Probing nonclassicality with matrices of phase-spacedistributionsMartin Bohmann1,2, Elizabeth Agudelo1, and Jan Sperling3

1Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna,

Austria2QSTAR, INO-CNR, and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy3Integrated Quantum Optics Group, Applied Physics, Paderborn University, 33098 Paderborn, Germany

We devise a method to certify nonclassicalfeatures via correlations of phase-space distri-butions by unifying the notions of quasipro-babilities and matrices of correlation func-tions. Our approach complements and ex-tends recent results that were based on Cheby-shev’s integral inequality [Phys. Rev. Lett.124, 133601 (2020)]. The method developedhere correlates arbitrary phase-space functionsat arbitrary points in phase space, includingmultimode scenarios and higher-order corre-lations. Furthermore, our approach providesnecessary and sufficient nonclassicality crite-ria, applies to phase-space functions beyonds-parametrized ones, and is accessible in ex-periments. To demonstrate the power ofour technique, the quantum characteristics ofdiscrete- and continuous-variable, single- andmultimode, as well as pure and mixed statesare certified only employing second-order cor-relations and Husimi functions, which alwaysresemble a classical probability distribution.Moreover, nonlinear generalizations of our ap-proach are studied. Therefore, a versatile andbroadly applicable framework is devised to un-cover quantum properties in terms of matricesof phase-space distributions.

1 IntroductionTelling classical and quantum features of a physicalsystem apart is a key challenge in quantum physics.Besides its fundamental importance, the notion of(quantum-optical) nonclassicality provides the basisfor many applications in photonic quantum technol-ogy and quantum information [1, 2, 3, 4, 5]. Nonclassi-cality is, for example, a resource in quantum networks[6], quantum metrology [7], boson sampling [8], or dis-tributed quantum computing [9]. The correspondingfree (i.e., classical) operations are passive linear opti-cal transformations and measurement. By exceedingsuch operations, protocols which utilize nonclassicalstates can be realized. Furthermore, nonclassicalityMartin Bohmann: martin.bohmann@oeaw.ac.at

is closely related to entanglement. Each entangledstate is nonclassical, and single-mode nonclassicalitycan be converted into two- and multi-mode entangle-ment [10, 11, 12].

Consequently, a plethora of techniques to detectnonclassical properties have been developed, eachcoming with its own operational meanings for appli-cations. For example, quantumness criteria which arebased on correlation functions and phase-space repre-sentations have been extensively studied in the con-text of nonclassical light [13, 14].

The description of physical systems using thephase-space formalism is one of the cornerstones ofmodern physics [15, 16, 17]. Beginning with ideas in-troduced by Wigner and others [18, 19, 20, 21], thenotion of a phase-space distribution for quantum sys-tems generalizes principles from classical statisticaltheories (including statistical mechanics, chaos the-ory, and thermodynamics) to the quantum domain.However, the nonnegativity condition of classicalprobabilities does not translate well to the quantumdomain. Rather, the notion of quasiprobabilities—i.e., normalised distributions that do not satisfy allaxioms of probability distributions and particularlycan attain negative values—was established and foundto be the eminent feature that separates classicalconcepts from genuine quantum effects. (See Refs.[22, 14] of thorough introductions to quasiproabili-ties.)

In particular, research in quantum optics signif-icantly benefited from the concept of phase-spacequasiprobability distributions, including prominentexamples, such as the Wigner function [19], theGlauber-Sudarshan P function [23, 24], and theHusimi Q function [25]. In fact, the very definitionof nonclassicality—the impossibility of describing thebehaviour of quantum light with classical statisticaloptics—is directly connected to negativities in suchquasiprobabilities, more specifically, the Glauber-Sudarshan P function [26, 27]. Because of the generalsuccess of quasiprobabilities, other phase-space distri-butions for light have been conceived [28, 29, 30], eachcoming with its own advantages and drawbacks. Forexample, squeezed states are represented by nonneg-ative (i.e., classical) Wigner functions although they

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form the basis for continuous-variable quantum infor-mation science and technology [31, 32, 33], also havinga paramount role for quantum metrology [34, 35].

Another way of revealing nonclassical effects is byusing correlation constraints which, when violated,witness nonclassicality. Typically, such conditions areformulated in terms of inequalities involving expec-tation values of different observables. Examples inoptics are photon anti-bunching [36, 37, 38] and sub-Poissonian photon-number distributions [39, 40], us-ing intensity correlations, as well as various squeez-ing criteria, being based on field-operator correlations[41, 42, 43, 44]. They can follow, for instance, fromapplying Cauchy-Schwartz inequalities [45] and un-certainty relations [46], as well as from other viola-tions of classical bounds [47, 48, 49]. Remarkably,many of these different criteria can be jointly de-scribed via so-called matrix of moments approaches[50, 51, 52, 53, 54]. However, each of the mentionedkinds of nonclassicality, such as squeezed and sub-Poissonian light, requires a different (sub-)matrix ofmoments, a hurdle we aim at overcoming.

Over the last two decades, there had been many at-tempts to unify matrix-of-moment-based criteria withquasiprobabilities. For example, the Fourier trans-form of the P function can be used, together withBochner’s theorem, to correlate such transformedphase-space distributions through determinants of amatrix [55, 56], being readily available in experimen-tal applications [57, 58, 59, 60], and further extendingto the Laplace transformation [61]. Furthermore, ajoint description of field-operator moments and trans-formed phase-space functions has been investigated aswell [62]. Rather than considering matrices of phase-space quasiproabilities, concepts like a matrix-valueddistributions enable us to analyzed nonclassical hy-brid systems [63, 64]. Very recently, a first successfulstrategy that truly unifies correlation functions andphase-space functions has been conceived [65]. How-ever, these first demonstrations of combining phase-space distributions and matrices of moments are stillrestricted to rather specific scenarios.

In this contribution, we formulate a general frame-work for uncovering quantum features through cor-relations in phase-space matrices which unifies thesetwo fundamental approaches to characterizing quan-tum systems. By combining matrix of momentsand quasiprobabilities, this method enables us toprobe nonclassical characteristics in different pointsin phase space, even using different phase-space dis-tributions at the same time. We specifically studyimplications from the resulting second- and higher-order phase-space distribution matrices for single- andmultimode quantum light. Furthermore, a directmeasurement scheme is proposed and non-Gaussianphase-space distributions are analyzed. To bench-mark our method, we consider a manifold of exam-ples, representing vastly different types of quantum

features. In particular, we show that our matrix-based approach can certify nonclassicality even ifthe underlying phase-space distribution is nonnega-tive. In summary, our approach renders it possi-ble test for nonclassicality by providing easily ac-cessible nonclassicality conditions. While previouslyderived phase-space-correlation conditions [65] wererestricted to single-mode scenarios, the present ap-proach straightforwardly extends to multimode cases.In addition, our phase-space matrix technique in-cludes nonclassicality-certification approaches basedon phase-space distributions and matrices of momentsas special cases, resulting in an overarching structurethat combines both previously separated techniques.

The paper is structured as follows. Some initialremarks are provided in Sec. 2. Our method is rig-orously derived and thoroughly discussed in Sec. 3.Section 4 concerns several generalizations and poten-tial implementations of our toolbox. Various exam-ples are analyzed in Sec. 5. Finally, we conclude inSec. 6.

2 PreliminariesIn their seminal papers [23, 24], Glauber and Su-darshan showed that all quantum states of light canbe represented diagonally in a coherent-state ba-sis through the Glauber-Sudarshan P distribution.Specifically, a single-mode quantum state can be ex-panded as

ρ =∫d2αP (α)|α〉〈α|, (1)

where |α〉 denotes a coherent state with a complexamplitude α. Then, classical states are identified asstatistical (i.e., incoherent) mixtures of pure coher-ent states, which resemble the behavior of a classicalharmonic oscillator most closely [66, 67]. For this di-agonal representation to exist for nonclassical statesas well, the Glauber-Sudarshan distribution has toexceed the class of classical probability distributions[26, 27], particularly violating the nonnegativity con-straint, P 0. This classification into states whichhave a classical correspondence and those which aregenuinely quantum is the common basis for certifyingnonclassical light.

As laid out in the introduction, nonclassicality is avital resource for utilizing quantum phenomena, rang-ing from fundamental to applied [6, 7, 8, 9]. In thiscontext, it is worth adding that, contrasting othernotions of quantumness, nonclassicality is based ona classical wave theory. That is, it is essential todiscern nonclassical coherence phenomena from thosewhich are accessible with classical statistical optics,as formalized through Eq. (1) with P ≥ 0. See, e.g.,Ref. [68] for a recent experiment that separates clas-sical and quantum interference effects in such a man-ner. For instance, free operations, i.e., those maps

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which preserve classical states, do include beam split-ter transformations, resulting in the generation of en-tanglement from single-mode nonclassical states viasuch a free operation [10, 11, 12] that is vital for manyquantum protocols.

2.1 Phase-space distributionsSince the Glauber-Sudarshan distribution can be ahighly singular distribution (see, e.g., Ref. [69]),generalized phase-space functions have been devised.Within the wide range of quantum-optical phase-space representations, the family of s-parametrizeddistributions [29, 30] is of particular interest. Suchdistributions can be expressed as

P (α;σ) = σ

π〈: exp (−σn(α)) :〉 , (2)

where colons indicate normal ordering [70] and n(α) =(a − α)†(a − α) is the displaced photon-number op-erator, written in terms of bosonic annihilation andcreation operators, a and a†, respectively. It is worthrecalling that the normal ordering acts on the expres-sion surrounded by the colons in such a way that cre-ation operators are arranged to the left of annihila-tion operators whilst ignoring commutation relations.Note that, for convenience, we parametrize distribu-tions via the width parameter σ, rather than using s.Both are related via

σ = 21− s . (3)

From this relation, we can identify the Husimi func-tion, Q(α) = P (α; 1), for s = −1 and σ = 1;the Wigner function, W (α) = P (α; 2), for s = 0and σ = 2; and the Glauber-Sudarshan function,P (α) = P (α;∞), for s = 1 and σ =∞.

Whenever a phase-space distribution contains anegative contribution, i.e., P (α;σ) < 0 for at leastone pair (α;σ), the underlying quantum state is non-classical [26, 27]. In such a case, the distributionP (α;σ) refers to as a quasiprobability distributionwhich is incompatible with classical probability the-ory. Nonetheless, for any σ ≥ 0 and any state, thisfunction represents a real-valued distribution which isnormalized, P (α;σ) = P (α;σ)∗ and

∫d2αP (α;σ) =

1. In addition, it is worth mentioning that the nor-malization of the state is guaranteed through the limit

limσ→0

π

σP (α;σ) = 〈: exp(0):〉 = 〈1〉 = 1. (4)

2.2 Matrix of moments approachBesides phase-space distributions, a second family ofnonclassicality criteria is based on correlation func-tions; see, e.g., Refs. [71, 72] for introductions. Forthis purpose, we can consider an operator functionf = f(a, a†). Then,

〈:f†f :〉 =∫d2αP (α)|f(α, α∗)|2

cl.≥ 0 (5)

holds true for all P ≥ 0. Now, one can expand f interms of a given set of operators, e.g., f =

∑i ciOi, re-

sulting in 〈:f†f :〉 =∑i,j c∗i cj〈:O

†i Oj :〉. Furthermore,

this expression is nonnegative [cf. Eq. (5)] iff the ma-trix (〈:O†i Oj :〉)i,j is positive semidefinite. This con-straint can, for example, be probed using Sylvester’scriterion [73] which states that a Hermitian matrix ispositive-definite if and only if all its leading princi-pal minors have a positive determinant. It is worthmentioning that Eq. (5) defines the notion of a non-classicality witness, where 〈:f†f :〉 < 0 certifies non-classicality.

The above observations form the basis for many ex-perimentally accessible nonclassicality criteria, suchas using basis operators which are powers of quadra-ture operators [50], photon-number operators [44],and general creation and annihilation operators [71,72]. See Refs. [13] for an overview of moment-based inequalities. In the following, we are going tocombine the phase-space distribution technique withthe method of matrices of moments to arrive at thesought-after unifying approach of both techniques.

3 Matrix of phase-space distributionsBoth phase-space distributions and matrices of mo-ments exhibit a rather dissimilar structure when itcomes to formulating constraints for classical light.Consequently, a full unification of both approachesis missing to date, excluding the few attempts men-tioned in Sec. 1. In this section, we bridge this gapand derive a matrix of phase-space distributions whichleads to previously unknown nonclassicality criteria,also overcoming the limitations of earlier methods.

3.1 Derivation

For the purpose of deriving our criteria, we consideran operator function f =

∑i ci exp[−σini(αi)]. Then,

the normally ordered expectation value of f†f can beexpanded as

〈:f†f :〉 =∑i,j

c∗i cj〈:e−σin(αi)e−σj n(αj):〉

=∑i,j

c∗jci exp[− σiσjσi + σj

|αi − αj |2]

×⟨

: exp[−(σi + σj) n

(σiαi + σjαjσi + σj

)]:⟩.

(6)

Based on the above relation, we may define two ma-trices, one for classical amplitudes,

M (c) =(

exp[− σiσjσi + σj

|αi − αj |2])

i,j

, (7)

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and one for the quantum-optical expectation values,

M (q) =(⟨

: exp[−(σi + σj) n

(σiαi + σjαjσi + σj

)]:⟩)

i,j

=(

π

σi + σjP

(σiαi + σjαjσi + σj

;σi + σj

))i,j

,

(8)

which can be expressed in terms of phase-space distri-butions using Eq. (2). Specifically, M (q) correspondsto a matrix of phase-space distributions. Moreover,the fact that the normally ordered expectation valueof f†f is nonnegative for classical light [Eq. (5)]is then identical to the entry-wise product (i.e., theHadamard product ) of both matrices being positivesemidefinite,

Mcl.≥ 0, with M = M (c) M (q) (9)

defining our phase-space matrix M .For classical light, all principal minors of M have

to be nonnegative according to Sylvester’s criterion.Conversely, the violation of this constraint certifies anonclassical state,

det(M) < 0, (10)

where M is defined through arbitrary small or largesets of parameters σi and σj and coherent ampli-tudes αi and αj . Therefore, inequality (10) enables usto formulate various nonclassicality conditions whichcorrelate distinct phase-space distributions as it typ-ically only done for matrix-of-moments-based tech-niques when using different kinds of observables. Wefinally remark that the expression in Eq. (10) resem-bles a nonlinear nonclassicality witnessing approach.

As a first example, we may explore the first-ordercriterion, i.e., a 1 × 1 matrix of quasiprobabilities.Selecting arbitrary σ-parameters and coherent ampli-tudes, i.e., (α1;σ1) = (α;σ), we find the followingrestriction for classical states [cf. Eq. (10)]:

π

2σP (α; 2σ)cl.≥ 0. (11)

This inequality reflects the fact that finding nega-tivities in a parametrized phase-space distributionP (α; 2σ) is sufficient to certify nonclassicality. Alsorecall that we retrieve the Glauber-Sudarshan distri-bution in the limit σ → ∞. Since the nonnegativityof this distribution defines the very notion of a non-classical state [26, 27], we can conclude from this ex-amples that our approach is necessary and sufficientfor certifying nonclassicality.

However, the Glauber-Sudarshan distribution hasthe disadvantage of being a highly singular for manyrelevant nonclassical states of light and, thus, hard toreconstruct from experimental data. Consequently,it is of practical importance (see Secs. 4 and 5) toconsider higher-order criteria beyond this trivial one.

3.2 Second-order criteriaWe begin our consideration with an interestingsecond-order case. We chose (α1;σ1) = (0; 0) and(α2;σ2) = (α;σ). This yields the 2 × 2 phase-spacematrix

M =(

1 〈: exp(−σn(α)):〉〈: exp(−σn(α)):〉 〈: exp(−2σn(α)):〉

). (12)

Up to a positive scaling, the determinant of this ma-trix results in the following nonclassicality criterion:

P (α; 2σ)− 2πσ

(P (α;σ))2< 0. (13)

In particular, we can set σ = 1 to relate this condi-tion to the Wigner and Husimi functions, leading toW (α)−2πQ(α)2 < 0. This special case of our generalapproach has been recently derived using a very dif-ferent approach, using Chebyshev’s integral inequal-ity [65]. There it was shown that, by applying theinequality (13) for σ = 1, it is possible to certify non-classicality even if the Wigner function of the stateunder study is nonnegative. In this context, remem-ber that the Husimi function, Q(α) = 〈α|ρ|α〉/π, isalways nonnegative, regardless of the state ρ.

Beyond this scenario, we now study a more gen-eral 2 × 2 phase-space matrix M . For an efficientdescription, it is convenient to redefine transformedparameters as

∆α = α2 − α1 and A = σ1α1 + σ2α2

σ1 + σ2, (14a)

σ = σ1σ2

σ1 + σ2and Σ = σ1 + σ2. (14b)

Note that these parameters relate to the two-bodyproblem. That is, the quantities in Eq. (14a) definethe relative position and barycenter in phase-space,respectively; and the two quantities in Eq. (14b) re-semble the reduced and total mass in a mechanicalsystem, respectively.

In this alternative parametrization, the two matri-ces, giving the total phase-space matrix M = M (c) M (q), read

M (c) =(

1 e−σ|∆α|2

e−σ|∆α|2 1

), and (15a)

M (q) =(〈:e−2σ1n(α1):〉 〈:e−Σn(A):〉〈:e−Σn(A):〉 〈:e−2σ2n(α2):〉

). (15b)

Hence, the determinant of the Hadamard product ofboth matrices then gives

det(M) = 〈:e−2σ1n(α1):〉〈:e−2σ2n(α2):〉

−e−2σ|∆α|2〈:e−Σn(A):〉2. (16)

If this determinant is negative for the state of lightunder study, its nonclassicality is proven. In terms of

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phase-space distributions, this condition can be alsorecast into the form

P (α1; 2σ1)P (α2; 2σ2)− 4σΣ

[e−σ|∆α|

2P (A; Σ)

]2< 0.(17)

Interestingly, this nonclassicality criterion correlatesdifferent points in phase space for different distribu-tions, P (α1; 2σ1) and P (α2; 2σ2), with a phase-spacedistribution with the total width Σ at the barycenterA of coherent amplitudes, P (A; Σ).

3.3 Higher-order cases

The next natural extension concerns the analysis ofhigher-order correlations. Clearly, one can obtain anincreasingly large set of nonclassicality tests with anincreasing dimensionality of M , determined by thenumber of pairs (αi;σi). In order to exemplify thispotential, let us focus on one specific 3 × 3 scenarioand more general scenarios for specific choices of pa-rameters.

Let us discuss the 3 × 3 case firstly, for which we are going to consider σ3 = 0. From this, one obtains thefollowing phase-space matrix:

M =

〈: exp(−2σ1n(α1)):〉 〈: exp(−σ1n(α1)− σ2n(α2)):〉 〈: exp(−σ1n(α1)):〉〈: exp(−σ1n(α1)− σ2n(α2)):〉 〈: exp(−2σ2n(α2)):〉 〈: exp(−σ2n(α2)):〉

〈: exp(−σ1n(α1)):〉 〈: exp(−σ2n(α2)):〉 1

. (18)

Again, directly expressing this matrix in terms of phase-space functions, as done previously, we get a third-ordernonclassicality criterion from its determinant [74]. It reads

det(M)π2 =

(P (α1; 2σ1)

2σ1− π

(P (α1;σ1)

σ1

)2)(

P (α2; 2σ2)2σ2

− π(P (α2;σ2)

σ2

)2)

−(

exp(−σ|∆α|2)P (A; Σ)Σ − πP (α1;σ1)

σ1

P (α2;σ2)σ2

)2< 0,

(19)

using the parameters defined in Eqs. (14a) and (14b). In fact, this condition combines the earlier derivedcriteria of the forms (13) and (16) in a manner similar to cross-correlations nonclassicality conditions knownfrom matrices of moments [61].

Another higher-order matrix scenario correspondsto having identical coherent amplitudes, i.e., αi = αfor all i. In this case, we find that the two Hadamard-product components of the matrix M simplify to

M (c) = (1)i,j and

M (q) =(

π

σi + σjP (α;σi + σj)

)i,j

,(20)

thus resulting in M = M (q). Therefore, we can for-mulate nonclassicality criteria which correlate an ar-bitrary number of different phase-space distributions,defined via σi, at the same point in phase-space, α.

Analogously, one can consider a scenario in whichall σ parameters are identical, σi = σ. Then, we get

M (c) =(e−σ|αi−αj |2/2)i,j and

M (q) =(π

2σP(αi + αj

2 ; 2σ))

i,j

.(21)

Consequently, we obtain nonclassicality criteria whichcorrelate an arbitrary number of different points inphase space, αi, for a single phase-space distribution,parametrized by σ.

3.4 Comparison with Chebyshev’s integral in-equality approachAs mentioned previously, a related method based onChebyshev’s integral inequality has been introducedrecently [65]. It also provides inequality conditions fordifferent phase-space distributions. The nonclassical-ity conditions based on Chebyshev’s integral inequal-ity take the form

P (α; Σ)− Σπ

D∏i=1

[π

σiP (α;σi)

]< 0, (22)

where Σ =∑Di=1 σi. To compare both approaches, let

us discuss their similarities and differences.In its simplest form, involving only σ1 and σ2, the

condition in Eq. (22) resembles the tests based on the2 × 2 matrix in Eq. (12). In particular, for the caseσ1 = σ2 = σ both methods yield the exact same con-ditions. For σ1 6= σ2 such an agreement of both meth-ods cannot be found because of the inherent symmetryof the phase-space matrix approach, M = M†, whichstems from its construction via a quadratic form; cf.Eq. (6). Also, for more general, higher-order condi-tions, i.e. D > 2, such similarities cannot be foundeither. Conditions of the form in Eq. (22) consist of

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only two summands. The first term is a single phase-space function with the width parameter Σ which isassociated to the highest σ parameter involved in theinequality. The second term is a product of D phase-space distributions, each individual distribution hasa width parameter σi, together bound by the condi-tion Σ =

∑Di=1 σi. By comparison, our phase-space

matrix approach yields, in general, a richer and morecomplex set of higher-order nonclassicality tests, suchas demonstrated in Sec. 3.3.

Let us point out further differences between thetwo approaches. Firstly, we observe that the inequal-ities based on Chebyshev’s integral inequality onlyapply to one single point in phase space. In con-trast, the phase-space matrix method devised hereincludes conditions that combine different points inphase space; cf. Eq. (6). Secondly, Chebyshev’s inte-gral inequality approach cannot be extended to mul-timode settings. Such a limitation does not exist forthe matrix approach either, as we show in the follow-ing Sec. 4.1. We conclude that both the technique inRef. [65] and our phase-space matrix approach for ob-taining phase-space inequalities yield similar second-order conditions but, in general, give rise to ratherdifferent nonclassicality criteria. In particular, thephase-space matrix framework offers a broader rangeof variables—be it coherent amplitudes or widths—that lead to a richer set of nonclassicality conditions.

3.5 Extended relations to nonclassicality crite-riaTo finalize our first discussions we now focus on therelation to matrices of moments. Previously, we haveshown that, already in the first order, our criteria arenecessary and sufficient to verify the nonclassicality,and we discussed our method in relation to Cheby-chev’s integral inequality. Furthermore, indirect tech-niques using transformed phase-space functions, suchas the characteristic function [62] and the two-sidedLaplace transform [61], have been previously relatedto moments. Thus, the question arises what the rela-tion of our direct technique to such matrices of mo-ments is.

For showing that our framework includes the ma-trix of moments technique, we may remind ourselvesthat derivatives can be understood as a linear combi-nation, specifically as a limit of a differential quotient,∂mz g(z) = limε→0 ε

−m∑mk=0

(mk

)(−1)m−kg(z + kε).

This enables us to write [75]

a†man = σ−(m+n) ∂mα ∂nα∗eσ|α|

2:e−σn(α):

∣∣∣α=0 and σ=0

,

(23)expressing arbitrary moments a†man via linear com-binations of the normally ordered operators thatrepresent σ-parametrized phase-space distributions.Thus, in the corresponding limits, we can iden-tify the operator f in Eq. (5) with f =

∑m,n cm,nσ

−(m+n)∂mα ∂nα∗eσ|α|

2 :e−σn(α):|α=0,σ=0 =∑m,n cm,na

†man. For such a choice f , 〈:f†f :〉 < 0 isin fact identical to the most general form of the matrixof moments criterion for nonclassicality [71, 72].

In conclusion, we find that our necessary and suf-ficient methodology not only includes nonclassicalitycriteria based on phase-space functions themselves [cf.Eq. (11)], but it also includes the technique of ma-trices of moments as a special case. In a hierarchicalpicture, this means that our family of nonclassicalitycriteria, including arbitrary orders of σ-parametrizedphase-space functions, encompasses both negativitiesof phase-space functions and matrices of moments.Because of the above relation, the order of momentsthat is required to certify nonclassicality also sets anupper bound to the size of the matrix of phase-spacedistributions so that it certifies nonclassicality. There-fore, our approach unifies and subsumes both earliertypes of nonclassicality conditions.

4 Generalizations and implementationIn this section, we generalize our approach to arbi-trary multimode nonclassical light and propose a mea-surement scheme to experimentally access the matrixof phase-space distributions. In addition, we showthat our approach applies to phase-space distributionswhich are no longer limited to σ parametrizations andrelate these findings to the response of nonlinear de-tection devices.

4.1 Multimode caseAfter our in-depth analysis of single-mode phase-space matrices, the multimode case follows almoststraightforwardly. For the purpose of such a gener-alization, we consider N optical modes, representedvia the annihilation operators am for m = 1, . . . , Nand extending to the displaced photon-number oper-ators nm(α(m)) = (am − α(m))†(am − α(m)). Now,σ-parametrized multimode phase-space functions canbe expressed as⟨

:e−σ(1)n1(α(1)) · · · e−σ

(N)nN (α(N)):⟩

= πN

σ(1) · · ·σ(N)P (α(1), . . . , α(N);σ(1), . . . , σ(N)),(24)

where we allow for different s parameters for eachmode, with s(m) = 1 − 2/σ(m) [Eq. (3)]. As in thesingle-mode case, we can now formulate a matrix Mof multimode phase-space functions,

M=

⟨:e−

N∑m=0

σ(m)i

nm(α(m)i

)e−

N∑m=0

σ(m)j

nm(α(m)j

):⟩

i,j

.

Consequently, this matrix of phase-space functionsalso has to be positive semidefinite if the underlying

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state of multimode light is classical. That is,

Mcl.≥ 0 (25)

holds true for classical light and for any dimension (ororder) of the multimode matrixM and any sigma andalpha values. Conversely, det(M) < 0 is a nonlinearwitness of multimode nonclassicality. Similarly to thesingle-mode case, an increasingly large matrixM withincreasingly dense sets of parameters for the variousalpha and sigma values then enables one to probe thenonclassicality of arbitrary multimode states.

Since we have already exemplified various scenariosfor single-mode phase-space correlations, in the fol-lowing, we restrict ourselves to a particular multimodecase. Specifically, we focus on two optical modes anda 3× 3 phase-space matrix M is, 1 πP (α(1);σ)

σπP (α(2);σ)

σπP (α(1);σ)

σπP (α(1);2σ)

2σπ2P (α(1),α(2);σ,σ)

σ2

πP (α(2);σ)σ

π2P (α(1),α(2);σ,σ)σ2

πP (α(1);2σ)2σ

,

where quasiprobabilities as a function of single-modeparameters indicate marginal phase-space distribu-tions. Adopting a notation of pairs of coherent am-plitudes and widths, M is thus defined via the fol-lowing two-mode parameters: (α(1)

1 , α(2)1 ;σ(1)

1 , σ(2)1 ) =

(0, 0; 0, 0), (α(1)2 , α

(2)2 ;σ(1)

2 , σ(2)2 ) = (α(1), 0;σ, 0), and

(α(1)3 , α

(2)3 ;σ(1)

3 , σ(2)3 ) = (0, α(2); 0, σ). In particular,

we can express the nonclassicality constraint from thedeterminant ofM [74] for σ = 1 via joint and marginalWigner and Husimi functions,

detMπ4 =

[W (α(1))

2π −Q(α(1))2] [

W (α(2))2π −Q(α(2))2

]−[Q(α(1), α(2))−Q(α(1))Q(α(2))

]2 cl.≥ 0.

(26)

Violating this inequality verifies the nonclassicality ofthe two-mode state under study.

4.2 Direct measurement schemeThe reconstruction of phase-space distributions canbe a challenging task [76]. For this reason, we aregoing to devise a directly accessible setup to inferthe phase-space matrix. See Fig. 1 for an outlinewhich is based on the approaches in Refs. [77, 72, 78].For convenience, we restrict ourselves to a singleoptical mode; the extension to multiple modes fol-lows straightforwardly. That is, each of the multiplemodes can be detected individually by a correlation-measurement setup as depicted in Fig. 1. Further-more, it is noteworthy that our phase-space matrixapproach is not limited to this specific measurementscheme proposed here and generally applies to any de-tection scenario which allows for a reconstruction ofquasiprobability distributions.

signal ρ

50:50

LOi

LOj

|t|2:|r|2 ×matrixentry Mi,j

Π(n)

Π(n)

Figure 1: Outline of phase-space matrix correlation mea-surement. The signal, i.e., the state ρ of the light field understudy, is split into two identical outputs at a 50:50 beam split-ter. Each of the resulting beams is combined with a local os-cillator (LO) on a |t|2 : |r|2 beam splitter and measured witha photon-number-based detector, represented through Π(n).The resulting correlations yield the entries of our phase-spacematrix M .

For the setup in Fig. 1, we begin our considera-tions with a coherent state |β〉, representing our sig-nal ρ = |β〉〈β|. Firstly, we split this signal equallyinto 2 modes, resulting in a two-mode coherent state|β/√

2, β/√

2〉. In addition, local oscillator states areprepared, |βi〉 and |βj〉 for each mode. Each of thetwo signals is then mixed with its local oscillator ona |t|2:|r|2 beam splitter, where |t|2 + |r|2 = 1. Oneoutput of each beam splitter is discarded, namely thelower and upper one for the top and bottom path inFig. 1, respectively. This results in the input-outputrelation

|β〉 7→∣∣∣∣t β√2

+ rβi, tβ√2

+ rβj

⟩, (27)

which is then detected as follows.Each of the resulting modes is measured with a

detector or detection scheme based on photon ab-sorption, thus being described by a positive operator-valued measure (POVM) which is diagonal in thephoton-number representation [79]. Consequently,one or a combination of detector outcomes (e.g., ina generating-function-type combination [80]) corre-sponds to a POVM element of the form Π(n) =:e−Γ(n):. Using |m〉〈m| = :e−nnm/m!: for anm-photon projector, this means that we identify∑∞m=0 πm|m〉〈m| = :e−n

∑m=0 πmn

m/m!: = Π(n) =:e−Γ(n):, where the eigenvalues πm corresponds tothe Taylor expansion coefficients of the function z 7→exp[z − Γ(z)]. Accordingly, the function Γ(n) modelsthe detector response [79, 70]. Finally, the correlationmeasurement of this response for our coherent signalstates takes the form

Mi,j = exp(−Γ(|t|2

2

∣∣∣∣β − r√

2βit

∣∣∣∣))× exp

(−Γ(|t|2

2

∣∣∣∣β + r√

2βjt

∣∣∣∣)) .(28)

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Now it is convenient to define Γ(n) = Γ(|t|2n/2)and

αi = −r√

2βit

, (29)

for all LO choices i and, similarly, for j. Further-more, we generalize this treatment to arbitrary states,ρ =

∫d2β P (β)|β〉〈β|, using the Glauber-Sudarshan

representation [Eq. (1)]. Therefore, the correlationsmeasured as described above [Eq. (28)] obey

Mi,j =⟨

:e−Γ(n(αi))e−Γ(n(αj)):⟩, (30)

which corresponds to a directly measured phase-spacematrix element, e.g., for a linear detector responseΓ(n) = σn. The other way around, we can choosef =

∑i ci exp(−Γ(n(αi))) for the general classicality

constraint in Eq. (5), even for nonlinear detector re-sponses. Then, the matrix of phase-space distributionapproach applies, regardless of a linear or nonlineardetection model. (See also Refs. [81, 80] in this con-text.)

As an example, we consider a case with two singleon-off click detectors (represented by Π(n) in Fig. 1)with a non-unit quantum efficiency ηdet and a non-vanishing dark-count rate δ [82], which represents re-alistic detectors in experiments. In addition, we canintroduce neutral density (ND) filters to attenuate thelight that impinges on each detector. The POVM el-ement for the no-click event in combination with theND filters then reads Π(n) = : exp(−(ηn+δ)):, where0 ≤ η ≤ ηdet is a controllable efficiency. The measuredcorrelation for this scenario takes the form

Mi,j = exp(−2δ)〈: exp (−ηin(αi)− ηj n(αj)) :〉. (31)

Therein, the adjustable efficiency ηi plays role of σi.Also, the positive factor that includes the dark countsis irrelevant because it does not change the sign of thedeterminant of M , i.e., the verified nonclassicality.

In summary, the measurement layout in Fig. 1 en-ables us to directly measure the entries of our phase-space matrix M . As an experimental setup, thisscheme also underlines the strong connection betweencorrelations and their measurements and phase-spacequasiprobabilities and their reconstruction. We mayemphasize that all experimental techniques and com-ponents that are used in the proposed setup are read-ily available; see, e.g., the related quantum state re-construction experiments reported in Refs. [83, 80].

4.3 Generalized phase-space functionsThe σ-parametrized phase-space distributions we con-sidered so far are related to each other via con-volutions with Gaussian distributions [28, 29, 30].However, there are additional means to represent astate without relying on Gaussian convolutions only.

Such generalized phase-space function can be ob-tained from the Glauber-Sudarshan P function via

PΩ(α) =∫d2α P (α) Ω(α; α, α∗) = 〈:Ω(α; a, a†):〉

(32)for a kernel Ω ≥ 0 [30, 84]. The construction of this so-called filter or regularizing function Ω can be done sothat the resulting distribution PΩ is regular (i.e., with-out the singular behavior known from the P function)and is positive semidefinite for all classical states [84].For instance, a non-Gaussian filter Ω has been used toexperimentally characterize squeezed states via regu-lar distributions which exhibit negativities in phasespace [85]; this cannot be done with s-parametrizedquasiprobability distributions, which are either non-negative or highly singular for squeezed states.

As done for the previously considered distributions,we can define an operator f =

∑i ciΩi(α; a, a†),

which leads to a phase-space matrix with the entries

Mi,j = 〈:Ωi(α; a, a†)Ωj(α; a, a†):〉 = PΩiΩj (α). (33)

This expression utilizes product of filtersΩ(α; α, α∗) = Ωi(α; α, α∗)Ωj(α; α, α∗) to be convo-luted with the P function. From this definition of aregularized phase-space matrix, we can proceed aswe did earlier to formulate nonclassicality criteria interms of phase-space functions.

Moreover, the non-Gaussian filter functions can beeven related to nonlinear detectors. For this purpose,we assume that Ω(α; α, α∗) = Ω(|α − α|2) (likewise,:Ω(α, a, a†): = :Ω(n(α)): in the normally ordered op-erator representation). In this form, the function isinvariant under rotations. As we did for the generalPOVM element Π(n), we can now identify

Γ(n) = − ln Ω(n). (34)

This enables us to associate non-Gaussian filters andnonlinear detectors and, by extension, generalizedphase-space matrices for certifying nonclassical statesof light. An example for this treatment is studied inSec. 5.5.

5 Examples and benchmarkingIn the following, we apply our method of phase-spacematrices to various examples and benchmark its per-formance. For the latter benchmark, we could con-sider different phase-space functions. Using the Pfunction would be impractical as it is often a highlysingular distribution. The Wigner function is regu-lar and can exhibit negativities. But error estima-tions from measured data can turn out to be ratherdifficult because it requires diverging pattern func-tions [86, 87] (see Ref. [88] for an in-depth analysis).Beyond those practical hurdles, we focus on the Qfunction here because, already in theory, it is always

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nonnegative. Thus, it is hard to verify nonclassicalfeatures based on this particular phase-space distribu-tion. Additionally, the Q function is easily accessiblein experiments and can be directly measured via thewidely-used double-homodyne (aka, eight-port homo-dyne) detection scheme [70].

Nonetheless, we are going to demonstrate that, withour method, it is already sufficient for many examplesto consider second-order correlations of Q functions.For this purpose, we use the condition in Eq. (17),which follows from the 2 × 2 matrix condition withσ1 = σ2 = 1/2. This special case of that conditionthen reads as

det(M) = Q(α1)Q(α2)− e−|α2−α1|2/2Q(α1+α2

2)2< 0.(35)

Meaning that, when the correlations from Q functionsat different points in phase space fall below the clas-sical limit zero, nonclassical light is certified with thenonnegative family of Q distributions.

Moreover, since Q functions are nonnegative, thesecond term in Eq. (35) is subtractive in nature.Thus, it is sufficient to find a point α1 in phase spacefor which Q(α1) = 0 holds true—together with anα2 with Q(α2/2) > 0, which has to exist becauseof normalization—in order to certify nonclassicalitythrough Eq. (35). Setting α1 = α, this leads to thesimple nonclassicality condition Q(α) = 0, which ap-plies to arbitrary quantum states. In Ref. [89], thisspecific condition has been independently verified asa nonclassical signature of non-Gaussian states. Here,we see that this nonclassical signature is indeed acorollary of our general approach. Furthermore, weremark that this condition only holds if theQ functionis exactly zero. In experimental scenarios, in whicherrors have to be accounted for, it is infeasible to getthis exact value. Therefore, the condition Eq. (35) ismore practical as it allows us to certify nonclassical-ity through a finite negative value. Furthermore, thiscondition is applicable even if Q(α) = 0 does not holdtrue.

5.1 Discrete-variable statesWe start our analysis of nonclassicality by consider-ing discrete-variable states for a single mode. In thecase of quantized harmonic oscillators, such as elec-tromagnetic fields, a family of discrete-variable statesthat are of particular importance are number states|n〉. They represent an n-fold excitation of the under-lying quantum field and show the particle nature ofsaid fields, thus being nonclassical when compared toclassical electromagnetic wave phenomena. However,photon-number states require Glauber-Sudarshan Pdistributions that are highly singular because theyinvolve up to 2nth-order derivatives of delta distri-butions [70]. On the other hand, the Q function of

Figure 2: Nonclassicality of number states |n〉 via Eq. (35)on a logarithmic scale. We choose α1 = 0 and determinedan optimal α2 =

√2n as points in phase space to correlate Q

functions. The largest certification of nonclassicality is foundfor a single photon, n = 1, and it decreases thereafter.

photon-number states,

Q|n〉(α) = |α|2n

πn! e−|α|2 , (36)

is an accessible and smooth, but nonnegative function.Thus, by itself, it cannot behave as a quasiprobabil-ity which includes negative contributions that uncovernonclassicality.

Except for vacuum, the Q|n〉 function is zero forα = 0 and positive for all other arguments α [Eq.(36)]. Consequently, we can apply Eq. (35) withα1 = 0 and α2 6= 0, yielding det(M) < 0. Fur-thermore, a straightforward optimization shows that|α2| =

√2n results in the minimal value det(M) =

−e−2n(n/2)2n/(πn!)2. Note that this family ofdiscrete-variable number states is rotationally invari-ant, rendering the phase of α2 irrelevant. In Fig. 2,we visualize the results of our analysis. For all numberstates, we observe a successful verification of nonclas-sicality in terms of inequality Eq. (35). The single-photon state shows the largest violation for this spe-cific nonclassicality test, and the negativity of det(M)decreases with the number of photons. A possibleexplanation for this behavior is that this conditionis most sensitive towards the particle nature of thequantum states, being most prominent in the singleexcitation of the quantized radiation field. Again, letus emphasize that we verified nonclassciality via amatrix M of classical (i.e., nonnegative) phase-spacefunctions.

5.2 Continuous-variable statesAfter studying essential examples of discrete-variable quantum states, we now divert ourattention to typical examples of continuous-variable states. For this reason, we considersqueezed vacuum states which are defined as |ξ〉 =(cosh r)−1/2∑∞

n=0(−eiϕ tanh[r]/2)n√

(2n)!|2n〉/n!,

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Figure 3: The maximally negative value for inequality (35)as a function of the squeezing parameter r is depicted, forthe choice α1 = 0. Because of det(M) < 0, nonclassicalityis certified via Q functions for all r 6= 0 [with Q(α) 0for all α]. A maximal violation of the classical constraintdet(M) ≥ 0 for the considered 2× 2 matrix M is found for4.95 dB [= 10 log10(e−2r) for r ≈ 0.57] squeezing.

for a squeezing parameter r = |ξ| and a phaseϕ = arg(ξ). Without a loss of generality, we setϕ = 0. Squeezed states are widely used in quan-tum optical experiments and provide the basis ofcontinuous-variable quantum information processing[31]. Their parametrized phase-space distributionsare known to be either highly singular or nonnegativeGaussian functions (see, e.g., Refs. [32, 69]). Forexample, the Q function of the states under studycan be written as

Q|ξ〉(α) =exp

[−|α|2 − tanh(r)Re(α2)

]π cosh(r) . (37)

In the context of earlier discussions, note that this Qfunction is not zero for α = 0, or anywhere else.

In Fig. 3, the left-hand side of inequality Eq. (35)is shown for the Q|ξ〉 function of a squeezing param-eter r. The points in phase space are determinedby choosing α1 = 0 and minimizing det(M), beingsolved for α2 = [(2/λ) ln[(1 +λ)/(1 +λ/2)]]1/2, whereλ = tanh(r). We observe negative values as a di-rect signature of the nonclassicality of squeezed states.Remarkably, this is achieved using the same criterionthat applies to photon-number states, typically vastlydifferent correlation functions are required (using ei-ther photon numbers [39] or quadratures [41]). Whileinequality Eq. (35) is violated for any squeezing pa-rameter r > 0, we see that there exists an optimalregion of squeezing values around r = 0.6 (likewise,5 dB of squeezing) for which the considered criterion isoptimal. In particular, this shows that this conditionworks optimally in a range of moderate squeezing val-ues and, thus, is compatible with typical experiments.We also want to recall that the Q|ξ〉 are a Gaussiandistributions which do not have any zeros in the phasespace. Thus, criteria based on the zeros of the Husimi

(a)

(b)

Figure 4: In plot (a), the two-modeQ function in Eq. (39) forthe mixed and weakly correlated state ρ is depicted for |λ|2 =1/2 and phase-space points with Im(α(1)) = Im(α(2)) = 0.Part (b) visualizes the application of the nonclassicality in-equality (40) to this state for N = 2 modes and for the pa-rameter pairs (α(1)

1 , α(2)1 ) = (α, 0) and (α(1)

2 , α(2)2 ) = (0, β).

Nonclassicality is verified because of det(M) < 0, and max-imized for |α| and |β| around one.

Q function [89] cannot detect nonclassicality in thisscenario. In contrast, our inequality condition caneven certify this Gaussian nonclassicality, hence pro-viding a more sensitive approach to detecting quan-tum light.

5.3 Mixed two-mode statesTo further challenge our approach, we now con-sider a bipartite mixed state. We begin witha two-mode squeezed vacuum state, |λ〉 =√

1− |λ|2∑∞n=0 λ

n|n, n〉. This state undergoes a fullphase diffusion, leading to the mixed state

ρ = 12π

2π∫0

dϕ |λeiϕ〉〈λeiϕ|

=∞∑n=0

(1−|λ|2)|λ|2n|n, n〉〈n, n|.

(38)

This state presents a particular challenge for nonclas-sicality verification because it shows only weak non-classicality and quantum correlations. Namely, thisstate is not entangled, has zero quantum discord, and

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Figure 5: Determinant (×104) of the multimode 2×2 phase-space matrix M of Q functions [det(M) in Eq. (40)] for theskew-symmetric, tripartite state |Ψ(−)

γ,3 〉, with α(m)1 = α1 and

α(m)2 = 0 for m = 1, 2, 3. Nonclassicality is verified for all

coherent amplitudes γ, which, without loss of generality, canbe chosen as a nonnegative number.

has classical marginal single-mode states (i.e., the par-tial traces tr1(ρ) = tr2(ρ) yield thermal states) [90].However, it shows nonclassical photon-photon corre-lations [91, 90, 92]. The state’s two-mode Q functioncan be computed using Gaussian functions and thephase averaging in Eq. (38), which gives

Qρ(α(1), α(2)) = 1−|λ|2

π2 e−|α(1)|2−|α(2)|2

× I0(2|λ||α(1)||α(2)|),(39)

where I0 denotes the zeroth modified Bessel functionof the first kind. See also Fig. 4(a) in this context.

To apply our approach, whilst using Q functionsonly, we can directly generalize our criterion in Eq.(35) to the multimode case (see also Sec. 4.1). For N

modes, this results in the nonclassicality criterion

det(M) =Q(α(1)1 , . . . , α

(N)1 )Q(α(1)

2 , . . . , α(N)2 )

− e−∑N

m=1|α(m)

2 −α(m)1 |2/2

×Q

(α

(1)1 +α(1)

22 , . . . ,

α(N)1 +α(N)

22

)2

< 0.

(40)

In Fig. 4(b), we apply the case N = 2 of this in-equality to identify the nonclassicality of ρ for |λ|2 =1/2. Again, the same approach as used in both single-mode scenarios enables us yet again to uncover thenonclassical behavior of this bipartite state for allnonzero choices of parameters |α| and |β|. Note inthis context that the phase of these parameters doesnot contribute because of the fully phase-randomizedstructure of the mixed state in Eq. (38).

5.4 Multimode superposition statesTo further exceed the previous, bipartite state, weconsider an N -mode state in this part. Specifically,we focus on a multimode superposition of coherentstates [93],

|Ψ(±)γ,N 〉 = |γ〉

⊗N ± | − γ〉⊗N√2(1± e−2N |γ|2

) , (41)

which consists of two N -fold tensor products of polaropposite coherent states, |±γ〉. Specifically, the skew-symmetric state |Ψ(−)

γ,N 〉 is of interest because it yieldsa GHZ state for |γ| → ∞ and W state for |γ| → 0,combining in an asymptotic manner two inequivalentforms of multipartite entanglement [94, 14].

The Q functions for the states in Eq. (41) can be straightforwardly computed; they read

Q|Ψ(±)γ,N〉(α

(1), . . . , α(N)) = e−N |γ|2e−|α

(1)|2 · · · e−|α(N)|2

2πN[1± e−2N |γ|2

] (cosh

[2 Re

(γ∗

N∑m=1

α(m)

)]± cos

[2 Im

(γ∗

N∑m=1

α(m)

)]).

(42)

To apply our criteria in Eq. (40), and for simplicity,we set α(m)

j = αj for all mode numbers m and pointsin phase space, αj . In Fig. 5, we exemplify the cer-tification of nonclassicality for the state |Ψ(−)

γ,3 〉 [Eq.(41)] as a function of α1 = Re(α1) and for a fixedα2 = 0. We remark that, for other mode numbersN , the plot looks quite similar. Most pronounced arenonclassical features for γ close to zero, relating toa W state in which a single photon is uniformly dis-tributed over three modes. For large γ values, relatingto a GHZ state, the negativities decrease, but det(M)

remains below zero. We reiterate that our relativelysimple, second-order correlations of Q functions ren-der it possible to certify the nonclassical properties ofmultimode, non-Gaussian states.

5.5 Generalized phase-space representationsand nonlinear detection modelFor demonstrating how our phase-space matrix ap-proach functions beyond s-parametrized distribu-tions, we consider an on-off detector that is based ontwo-photon absorption [95]. In this case, the POVM

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Figure 6: Application of the nonclassicality criterion in Eq.(44) as a function of Re(α1) and Im(α2), while fixingIm(α1) = Re(α2) = 0. Nonlinear detectors—thus, a nonlin-ear Q function—with η = σ = 1 and χ = 0.01 are used, Eq.(43). Because of the negativities for the considered single-mode, symmetric state [cf. Eq. (41) for N = 1 and γ = 1],this state is shown to be nonclassical.

element for no click is approximated by

Π = :e−ηn+χn2: =

∞∑n=0

(2n)!n!

(χ

η2

)n: (ηn)2n

(2n)! e−ηn:,

(43)

where :(ηn)2ne−ηn/(2n)!: describes a measurementoperator for 2n-photon states with a linear quantumefficiency η. In this context, it is worth mentioningthat χ [eη2]/[4n] has to be satisfied to ensure thatthe approximated POVM element correctly appliesfor photon numbers up to 2n [96]. The parameterχ relates to the nonlinear absorption efficiency.

Based on such a nonlinear detector, we thendefine the non-Gaussian operator Ω(α; η, χ) =:e−ηn(α)+χn(α)2 :, as described in Secs. 4.2 and 4.3.For a correlation measurement with two detectors (seeFig. 1), this then results in the correlation matrix el-ements 〈:Ω(αi; ηi, χi)Ω(αj ; ηj , χj):〉. For specific pa-rameters and up to a scaling with π, this correla-tion function also results in the nonlinear QΩ(α) =〈:Ω(α; 1, χ)Ω(0; 0, 0):〉 function (cf. Sec. 4.3 for thesimilarly defined PΩ), where σ = η = 1. By extension,and using χ = χ′ and η = 1 = η′, these phase-spacecorrelation functions also provide the entries requiredfor the nonclassicality criterion. Here, it reads

〈:Ω(α1; 1, χ)Ω(α1; 1, χ):〉〈:Ω(α2; 1, χ)Ω(α2; 1, χ):〉−〈:Ω(α1; 1, χ)Ω(α2; 1, χ):〉2 < 0,

(44)

which applies to the nonlinear detection scenario un-der study.

In Fig. 6, we apply this approach and consider thesingle-mode even coherent state |Ψ(+)

γ,1 〉 [cf. Eq. (41)for N = 1], which is a non-Gaussian state, becausewe focused on the odd coherent state in the previous

example. It is worth emphasizing that other meth-ods to infer nonclassical light (e.g., the Chebyshevapproach from Ref. [65]) are incapable to detect thisstate’s quantum features. Here, we can directly cer-tify nonclassicality of this non-Gaussian state despitethe challenge of also having a non-Gaussian detectionmodel.

6 ConclusionIn summary, we devised a generally applicable methodthat unifies nonclassicality criteria from correla-tion functions with quasiprobability distributions.Thereby, we created an advanced toolbox of non-classicality tests which exploit the capabilities ofboth phase-space distributions and matrices of mo-ments to probe for nonclassical effects. Furthermore,our framework is applicable to an arbitrary num-ber of modes, arbitrary orders of correlation, andeven phase-space functions perturbed through convo-lutions with non-Gaussian kernels. A measurementscheme was proposed to directly determine the ele-ments of the phase-space matrix, the underlying keyquantities of our method. In addition, we showedand discussed in detail that our treatment includesprevious findings as special cases, is experimentallyaccessible even if other methods are not, and over-comes challenges of previous techniques when identi-fying nonclassicality.

The phase-space-matrix approach incorporatesnonclassicality tests based on negativities of thephase-space distributions, including the Glauber-Sudarshan P function, and the matrix-of-momentsapproach as special cases. Thus, we were able to unifytwo major techniques for certification of nonclassical-ity. As the P function and the matrix of momentsthemselves are already necessary and sufficient con-ditions for the detection of nonclassicality, the intro-duced phase-space-matrix approach obeys the sameuniversal feature. In other words, for any nonclassi-cal state there exists a phase-space matrix conditionwhich certifies its nonclassicality.

By applying our nonclassicality criteria to a di-verse set of examples, we further demonstrated thepower and versatility of our method. These exam-ples covered discrete- and continuous-variable, single-and multimode, Gaussian and non-Gaussian, as wellas pure and mixed quantum states of light. Remark-ably, we used for all these states only the family ofsecond-order correlations and phase-space distribu-tions which are always nonnegative. Nevertheless,these basic criteria were already sufficient to certifydistinct nonclassical effects on one common ground,further demonstrating the strength of our method.When compared to matrices of moments, the kindsof nonclassicality under study would require very dif-ferent moments for determining the states’ distinctquantum properties. Finally, we put forward an ex-

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perimental scheme, only relying on readily availableoptical components, to directly measure the quanti-ties required to apply our method. This scheme ap-plies even for imperfect detectors with a nonlinear re-sponse. Furthermore, we want to add that the practi-cality and strength of the matrix of phase-space distri-butions in certifying nonclassicality of lossy and noisyquantum state can be experimentally demonstrated[97].

Here, we focused on nonclassical effects of light, ow-ing to their relevance for photonic quantum compu-tation and optical quantum communication. The in-troduced approach may be further developed for thecertification of other quantum features, such as non-Gaussianity. Currently, our method detects nonclas-sicality for Gaussian and non-Gaussian states equally,which could be further developed for a more fine-grained quantumness analysis. However, instead ofapplying normal ordering, the construction of linearand nonlinear witnesses has to be adapted for thispurpose. Furthermore, other kinds of quantum ef-fects, such as entanglement, can be interpreted interms of quasiprobabilities [22] and are similarly wit-nessed through correlations [98]. Thus, an extensionto entanglement might be feasible as well. Therefore,our findings may provide the starting point for un-covering quantum characteristics through matrices ofquasiprobabilities in other physical systems. Addi-tionally, the derived framework can be utilized in thecontext of quantum information theory, such as inrecently formulated resource theories of nonclassical-ity [6, 7] and other measures of nonclassicality [99],which employ the phase-space formalism [22], thuspotentially benefiting from our phase-space correla-tion conditions for future applications.

Note added. After finalizing this work, we havebeen made aware of a related work in preparation byJ. Park, J. Lee, and H. Nha [100].

Acknowledgments

M.B. acknowledges financial support by the Leopold-ina Fellowship Programme of the German NationalAcademy of Science (LPDS 2019-01) and by the ErwinSchrödinger International Institute for Mathematicsand Physics (ESI), Vienna (Austria) through the the-matic programme on Quantum Simulation - fromTheory to Application. E.A. acknowledges fundingfrom the European Union’s Horizon 2020 research andinnovation programme under the Marie Skłodowska-Curie IF InDiQE (EU project 845486). The authorsthank J. Park, J. Lee, and H. Nha for valuable com-ments.

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