, in some cases, the resulting degrees of quantumness order states differently, from whence ensues, at the very least, puzzlement. This is often related to the first drawback: in naively applying bespoke principles to other systems, one may be surprised that they disagree as to which states are the most quantum. We, therefore, advocate for the removal of a privileged example (such the harmonic oscillator) from a crowning of the most quantum states.

In this paper, we investigate the indicators of quantumness, which are of universal application and are not restricted to a particular symmetry. The phase-space approach is germane to these purposes. Presenting quantum theory as a statistical theory on a classical phase space has attracted a lot of attention since the dawn of the former, providing clear insights into the classical-quantum boundary. In addition, this setting avoids the operator formalism, thereby freeing quantization from the burden of Hilbert space.²⁶ The extensive literature on phase-space approaches to quantum theory can be found in books^27–29 and reviews.^30–34 

The phase-space approach maps the state of a system to a function on the phase space. However, this mapping, first suggested by Weyl³⁵ and later put on solid grounds by Stratonovich,³⁶ is not unique: a whole family of s-parametrized quasiprobabilities can be assigned to each state. The most common choices of s are +1, 0, and −1, which correspond to the P (Glauber–Sudarshan),^37,38 W (Wigner),³⁹ and Q (Husimi–Kano)^40,41 functions, respectively. For continuous variables, such as Cartesian position and momentum, the textbook example that triggered interest in this field, the parameter s defines different orderings of the basic variables.

We stress that all of these quasiprobability functions contain complete information about the state; there is no privileged status for any of them. The choice of a particular instance depends exclusively on its convenience for the problem at hand. In this paper, we are unabashedly pro-Husimi for it is positive and always exists regardless of the choice of phase space, revealing the quantumness of a state in a most conspicuous manner. Moreover, its moments are straightforward to determine experimentally, allowing one to directly build information about a quantum state in a logical manner.

More concretely, our aim is to use the Husimi Q-function as the basic ingredient to formulate sensible extremal principles, so we can determine in a consistent manner the least and the most quantum state. Extremal principles are of paramount importance in physics, of which Fermat least-time, Hamilton least-action, and minimal entropy principles are well known examples. In some sense, extremal principles are guided by the Aristotelian notion of economy in nature. In this respect, it is important to emphasize that even if the guiding quantities, such as action or entropy, sometimes have no simple operational definition, the resulting extremal states have complete physical significance. Following this approach, we explore the states that are likely to play a major role in demonstrating true quantum advantages.

A final word of caution: for composite systems, quantum correlations⁴² and entanglement measures^43,44 provide powerful tools for inspecting quantumness. In addition, on physical grounds, the quantum properties are largely revealed through the projection associated with the measurement process⁴⁵ and the concomitant modification of the state of the system. Therefore, to evaluate quantumness, one needs several sequential measurements on the same system. This calls for correlating two or more measurements: then, there are more subtle tests such as looking for variances of measurements, which concern the statistics of several independent single measurements.⁴⁶ These questions are certainly enthralling, but they lie outside the scope of our present work.

In the continuous-variable (CV) setting, quantum information is encoded in degrees of freedom with continuous spectra.^47,48 The best-known example of this is the quantized harmonic oscillator, which can be described by the position and momentum. The CV setting is an excellent tool to represent, among other examples, modes of the electromagnetic field, Josephson junctions, and Bose–Einstein condensates. In this section, we present our quantumness indicators for this outstanding case.

We start by briefly recalling the structures needed to set up a proper phase-space description of the Cartesian quantum mechanics of a harmonic oscillator. For simplicity, we choose one degree of freedom, so the associated phase space is the complex plane $C$⁠. The relevant observables are the Hermitian coordinate and momentum operators $x̂$ and $p̂$⁠, with canonical commutation relation (⁠$ℏ=1$ throughout)

These operators are the generators of the Heisenberg–Weyl algebra,⁴⁹ which is the hallmark of noncommutativity in quantum theory. In general, the elements of this algebra can be written as linear combinations of the three generators, i.e., $is1̂+i(px̂−xp̂)$⁠. The elements of the group corresponding to the Lie algebra [Eq. (1)] are obtained, as usual, by exponentiation: $ĝ(s,x,p)=eis exp [i(px̂−xp̂)]$⁠. To interpret the action of this element, we recall that the action of the basic subgroups $Û(x)=exp (−ix p̂)$ and $V̂(p)=exp (ip x̂)$ in the bases of eigenvectors of position and momentum is

so they represent displacements along the corresponding coordinate axes. Therefore, the operator $D̂(x,p)=eixp Û(x)V̂(p)$ corresponds to a displacement operator in phase space.

Instead of the variables $x̂$ and $p̂$⁠, another pair of operators is often more suitable: they are defined by

so that their commutation relation is

The displacement operator can, then, be written as

where the complex number α is $α=(x+ip)/2$⁠. Note that $D̂†(α)=D̂(−α)$⁠, while $D̂(0)=1̂$⁠. Displacement operators obey the simple composition law

The displacements constitute a basic ingredient for the definition of coherent states. Let us choose a fixed, normalized reference state $|Ψ0⟩$⁠. For any $α∈ℂ$⁠, we can define the coherent state $|α⟩$ as^50,51

so that the resulting set of states are parametrized by phase-space points and inherit properties from those of $D̂(α)$⁠. The standard choice for the fiducial vector $|Ψ0⟩$ is the vacuum $|0⟩$⁠. This has quite a number of relevant properties, which make any coherent state a minimum-uncertainty state, namely,

where $Varϱ(Â)=⟨Â2⟩ϱ̂−⟨Â⟩ϱ̂2$ is the variance of the operator $Â$⁠.

Coherent states are not mutually orthogonal; their overlap is

but they can be considered as approximately orthogonal for sufficiently different values of α and β. Still, coherent states form an overcomplete basis in the space of states, so we have the resolution of unity

in terms of nonorthogonal projectors. This completeness relation allows us to expand an arbitrary pure state $|ψ⟩$ as

where $ψ(α∗)=⟨α|ψ⟩$ is the wave function in the coherent-state representation, which completely determines the state $|ψ⟩$⁠. If we take into account that the expansion on the Fock basis ${|n⟩}$ of a coherent state reads

we can write

where $fψ(α)$ is an entire analytic function called the Bargmann–Segal representation of the state.^52,53

Finally, we mention that coherent states are quite often singled out as quasi-classical, in the sense that the mean values of the position and momentum operators oscillate in agreement with classical laws. However, this is misleading because, according to the Ehrenfest theorem,⁵⁴ this holds true for any state under the dynamics of the harmonic oscillator.

The attempts to find a description of quantum states that are similar to classical states have given rise to several quasiprobability distributions, as discussed in Sec. I.

The Wigner function permits a direct comparison between classical and quantum dynamics. For coherent and squeezed states, the Wigner function takes the form of a Gaussian. Generally, however, it is not a positive function and thus not a probability distribution, even though it is normalized. To overcome this shortcoming, the Husimi Q-function is defined as

It is everywhere non-negative and properly normalized,

and can, therefore, be interpreted as a genuine probability distribution. Nevertheless, unlike classical probability distributions, which obey no further constraints, it is also bounded from above by $1/π$⁠. In addition, it is a distribution of a somewhat peculiar kind since it is not a probability density for mutually exclusive events. Instead, $Qϱ(α)$ is the probability that the system, if measured, would be found in a coherent state whose probability density has its mean at α. Such events are not mutually exclusive because the coherent states overlap with each other.⁵⁵ 

Compared to the Wigner function, the Q-function has the disadvantage that one does not recover the correct marginals. Moreover, its definition depends on the definition of the coherent states and hence on some special reference state in the Hilbert space. Yet, this oddity can be turned to an advantage. The Q-function is the pertinent quasiprobability distribution to employ when the measurement device introduces noise that can be modeled by the reference state used to define the Q-function.⁵⁶ 

For a pure state $|ψ⟩$⁠, the Q-function reduces to

The zeros of the Husimi Q function are, then, the complex conjugates of the zeros of the Bargmann function $fψ$⁠: they constitute the so-called stellar representation of the state⁵⁷ (also called the Majorana constellation⁵⁸). Two states with the same constellation are the same up to a global phase. The function $fψ(α)$ can, thus, be aptly called the stellar function.⁵⁹ Since this function is an analytical function, its zeros form a discrete set.⁶⁰ 

Any single-mode Gaussian pure state may be obtained from the vacuum by way of a displacement $D̂(β)=exp (βâ†−β∗â)$ and a squeezing operation $Ŝ(ξ)=exp [12(ξâ2−ξ∗â†2)]$⁠. The resulting state $|β,ξ⟩$ is the most general Gaussian pure state, and its corresponding stellar function is⁶¹ 

where

and $ξ=r eiθ$ (with $r≥0$⁠) is the squeezing parameter. Setting ξ = 0 yields the stellar function for the coherent state $|β⟩$⁠, whereas setting β  = 0 yields the stellar function for the squeezed vacuum $|ξ⟩$⁠. It is clear that the stellar function of a Gaussian state does not have zeros and the associated constellation is void.

At the opposite extreme, we have the number states, with stellar functions

so the constellations reduce to the origin, with multiplicity n.

We can now use antistereographic projection to represent the roots as points on the unit sphere whose south pole touches the origin of the complex plane. It is straightforward to prove geometrically that the line that connects the north pole with a point $α=tan (θ/2)exp (iϕ)$ in the complex plane will intersect the unit sphere at a point with spherical coordinates $(θ,ϕ)$⁠; this point is the associated Majorana star. In this picture, Gaussian states collapse to the north pole, while the south pole corresponds to number states. In between, we have states represented by a set of points on the sphere. As an intermediate example, we consider the cat states

with $N±=2[1±exp (−2|β|2)]$⁠, which have also been called even (+) and odd (−) coherent states.⁶² For the odd case $|cat−⟩$ and $β=iy$⁠, we have

and the constellation is now discrete but with infinite points. One can also find core states, as termed in Ref. 63, which are single-mode normalized states that have a polynomial stellar function (and so a discrete constellation). This evidence seems to suggest that quantumness can be linked to the geometrical properties of a state's constellation.

The moments of $x̂$ and $p̂$ are often the most experimentally accessible pieces of information. For Gaussian states, the first two moments suffice, but, in general, higher-order moments contribute valuable information. Since the dynamical variables are noncommuting, higher-order moments make unambiguous sense only after we commit to a particular operator ordering scheme or, equivalently, to a particular quasiprobability. The case of Weyl (or symmetric) ordering, corresponding to the Wigner function, has been carefully examined;⁶⁴ here, we solve the problem from the Husimi perspective.

To elucidate the question, we first briefly recall the so-called s-parametrized Stratonovich–Weyl map,⁶⁵ 

which maps each observable $Â$ onto a function on the complex plane $ℂ$⁠. The corresponding kernels $ŵ(s)$ were worked out by Cahill and Glauber,⁶⁶ 

The function $WA(s)(α)$ is called the symbol of the operator. The value s = 0 corresponds to the Weyl symbol (symmetric ordering), whereas s = 1 and s = −1 lead to the contravariant P-symbol (normal ordering) and covariant Q-symbol (antinormal ordering), respectively. In particular, the symbols of the density operator are the traditional quasiprobability distributions: they are all covariant under displacements and provide a basic overlap relation

From this viewpoint, the Wigner symbol is self-dual, while the P and Q symbols are dual to each other. To stress this point, we will use the notation $ŵ(+1)(α)=p̂(α)$ and $ŵ(−1)=q̂(α)$ so that

Using the definition [Eq. (23)], the $q̂$-kernel can be explicitly written as

with

where $Ln(k)(x)$ are the associated Laguerre polynomials. Equation (26) can be rewritten in the following compact form:

where $K=0, 1/2, 1,…$⁠, and we have introduced the operators

which constitute a basis in the algebra of monomials. These homogeneous polynomials form a set and transform linearly among themselves under symplectic transformations in Sp(2, $ℝ$⁠), i.e., transformations that preserve the Poisson bracket for classical descriptions or the commutator [Eq. (1)] for quantum ones.^67,68 In this way, we can expand an arbitrary density operator as

where the coefficients can be read off from

These coefficients can be called the state multipoles, in close analogy with the established techniques that we will use in Sec. III C.

The non-negativity of the density operator of a state is faithfully encoded in the Husimi Q function, which, then, constrains the moments of the Husimi distribution. These constraints appear in a canonically covariant form when expressed in terms of the multipoles. Actually, the cumulative multipole distribution

can be interpreted as a generalized uncertainty principle.⁶⁴ This will play a crucial role in Sec. III as a good indicator of quantumness. It is reasonable that coherent states maximize the quantity $AM$⁠, at least for large M and, again, are the least quantum states.

The states that minimize $AM$ can be seen as the most quantum ones. For a spin, they will be called the Kings of Quantumness. We, thus, look for the constraints on the density operator $ϱ̂$⁠, which allow the cumulative multipoles to vanish to a given order. This imposes that all $ϱKq=0$ for $K≤M$ for some M.

We first examine the case q = 0. According to Eq. (31), we have

where $LK(x)$ are the Laguerre polynomials. The function $P(α)$ can be formally expressed as⁶⁹ 

where $ρnm$ are the matrix elements of the density operator in the Fock basis, and we have expressed the complex amplitude in polar coordinates $α=r exp (iθ)$⁠. Since $qKK(α)=qKK(r)$⁠, the integral over θ is readily carried out. The radial part can be integrated by parts, with the result

For $ϱK0=0$ to hold for all integers $K≤M$⁠, we must have all $ρnn$ be equal for $n≤M$⁠. If these components are nonzero, then the state must be mixed. All states whose diagonal components $ρnn$ are only nonzero for $n>K$ have $ϱK0=0$⁠. The lowest-energy state with this property is clearly the Fock state $|K+1⟩$⁠. This state is pure.

For $q≠0$⁠, a similar calculation shows that

and an analogous expression for q < 0. From here, one can check that all of the off-diagonal components smaller than M must vanish for a state to have $AM=0$⁠. Borrowing the terminology of spins, we refer to these states as Mth order unpolarized.

We, thus, conclude that to be unpolarized to order M, a state can either have no components with energy less than or equal to M or can be proportional to the identity on that subspace. The lowest-energy pure state that is unpolarized to order M is, thus, the lowest-energy pure state that only has components with more energy than M: the Fock state $|M+1⟩$⁠.

The standard von Neumann definition of the quantum-mechanical entropy

vanishes for all pure states. Thus, it cannot distinguish between various pure states, and it is rather a measure of the purity of quantum states. A different definition of classical-like entropy associated with a quantum state was proposed by Wehrl,⁵⁹ 

The Wehrl entropy can loosely be interpreted as an information measure for a joint noisy measurement of x and p. The Q-function can never be so concentrated that $SW$ becomes negative. On the contrary, classical distributions can be arbitrarily concentrated in phase space, and classical entropies can take on negative values.⁷⁰ They may even tend to minus infinity if their distributions tend to delta functions. Moreover, Wehrl proved the even stronger relationship,

which establishes a connection between the Wehrl and the von Neumann entropies of a given state.

However, the most important property of this classical-like entropy is, perhaps, the following inequality:

Lieb,⁷¹ using sophisticated techniques from functional analysis, was able to demonstrate that the equality holds if, and only if, the considered state is a coherent state. In other words, coherent states minimize the Wehrl entropy.

We observe that the Wehrl entropy is a good measure of the strength of the coherent component. That is, it measures how “close” a given state is to a coherent state. For any coherent state, the Wehrl entropy is identically equal to one. The Wehrl entropy does not change when a state undergoes a displacement operation: for general Gaussian states, we have

Thus, the Wehrl entropy of squeezed states with a nonzero coherent component is smaller than the squeezed vacuum with the same average photon number. Observe that the Wehrl entropy is independent from the coherent component of the state.

The natural question is what are the states maximize the Wehrl entropy. If we fix the average photon number $n¯=⟨N̂⟩$⁠, a simple calculus of variations gives⁷⁰ 

which corresponds to single-mode thermal radiation. However, this is a mixed state. A more fair way of proceeding would be to restrict ourselves to pure states, as this is the case for coherent states. Our physical intuition suggests that we first assess number states; a direct calculation gives

where $ψ(n)$ is the digamma function.⁷² 

Interpolating between coherent states and number states are the photon-added coherent states,⁷³ 

These states offer the opportunity to closely follow the smooth transition between the particlelike and the wavelike behavior of light.⁷⁴ Their mean number of photons is given by

which increases with $|β|$ and m.

The Wehrl entropy for these states can be analytically computed. The final result looks daunting. viz.,

where $F11(a;b;z)$ is the Gauss confluent hypergeometric function. However, we recover the proper limits: for small $|β|$⁠, the Wehrl entropy becomes

which decreases with $|β|$⁠, whereas for large $|β|$⁠, $SW(|β,m⟩)$ decreases to 1.

Even though displacing a state has no effect on its Wehrl entropy, it does have an effect on its average photon number $n¯$⁠. This means that there exist states with the same Wehrl entropy as $|β,m⟩$ but with a smaller average number of photons. Displacing a Fock state always increases the average photon number; similarly, it is easy to find γ such that $D̂†(γ)|β,m⟩$ has minimal $n¯$⁠. Nonetheless, Fock states with this minimum average photon number still have greater Wehrl entropy than $|β,m⟩$ (see Fig.1). Similar considerations help motivate our conjecture that Fock states maximize the Wehrl entropy for a given average photon number.

The von Neumann entropy S_Q, containing the factor $ln ϱ̂$⁠, involves all the powers of the density operator, which complicates its evaluation. For this reason, it is useful to linearize the logarithm as $ln ϱ̂=ln [1−(1−ϱ̂)]≃−(1−ϱ̂)$⁠, and then, S_Q reduces to $SL=1−Tr(ϱ̂2)$⁠. The resulting linear entropy is just as good as the von Neumann entropy for quantifying the degree of purity of a state. However, S_L is easier to calculate than S_Q, given that it does not require diagonalization of the density matrix and hence its appeal for quantum information.

The same linearization can be applied to the Wehrl entropy. The resulting measure

might be called the linear Wehrl entropy. Interestingly, the integral of the square of the Husimi distribution appearing in the definition of $SW,lin$⁠,

does not reduce to the purity, as would happen if one would calculate the square of the Wigner function. (Notice that with the Husimi Q function being a probability density, its moments must be understood as in Sec. II C. However, one often finds in the literature the term moment^75,76 referring to quantities such as M₂.)

For a pure state with amplitudes ψ_n in the Fock basis, $M2(ϱ̂)$ can be expressed as

The integral can be directly computed to yield the compact result⁷⁵ 

with

Explicitly related to $SW,lin$ is the quantity

known as the inverse participation ratio, which has been widely used as a measure of the localization of a state in phase space.^77–79 

For a coherent state $|β⟩$⁠, we have

and this corresponds to the most localized state. Squeezing delocalizes the state

From this perspective, quantumness is loosely associated with delocalization in phase space.

Again, we assess the question for number states: their only nonzero element is $|BK|2=δK,2n2−K(KK/2)$⁠, and consequently,

This is the same scaling with $n¯$ as for squeezed states [Eq. (55)], but number states are less localized by a factor of at least $π$⁠.

We can also look at intermediate cases, such as the states

with average photon number $n¯=n+(p−1)δ$⁠. Their moments decrease at the same order of $n¯$ as squeezed and number states but with a slower prefactor, regardless of the sign of δ, which only contributes on the order of $n¯−3/2$⁠,

The participation ratio is smaller for $|δ⟩$ than for $|n⟩$ for the simple reason that the former has more nonzero coefficients ψ_n. Since $1+2p(1−p)<π$⁠, the states $|δ⟩$ are less localized than squeezed states for all p.

The inverse participation ratio measures the localization of the Husimi function through its second-order moment. Another sensible way of quantifying localization is the maximum value attained by the Q function, which is a proxy for determining the infinite-order moment. Therefore, if we restrict for simplicity to pure states, we consider the quantity⁸⁰ 

This quantity is a proper distance measure for pure states and is invariant under arbitrary displacements. It can be related to the Hilbert–Schmidt distance between a state $|ψ⟩$ and the convex set of coherent states, where the latter serves as reference for the most localized states.¹² The states that minimize $M∞$ are seen to be the most quantum ones. For a spin, when this distance is taken to be that of the closest mixture of coherent states to a given pure state, the most quantum states will be called the Queens of Quantumness.

Coherent states clearly attain the maximal value

For more general Gaussian states, we can restrict ourselves to the squeezed vacuum since the displacement of states does not influence their values of $Qmax$⁠. We, then, have

showcasing their high degrees of quantumness.

For number states, a direct calculation gives

The upper limit value of quantumness, corresponding to the lowest value of $M∞$⁠, is reached when $n≫1$⁠.

As before, we can look at the intermediate photon-added coherent states $|β,m⟩$⁠. Since adding m photons always scales the Q function by $|α|2m$ and a normalization constant,⁸⁰ one can readily compute

The Laguerre polynomials decrease with $|β|2$ and decrease with m; this immediately allows us to identify Fock states, with n = m and $|β|2=0$⁠, as having the highest quantumness.

As a final example, we return to the cat states [Eq. (20)]. Now, for the even cat states, we find

and one sees that the quantumness grows continuously as β grows but is eventually saturated. This accords with the decoherence rate for superpositions of coherent states increasing with increased separation between the components of the superposition (in the phase space).^20,81 Similar conclusions can be found by numerically calculating $M∞$ for almost all of the Yurke–Stoler states,⁸² 

where for all $θ≠(2k+1)π$⁠, the quantumness grows from $M∞=1$ (for β = 0) and asymptotically reaches $M∞=12$ in the limit $β→∞$⁠. The odd cat states, in contrast, already possessing some quantumness in the limit $β→0$ because their vacuum components vanish, have the $M∞$ range monotonically from $exp (−1)≃0.37$ to $12$ as β increases from 0 to infinity, with quantumness decreasing with increased β.²⁰ 

The field of quantum metrology, in the broadest sense, concerns itself with inventing useful nonclassical states and extracting from them some requisite metrological content. Loosely speaking, the metrological power of a quantum state is the metrological advantage that can be attributed to that state relative to classical limits.^83–85 

The basic elements of parameter estimation can be found in a number of comprehensive reviews.^86–93 The most fundamental element of quantum metrology is the probe state, which is represented by some density operator $ϱ̂$⁠. We are often interested in estimating a single parameter λ encoded via some quantum channel $Eλ$⁠. Information about λ is extracted by passing a state $ϱ̂$ through the quantum channel $Eλ$⁠, resulting in the state transformation $Eλ(ϱ̂)=ϱ̂λ$⁠. For simplicity, we will restrict our attention to unitary channels so that $ϱ̂λ=Ûλ ϱ̂Ûλ†$⁠. The unitary operator $Ûλ$ can be written as $Ûλ=exp (−iλĜ)$⁠, where $Ĝ$ is a Hermitian operator that is called the generator of the unitary transformation.

The information about λ is, thereby, encoded onto the probe $ϱ̂$⁠. In order to estimate λ, we perform a measurement on $ϱ̂$⁠, which is represented by some set of positive operator-valued measures (POVMs),⁹⁴ $M̂={Π̂x}$⁠. The latter comprise a set of positive semidefinite Hermitian operators that resolve the identity, that is,

By performing a measurement, we obtain a statistical distribution $P(x|λ)=Tr(ϱ̂λ Π̂x)$⁠. Afterward, what remains is to obtain the best estimate of λ given $P(x|λ)$⁠, which is done using an estimator $λˇ$⁠. Since we are trying to estimate the value of a fixed parameter λ, our guess should be correct on average if we repeat the experiment enough times. This means that we should have

When this happens, the estimator $λˇ$ is called unbiased.

There are an infinite number of possible POVMs that we can consider in quantum mechanics. It is, therefore, natural to ask whether there is an optimal measurement $M̂$ that we should perform on the state $ϱ̂λ$⁠. Helstrom, in his classical work,⁹⁵ demonstrated that this question can actually be answered using fairly elementary arguments. To this end, we introduce the quantum Fisher information by⁹⁶ 

where $Dλ2$ is the so-called symmetric logarithmic derivative, defined implicitly via

where ${·,·}$ stands for the anticommutator ${Â,B̂}=ÂB̂+B̂Â$⁠.

For unitary processes, with generator $Ĝ$⁠, the quantum Fisher information simplifies to⁸⁷ 

where p_i and $|i⟩$ are the eigenvalues and eigenvectors of $ϱ̂$⁠, respectively.

The quantum Fisher information $F(ϱ̂,λ)$ is, then, optimized over all the measurements. As such, it depends only on the state. The crux of quantum metrology is the quantum version of the time-honored Cramér–Rao lower bound:^97,98 for any unbiased estimator $λˇ$⁠, we have

per single-detection event. This quantum Cramer–Rao bound may always be saturated, at least in principle. Recall that the symmetric logarithmic derivative $D$ is a Hermitian matrix, which is diagonalizable. As such, we can consider its spectral decomposition $D=∑xdλ,x|φλ,x⟩⟨φλ,x|$⁠. Since the bound can be saturated, this suggests that the quantum Fisher information quantifies the usefulness of a probe $ϱ̂$ for the measurement of a given λ.

For the case of a pure state, we have that

and in consequence, the most quantum states, from this metrological point of view, are those with maximal quantum Fisher information. For unitary dynamics where the eigenvalues of $Ĝ$ are bounded, one may immediately identify the extremal quantum states: they are⁹⁹ 

where $|rmax⟩$ and $|rmin⟩$ are the eigenvectors of $ϱ̂$ with maximal and minimal eigenvalues, respectively, and χ is an arbitrary phase.

Later on, we will be interested in estimating rotations. The natural counterpart in the case of a single mode is estimating a linear displacement. Let us consider the displacement

where $x̂θ=(eiθâ+e−iθâ†)/2$ represents a general rotated quadrature of the field. This has the general form of the displacement operator $D̂(α)$ with $α=reiθ/2$⁠. Now, imagine we want to estimate the magnitude of the displacement r; this is equivalent to choosing $x̂θ$ as the generator of the transformation. Consequently, we have

where $Covψ(Â,B̂)=12⟨{Â,B̂}⟩ψ−⟨Â⟩ψ⟨B̂⟩ψ$⁠. Since the information about the phase θ is irrelevant, we can average over that variable; the result is

where $F(ϱ̂,x)$ and $F(ϱ̂,p)$ are the quantum Fisher information for the corresponding quadratures. For pure states, this becomes

This reduces to a number of other quantumness measures of a state, including the total noise,^100,101 total variance,²⁵ mean quadrature variance,⁸⁴ and quadrature coherence scale.¹⁰² For a coherent state $F¯(|α⟩)=2$ and for any classical state, the inequality $F¯(ϱ̂cl)≤2$ holds.

Clearly, the most quantum states are those maximizing $F¯(ϱ̂)$⁠. For a given $n¯=⟨â†â⟩$⁠, any state with $⟨â⟩=0$ maximizes $F¯$⁠, that is, the states with the greatest metrological power are those with

The first-order moments of these states vanish, which makes the latter treat all quadratures equally. In turn, higher-order moments of the states govern the metrological power through $n¯$⁠. This is a recurring theme that we will again see with the $su$(2) algebra. We see that coherent states have the least metrological power, while states with vanishing first-order moments have the greatest metrological power.

There is more to the metrological-power story than averaging the quantum Fisher information over different experimental configurations. It is insightful to average the variances of all estimates of the displacement r by averaging the quantum Cramér–Rao bound over all phases θ,

Since variances are always positive, $1/F(ϱ̂,r,θ)$ is a convex functional of $F(ϱ̂,r,θ)$⁠, allowing the use of Jensen's inequality¹⁰³ 

Our convex functional is nonlinear; Jensen's inequality is, then, saturated if and only if $F(ϱ̂,r,θ)$ is independent of the direction θ. This is only the case when the covariance matrix of $x̂$ and $p̂$ found in Eq. (75) is isotropic (i.e., proportional to the identity matrix). The isotropy of the covariance matrix is equivalent to $Varψ(â)=0$⁠, which is satisfied by coherent states, number states, and more.

Even though the second-order moments of both coherent states and number states are isotropic, the former have much less metrological power because their first-order moments are maximal. This may be seen by directly integrating [Eq. (80)], which can be formulated in a number of insightful ways

The Schrödinger–Robertson uncertainty relation ensures that¹⁰⁴ 

with equality if and only if $|ψ⟩$ is an eigenstate of $x̂+γp̂$ for some complex constant γ. The only possible normalizable states with this condition are Gaussian states, i.e., minimizing metrological power is equivalent to being a squeezed coherent state. We can also explicitly see that maximizing the metrological power is equivalent to setting the first-order moments $⟨x̂⟩ψ$ and $⟨p̂⟩ψ$ to zero and making the second-order moments isotropic,

This is in turn equivalent to the vanishing of the first- and second-order moments of $â$⁠,

From this, it is clear that states with maximal first-order moment $⟨â⟩ψ$ (i.e., coherent states) have the least metrological power, whereas states with vanishing first- and second-order moments of $â$ have the greatest metrological power. Moreover, while a naive averaging of the Fisher information implies the large metrological power of squeezed states, an average over the Cramér–Rao bound shows that their metrological advantage for particular measurements is perfectly balanced by their disadvantage for other measurements, making them only as useful as coherent states.

One way of identifying states with the greatest metrological power is from the symmetries of their Husimi functions. If the Husimi function is unchanged by a rotation in the complex plane $Qψ(α)=Qψ(e−iϕα)$⁠, its moments must obey

The kth moment must vanish unless $eikϕ=1$⁠. The rotational symmetries given by $ϕ$ determine the metrological power of a state.

Cat states [Eq. (20)], for example, have Husimi functions that are symmetric under $α→−α$⁠; they, thus, have vanishing first-order moments but nonzero k = 2 moments. Similarly, the more sophisticated four-component compass state¹⁰⁵ $∝|α⟩+|−α⟩+|iα⟩+|−iα⟩$ has a rotational symmetry with $ϕ=π/2$⁠, and so, all of its moments up until k = 4 vanish.

Number states, at the far extreme, have all of their moments vanish

due to the continuous polar symmetry $Qn(α)=Qn(|α|)$ of their Husimi functions. A different kind of probe may be useful to showcase the advantages of states whose higher-order moments are also isotropic.

Quantum information can also be encoded in discrete degrees of freedom. The focus for this discrete-variable (DV) approach is on exploiting two-level systems—qubits—to generate a quantum analogue of the classical bit.¹⁰⁶ Common examples involve single photons living in a finite-dimensional space spanned, for instance, by orthogonal polarization modes, orbital angular momentum, or the two paths followed in an interferometer. From a mathematical perspective, these cases can be understood within the formalism of SU(2), and so, we will talk here about spin variables. In this section, we present our quantumness indicators paralleling our CV treatment as much as possible, portraying the nuanced similarities and differences between these two worlds.

In this section, we derive the spin or Bloch coherent states, in a fashion similar to what we did for the canonical coherent states. Our Hilbert space will be any finite-dimensional Hilbert space in which SU(2) acts irreducibly.

We consider a system whose dynamical group is SU(2). The corresponding Lie algebra $su(2)$ is spanned by the operators ${Ŝx, Ŝy, Ŝz}$⁠, satisfying the commutation relation

and cyclic permutations thereof. The Casimir operator is $Ŝ2=Ŝx2+Ŝy2+Ŝz2=S(S+1)1$⁠, so the eigenvalue S, which is a non-negative integer or half integer, labels the irreducible representations (irreps).

We employ a fixed irrep of spin S, with a $(2S+1)$-dimensional carrier space $HS$ spanned by the standard angular momentum basis ${|S,m⟩|m=−S,…,S}$⁠, whose elements are simultaneous eigenstates of $Ŝ2$ and $Ŝz$⁠,