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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality [65]. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyond $s$-parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.

The intuitively accessible representation of quantum effects via quasiprobabilities, defying the nonnegativity requirement of classical probabilities, is a common technique to identify quantum features. However, the complexity of the reconstruction of such distributions increases with their sensitivity to uncover nonclassical signatures. Conversely, approaches based on correlation functions are experimentally available but less intuitive.

The method devised in our paper overcomes such disadvantageous features by unifying both aforementioned techniques. That is, quasiprobabilities can be correlated to unveil nonclassical properties even if the individual distributions are not sensitive enough to identify quantum properties. For example, it is shown that this necessary and sufficient approach applies to discrete- and continuous-variable, single- and multimode, pure and mixed states of light using phase-space distributions that can never become negative.

Thereby, we demonstrate the usefulness of our novel method to certify quantum characteristics in a practical manner that formthe basis for current and future quantum technologies.

► BibTeX data

► References

[1] E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature (London) 409, 46 (2001).
https:/​/​doi.org/​10.1038/​35051009

[2] T. C. Ralph and P. K. Lam, A bright future for quantum communications, Nat. Photonics 3, 671 (2009).
https:/​/​doi.org/​10.1038/​nphoton.2009.222

[3] J. L. O'Brien, A. Furusawa, and J. Vučković, Photonic quantum technologies, Nat. Photonics 3, 687 (2009).
https:/​/​doi.org/​10.1038/​nphoton.2009.229

[4] M. Krenn, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, Quantum communication with photons, in Optics in Our Time (Springer, Cham, 2016), pp. 455–482.

[5] S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Appl. Phys. Rev. 6, 041303 (2019).
https:/​/​doi.org/​10.1063/​1.5115814

[6] B. Yadin, F. C. Binder, J. Thompson, V. Narasimhachar, M. Gu, and M. S. Kim, Operational Resource Theory of Continuous-Variable Nonclassicality, Phys. Rev. X 8, 041038 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041038

[7] H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Nonclassicality as a Quantifiable Resource for Quantum Metrology, Phys. Rev. Lett. 122, 040503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.040503

[8] F. Shahandeh, A. P. Lund, and T. C. Ralph, Quantum Correlations in Nonlocal Boson Sampling, Phys. Rev. Lett. 119, 120502 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.120502

[9] F. Shahandeh, A. P. Lund, and T. C. Ralph, Quantum correlations and global coherence in distributed quantum computing, Phys. Rev. A 99, 052303 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.052303

[10] M. S. Kim, W. Son, V. Bužek, and P. L. Knight, Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement, Phys. Rev. A 65, 032323 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.032323

[11] W. Vogel and J. Sperling, Unified quantification of nonclassicality and entanglement, Phys. Rev. A 89, 052302 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.052302

[12] N. Killoran, F. E. S. Steinhoff, and M. B. Plenio, Converting Nonclassicality into Entanglement, Phys. Rev. Lett. 116, 080402 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.080402

[13] A. Miranowicz, M. Bartkowiak, X. Wang, Y.-x. Liu, and F. Nori, Testing nonclassicality in multimode fields: A unified derivation of classical inequalities, Phys. Rev. A 82, 013824 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.82.013824

[14] J. Sperling and W. Vogel, Quasiprobability distributions for quantum-optical coherence and beyond, Phys. Scr. 95, 034007 (2020).
https:/​/​doi.org/​10.1088/​1402-4896/​ab5501

[15] W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).

[16] C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space (World Scientific, Singapore, 2005).

[17] D. D. Nolte, The tangled tale of phase space, Phys. Today 63, 33 (2010).
https:/​/​doi.org/​10.1063/​1.3397041

[18] H. Weyl, Quantenmechanik und Gruppentheorie, Z. Phys. 46, 1 (1927).
https:/​/​doi.org/​10.1007/​BF02055756

[19] E. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Phys. Rev. 40, 749 (1932).
https:/​/​doi.org/​10.1103/​PhysRev.40.749

[20] H. J. Groenewold, On the principles of elementary quantum mechanics, Physica 12, 405 (1946).
https:/​/​doi.org/​10.1016/​S0031-8914(46)80059-4

[21] J. Moyal, Quantum mechanics as a statistical theory, Math. Proc. Camb. Philos. Soc. 45, 99 (1949).
https:/​/​doi.org/​10.1017/​S0305004100000487

[22] J. Sperling and I. A. Walmsley, Quasiprobability representation of quantum coherence, Phys. Rev. A 97, 062327 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.062327

[23] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131, 2766 (1963).
https:/​/​doi.org/​10.1103/​PhysRev.131.2766

[24] E. C. G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Phys. Rev. Lett. 10, 277 (1963).
https:/​/​doi.org/​10.1103/​PhysRevLett.10.277

[25] K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940).
https:/​/​doi.org/​10.11429/​ppmsj1919.22.4_264

[26] U. M. Titulaer and R. J. Glauber, Correlation functions for coherent fields, Phys. Rev. 140, B676 (1965).
https:/​/​doi.org/​10.1103/​PhysRev.140.B676

[27] L. Mandel, Non-classical states of the electromagnetic field, Phys. Scr. T 12, 34 (1986).
https:/​/​doi.org/​10.1088/​0031-8949/​1986/​T12/​005

[28] L. Cohen, Generalized Phase-Space Distribution Functions, J. Math. Phys. 7, 781 (1966).
https:/​/​doi.org/​10.1063/​1.1931206

[29] K. E. Cahill and R. J. Glauber, Density Operators and Quasiprobability Distributions, Phys. Rev. 177, 1882 (1969).
https:/​/​doi.org/​10.1103/​PhysRev.177.1882

[30] G. S. Agarwal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space, Phys. Rev. D 2, 2187 (1970).
https:/​/​doi.org/​10.1103/​PhysRevD.2.2187

[31] S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
https:/​/​doi.org/​10.1103/​RevModPhys.77.513

[32] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
https:/​/​doi.org/​10.1103/​RevModPhys.84.621

[33] G. Adesso, S. Ragy, and A. R. Lee, Continuous Variable Quantum Information: Gaussian States and Beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014).
https:/​/​doi.org/​10.1142/​S1230161214400010

[34] H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, First Long-Term Application of Squeezed States of Light in a Gravitational-Wave Observatory, Phys. Rev. Lett. 110, 181101 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.181101

[35] M. Tse et al., Quantum-Enhanced Advanced LIGO Detectors in the Era of Gravitational-Wave Astronomy, Phys. Rev. Lett. 123, 231107 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.231107

[36] H. J. Carmichael and D. F. Walls, Proposal for the measurement of the resonant Stark effect by photon correlation techniques, J. Phys. B 9, L43 (1976).
https:/​/​doi.org/​10.1088/​0022-3700/​9/​4/​001

[37] H. J. Kimble and L. Mandel, Theory of resonance fluorescence, Phys. Rev. A 13, 2123 (1976).
https:/​/​doi.org/​10.1103/​PhysRevA.13.2123

[38] H. J. Kimble, M. Dagenais, and L. Mandel, Photon Antibunching in Resonance Fluorescence, Phys. Rev. Lett. 39, 691 (1977).
https:/​/​doi.org/​10.1103/​PhysRevLett.39.691

[39] L. Mandel, Sub-Poissonian photon statistics in resonance fluorescence, Opt. Lett. 4, 205 (1979).
https:/​/​doi.org/​10.1364/​OL.4.000205

[40] X. T. Zou and L. Mandel, Photon-antibunching and sub-Poissonian photon statistics, Phys. Rev. A 41, 475 (1990).
https:/​/​doi.org/​10.1103/​PhysRevA.41.475

[41] H. P. Yuen, Two-photon coherent states of the radiation field, Phys. Rev. A 13, 2226 (1976).
https:/​/​doi.org/​10.1103/​PhysRevA.13.2226

[42] D. F. Walls, Squeezed states of light, Nature (London) 306, 141 (1983).
https:/​/​doi.org/​10.1038/​306141a0

[43] R. Loudon and P. Knight, Squeezed Light, J. Mod. Opt. 34, 709 (1987).
https:/​/​doi.org/​10.1080/​09500348714550721

[44] G. Agarwal, Nonclassical characteristics of the marginals for the radiation field, Opt. Commun. 95, 109 (1993).
https:/​/​doi.org/​10.1016/​0030-4018(93)90059-E

[45] G. S. Agarwal, Nonclassical statistics of fields in pair coherent states, J. Opt. Soc. Am. B 5, 1940 (1988).
https:/​/​doi.org/​10.1364/​JOSAB.5.001940

[46] M. Hillery, Amplitude-squared squeezing of the electromagnetic field, Phys. Rev. A 36, 3796 (1987).
https:/​/​doi.org/​10.1103/​PhysRevA.36.3796

[47] D. N. Klyshko, The nonclassical light, Phys.-Uspekhi 39, 573 (1996).
https:/​/​doi.org/​10.1070/​PU1996v039n06ABEH000149

[48] Á. Rivas and A. Luis, Nonclassicality of states and measurements by breaking classical bounds on statistics, Phys. Rev. A 79, 042105 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.042105

[49] M. Bohmann, L. Qi, W. Vogel, and M. Chekhova, Detection-device-independent verification of nonclassical light, Phys. Rev. Res. 1, 033178 (2019).
https:/​/​doi.org/​10.1103/​PhysRevResearch.1.033178

[50] G. S. Agarwal and K. Tara, Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics, Phys. Rev. A 46 485 (1992).
https:/​/​doi.org/​10.1103/​PhysRevA.46.485

[51] E. Shchukin and W. Vogel, Inseparability Criteria for Continuous Bipartite Quantum States, Phys. Rev. Lett. 95, 230502 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.230502

[52] E. Shchukin and W. Vogel, Conditions for multipartite continuous-variable entanglement, Phys. Rev. A 74, 030302(R) (2006).
https:/​/​doi.org/​10.1103/​PhysRevA.74.030302

[53] A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, Inseparability criteria based on matrices of moments, Phys. Rev. A 80, 052303 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.052303

[54] E. Shchukin and W. Vogel, Universal Measurement of Quantum Correlations of Radiation, Phys. Rev. Lett. 96, 200403 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.200403

[55] W. Vogel, Nonclassical states: An observable criterion, Phys. Rev. Lett. 84, 1849 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.84.1849

[56] T. Richter and W. Vogel, Nonclassicality of quantum states: A hierarchy of observable conditions, Phys. Rev. Lett. 89, 283601 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.283601

[57] A. I. Lvovsky and J. H. Shapiro, Nonclassical character of statistical mixtures of the single-photon and vacuum optical states, Phys. Rev. A 65, 033830 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.033830

[58] A. Zavatta, V. Parigi, and M. Bellini, Experimental nonclassicality of single-photon-added thermal light states, Phys. Rev. A 75, 052106 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.052106

[59] T. Kiesel, W. Vogel, B. Hage, J. DiGuglielmo, A. Samblowski, and R. Schnabel, Experimental test of nonclassicality criteria for phase-diffused squeezed states, Phys. Rev. A 79, 022122 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.022122

[60] A. Mari, K. Kieling, B. M. Nielsen, E. S. Polzik, and J. Eisert, Directly estimating nonclassicality, Phys. Rev. Lett. 106, 010403 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.010403

[61] J. Sperling, W. Vogel, and G. S. Agarwal, Operational definition of quantum correlations of light, Phys. Rev. A 94, 013833 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.013833

[62] S. Ryl, J. Sperling, E. Agudelo, M. Mraz, S. Köhnke, B. Hage, and W. Vogel, Unified nonclassicality criteria, Phys. Rev. A 92, 011801(R) (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.011801

[63] S. Wallentowitz, R. L. de Matos Filho, and W. Vogel, Determination of entangled quantum states of a trapped atom, Phys. Rev. A 56, 1205 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.56.1205

[64] E. Agudelo, J. Sperling, L. S. Costanzo, M. Bellini, A. Zavatta, and W. Vogel, Conditional Hybrid Nonclassicality, Phys. Rev. Lett. 119, 120403 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.120403

[65] M. Bohmann and E. Agudelo, Phase-space inequalities beyond negativities, Phys. Rev. Lett. 124, 133601 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.133601

[66] E. Schrödinger, Der stetige Übergang von der Mikro- zur Makromechanik, Naturwiss. 14, 664 (1926).
https:/​/​doi.org/​10.1007/​BF01507634

[67] M. Hillery, Classical Pure States are Coherent States, Phys. Lett. 111, 409 (1985).
https:/​/​doi.org/​10.1016/​0375-9601(85)90483-9

[68] M. Rezai, J. Sperling, and I. Gerhardt, What can single photons do what lasers cannot do?, Quantum Sci. Technol. 4, 045008 (2019).
https:/​/​doi.org/​10.1088/​2058-9565/​ab3d56

[69] J. Sperling, Characterizing maximally singular phase-space distributions, Phys. Rev. A 94, 013814 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.013814

[70] W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, Weinheim, 2006).

[71] E. Shchukin, T. Richter, and W. Vogel, Nonclassicality criteria in terms of moments, Phys. Rev. A 71, 011802(R) (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.011802

[72] E. Shchukin and W. Vogel, Nonclassical moments and their measurement, Phys. Rev. A 72, 043808 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.043808

[73] R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).

[74] The determinant of a $3\times 3$ matrix $X=\left(\begin{smallmatrix} \mu & u & v u & U & \chi v & \chi & V \end{smallmatrix}\right)$ takes the general form $\det X=[(\mu U-u^2)(\mu V-v^2)-(\mu\chi-uv)^2]/​\mu$, which is particularly interesting for the case $\mu=1$ because it relates to cross-correlation functions.

[75] It is worth noting that, in quantum optics, the partial derivative with respect to a complex amplitude $\alpha$ is given in terms of partial derivatives of the real and imaginary part, $\partial_\alpha=(\partial_{\mathrm{Re}(\alpha)}+i\partial_{\mathrm{Im}(\alpha)})/​2$ and $\partial_{\alpha^\ast}=(\partial_{\mathrm{Re}(\alpha)}-i\partial_{\mathrm{Im}(\alpha)})/​2$.

[76] A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.299

[77] S. Wallentowitz and W. Vogel, Unbalanced homodyning for quantum state measurements, Phys. Rev. A 53, 4528 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.53.4528

[78] K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Direct measurement of the Wigner function by photon counting, Phys. Rev. A 60, 674 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.60.674

[79] P. L. Kelley and W. H. Kleiner, Theory of electromagnetic field measurement and photoelectron counting, Phys. Rev. 136, A316 (1964).
https:/​/​doi.org/​10.1103/​PhysRev.136.A316

[80] J. Sperling et al., Detector-Agnostic Phase-Space Distributions, Phys. Rev. Lett. 124, 013605 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.013605

[81] G. S. Agarwal, M. O. Scully, and H. Walther, Phase narrowing a coherent state via repeated measures: only the no counts count, Phys. Scr. T 48, 128 (1993).
https:/​/​doi.org/​10.1088/​0031-8949/​1993/​T48/​020

[82] For simplicity, we assume an equal dark-count rate $\delta$ for both detectors. However, one can readily generalized this to different dark-count rates for each detector, as $\det (M)<0$ remains a sufficient nonclassicality condition.

[83] M. Bohmann, J. Tiedau, T. Bartley, J. Sperling, C. Silberhorn, and W. Vogel, Incomplete Detection of Nonclassical Phase-Space Distributions, Phys. Rev. Lett. 120, 063607 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.063607

[84] T. Kiesel and W. Vogel, Nonclassicality filters and quasi-probabilities, Phys. Rev. A 82, 032107 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.82.032107

[85] T. Kiesel, W. Vogel, B. Hage, and R. Schnabel, Direct sampling of negative quasiprobabilities of a squeezed state, Phys. Rev. Lett. 107 113604 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.113604

[86] T. Richter, Pattern functions used in tomographic reconstruction of photon statistics revisited, Phys. Lett. A 211, 327 (1996).
https:/​/​doi.org/​10.1016/​0375-9601(96)00029-1

[87] U. Leonhard, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, Sampling of photon statistics and density matrix using homodyne detection, Opt. Commun. 127, 144 (1996).
https:/​/​doi.org/​10.1016/​0030-4018(96)00061-2

[88] E. Agudelo, J. Sperling, W. Vogel, S. Köhnke, M. Mraz, and B. Hage, Continuous sampling of the squeezed-state nonclassicality, Phys. Rev. A 92, 033837 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.033837

[89] N. Lütkenhaus and S. M. Barnett, Nonclassical effects in phase space, Phys. Rev. A 51, 3340 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.51.3340

[90] E. Agudelo, J. Sperling, and W. Vogel, Quasiprobabilities for multipartite quantum correlations of light, Phys. Rev. A 87, 033811 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.033811

[91] A. Ferraro and M. G. A. Paris, Nonclassicality Criteria from Phase-Space Representations and Information-Theoretical Constraints Are Maximally Inequivalent, Phys. Rev. Lett. 108, 260403 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.260403

[92] J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, Uncovering Quantum Correlations with Time-Multiplexed Click Detection, Phys. Rev. Lett. 115, 023601 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.023601

[93] V. V. Dodonov, I. A. Malkin, and V. I. Manko, Even and odd coherent states and excitations of a singular oscillator, Physica (Amsterdam) 72, 597 (1974).
https:/​/​doi.org/​10.1016/​0031-8914(74)90215-8

[94] W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.62.062314

[95] A. K. Jaiswal and G. S. Agarwal, Photoelectric detection with Two-Photon Absorption, J. Opt. Soc. Am. 59, 1446 (1969).
https:/​/​doi.org/​10.1364/​JOSA.59.001446

[96] The approximate POVM element in Eq. (43) has a decomposition in terms of lossy even photon-number operators with the expansion coefficients $[(2n)!/​n!](\chi/​\eta^2)^n$, which diverge for $n\to\infty$. Using the bounds $\sqrt{2\pi}m^{m+1/​2}e^{-m}\leq m!\leq e m^{m+1/​2}e^{-m}$, one finds the bound $\chi\ll e\eta^2/​[4n]$ to satisfy $[(2n)!/​n!](\chi/​\eta^2)^n\leq [e/​\sqrt\pi]([4n\chi]/​[e\eta^2])^n\leq 1$ for correctly applying this approximation for upto $2n$ photons. Also note that for coherent states, one obtains the nonnegative function $\langle\alpha|\hat\Pi|\alpha\rangle=\exp(-\eta|\alpha|^2+\chi|\alpha|^4)\geq0$, representing the non-Gaussian integration kernel $\Omega$.

[97] N. Biagi, M. Bohmann, E. Agudelo, M. Bellini, and A. Zavatta, Experimental certification of nonclassicality via phase-space inequalities, arXiv:2010.00259 [quant-ph].
arXiv:2010.00259

[98] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[99] K. C. Tan, S. Choi, and H. Jeong, Negativity of Quasiprobability Distributions as a Measure of Nonclassicality, Phys. Rev. Lett. 124, 110404 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.110404

[100] J. Park, J. Lee, and H. Nha Verifying nonclassicality beyond negativity in phase space, arXiv:2005.05739 [quant-ph]; J. Park and H. Nha, Efficient and faithful criteria on nonclassicality for continuous variables, presented at 15th International Conference on Squeezed States and Uncertainty Relations, Jeju, South Korea, 2017.
arXiv:2005.05739

Cited by

[1] Jiyong Park, Jaehak Lee, and Hyunchul Nha, "Verifying nonclassicality beyond negativity in phase space", arXiv:2005.05739.

[2] Nicola Biagi, Martin Bohmann, Elizabeth Agudelo, Marco Bellini, and Alessandro Zavatta, "Experimental certification of nonclassicality via phase-space inequalities", arXiv:2010.00259.

The above citations are from SAO/NASA ADS (last updated successfully 2020-10-19 14:36:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-10-19 14:36:44).