Probing Nonclassicality with Matrices of Phase-Space Distributions Martin Bohmann1,2, Elizabeth Agudelo1, and Jan Sperling3

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Probing nonclassicality with matrices of phase-space distributions Martin Bohmann1,2, Elizabeth Agudelo1, and Jan Sperling3

1Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria 2QSTAR, INO-CNR, and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy 3Integrated Quantum Optics Group, Applied Physics, Paderborn University, 33098 Paderborn, Germany

We devise a method to certify nonclassical is closely related to entanglement. Each entangled features via correlations of phase-space distri- state is nonclassical, and single-mode nonclassicality butions by unifying the notions of quasipro- can be converted into two- and multi-mode entangle- babilities and matrices of correlation func- ment [10, 11, 12]. tions. Our approach complements and ex- Consequently, a plethora of techniques to detect tends recent results that were based on Cheby- nonclassical properties have been developed, each shev’s integral inequality [Phys. Rev. Lett. coming with its own operational meanings for appli- 124, 133601 (2020)]. The method developed cations. For example, quantumness criteria which are here correlates arbitrary phase-space functions based on correlation functions and phase-space repre- at arbitrary points in phase space, including sentations have been extensively studied in the con- multimode scenarios and higher-order corre- text of nonclassical light [13, 14]. lations. Furthermore, our approach provides The description of physical systems using the necessary and sufficient nonclassicality crite- phase-space formalism is one of the cornerstones of ria, applies to phase-space functions beyond modern physics [15, 16, 17]. Beginning with ideas in- s-parametrized ones, and is accessible in ex- troduced by Wigner and others [18, 19, 20, 21], the periments. To demonstrate the power of notion of a phase-space distribution for quantum sys- our technique, the quantum characteristics of tems generalizes principles from classical statistical discrete- and continuous-variable, single- and theories (including statistical mechanics, chaos the- multimode, as well as pure and mixed states ory, and thermodynamics) to the quantum domain. are certified only employing second-order cor- However, the nonnegativity condition of classical relations and Husimi functions, which always probabilities does not translate well to the quantum resemble a classical probability distribution. domain. Rather, the notion of quasiprobabilities— Moreover, nonlinear generalizations of our ap- i.e., normalised distributions that do not satisfy all proach are studied. Therefore, a versatile and axioms of probability distributions and particularly broadly applicable framework is devised to un- can attain negative values—was established and found cover quantum properties in terms of matrices to be the eminent feature that separates classical of phase-space distributions. concepts from genuine quantum effects. (See Refs. [22, 14] of thorough introductions to quasiproabilities.) 1 Introduction In particular, research in quantum optics signif- icantly benefited from the concept of phase-space Telling classical and quantum features of a physical quasiprobability distributions, including prominent system apart is a key challenge in quantum physics. examples, such as the Wigner function [19], the Besides its fundamental importance, the notion of arXiv:2003.11031v3 [quant-ph] 12 Oct 2020 Glauber-Sudarshan P function [23, 24], and the (quantum-optical) nonclassicality provides the basis Husimi Q function [25]. In fact, the very definition for many applications in photonic quantum technol- of nonclassicality—the impossibility of describing the ogy and quantum information [1,2,3,4,5]. Nonclassi- behaviour of quantum light with classical statistical cality is, for example, a resource in quantum networks optics—is directly connected to negativities in such [6], quantum metrology [7], boson sampling [8], or dis- quasiprobabilities, more specifically, the Glauber- tributed quantum computing [9]. The corresponding Sudarshan P function [26, 27]. Because of the general free (i.e., classical) operations are passive linear opti- success of quasiprobabilities, other phase-space distri- cal transformations and measurement. By exceeding butions for light have been conceived [28, 29, 30], each such operations, protocols which utilize nonclassical coming with its own advantages and drawbacks. For states can be realized. Furthermore, nonclassicality example, squeezed states are represented by nonneg- Martin Bohmann: [email protected] ative (i.e., classical) Wigner functions although they

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 1 form the basis for continuous-variable quantum infor- features. In particular, we show that our matrix- mation science and technology [31, 32, 33], also having based approach can certify nonclassicality even if a paramount role for quantum metrology [34, 35]. the underlying phase-space distribution is nonnega- Another way of revealing nonclassical effects is by tive. In summary, our approach renders it possi- using correlation constraints which, when violated, ble test for nonclassicality by providing easily ac- witness nonclassicality. Typically, such conditions are cessible nonclassicality conditions. While previously formulated in terms of inequalities involving expec- derived phase-space-correlation conditions [65] were tation values of different observables. Examples in restricted to single-mode scenarios, the present ap- optics are photon anti-bunching [36, 37, 38] and sub- proach straightforwardly extends to multimode cases. Poissonian photon-number distributions [39, 40], us- In addition, our phase-space matrix technique in- ing intensity correlations, as well as various squeez- cludes nonclassicality-certification approaches based ing criteria, being based on field-operator correlations on phase-space distributions and matrices of moments [41, 42, 43, 44]. They can follow, for instance, from as special cases, resulting in an overarching structure applying Cauchy-Schwartz inequalities [45] and un- that combines both previously separated techniques. certainty relations [46], as well as from other viola- The paper is structured as follows. Some initial tions of classical bounds [47, 48, 49]. Remarkably, remarks are provided in Sec.2. Our method is rig- many of these different criteria can be jointly de- orously derived and thoroughly discussed in Sec.3. scribed via so-called matrix of moments approaches Section4 concerns several generalizations and poten- [50, 51, 52, 53, 54]. However, each of the mentioned tial implementations of our toolbox. Various exam- kinds of nonclassicality, such as squeezed and sub- ples are analyzed in Sec.5. Finally, we conclude in Poissonian light, requires a different (sub-)matrix of Sec.6. moments, a hurdle we aim at overcoming. Over the last two decades, there had been many at- 2 Preliminaries tempts to unify matrix-of-moment-based criteria with quasiprobabilities. For example, the Fourier trans- In their seminal papers [23, 24], Glauber and Su- form of the P function can be used, together with darshan showed that all quantum states of light can Bochner’s theorem, to correlate such transformed be represented diagonally in a coherent-state ba- phase-space distributions through determinants of a sis through the Glauber-Sudarshan P distribution. matrix [55, 56], being readily available in experimen- Specifically, a single-mode quantum state can be ex- tal applications [57, 58, 59, 60], and further extending panded as to the Laplace transformation [61]. Furthermore, a Z joint description of field-operator moments and trans- ρˆ = d2α P (α)|αihα|, (1) formed phase-space functions has been investigated as well [62]. Rather than considering matrices of phase- where |αi denotes a coherent state with a complex space quasiproabilities, concepts like a matrix-valued amplitude α. Then, classical states are identified as distributions enable us to analyzed nonclassical hy- statistical (i.e., incoherent) mixtures of pure coher- brid systems [63, 64]. Very recently, a first successful ent states, which resemble the behavior of a classical strategy that truly unifies correlation functions and harmonic oscillator most closely [66, 67]. For this di- phase-space functions has been conceived [65]. How- agonal representation to exist for nonclassical states ever, these first demonstrations of combining phase- as well, the Glauber-Sudarshan distribution has to space distributions and matrices of moments are still exceed the class of classical probability distributions restricted to rather specific scenarios. [26, 27], particularly violating the nonnegativity con- In this contribution, we formulate a general frame- straint, P 0. This classification into states which work for uncovering quantum features through cor- have a classical correspondence and those which are relations in phase-space matrices which unifies these genuinely quantum is the common basis for certifying two fundamental approaches to characterizing quan- nonclassical light. tum systems. By combining matrix of moments As laid out in the introduction, nonclassicality is a and quasiprobabilities, this method enables us to vital resource for utilizing quantum phenomena, rang- probe nonclassical characteristics in different points ing from fundamental to applied [6,7,8,9]. In this in phase space, even using different phase-space dis- context, it is worth adding that, contrasting other tributions at the same time. We specifically study notions of quantumness, nonclassicality is based on implications from the resulting second- and higher- a classical wave theory. That is, it is essential to order phase-space distribution matrices for single- and discern nonclassical coherence phenomena from those multimode quantum light. Furthermore, a direct which are accessible with classical statistical optics, measurement scheme is proposed and non-Gaussian as formalized through Eq. (1) with P ≥ 0. See, e.g., phase-space distributions are analyzed. To bench- Ref. [68] for a recent experiment that separates clas- mark our method, we consider a manifold of exam- sical and quantum interference effects in such a man- ples, representing vastly different types of quantum ner. For instance, free operations, i.e., those maps

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 2 which preserve classical states, do include beam split- holds true for all P ≥ 0. Now, one can expand fˆ in ˆ P ˆ ter transformations, resulting in the generation of en- terms of a given set of operators, e.g., f = i ciOi, re- tanglement from single-mode nonclassical states via ˆ† ˆ P ∗ ˆ† ˆ sulting in h:f f:i = i,j ci cjh:Oi Oj:i. Furthermore, such a free operation [10, 11, 12] that is vital for many this expression is nonnegative [cf. Eq. (5)] iff the ma- quantum protocols. ˆ† ˆ trix (h:Oi Oj:i)i,j is positive semidefinite. This constraint can, for example, be probed using Sylvester’s 2.1 Phase-space distributions criterion [73] which states that a Hermitian matrix is positive-definite if and only if all its leading princi- Since the Glauber-Sudarshan distribution can be a pal minors have a positive determinant. It is worth highly singular distribution (see, e.g., Ref. [69]), mentioning that Eq. (5) defines the notion of a non- generalized phase-space functions have been devised. classicality witness, where h:fˆ†fˆ:i < 0 certifies non- Within the wide range of quantum-optical phase- classicality. space representations, the family of s-parametrized distributions [29, 30] is of particular interest. Such The above observations form the basis for many ex- distributions can be expressed as perimentally accessible nonclassicality criteria, such σ as using basis operators which are powers of quadra- P (α; σ) = h: exp (−σnˆ(α)) :i , (2) ture operators [50], photon-number operators [44], π and general creation and annihilation operators [71, where colons indicate normal ordering [70] and nˆ(α) = 72]. See Refs. [13] for an overview of moment- † (â − α) (â − α) is the displaced photon-number op- based inequalities. In the following, we are going to erator, written in terms of bosonic annihilation and combine the phase-space distribution technique with † creation operators, aˆ and aˆ , respectively. It is worth the method of matrices of moments to arrive at the recalling that the normal ordering acts on the expres- sought-after unifying approach of both techniques. sion surrounded by the colons in such a way that creation operators are arranged to the left of annihilation operators whilst ignoring commutation relations. Note that, for convenience, we parametrize distribu- 3 Matrix of phase-space distributions tions via the width parameter σ, rather than using s. Both are related via Both phase-space distributions and matrices of mo- 2 ments exhibit a rather dissimilar structure when it σ = . (3) 1 − s comes to formulating constraints for classical light. Consequently, a full unification of both approaches From this relation, we can identify the Husimi func- is missing to date, excluding the few attempts men- tion, Q(α) = P (α; 1), for s = −1 and σ = 1; tioned in Sec.1. In this section, we bridge this gap the Wigner function, W (α) = P (α; 2), for s = 0 and derive a matrix of phase-space distributions which and σ = 2; and the Glauber-Sudarshan function, leads to previously unknown nonclassicality criteria, P (α) = P (α; ∞), for s = 1 and σ = ∞. also overcoming the limitations of earlier methods. Whenever a phase-space distribution contains a negative contribution, i.e., P (α; σ) < 0 for at least one pair (α; σ), the underlying quantum state is nonclassical [26, 27]. In such a case, the distribution 3.1 Derivation P (α; σ) refers to as a quasiprobability distribution For the purpose of deriving our criteria, we consider which is incompatible with classical probability the- ˆ P an operator function f = ci exp[−σinî(αi)]. Then, ory. Nonetheless, for any σ ≥ 0 and any state, this i the normally ordered expectation value of fˆ†fˆ can be function represents a real-valued distribution which is expanded as normalized, P (α; σ) = P (α; σ)∗ and R d2α P (α; σ) = 1. In addition, it is worth mentioning that the nor- ˆ† ˆ X ∗ −σinˆ(αi) −σj nˆ(αj ) malization of the state is guaranteed through the limit h:f f:i = ci cjh:e e :i π i,j lim P (α; σ) = h: exp(0):i = h1î = 1. (4) X σiσj σ→0 σ = c∗c exp − |α − α |2 j i σ + σ i j (6) i,j i j 2.2 Matrix of moments approach σiαi + σjαj × : exp −(σi + σj)n ˆ : . Besides phase-space distributions, a second family of σi + σj nonclassicality criteria is based on correlation functions; see, e.g., Refs. [71, 72] for introductions. For Based on the above relation, we may define two ma- this purpose, we can consider an operator function trices, one for classical amplitudes, fˆ = f(â, aˆ†). Then, Z cl. (c) σiσj 2 h:fˆ†fˆ:i = d2α P (α)|f(α, α∗)|2 ≥ 0 (5) M = exp − |αi − αj| , (7) σi + σj i,j

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 3 and one for the quantum-optical expectation values, 3.2 Second-order criteria (q) σiαi + σjαj We begin our consideration with an interesting M = : exp −(σi + σj)n ˆ : second-order case. We chose (α1; σ1) = (0; 0) and σi + σj i,j (α2; σ2) = (α; σ). This yields the 2 × 2 phase-space π σiαi + σjαj = P ; σi + σj , matrix σi + σj σi + σj i,j 1 h: exp(−σnˆ(α)):i (8) M = . (12) h: exp(−σnˆ(α)):i h: exp(−2σnˆ(α)):i which can be expressed in terms of phase-space distributions using Eq. (2). Specifically, M (q) corresponds Up to a positive scaling, the determinant of this ma- to a matrix of phase-space distributions. Moreover, trix results in the following nonclassicality criterion: the fact that the normally ordered expectation value ˆ† ˆ 2π of f f is nonnegative for classical light [Eq. (5)] P (α; 2σ) − (P (α; σ))2 < 0. (13) is then identical to the entry-wise product (i.e., the σ Hadamard product ◦) of both matrices being positive In particular, we can set σ = 1 to relate this condi- semidefinite, tion to the Wigner and Husimi functions, leading to 2 cl. W (α)−2πQ(α) < 0. This special case of our general (c) (q) M ≥ 0, with M = M ◦ M (9) approach has been recently derived using a very different approach, using Chebyshev’s integral inequal- defining our phase-space matrix M. ity [65]. There it was shown that, by applying the For classical light, all principal minors of M have inequality (13) for σ = 1, it is possible to certify non- to be nonnegative according to Sylvester’s criterion. classicality even if the Wigner function of the state Conversely, the violation of this constraint certifies a under study is nonnegative. In this context, remem- nonclassical state, ber that the Husimi function, Q(α) = hα|ρˆ|αi/π, is det(M) < 0, (10) always nonnegative, regardless of the state ρˆ. Beyond this scenario, we now study a more gen- where M is defined through arbitrary small or large eral 2 × 2 phase-space matrix M. For an efficient sets of parameters σi and σj and coherent ampli- description, it is convenient to redefine transformed tudes αi and αj. Therefore, inequality (10) enables us parameters as to formulate various nonclassicality conditions which σ1α1 + σ2α2 correlate distinct phase-space distributions as it typ- ∆α = α2 − α1 and A = , (14a) ically only done for matrix-of-moments-based tech- σ1 + σ2 σ1σ2 niques when using different kinds of observables. We σ˜ = and Σ = σ1 + σ2. (14b) finally remark that the expression in Eq. (10) resem- σ1 + σ2 bles a nonlinear nonclassicality witnessing approach. Note that these parameters relate to the two-body As a first example, we may explore the first-order problem. That is, the quantities in Eq. (14a) define criterion, i.e., a 1 × 1 matrix of quasiprobabilities. the relative position and barycenter in phase-space, Selecting arbitrary σ-parameters and coherent ampli- respectively; and the two quantities in Eq. (14b) re- tudes, i.e., (α1; σ1) = (α; σ), we find the following semble the reduced and total mass in a mechanical restriction for classical states [cf. Eq. (10)]: system, respectively. In this alternative parametrization, the two matri- π cl. P (α; 2σ) ≥ 0. (11) ces, giving the total phase-space matrix M = M (c) ◦ 2σ M (q), read This inequality reflects the fact that finding nega- −σ˜|∆α|2 ! tivities in a parametrized phase-space distribution (c) 1 e M = 2 , and (15a) P (α; 2σ) is sufficient to certify nonclassicality. Also e−σ˜|∆α| 1 recall that we retrieve the Glauber-Sudarshan distri- h:e−2σ1nˆ(α1):i h:e−Σˆn(A):i bution in the limit σ → ∞. Since the nonnegativity M (q) = . (15b) of this distribution defines the very notion of a non- h:e−Σˆn(A):i h:e−2σ2nˆ(α2):i classical state [26, 27], we can conclude from this examples that our approach is necessary and sufficient Hence, the determinant of the Hadamard product of for certifying nonclassicality. both matrices then gives However, the Glauber-Sudarshan distribution has −2σ1nˆ(α1) −2σ2nˆ(α2) the disadvantage of being a highly singular for many det(M) = h:e :ih:e :i 2 relevant nonclassical states of light and, thus, hard to −e−2˜σ|∆α| h:e−Σˆn(A):i2. (16) reconstruct from experimental data. Consequently, it is of practical importance (see Secs.4 and5) to If this determinant is negative for the state of light consider higher-order criteria beyond this trivial one. under study, its nonclassicality is proven. In terms of

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 4 phase-space distributions, this condition can be also 3.3 Higher-order cases recast into the form

2 4˜σ h −σ˜|∆α|2 i P (α1; 2σ1)P (α2; 2σ2) − e P (A; Σ) < 0. The next natural extension concerns the analysis of Σ (17) higher-order correlations. Clearly, one can obtain an increasingly large set of nonclassicality tests with an Interestingly, this nonclassicality criterion correlates increasing dimensionality of M, determined by the different points in phase space for different distribu- number of pairs (αi; σi). In order to exemplify this tions, P (α1; 2σ1) and P (α2; 2σ2), with a phase-space potential, let us focus on one specific 3 × 3 scenario distribution with the total width Σ at the barycenter and more general scenarios for specific choices of pa- A of coherent amplitudes, P (A; Σ). rameters.

Let us discuss the 3 × 3 case ﬁrstly, for which we are going to consider σ3 = 0. From this, one obtains the following phase-space matrix:   h: exp(−2σ1nˆ(α1)):i h: exp(−σ1nˆ(α1) − σ2nˆ(α2)):i h: exp(−σ1nˆ(α1)):i M = h: exp(−σ1nˆ(α1) − σ2nˆ(α2)):i h: exp(−2σ2nˆ(α2)):i h: exp(−σ2nˆ(α2)):i . (18) h: exp(−σ1nˆ(α1)):i h: exp(−σ2nˆ(α2)):i 1

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

2! 2! det(M) P (α1; 2σ1) P (α1; σ1) P (α2; 2σ2) P (α2; σ2) 2 = − π − π π 2σ1 σ1 2σ2 σ2 (19) P (A; Σ) P (α ; σ ) P (α ; σ )2 − exp(−σ˜|∆α|2) − π 1 1 2 2 < 0, Σ σ1 σ2 using the parameters deﬁned in Eqs. (14a) and (14b). In fact, this condition combines the earlier derived criteria of the forms (13) and (16) in a manner similar to cross-correlations nonclassicality conditions known from matrices of moments [61].

Another higher-order matrix scenario corresponds 3.4 Comparison with Chebyshev’s integral in- to having identical coherent amplitudes, i.e., αi = α equality approach for all i. In this case, we find that the two Hadamard- product components of the matrix M simplify to As mentioned previously, a related method based on Chebyshev’s integral inequality has been introduced M (c) = (1) and recently [65]. It also provides inequality conditions for i,j different phase-space distributions. The nonclassical- (20) (q) π ity conditions based on Chebyshev’s integral inequal- M = P (α; σi + σj) , σi + σj i,j ity take the form D (q) Σ Y π thus resulting in M = M . Therefore, we can for- P (α; Σ) − P (α; σ ) < 0, (22) π σ i mulate nonclassicality criteria which correlate an ar- i=1 i bitrary number of different phase-space distributions, PD defined via σi, at the same point in phase-space, α. where Σ = i=1 σi. To compare both approaches, let Analogously, one can consider a scenario in which us discuss their similarities and differences. In its simplest form, involving only σ and σ , the all σ parameters are identical, σi = σ. Then, we get 1 2 condition in Eq. (22) resembles the tests based on the 2 (c) −σ|αi−αj | /2 2 × 2 matrix in Eq. (12). In particular, for the case M =(e )i,j and σ1 = σ2 = σ both methods yield the exact same con- π α + α (21) M (q) = P i j ; 2σ . ditions. For σ1 6= σ2 such an agreement of both meth- 2σ 2 i,j ods cannot be found because of the inherent symmetry of the phase-space matrix approach, M = M †, which Consequently, we obtain nonclassicality criteria which stems from its construction via a quadratic form; cf. correlate an arbitrary number of different points in Eq. (6). Also, for more general, higher-order condi- phase space, αi, for a single phase-space distribution, tions, i.e. D > 2, such similarities cannot be found parametrized by σ. either. Conditions of the form in Eq. (22) consist of

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 5 P −(m+n) m n σ|α|2 −σnˆ(α) only two summands. The first term is a single phase- m,n cm,nσ ∂α ∂α∗ e :e :|α=0,σ=0 = space function with the width parameter Σ which is P †m n ˆ ˆ† ˆ m,n cm,naˆ aˆ . For such a choice f, h:f f:i < 0 is associated to the highest σ parameter involved in the in fact identical to the most general form of the matrix inequality. The second term is a product of D phase- of moments criterion for nonclassicality [71, 72]. space distributions, each individual distribution has In conclusion, we find that our necessary and suf- a width parameter σi, together bound by the condi- ficient methodology not only includes nonclassicality PD tion Σ = i=1 σi. By comparison, our phase-space criteria based on phase-space functions themselves [cf. matrix approach yields, in general, a richer and more Eq. (11)], but it also includes the technique of ma- complex set of higher-order nonclassicality tests, such trices of moments as a special case. In a hierarchical as demonstrated in Sec. 3.3. picture, this means that our family of nonclassicality Let us point out further differences between the criteria, including arbitrary orders of σ-parametrized two approaches. Firstly, we observe that the inequal- phase-space functions, encompasses both negativities ities based on Chebyshev’s integral inequality only of phase-space functions and matrices of moments. apply to one single point in phase space. In con- Because of the above relation, the order of moments trast, the phase-space matrix method devised here that is required to certify nonclassicality also sets an includes conditions that combine different points in upper bound to the size of the matrix of phase-space phase space; cf. Eq. (6). Secondly, Chebyshev’s inte- distributions so that it certifies nonclassicality. There- gral inequality approach cannot be extended to mul- fore, our approach unifies and subsumes both earlier timode settings. Such a limitation does not exist for types of nonclassicality conditions. the matrix approach either, as we show in the following Sec. 4.1. We conclude that both the technique in Ref. [65] and our phase-space matrix approach for ob- 4 Generalizations and implementation taining phase-space inequalities yield similar second- order conditions but, in general, give rise to rather In this section, we generalize our approach to arbi- different nonclassicality criteria. In particular, the trary multimode nonclassical light and propose a mea- phase-space matrix framework offers a broader range surement scheme to experimentally access the matrix of variables—be it coherent amplitudes or widths— of phase-space distributions. In addition, we show that lead to a richer set of nonclassicality conditions. that our approach applies to phase-space distributions which are no longer limited to σ parametrizations and relate these findings to the response of nonlinear de- 3.5 Extended relations to nonclassicality crite- tection devices. ria To finalize our first discussions we now focus on the 4.1 Multimode case relation to matrices of moments. Previously, we have After our in-depth analysis of single-mode phase- shown that, already in the first order, our criteria are space matrices, the multimode case follows almost necessary and sufficient to verify the nonclassicality, straightforwardly. For the purpose of such a gener- and we discussed our method in relation to Cheby- alization, we consider N optical modes, represented chev’s integral inequality. Furthermore, indirect tech- via the annihilation operators aˆm for m = 1,...,N niques using transformed phase-space functions, such and extending to the displaced photon-number oper- as the characteristic function [62] and the two-sided (m) (m) † (m) ators nˆm(α ) = (âm − α ) (âm − α ). Now, Laplace transform [61], have been previously related σ-parametrized multimode phase-space functions can to moments. Thus, the question arises what the rela- be expressed as tion of our direct technique to such matrices of mo- D (1) (1) (N) (N) E ments is. :e−σ nˆ1(α ) ··· e−σ nˆN (α ): For showing that our framework includes the ma- (24) πN trix of moments technique, we may remind ourselves = P (α(1), . . . , α(N); σ(1), . . . , σ(N)), that derivatives can be understood as a linear combi- σ(1) ··· σ(N) nation, specifically as a limit of a differential quotient, where we allow for different s parameters for each ∂mg(z) = lim −m Pm m(−1)m−kg(z + k). z →0 k=0 k mode, with s(m) = 1 − 2/σ(m) [Eq. (3)]. As in the This enables us to write [75] single-mode case, we can now formulate a matrix M of multimode phase-space functions, †m n −(m+n) m n σ|α|2 −σnˆ(α) aˆ aˆ = σ ∂α ∂α∗ e :e : , α=0 and σ=0  N N  * P (m) (m) P (m) (m) + (23) − σi nˆm(αi ) − σj nˆm(αj ) †m n  m=0 m=0  expressing arbitrary moments aˆ aˆ via linear com- M=  :e e :  . binations of the normally ordered operators that represent σ-parametrized phase-space distributions. i,j Thus, in the corresponding limits, we can iden- Consequently, this matrix of phase-space functions tify the operator fˆ in Eq. (5) with fˆ = also has to be positive semidefinite if the underlying

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 6 state of multimode light is classical. That is, LOi Π(n̂) cl. signal ρ̂ M ≥ 0 (25) holds true for classical light and for any dimension (or 2 2 matrix order) of the multimode matrix M and any sigma and |t| :|r| × entry Mi,j alpha values. Conversely, det(M) < 0 is a nonlinear witness of multimode nonclassicality. Similarly to the 50:50 single-mode case, an increasingly large matrix M with Π increasingly dense sets of parameters for the various (n̂) LO alpha and sigma values then enables one to probe the j nonclassicality of arbitrary multimode states. Figure 1: Outline of phase-space matrix correlation mea- Since we have already exemplified various scenarios surement. The signal, i.e., the state ρˆ of the light field under for single-mode phase-space correlations, in the fol- study, is split into two identical outputs at a 50:50 beam split- lowing, we restrict ourselves to a particular multimode ter. Each of the resulting beams is combined with a local os- case. Specifically, we focus on two optical modes and cillator (LO) on a |t|2 : |r|2 beam splitter and measured with a 3 × 3 phase-space matrix M is, a photon-number-based detector, represented through Π(ˆn). The resulting correlations yield the entries of our phase-space  πP (α(1);σ) πP (α(2);σ)  matrix M. 1 σ σ  πP (α(1);σ) πP (α(1);2σ) π2P (α(1),α(2);σ,σ)  ,  σ 2σ σ2  (2) 2 (1) (2) (1) πP (α ;σ) π P (α ,α ;σ,σ) πP (α ;2σ) For the setup in Fig.1, we begin our considera- σ σ2 2σ tions with a coherent state |βi, representing our sig- where quasiprobabilities as a function of single-mode nal ρˆ = |βihβ|. Firstly, we split this signal equally parameters indicate marginal phase-space distribu- into√2 modes,√ resulting in a two-mode coherent state tions. Adopting a notation of pairs of coherent am- |β/ 2, β/ 2i. In addition, local oscillator states are plitudes and widths, M is thus defined via the fol- prepared, |βii and |βji for each mode. Each of the (1) (2) (1) (2) lowing two-mode parameters: (α1 , α1 ; σ1 , σ1 ) = two signals is then mixed with its local oscillator on (1) (2) (1) (2) (1) 2 2 2 2 (0, 0; 0, 0), (α2 , α2 ; σ2 , σ2 ) = (α , 0; σ, 0), and a |t| :|r| beam splitter, where |t| + |r| = 1. One (1) (2) (1) (2) (2) output of each beam splitter is discarded, namely the (α3 , α3 ; σ3 , σ3 ) = (0, α ; 0, σ). In particular, we can express the nonclassicality constraint from the lower and upper one for the top and bottom path in determinant of M [74] for σ = 1 via joint and marginal Fig.1, respectively. This results in the input-output Wigner and Husimi functions, relation β β det M h (1) i h (2) i = W (α ) − Q(α(1))2 W (α ) − Q(α(2))2 |βi 7→ t√ + rβi, t√ + rβj , (27) π4 2π 2π 2 2 h i2 cl. − Q(α(1), α(2)) − Q(α(1))Q(α(2)) ≥ 0. which is then detected as follows. Each of the resulting modes is measured with a (26) detector or detection scheme based on photon ab- Violating this inequality verifies the nonclassicality of sorption, thus being described by a positive operator- the two-mode state under study. valued measure (POVM) which is diagonal in the photon-number representation [79]. Consequently, 4.2 Direct measurement scheme one or a combination of detector outcomes (e.g., in a generating-function-type combination [80]) corre- The reconstruction of phase-space distributions can sponds to a POVM element of the form Π(ˆn) = be a challenging task [76]. For this reason, we are :e−Γ(ˆn):. Using |mihm| = :e−nˆ nˆm/m!: for an going to devise a directly accessible setup to infer m-photon projector, this means that we identify P∞ −nˆ P m the phase-space matrix. See Fig.1 for an outline m=0 πm|mihm| = :e m=0 πmnˆ /m!: = Π(ˆn) = −Γ(ˆn) which is based on the approaches in Refs. [77, 72, 78]. :e :, where the eigenvalues πm corresponds to For convenience, we restrict ourselves to a single the Taylor expansion coefficients of the function z 7→ optical mode; the extension to multiple modes fol- exp[z − Γ(z)]. Accordingly, the function Γ(ˆn) models lows straightforwardly. That is, each of the multiple the detector response [79, 70]. Finally, the correlation modes can be detected individually by a correlation- measurement of this response for our coherent signal measurement setup as depicted in Fig.1. Further- states takes the form √ more, it is noteworthy that our phase-space matrix 2 |t| r 2βi approach is not limited to this specific measurement Mi,j = exp −Γ β − 2 t scheme proposed here and generally applies to any de- √ (28) 2 tection scenario which allows for a reconstruction of |t| r 2βj × exp −Γ β + . quasiprobability distributions. 2 t

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 7 Now it is convenient to define Γ(ˆ˜ n) = Γ(|t|2n/ˆ 2) Such generalized phase-space function can be ob- and √ tained from the Glauber-Sudarshan P function via r 2βi Z αi = − , (29) 2 ∗ † t PΩ(α) = d α˜ P (˜α) Ω(α;α, ˜ α˜ ) = h:Ω(α;a, ˆ aˆ ):i for all LO choices i and, similarly, for j. Further- (32) more, we generalize this treatment to arbitrary states, for a kernel Ω ≥ 0 [30, 84]. The construction of this so- ρˆ = R d2β P (β)|βihβ|, using the Glauber-Sudarshan called filter or regularizing function Ω can be done so representation [Eq. (1)]. Therefore, the correlations that the resulting distribution PΩ is regular (i.e., with- measured as described above [Eq. (28)] obey out the singular behavior known from the P function) and is positive semidefinite for all classical states [84]. D E −Γ(ˆ˜ n(αi)) −Γ(ˆ˜ n(αj )) For instance, a non-Gaussian filter Ω has been used to Mi,j = :e e : , (30) experimentally characterize squeezed states via regular distributions which exhibit negativities in phase which corresponds to a directly measured phase-space space [85]; this cannot be done with s-parametrized matrix element, e.g., for a linear detector response quasiprobability distributions, which are either non- Γ(ˆ˜ n) = σnˆ. The other way around, we can choose negative or highly singular for squeezed states. fˆ = P c exp(−Γ(ˆ˜ n(α ))) for the general classicality i i i As done for the previously considered distributions, constraint in Eq. (5), even for nonlinear detector re- ˆ P † we can define an operator f = i ciΩi(α;a, ˆ aˆ ), sponses. Then, the matrix of phase-space distribution which leads to a phase-space matrix with the entries approach applies, regardless of a linear or nonlinear detection model. (See also Refs. [81, 80] in this con- † † Mi,j = h:Ωi(α;a, ˆ aˆ )Ωj(α;a, ˆ aˆ ):i = PΩ Ω (α). (33) text.) i j As an example, we consider a case with two single This expression utilizes product of filters ∗ ∗ ∗ on-off click detectors (represented by Π(ˆn) in Fig.1) Ω(α;α, ˜ α˜ ) = Ωi(α;α, ˜ α˜ )Ωj(α;α, ˜ α˜ ) to be convo- with a non-unit quantum efficiency ηdet and a non- luted with the P function. From this definition of a vanishing dark-count rate δ [82], which represents re- regularized phase-space matrix, we can proceed as alistic detectors in experiments. In addition, we can we did earlier to formulate nonclassicality criteria in introduce neutral density (ND) filters to attenuate the terms of phase-space functions. light that impinges on each detector. The POVM el- Moreover, the non-Gaussian filter functions can be ement for the no-click event in combination with the even related to nonlinear detectors. For this purpose, ND filters then reads Π(ˆˆ n) = : exp(−(ηnˆ + δ)):, where we assume that Ω(α;α, ˜ α˜∗) = Ω(|α − α˜|2) (likewise, † 0 ≤ η ≤ ηdet is a controllable efficiency. The measured :Ω(α, a,ˆ aˆ ): = :Ω(ˆn(α)): in the normally ordered op- correlation for this scenario takes the form erator representation). In this form, the function is invariant under rotations. As we did for the general Mi,j = exp(−2δ)h: exp (−ηinˆ(αi) − ηjnˆ(αj)) :i. (31) POVM element Π(ˆn), we can now identify

Therein, the adjustable efficiency ηi plays role of σi. Γ(ˆn) = − ln Ω(ˆn). (34) Also, the positive factor that includes the dark counts is irrelevant because it does not change the sign of the This enables us to associate non-Gaussian filters and determinant of M, i.e., the verified nonclassicality. nonlinear detectors and, by extension, generalized In summary, the measurement layout in Fig.1 en- phase-space matrices for certifying nonclassical states ables us to directly measure the entries of our phase- of light. An example for this treatment is studied in space matrix M. As an experimental setup, this Sec. 5.5. scheme also underlines the strong connection between correlations and their measurements and phase-space quasiprobabilities and their reconstruction. We may 5 Examples and benchmarking emphasize that all experimental techniques and com- In the following, we apply our method of phase-space ponents that are used in the proposed setup are read- matrices to various examples and benchmark its per- ily available; see, e.g., the related quantum state re- formance. For the latter benchmark, we could con- construction experiments reported in Refs. [83, 80]. sider different phase-space functions. Using the P function would be impractical as it is often a highly 4.3 Generalized phase-space functions singular distribution. The Wigner function is regular and can exhibit negativities. But error estima- The σ-parametrized phase-space distributions we con- tions from measured data can turn out to be rather sidered so far are related to each other via con- difficult because it requires diverging pattern func- volutions with Gaussian distributions [28, 29, 30]. tions [86, 87] (see Ref. [88] for an in-depth analysis). However, there are additional means to represent a Beyond those practical hurdles, we focus on the Q state without relying on Gaussian convolutions only. function here because, already in theory, it is always

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 8 nonnegative. Thus, it is hard to verify nonclassical features based on this particular phase-space distribution. Additionally, the Q function is easily accessible in experiments and can be directly measured via the widely-used double-homodyne (aka, eight-port homodyne) detection scheme [70]. Nonetheless, we are going to demonstrate that, with our method, it is already sufficient for many examples to 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 with σ1 = σ2 = 1/2. This special case of that condition then reads as Figure 2: Nonclassicality of number states |ni via Eq. (35) −|α −α |2/2 α +α 2 2 1 1 2 on a logarithmic scale. We choose α1 = 0 and determined det(M) = Q(α1)Q(α2) − e Q 2 < 0. √ (35) an optimal α2 = 2n as points in phase space to correlate Q functions. The largest certification of nonclassicality is found Meaning that, when the correlations from Q functions for a single photon, n = 1, and it decreases thereafter. at different points in phase space fall below the classical limit zero, nonclassical light is certified with the photon-number states, nonnegative family of Q distributions. 2n Moreover, since Q functions are nonnegative, the |α| 2 Q (α) = e−|α| , (36) second term in Eq. (35) is subtractive in nature. |ni πn! Thus, it is sufficient to find a point α1 in phase space is an accessible and smooth, but nonnegative function. for which Q(α1) = 0 holds true—together with an Thus, by itself, it cannot behave as a quasiprobabil- α2 with Q(α2/2) > 0, which has to exist because of normalization—in order to certify nonclassicality ity which includes negative contributions that uncover nonclassicality. through Eq. (35). Setting α1 = α, this leads to the simple nonclassicality condition Q(α) = 0, which ap- Except for vacuum, the Q|ni function is zero for plies to arbitrary quantum states. In Ref. [89], this α = 0 and positive for all other arguments α [Eq. specific condition has been independently verified as (36)]. Consequently, we can apply Eq. (35) with a nonclassical signature of non-Gaussian states. Here, α1 = 0 and α2 6= 0, yielding det(M) < 0. Fur- we see that this nonclassical signature is indeed a thermore,√ a straightforward optimization shows that corollary of our general approach. Furthermore, we |α2| = 2n results in the minimal value det(M) = −2n 2n 2 remark that this condition only holds if the Q function −e (n/2) /(πn!) . Note that this family of is exactly zero. In experimental scenarios, in which discrete-variable number states is rotationally invari- errors have to be accounted for, it is infeasible to get ant, rendering the phase of α2 irrelevant. In Fig.2, this exact value. Therefore, the condition Eq. (35) is we visualize the results of our analysis. For all number more practical as it allows us to certify nonclassical- states, we observe a successful verification of nonclas- ity through a finite negative value. Furthermore, this sicality in terms of inequality Eq. (35). The single- condition is applicable even if Q(α) = 0 does not hold photon state shows the largest violation for this spe- true. cific nonclassicality test, and the negativity of det(M) decreases with the number of photons. A possible explanation for this behavior is that this condition 5.1 Discrete-variable states is most sensitive towards the particle nature of the quantum states, being most prominent in the single We start our analysis of nonclassicality by consider- excitation of the quantized radiation field. Again, let ing discrete-variable states for a single mode. In the us emphasize that we verified nonclassciality via a case of quantized harmonic oscillators, such as elec- matrix M of classical (i.e., nonnegative) phase-space tromagnetic fields, a family of discrete-variable states functions. that are of particular importance are number states |ni. They represent an n-fold excitation of the under- 5.2 Continuous-variable states lying quantum field and show the particle nature of said fields, thus being nonclassical when compared to After studying essential examples of discrete- classical electromagnetic wave phenomena. However, variable quantum states, we now divert our photon-number states require Glauber-Sudarshan P attention to typical examples of continuous- distributions that are highly singular because they variable states. For this reason, we consider involve up to 2nth-order derivatives of delta distri- squeezed vacuum states which are defined as |ξi = −1/2 P∞ iϕ np butions [70]. On the other hand, the Q function of (cosh r) n=0(−e tanh[r]/2) (2n)!|2ni/n!,

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 9 (a)

(b)

Figure 3: The maximally negative value for inequality (35) as a function of the squeezing parameter r is depicted, for the choice α1 = 0. Because of det(M) < 0, nonclassicality is certified via Q functions for all r 6= 0 [with Q(α) 0 for all α]. A maximal violation of the classical constraint det(M) ≥ 0 for the considered 2 × 2 matrix M is found for −2r 4.95 dB [= 10 log10(e ) for r ≈ 0.57] squeezing. for a squeezing parameter r = |ξ| and a phase ϕ = arg(ξ). Without a loss of generality, we set ϕ = 0. Squeezed states are widely used in quantum optical experiments and provide the basis of Figure 4: In plot (a), the two-mode Q function in Eq. (39) for 2 continuous-variable quantum information processing the mixed and weakly correlated state ρˆ is depicted for |λ| = (1) (2) [31]. Their parametrized phase-space distributions 1/2 and phase-space points with Im(α ) = Im(α ) = 0. Part (b) visualizes the application of the nonclassicality in- are known to be either highly singular or nonnegative equality (40) to this state for N = 2 modes and for the pa- Gaussian functions (see, e.g., Refs. [32, 69]). For rameter pairs (α(1), α(2)) = (α, 0) and (α(1), α(2)) = (0, β). example, the Q function of the states under study 1 1 2 2 Nonclassicality is verified because of det(M) < 0, and max- can be written as imized for |α| and |β| around one. exp −|α|2 − tanh(r)Re(α2) Q|ξi(α) = . (37) π cosh(r) Q function [89] cannot detect nonclassicality in this In the context of earlier discussions, note that this Q scenario. In contrast, our inequality condition can function is not zero for α = 0, or anywhere else. even certify this Gaussian nonclassicality, hence pro- In Fig.3, the left-hand side of inequality Eq. (35) viding a more sensitive approach to detecting quantum light. is shown for the Q|ξi function of a squeezing parameter r. The points in phase space are determined by choosing α1 = 0 and minimizing det(M), being 5.3 Mixed two-mode states 1/2 solved for α2 = [(2/λ) ln[(1 + λ)/(1 + λ/2)]] , where λ = tanh(r). We observe negative values as a di- To further challenge our approach, we now con- rect signature of the nonclassicality of squeezed states. sider a bipartite mixed state. We begin with Remarkably, this is achieved using the same criterion a two-mode squeezed vacuum state, |λi = p1 − |λ|2 P∞ λn|n, ni. This state undergoes a full that applies to photon-number states, typically vastly n=0 different correlation functions are required (using ei- phase diffusion, leading to the mixed state ther photon numbers [39] or quadratures [41]). While 2π inequality Eq. (35) is violated for any squeezing pa- 1 Z ρˆ = dϕ |λeiϕihλeiϕ| rameter r > 0, we see that there exists an optimal 2π region of squeezing values around r = 0.6 (likewise, 0 (38) ∞ 5 dB of squeezing) for which the considered criterion is X = (1−|λ|2)|λ|2n|n, nihn, n|. optimal. In particular, this shows that this condition n=0 works optimally in a range of moderate squeezing values and, thus, is compatible with typical experiments. This state presents a particular challenge for nonclas- We also want to recall that the Q|ξi are a Gaussian sicality verification because it shows only weak non- distributions which do not have any zeros in the phase classicality and quantum correlations. Namely, this space. Thus, criteria based on the zeros of the Husimi state is not entangled, has zero quantum discord, and

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 10 modes, this results in the nonclassicality criterion

(1) (N) (1) (N) det(M) =Q(α1 , . . . , α1 )Q(α2 , . . . , α2 ) N − P |α(m)−α(m)|2/2 − e m=1 2 1 !2 α(1)+α(1) α(N)+α(N) × Q 1 2 ,..., 1 2 < 0. 2 2 (40)

In Fig.4(b), we apply the case N = 2 of this inequality to identify the nonclassicality of ρˆ for |λ|2 = 1/2. Again, the same approach as used in both single- Figure 5: Determinant (×104) of the multimode 2×2 phase- mode scenarios enables us yet again to uncover the space matrix M of Q functions [det(M) in Eq. (40)] for the nonclassical behavior of this bipartite state for all (−) (m) nonzero choices of parameters |α| and |β|. Note in skew-symmetric, tripartite state |Ψγ,3 i, with α1 = α1 and (m) this context that the phase of these parameters does α2 = 0 for m = 1, 2, 3. Nonclassicality is verified for all coherent amplitudes γ, which, without loss of generality, can not contribute because of the fully phase-randomized be chosen as a nonnegative number. structure of the mixed state in Eq. (38). has classical marginal single-mode states (i.e., the par- 5.4 Multimode superposition states tial traces tr1(ˆρ) = tr2(ˆρ) yield thermal states) [90]. To further exceed the previous, bipartite state, we However, it shows nonclassical photon-photon corre- consider an N-mode state in this part. Specifically, lations [91, 90, 92]. The state’s two-mode Q function we focus on a multimode superposition of coherent can be computed using Gaussian functions and the states [93], phase averaging in Eq. (38), which gives |γi⊗N ± | − γi⊗N 2 |Ψ(±) i = , (41) (1) (2) 1−|λ| −|α(1)|2−|α(2)|2 γ,N q Q (α , α ) = e −2N|γ|2 ρˆ π2 (39) 2 1 ± e (1) (2) × I0(2|λ||α ||α |), which consists of two N-fold tensor products of polar where I0 denotes the zeroth modified Bessel function opposite coherent states, |±γi. Specifically, the skew- (−) of the first kind. See also Fig.4(a) in this context. symmetric state |Ψγ,N i is of interest because it yields To apply our approach, whilst using Q functions a GHZ state for |γ| → ∞ and W state for |γ| → 0, only, we can directly generalize our criterion in Eq. combining in an asymptotic manner two inequivalent (35) to the multimode case (see also Sec. 4.1). For N forms of multipartite entanglement [94, 14].

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

−N|γ|2 −|α(1)|2 −|α(N)|2 " N !# " N !#! (1) (N) e e ··· e ∗ X (m) ∗ X (m) Q (±) (α , . . . , α ) = cosh 2 Re γ α ± cos 2 Im γ α . |Ψ i N −2N|γ|2 γ,N 2π 1 ± e m=1 m=1 (42)

To apply our criteria in Eq. (40), and for simplicity, remains below zero. We reiterate that our relatively (m) simple, second-order correlations of Q functions ren- we set αj = αj for all mode numbers m and points in phase space, αj. In Fig.5, we exemplify the cer- der it possible to certify the nonclassical properties of (−) multimode, non-Gaussian states. tification of nonclassicality for the state |Ψγ,3 i [Eq. (41)] as a function of α1 = Re(α1) and for a fixed α2 = 0. We remark that, for other mode numbers 5.5 Generalized phase-space representations N, the plot looks quite similar. Most pronounced are and nonlinear detection model nonclassical features for γ close to zero, relating to a W state in which a single photon is uniformly dis- For demonstrating how our phase-space matrix ap- tributed over three modes. For large γ values, relating proach functions beyond s-parametrized distribu- to a GHZ state, the negativities decrease, but det(M) tions, we consider an on-off detector that is based on two-photon absorption [95]. In this case, the POVM

Accepted in Quantum 2020-10-04, click title to verify. Published under CC-BY 4.0. 11 example. It is worth emphasizing that other methods to infer nonclassical light (e.g., the Chebyshev approach from Ref. [65]) are incapable to detect this state’s quantum features. Here, we can directly certify nonclassicality of this non-Gaussian state despite the challenge of also having a non-Gaussian detection model.

6 Conclusion

In summary, we devised a generally applicable method that unifies nonclassicality criteria from correlation functions with quasiprobability distributions. Figure 6: Application of the nonclassicality criterion in Eq. Thereby, we created an advanced toolbox of non- (44) as a function of Re(α1) and Im(α2), while fixing classicality tests which exploit the capabilities of Im(α1) = Re(α2) = 0. Nonlinear detectors—thus, a nonlinear Q function—with η = σ = 1 and χ = 0.01 are used, Eq. both phase-space distributions and matrices of mo- (43). Because of the negativities for the considered single- ments to probe for nonclassical effects. Furthermore, mode, symmetric state [cf. Eq. (41) for N = 1 and γ = 1], our framework is applicable to an arbitrary num- this state is shown to be nonclassical. ber of modes, arbitrary orders of correlation, and even phase-space functions perturbed through convolutions with non-Gaussian kernels. A measurement element for no click is approximated by scheme was proposed to directly determine the ele- ∞ n 2n ments of the phase-space matrix, the underlying key 2 X (2n)! χ (ηnˆ) Πˆ = :e−ηnˆ+χnˆ : = : e−ηnˆ :, quantities of our method. In addition, we showed n! η2 (2n)! n=0 and discussed in detail that our treatment includes (43) previous findings as special cases, is experimentally accessible even if other methods are not, and over- 2n −ηnˆ where :(ηnˆ) e /(2n)!: describes a measurement comes challenges of previous techniques when identi- operator for 2n-photon states with a linear quantum fying nonclassicality. efficiency η. In this context, it is worth mentioning The phase-space-matrix approach incorporates 2 that χ [eη ]/[4n] has to be satisfied to ensure that nonclassicality tests based on negativities of the the approximated POVM element correctly applies phase-space distributions, including the Glauber- for photon numbers up to 2n [96]. The parameter Sudarshan P function, and the matrix-of-moments χ relates to the nonlinear absorption efficiency. approach as special cases. Thus, we were able to unify Based on such a nonlinear detector, we then two major techniques for certification of nonclassical- define the non-Gaussian operator Ω(ˆ α; η, χ) = 2 ity. As the P function and the matrix of moments :e−ηnˆ(α)+χnˆ(α) :, as described in Secs. 4.2 and 4.3. themselves are already necessary and sufficient con- For a correlation measurement with two detectors (see ditions for the detection of nonclassicality, the intro- Fig.1), this then results in the correlation matrix el- duced phase-space-matrix approach obeys the same ˆ ˆ ements h:Ω(αi; ηi, χi)Ω(αj; ηj, χj):i. For specific pa- universal feature. In other words, for any nonclassi- rameters and up to a scaling with π, this correla- cal state there exists a phase-space matrix condition tion function also results in the nonlinear QΩ(α) = which certifies its nonclassicality. h:Ω(ˆ α; 1, χ)Ω(0;ˆ 0, 0):i function (cf. Sec. 4.3 for the By applying our nonclassicality criteria to a di- similarly defined PΩ), where σ = η = 1. By extension, verse set of examples, we further demonstrated the and using χ = χ0 and η = 1 = η0, these phase-space power and versatility of our method. These exam- correlation functions also provide the entries required ples covered discrete- and continuous-variable, single- for the nonclassicality criterion. Here, it reads and multimode, Gaussian and non-Gaussian, as well as pure and mixed quantum states of light. Remark- ˆ ˆ ˆ ˆ h:Ω(α1; 1, χ)Ω(α1; 1, χ):ih:Ω(α2; 1, χ)Ω(α2; 1, χ):i ably, we used for all these states only the family of ˆ ˆ 2 −h:Ω(α1; 1, χ)Ω(α2; 1, χ):i < 0, second-order correlations and phase-space distribu- (44) tions which are always nonnegative. Nevertheless, these basic criteria were already sufficient to certify which applies to the nonlinear detection scenario un- distinct nonclassical effects on one common ground, der study. further demonstrating the strength of our method. In Fig.6, we apply this approach and consider the When compared to matrices of moments, the kinds (+) single-mode even coherent state |Ψγ,1 i [cf. Eq. (41) of nonclassicality under study would require very dif- for N = 1], which is a non-Gaussian state, because ferent moments for determining the states’ distinct we focused on the odd coherent state in the previous quantum properties. Finally, we put forward an ex-

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