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 quasiproabili- ties.) 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