{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T03:06:12Z","timestamp":1781665572813,"version":"3.54.5"},"reference-count":100,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,10,15]],"date-time":"2020-10-15T00:00:00Z","timestamp":1602720000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>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 \\cite{BA19}. 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<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>s<\/mml:mi><\/mml:math>-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.<\/jats:p>","DOI":"10.22331\/q-2020-10-15-343","type":"journal-article","created":{"date-parts":[[2020,10,15]],"date-time":"2020-10-15T14:32:39Z","timestamp":1602772359000},"page":"343","source":"Crossref","is-referenced-by-count":30,"title":["Probing nonclassicality with matrices of phase-space distributions"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3857-4555","authenticated-orcid":false,"given":"Martin","family":"Bohmann","sequence":"first","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"},{"name":"QSTAR, INO-CNR, and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5604-9407","authenticated-orcid":false,"given":"Elizabeth","family":"Agudelo","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5844-3205","authenticated-orcid":false,"given":"Jan","family":"Sperling","sequence":"additional","affiliation":[{"name":"Integrated Quantum Optics Group, Applied Physics, Paderborn University, 33098 Paderborn, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2020,10,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature (London) 409, 46 (2001).","DOI":"10.1038\/35051009"},{"key":"1","doi-asserted-by":"publisher","unstructured":"T. C. Ralph and P. K. Lam, A bright future for quantum communications, Nat. Photonics 3, 671 (2009).","DOI":"10.1038\/nphoton.2009.222"},{"key":"2","doi-asserted-by":"publisher","unstructured":"J. L. O'Brien, A. Furusawa, and J. Vu\u010dkovi\u0107, Photonic quantum technologies, Nat. Photonics 3, 687 (2009).","DOI":"10.1038\/nphoton.2009.229"},{"key":"3","doi-asserted-by":"crossref","unstructured":"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\u2013482.","DOI":"10.1007\/978-3-319-31903-2_18"},{"key":"4","doi-asserted-by":"publisher","unstructured":"S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Appl. Phys. Rev. 6, 041303 (2019).","DOI":"10.1063\/1.5115814"},{"key":"5","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevX.8.041038"},{"key":"6","doi-asserted-by":"publisher","unstructured":"H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Nonclassicality as a Quantifiable Resource for Quantum Metrology, Phys. Rev. Lett. 122, 040503 (2019).","DOI":"10.1103\/PhysRevLett.122.040503"},{"key":"7","doi-asserted-by":"publisher","unstructured":"F. Shahandeh, A. P. Lund, and T. C. Ralph, Quantum Correlations in Nonlocal Boson Sampling, Phys. Rev. Lett. 119, 120502 (2017).","DOI":"10.1103\/PhysRevLett.119.120502"},{"key":"8","doi-asserted-by":"publisher","unstructured":"F. Shahandeh, A. P. Lund, and T. C. Ralph, Quantum correlations and global coherence in distributed quantum computing, Phys. Rev. A 99, 052303 (2019).","DOI":"10.1103\/PhysRevA.99.052303"},{"key":"9","doi-asserted-by":"publisher","unstructured":"M. S. Kim, W. Son, V. Bu\u017eek, and P. L. Knight, Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement, Phys. Rev. A 65, 032323 (2002).","DOI":"10.1103\/PhysRevA.65.032323"},{"key":"10","doi-asserted-by":"publisher","unstructured":"W. Vogel and J. Sperling, Unified quantification of nonclassicality and entanglement, Phys. Rev. A 89, 052302 (2014).","DOI":"10.1103\/PhysRevA.89.052302"},{"key":"11","doi-asserted-by":"publisher","unstructured":"N. Killoran, F. E. S. Steinhoff, and M. B. Plenio, Converting Nonclassicality into Entanglement, Phys. Rev. Lett. 116, 080402 (2016).","DOI":"10.1103\/PhysRevLett.116.080402"},{"key":"12","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevA.82.013824"},{"key":"13","doi-asserted-by":"publisher","unstructured":"J. Sperling and W. Vogel, Quasiprobability distributions for quantum-optical coherence and beyond, Phys. Scr. 95, 034007 (2020).","DOI":"10.1088\/1402-4896\/ab5501"},{"key":"14","doi-asserted-by":"crossref","unstructured":"W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).","DOI":"10.1002\/3527602976"},{"key":"15","doi-asserted-by":"crossref","unstructured":"C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space (World Scientific, Singapore, 2005).","DOI":"10.1142\/5287"},{"key":"16","doi-asserted-by":"publisher","unstructured":"D. D. Nolte, The tangled tale of phase space, Phys. Today 63, 33 (2010).","DOI":"10.1063\/1.3397041"},{"key":"17","doi-asserted-by":"publisher","unstructured":"H. Weyl, Quantenmechanik und Gruppentheorie, Z. Phys. 46, 1 (1927).","DOI":"10.1007\/BF02055756"},{"key":"18","doi-asserted-by":"publisher","unstructured":"E. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Phys. Rev. 40, 749 (1932).","DOI":"10.1103\/PhysRev.40.749"},{"key":"19","doi-asserted-by":"publisher","unstructured":"H. J. Groenewold, On the principles of elementary quantum mechanics, Physica 12, 405 (1946).","DOI":"10.1016\/S0031-8914(46)80059-4"},{"key":"20","doi-asserted-by":"publisher","unstructured":"J. Moyal, Quantum mechanics as a statistical theory, Math. Proc. Camb. Philos. Soc. 45, 99 (1949).","DOI":"10.1017\/S0305004100000487"},{"key":"21","doi-asserted-by":"publisher","unstructured":"J. Sperling and I. A. Walmsley, Quasiprobability representation of quantum coherence, Phys. Rev. A 97, 062327 (2018).","DOI":"10.1103\/PhysRevA.97.062327"},{"key":"22","doi-asserted-by":"publisher","unstructured":"R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131, 2766 (1963).","DOI":"10.1103\/PhysRev.131.2766"},{"key":"23","doi-asserted-by":"publisher","unstructured":"E. C. G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Phys. Rev. Lett. 10, 277 (1963).","DOI":"10.1103\/PhysRevLett.10.277"},{"key":"24","doi-asserted-by":"publisher","unstructured":"K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940).","DOI":"10.11429\/ppmsj1919.22.4_264"},{"key":"25","doi-asserted-by":"publisher","unstructured":"U. M. Titulaer and R. J. Glauber, Correlation functions for coherent fields, Phys. Rev. 140, B676 (1965).","DOI":"10.1103\/PhysRev.140.B676"},{"key":"26","doi-asserted-by":"publisher","unstructured":"L. Mandel, Non-classical states of the electromagnetic field, Phys. Scr. T 12, 34 (1986).","DOI":"10.1088\/0031-8949\/1986\/T12\/005"},{"key":"27","doi-asserted-by":"publisher","unstructured":"L. Cohen, Generalized Phase-Space Distribution Functions, J. Math. Phys. 7, 781 (1966).","DOI":"10.1063\/1.1931206"},{"key":"28","doi-asserted-by":"publisher","unstructured":"K. E. Cahill and R. J. Glauber, Density Operators and Quasiprobability Distributions, Phys. Rev. 177, 1882 (1969).","DOI":"10.1103\/PhysRev.177.1882"},{"key":"29","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevD.2.2187"},{"key":"30","doi-asserted-by":"publisher","unstructured":"S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).","DOI":"10.1103\/RevModPhys.77.513"},{"key":"31","doi-asserted-by":"publisher","unstructured":"C. Weedbrook, S. Pirandola, R. Garc\u00eda-Patr\u00f3n, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).","DOI":"10.1103\/RevModPhys.84.621"},{"key":"32","doi-asserted-by":"publisher","unstructured":"G. Adesso, S. Ragy, and A. R. Lee, Continuous Variable Quantum Information: Gaussian States and Beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014).","DOI":"10.1142\/S1230161214400010"},{"key":"33","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevLett.110.181101"},{"key":"34","doi-asserted-by":"publisher","unstructured":"M. Tse et al., Quantum-Enhanced Advanced LIGO Detectors in the Era of Gravitational-Wave Astronomy, Phys. Rev. Lett. 123, 231107 (2019).","DOI":"10.1103\/PhysRevLett.123.231107"},{"key":"35","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1088\/0022-3700\/9\/4\/001"},{"key":"36","doi-asserted-by":"publisher","unstructured":"H. J. Kimble and L. Mandel, Theory of resonance fluorescence, Phys. Rev. A 13, 2123 (1976).","DOI":"10.1103\/PhysRevA.13.2123"},{"key":"37","doi-asserted-by":"publisher","unstructured":"H. J. Kimble, M. Dagenais, and L. Mandel, Photon Antibunching in Resonance Fluorescence, Phys. Rev. Lett. 39, 691 (1977).","DOI":"10.1103\/PhysRevLett.39.691"},{"key":"38","doi-asserted-by":"publisher","unstructured":"L. Mandel, Sub-Poissonian photon statistics in resonance fluorescence, Opt. Lett. 4, 205 (1979).","DOI":"10.1364\/OL.4.000205"},{"key":"39","doi-asserted-by":"publisher","unstructured":"X. T. Zou and L. Mandel, Photon-antibunching and sub-Poissonian photon statistics, Phys. Rev. A 41, 475 (1990).","DOI":"10.1103\/PhysRevA.41.475"},{"key":"40","doi-asserted-by":"publisher","unstructured":"H. P. Yuen, Two-photon coherent states of the radiation field, Phys. Rev. A 13, 2226 (1976).","DOI":"10.1103\/PhysRevA.13.2226"},{"key":"41","doi-asserted-by":"publisher","unstructured":"D. F. Walls, Squeezed states of light, Nature (London) 306, 141 (1983).","DOI":"10.1038\/306141a0"},{"key":"42","doi-asserted-by":"publisher","unstructured":"R. Loudon and P. Knight, Squeezed Light, J. Mod. Opt. 34, 709 (1987).","DOI":"10.1080\/09500348714550721"},{"key":"43","doi-asserted-by":"publisher","unstructured":"G. Agarwal, Nonclassical characteristics of the marginals for the radiation field, Opt. Commun. 95, 109 (1993).","DOI":"10.1016\/0030-4018(93)90059-E"},{"key":"44","doi-asserted-by":"publisher","unstructured":"G. S. Agarwal, Nonclassical statistics of fields in pair coherent states, J. Opt. Soc. Am. B 5, 1940 (1988).","DOI":"10.1364\/JOSAB.5.001940"},{"key":"45","doi-asserted-by":"publisher","unstructured":"M. Hillery, Amplitude-squared squeezing of the electromagnetic field, Phys. Rev. A 36, 3796 (1987).","DOI":"10.1103\/PhysRevA.36.3796"},{"key":"46","doi-asserted-by":"publisher","unstructured":"D. N. Klyshko, The nonclassical light, Phys.-Uspekhi 39, 573 (1996).","DOI":"10.1070\/PU1996v039n06ABEH000149"},{"key":"47","doi-asserted-by":"publisher","unstructured":"\u00c1. Rivas and A. Luis, Nonclassicality of states and measurements by breaking classical bounds on statistics, Phys. Rev. A 79, 042105 (2009).","DOI":"10.1103\/PhysRevA.79.042105"},{"key":"48","doi-asserted-by":"publisher","unstructured":"M. Bohmann, L. Qi, W. Vogel, and M. Chekhova, Detection-device-independent verification of nonclassical light, Phys. Rev. Res. 1, 033178 (2019).","DOI":"10.1103\/PhysRevResearch.1.033178"},{"key":"49","doi-asserted-by":"publisher","unstructured":"G. S. Agarwal and K. Tara, Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics, Phys. Rev. A 46 485 (1992).","DOI":"10.1103\/PhysRevA.46.485"},{"key":"50","doi-asserted-by":"publisher","unstructured":"E. Shchukin and W. Vogel, Inseparability Criteria for Continuous Bipartite Quantum States, Phys. Rev. Lett. 95, 230502 (2005).","DOI":"10.1103\/PhysRevLett.95.230502"},{"key":"51","doi-asserted-by":"publisher","unstructured":"E. Shchukin and W. Vogel, Conditions for multipartite continuous-variable entanglement, Phys. Rev. A 74, 030302(R) (2006).","DOI":"10.1103\/PhysRevA.74.030302"},{"key":"52","doi-asserted-by":"publisher","unstructured":"A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, Inseparability criteria based on matrices of moments, Phys. Rev. A 80, 052303 (2009).","DOI":"10.1103\/PhysRevA.80.052303"},{"key":"53","doi-asserted-by":"publisher","unstructured":"E. Shchukin and W. Vogel, Universal Measurement of Quantum Correlations of Radiation, Phys. Rev. Lett. 96, 200403 (2006).","DOI":"10.1103\/PhysRevLett.96.200403"},{"key":"54","doi-asserted-by":"publisher","unstructured":"W. Vogel, Nonclassical states: An observable criterion, Phys. Rev. Lett. 84, 1849 (2000).","DOI":"10.1103\/PhysRevLett.84.1849"},{"key":"55","doi-asserted-by":"publisher","unstructured":"T. Richter and W. Vogel, Nonclassicality of quantum states: A hierarchy of observable conditions, Phys. Rev. Lett. 89, 283601 (2002).","DOI":"10.1103\/PhysRevLett.89.283601"},{"key":"56","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevA.65.033830"},{"key":"57","doi-asserted-by":"publisher","unstructured":"A. Zavatta, V. Parigi, and M. Bellini, Experimental nonclassicality of single-photon-added thermal light states, Phys. Rev. A 75, 052106 (2007).","DOI":"10.1103\/PhysRevA.75.052106"},{"key":"58","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevA.79.022122"},{"key":"59","doi-asserted-by":"publisher","unstructured":"A. Mari, K. Kieling, B. M. Nielsen, E. S. Polzik, and J. Eisert, Directly estimating nonclassicality, Phys. Rev. Lett. 106, 010403 (2011).","DOI":"10.1103\/PhysRevLett.106.010403"},{"key":"60","doi-asserted-by":"publisher","unstructured":"J. Sperling, W. Vogel, and G. S. Agarwal, Operational definition of quantum correlations of light, Phys. Rev. A 94, 013833 (2016).","DOI":"10.1103\/PhysRevA.94.013833"},{"key":"61","doi-asserted-by":"publisher","unstructured":"S. Ryl, J. Sperling, E. Agudelo, M. Mraz, S. K\u00f6hnke, B. Hage, and W. Vogel, Unified nonclassicality criteria, Phys. Rev. A 92, 011801(R) (2015).","DOI":"10.1103\/PhysRevA.92.011801"},{"key":"62","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevA.56.1205"},{"key":"63","doi-asserted-by":"publisher","unstructured":"E. Agudelo, J. Sperling, L. S. Costanzo, M. Bellini, A. Zavatta, and W. Vogel, Conditional Hybrid Nonclassicality, Phys. Rev. Lett. 119, 120403 (2017).","DOI":"10.1103\/PhysRevLett.119.120403"},{"key":"64","doi-asserted-by":"publisher","unstructured":"M. Bohmann and E. Agudelo, Phase-space inequalities beyond negativities, Phys. Rev. Lett. 124, 133601 (2020).","DOI":"10.1103\/PhysRevLett.124.133601"},{"key":"65","doi-asserted-by":"publisher","unstructured":"E. Schr\u00f6dinger, Der stetige \u00dcbergang von der Mikro- zur Makromechanik, Naturwiss. 14, 664 (1926).","DOI":"10.1007\/BF01507634"},{"key":"66","doi-asserted-by":"publisher","unstructured":"M. Hillery, Classical Pure States are Coherent States, Phys. Lett. 111, 409 (1985).","DOI":"10.1016\/0375-9601(85)90483-9"},{"key":"67","doi-asserted-by":"publisher","unstructured":"M. Rezai, J. Sperling, and I. Gerhardt, What can single photons do what lasers cannot do?, Quantum Sci. Technol. 4, 045008 (2019).","DOI":"10.1088\/2058-9565\/ab3d56"},{"key":"68","doi-asserted-by":"publisher","unstructured":"J. Sperling, Characterizing maximally singular phase-space distributions, Phys. Rev. A 94, 013814 (2016).","DOI":"10.1103\/PhysRevA.94.013814"},{"key":"69","doi-asserted-by":"crossref","unstructured":"W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, Weinheim, 2006).","DOI":"10.1002\/3527608524"},{"key":"70","doi-asserted-by":"publisher","unstructured":"E. Shchukin, T. Richter, and W. Vogel, Nonclassicality criteria in terms of moments, Phys. Rev. A 71, 011802(R) (2005).","DOI":"10.1103\/PhysRevA.71.011802"},{"key":"71","doi-asserted-by":"publisher","unstructured":"E. Shchukin and W. Vogel, Nonclassical moments and their measurement, Phys. Rev. A 72, 043808 (2005).","DOI":"10.1103\/PhysRevA.72.043808"},{"key":"72","doi-asserted-by":"crossref","unstructured":"R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).","DOI":"10.1017\/CBO9780511810817"},{"key":"73","unstructured":"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."},{"key":"74","unstructured":"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 $\\partial_{\\alpha^\\ast}=(\\partial_{\\mathrm{Re}(\\alpha)}-i\\partial_{\\mathrm{Im}(\\alpha)})\/2$."},{"key":"75","doi-asserted-by":"publisher","unstructured":"A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).","DOI":"10.1103\/RevModPhys.81.299"},{"key":"76","doi-asserted-by":"publisher","unstructured":"S. Wallentowitz and W. Vogel, Unbalanced homodyning for quantum state measurements, Phys. Rev. A 53, 4528 (1996).","DOI":"10.1103\/PhysRevA.53.4528"},{"key":"77","doi-asserted-by":"publisher","unstructured":"K. Banaszek, C. Radzewicz, K. W\u00f3dkiewicz, and J. S. Krasi\u0144ski, Direct measurement of the Wigner function by photon counting, Phys. Rev. A 60, 674 (1999).","DOI":"10.1103\/PhysRevA.60.674"},{"key":"78","doi-asserted-by":"publisher","unstructured":"P. L. Kelley and W. H. Kleiner, Theory of electromagnetic field measurement and photoelectron counting, Phys. Rev. 136, A316 (1964).","DOI":"10.1103\/PhysRev.136.A316"},{"key":"79","doi-asserted-by":"publisher","unstructured":"J. Sperling et al., Detector-Agnostic Phase-Space Distributions, Phys. Rev. Lett. 124, 013605 (2020).","DOI":"10.1103\/PhysRevLett.124.013605"},{"key":"80","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1088\/0031-8949\/1993\/T48\/020"},{"key":"81","unstructured":"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."},{"key":"82","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevLett.120.063607"},{"key":"83","doi-asserted-by":"publisher","unstructured":"T. Kiesel and W. Vogel, Nonclassicality filters and quasi-probabilities, Phys. Rev. A 82, 032107 (2010).","DOI":"10.1103\/PhysRevA.82.032107"},{"key":"84","doi-asserted-by":"publisher","unstructured":"T. Kiesel, W. Vogel, B. Hage, and R. Schnabel, Direct sampling of negative quasiprobabilities of a squeezed state, Phys. Rev. Lett. 107 113604 (2011).","DOI":"10.1103\/PhysRevLett.107.113604"},{"key":"85","doi-asserted-by":"publisher","unstructured":"T. Richter, Pattern functions used in tomographic reconstruction of photon statistics revisited, Phys. Lett. A 211, 327 (1996).","DOI":"10.1016\/0375-9601(96)00029-1"},{"key":"86","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1016\/0030-4018(96)00061-2"},{"key":"87","doi-asserted-by":"publisher","unstructured":"E. Agudelo, J. Sperling, W. Vogel, S. K\u00f6hnke, M. Mraz, and B. Hage, Continuous sampling of the squeezed-state nonclassicality, Phys. Rev. A 92, 033837 (2015).","DOI":"10.1103\/PhysRevA.92.033837"},{"key":"88","doi-asserted-by":"publisher","unstructured":"N. L\u00fctkenhaus and S. M. Barnett, Nonclassical effects in phase space, Phys. Rev. A 51, 3340 (1995).","DOI":"10.1103\/PhysRevA.51.3340"},{"key":"89","doi-asserted-by":"publisher","unstructured":"E. Agudelo, J. Sperling, and W. Vogel, Quasiprobabilities for multipartite quantum correlations of light, Phys. Rev. A 87, 033811 (2013).","DOI":"10.1103\/PhysRevA.87.033811"},{"key":"90","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevLett.108.260403"},{"key":"91","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevLett.115.023601"},{"key":"92","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1016\/0031-8914(74)90215-8"},{"key":"93","doi-asserted-by":"publisher","unstructured":"W. D\u00fcr, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000).","DOI":"10.1103\/PhysRevA.62.062314"},{"key":"94","doi-asserted-by":"publisher","unstructured":"A. K. Jaiswal and G. S. Agarwal, Photoelectric detection with Two-Photon Absorption, J. Opt. Soc. Am. 59, 1446 (1969).","DOI":"10.1364\/JOSA.59.001446"},{"key":"95","unstructured":"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 $\\Omega$."},{"key":"96","unstructured":"N. Biagi, M. Bohmann, E. Agudelo, M. Bellini, and A. Zavatta, Experimental certification of nonclassicality via phase-space inequalities, arXiv:2010.00259 [quant-ph]."},{"key":"97","doi-asserted-by":"publisher","unstructured":"R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).","DOI":"10.1103\/RevModPhys.81.865"},{"key":"98","doi-asserted-by":"publisher","unstructured":"K. C. Tan, S. Choi, and H. Jeong, Negativity of Quasiprobability Distributions as a Measure of Nonclassicality, Phys. Rev. Lett. 124, 110404 (2020).","DOI":"10.1103\/PhysRevLett.124.110404"},{"key":"99","unstructured":"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."}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2020-10-15-343\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T12:32:32Z","timestamp":1669206752000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2020-10-15-343\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,15]]},"references-count":100,"URL":"https:\/\/doi.org\/10.22331\/q-2020-10-15-343","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,15]]},"article-number":"343"}}