# Witnessing Wigner Negativity

1Institute for Quantum Information and Matter, Caltech
2Université de Paris, IRIF, CNRS, France
3Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

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### Abstract

Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.

Continuous-variable quantum information uses information encoded in the continuous degrees of freedom of quantum systems and is a promising candidate for quantum computing. In continuous variables, quantum states may be represented equivalently in phase space via their Wigner function.

The negativity of the Wigner function is a sign of non-classicality and a necessary resource for any quantum computational speedup. Detecting this negativity for experimental quantum states is therefore crucial for the development of continuous-variable quantum technologies. However, this detection can be very difficult as it usually relies on quantum state tomography, which requires an exponential number of samples compared to the system size.

In this work, we propose an alternative, more efficient, approach which introduces specific observables equipped with threshold values, such that if the expectation value of an observable with an unknown quantum state exceeds its threshold value then the state is certified to exhibit Wigner negativity.

Our results pave the way for the characterisation of non-classical quantum states, with direct applications in quantum optics experiments. The mathematical aspects of our work also motivate further study of the set of quantum states with positive Wigner function.

### ► References

[1] S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables,'' in Quantum Information with Continuous Variables, pp. 9–17. Springer, 1999.
https:/​/​doi.org/​10.1007/​978-94-015-1258-9_2

[2] S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,'' Nature Photonics 7, 982 (2013).
https:/​/​doi.org/​10.1038/​nphoton.2013.287

[3] U. Leonhardt, Essential Quantum Optics,''. Cambridge University Press, Cambridge, UK, 1st ed., 2010.
https:/​/​doi.org/​10.1017/​CBO9780511806117

[4] J. E. Moyal, Quantum mechanics as a statistical theory,'' in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 45, pp. 99–124, Cambridge University Press. 1949.
https:/​/​doi.org/​10.1017/​S0305004100000487

[5] E. P. Wigner, On the quantum correction for thermodynamic equilibrium,'' in Part I: Physical Chemistry. Part II: Solid State Physics, pp. 110–120. Springer, 1997.
https:/​/​doi.org/​10.1007/​978-3-642-59033-7_8

[6] C. T. Lee, Measure of the nonclassicality of nonclassical states,'' Physical Review A 44, R2775 (1991).
https:/​/​doi.org/​10.1103/​PhysRevA.44.R2775

[7] G. Giedke and J. I. Cirac, Characterization of Gaussian operations and distillation of Gaussian states,'' Physical Review A 66, 032316 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.66.032316

[8] J. Eisert, S. Scheel, and M. B. Plenio, Distilling Gaussian states with Gaussian operations is impossible,'' Physical Review Letters 89, 137903 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.137903

[9] J. Fiurášek, Gaussian transformations and distillation of entangled Gaussian states,'' Physical Review Letters 89, 137904 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.137904

[10] J. Niset, J. Fiurášek, and N. J. Cerf, No-go theorem for Gaussian quantum error correction,'' Physical Review Letters 102, 120501 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.120501

[11] S. Ghose and B. C. Sanders, Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps,'' Journal of Modern Optics 54, 855–869 (2007).
https:/​/​doi.org/​10.1080/​09500340601101575

[12] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Efficient Classical Simulation of Continuous Variable Quantum Information Processes,'' Physical Review Letters 88, 097904 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.097904

[13] U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, Classical simulation of Gaussian quantum circuits with non-Gaussian input states,'' arXiv:2010.14363.
arXiv:2010.14363

[14] R. L. Hudson, When is the Wigner quasi-probability density non-negative?,'' Reports on Mathematical Physics 6, 249–252 (1974).
https:/​/​doi.org/​10.1016/​0034-4877(74)90007-X

[15] F. Soto and P. Claverie, When is the Wigner function of multidimensional systems nonnegative?,'' Journal of Mathematical Physics 24, 97–100 (1983).
https:/​/​doi.org/​10.1063/​1.525607

[16] A. Mandilara, E. Karpov, and N. Cerf, Extending Hudson’s theorem to mixed quantum states,'' Physical Review A 79, 062302 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.062302

[17] R. Filip and L. Mišta Jr, Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,'' Physical Review Letters 106, 200401 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.200401

[18] K. C. Tan, S. Choi, and H. Jeong, Negativity of quasiprobability distributions as a measure of nonclassicality,'' Physical review letters 124, 110404 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.110404

[19] U. Titulaer and R. Glauber, Correlation functions for coherent fields,'' Physical Review 140, B676 (1965).
https:/​/​doi.org/​10.1103/​PhysRev.140.B676

[20] A. Kenfack and K. Życzkowski, Negativity of the Wigner function as an indicator of non-classicality,'' Journal of Optics B: Quantum and Semiclassical Optics 6, 396 (2004).
https:/​/​doi.org/​10.1088/​1464-4266/​6/​10/​003

[21] A. Mari and J. Eisert, Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient,'' Physical Review Lett. 109, 230503 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.230503

[22] L. García-Álvarez, C. Calcluth, A. Ferraro, and G. Ferrini, Efficient simulatability of continuous-variable circuits with large Wigner negativity,'' arXiv:2005.12026.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043322
arXiv:2005.12026

[23] J. Preskill, Quantum Computing in the NISQ era and beyond,'' Quantum 2, 79 (2018).
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[24] J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking,'' Nature Reviews Physics 2, 382–390 (2020).
https:/​/​doi.org/​10.1038/​s42254-020-0186-4

[25] G. M. D'Ariano, M. G. Paris, and M. F. Sacchi, Quantum tomography,'' Advances in Imaging and Electron Physics 128, 206–309 (2003), arXiv:quant-ph/​0302028.
arXiv:quant-ph/0302028

[26] A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography,'' Reviews of Modern Physics 81, 299 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.299

[27] U. Chabaud, T. Douce, F. Grosshans, E. Kashefi, and D. Markham, Building Trust for Continuous Variable Quantum States,'' in 15th Conference on the Theory of Quantum Computation, Communication and Cryptography. 2020.
https:/​/​doi.org/​10.4230/​LIPIcs.TQC.2020.3

[28] B. M. Terhal, A family of indecomposable positive linear maps based on entangled quantum states,'' Linear Algebra and its Applications 323, 61–73 (2001).
https:/​/​doi.org/​10.1016/​S0024-3795(00)00251-2

[29] M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Optimization of entanglement witnesses,'' Physical Review A 62, 052310 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.62.052310

[30] A. Mari, K. Kieling, B. M. Nielsen, E. Polzik, and J. Eisert, Directly estimating nonclassicality,'' Physical Review Letters 106, 010403 (2011).
https:/​/​doi.org/​10.1103/​physrevlett.106.010403

[31] T. Kiesel and W. Vogel, Universal nonclassicality witnesses for harmonic oscillators,'' Physical Review A 85, 062106 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.062106

[32] U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, Certification of non-Gaussian states with operational measurements,'' arXiv:2011.04320.
arXiv:2011.04320

[33] J.-B. Lasserre, Global optimization with polynomials and the problem of moments,'' SIAM Journal on optimization 11, 796–817 (2001).
https:/​/​doi.org/​10.1137/​S1052623400366802

[34] P. A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000.
https:/​/​doi.org/​10.7907/​2K6Y-CH43

[35] J. B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization,'' SIAM Journal on Optimization 21, 864–885 (2011).
https:/​/​doi.org/​10.1137/​100806990

[36] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition,''. Cambridge University Press, New York, NY, USA, 10th ed., 2011.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[37] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information,'' Reviews of Modern Physics 84, 621 (2012).
https:/​/​doi.org/​10.1103/​RevModPhys.84.621

[38] A. Wünsche, Laguerre 2D-functions and their application in quantum optics,'' Journal of Physics A: Mathematical and General 31, 8267 (1998).
https:/​/​doi.org/​10.1088/​0305-4470/​31/​40/​017

[39] A. Royer, Wigner function as the expectation value of a parity operator,'' Physical Review A 15, 449 (1977).
https:/​/​doi.org/​10.1103/​PhysRevA.15.449

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

[41] K. E. Cahill and R. J. Glauber, Density operators and quasiprobability distributions,'' Physical Review 177, 1882 (1969).
https:/​/​doi.org/​10.1103/​PhysRev.177.1882

[42] K. Husimi, Some formal properties of the density matrix,'' Proceedings of the Physico-Mathematical Society of Japan. 3rd Series 22, 264–314 (1940).
https:/​/​doi.org/​10.11429/​ppmsj1919.22.4_264

[43] T. Richter, Determination of photon statistics and density matrix from double homodyne detection measurements,'' Journal of Modern Optics 45, 1735–1749 (1998).
https:/​/​doi.org/​10.1080/​09500349808230666

[44] U. Chabaud, F. Grosshans, E. Kashefi, and D. Markham, Efficient verification of Boson Sampling,'' arXiv:2006.03520.
arXiv:2006.03520

[45] A. Ferraro, S. Olivares, and M. G. Paris, Gaussian states in continuous variable quantum information,'' arXiv:quant-ph/​0503237.
arXiv:quant-ph/0503237

[46] F. Albarelli, M. G. Genoni, M. G. Paris, and A. Ferraro, Resource theory of quantum non-Gaussianity and Wigner negativity,'' Physical Review A 98, 052350 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.052350

[47] R. Takagi and Q. Zhuang, Convex resource theory of non-Gaussianity,'' Physical Review A 97, 062337 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.062337

[48] Q. Zhuang, P. W. Shor, and J. H. Shapiro, Resource theory of non-Gaussian operations,'' Physical Review A 97, 052317 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.052317

[49] U. Chabaud, D. Markham, and F. Grosshans, Stellar representation of non-Gaussian quantum states,'' Physical Review Letters 124, 063605 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.063605

[50] L. Vandenberghe and S. Boyd, Semidefinite programming,'' SIAM review 38, 49–95 (1996).
https:/​/​doi.org/​10.1137/​1038003

[51] J. Fiurášek and M. Ježek, Witnessing negativity of Wigner function by estimating fidelities of catlike states from homodyne measurements,'' Physical Review A 87, 062115 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.062115

[52] M. Walschaers, C. Fabre, V. Parigi, and N. Treps, Entanglement and Wigner Function Negativity of Multimode Non-Gaussian States,'' Physical Review Letters 119, 183601 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.183601

[53] U. Chabaud and P.-E. Emeriau, Zeilberger's algorithm and Hierarchy of semidefinite programs.'' Software Heritage repository swh:1:dir:d98f70e386783ef69 bf8c2ecafdb7b328b19b7ec containing the numerical tools developed for this article.
https:/​/​archive.softwareheritage.org/​swh:1:dir:d98f70e386783ef69bf8c2ecafdb7b328b19b7ec/​

[54] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, Generating optical Schrödinger kittens for quantum information processing,'' Science 312, 83–86 (2006).
https:/​/​doi.org/​10.1126/​science.1122858

[55] B. C. Sanders, Entangled coherent states,'' Physical Review A 45, 6811 (1992).
https:/​/​doi.org/​10.1103/​PhysRevA.45.6811

[56] W. H. Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence,'' Nature 412, 712–717 (2001).
https:/​/​doi.org/​10.1038/​35089017

[57] G. Sagnol and M. Stahlberg, Picos, a python interface to conic optimization solvers,'' in Proceedings of the in 21st International Symposium on Mathematical Programming. 2012.

[58] M. ApS, MOSEK Optimizer API for Python 9.2.36, 2019. https:/​/​docs.mosek.com/​9.2/​pythonapi/​index.html.
https:/​/​docs.mosek.com/​9.2/​pythonapi/​index.html

[59] M. Nakata, A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP,-QD and-DD.,'' in 2010 IEEE International Symposium on Computer-Aided Control System Design, pp. 29–34, IEEE. 2010.
https:/​/​doi.org/​10.1109/​CACSD.2010.5612693

[60] K. Fujisawa, M. Kojima, K. Nakata, and M. Yamashita, SDPA (SemiDefinite Programming Algorithm) User’s Manual—Version 6.2. 0, 2002.

[61] A. Barvinok, A course in convexity,'', vol. 54 of Graduate Studies in Mathematics. American Mathematical Society, 2002.
https:/​/​doi.org/​10.1090/​gsm/​054

[62] G. Szegö, Orthogonal Polynomials, revised ed,'' in American Mathematical Society Colloquium Publications, vol. 23. 1959.
https:/​/​doi.org/​10.1090/​coll/​023

[63] O. Nikodym, Sur une généralisation des intégrales de M. J. Radon,'' Fundamenta Mathematicae 15, 131–179 (1930).
https:/​/​doi.org/​10.4064/​fm-15-1-131-179

[64] M. Guillemot-Teissier, Développements des distributions en séries de fonctions orthogonales. Séries de Legendre et de Laguerre,'' Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 25, 519–573 (1971).

[65] M. Reed and B. Simon, II: Fourier Analysis, Self-Adjointness,'', vol. 2. Elsevier, 1975.

[66] M. Riesz, Sur le problème des moments, Troisième Note,'' Ark. Mat. Fys 16, 1–52 (1923).

[67] E. Haviland, On the momentum problem for distribution functions in more than one dimension. II,'' American Journal of Mathematics 58, 164–168 (1936).
https:/​/​doi.org/​10.2307/​2371063

[68] D. Hilbert, Über die darstellung definiter formen als summe von formenquadraten,'' Mathematische Annalen 32, 342–350 (1888).
https:/​/​doi.org/​10.1007/​BF01443605

[69] H. W. Gould, Combinatorial Identities: A standardized set of tables listing 500 binomial coefficient summations,''. Morgantown, W Va, 1972.

[70] D. Zeilberger, The method of creative telescoping,'' Journal of Symbolic Computation 11, 195–204 (1991).
https:/​/​doi.org/​10.1016/​S0747-7171(08)80044-2

[71] U. Leonhardt, Quantum-state tomography and discrete Wigner function,'' Physical Review Letters 74, 4101 (1995).
https:/​/​doi.org/​10.1103/​PhysRevLett.74.4101

[72] D. Gross, Hudson’s theorem for finite-dimensional quantum systems,'' Journal of mathematical physics 47, 122107 (2006).
https:/​/​doi.org/​10.1063/​1.2393152

[73] R. W. Spekkens, Negativity and contextuality are equivalent notions of nonclassicality,'' Physical Review Letters 101, 020401 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.020401

[74] N. Delfosse, C. Okay, J. Bermejo-Vega, D. E. Browne, and R. Raussendorf, Equivalence between contextuality and negativity of the Wigner function for qudits,'' New Journal of Physics 19, 123024 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa8fe3

[75] R. Raussendorf, D. E. Browne, N. Delfosse, C. Okay, and J. Bermejo-Vega, Contextuality and Wigner-function negativity in qubit quantum computation,'' Physical Review A 95, 052334 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.052334

[76] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the magic' for quantum computation,'' Nature 510, 351 (2014).
https:/​/​doi.org/​10.1038/​nature13460

[77] J. Bermejo-Vega, N. Delfosse, D. E. Browne, C. Okay, and R. Raussendorf, Contextuality as a resource for models of quantum computation with qubits,'' Physical Review Letters 119, 120505 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.120505

[78] R. S. Barbosa, T. Douce, P.-E. Emeriau, E. Kashefi, and S. Mansfield, Continuous-variable nonlocality and contextuality,'' arXiv:1905.08267.
arXiv:1905.08267

[79] M. Navascués, S. Pironio, and A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,'' New Journal of Physics 10, 073013 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​7/​073013

[80] R. E. Curto and L. A. Fialkow, An analogue of the Riesz–Haviland theorem for the truncated moment problem,'' Journal of Functional Analysis 255, 2709–2731 (2008).
https:/​/​doi.org/​10.1016/​j.jfa.2008.09.003

[81] D. Henrion and M. Korda, Convex computation of the region of attraction of polynomial control systems,'' IEEE Transactions on Automatic Control 59, 297–312 (2014).
https:/​/​doi.org/​10.1109/​TAC.2013.2283095

[82] J.-B. Lasserre, `Moments, positive polynomials and their applications,'' in Series on Optimization and its Applications, vol. 1. Imperial College Press, 2009.
https:/​/​doi.org/​10.1142/​p665

### Cited by

[1] Ulysse Chabaud, Ganaël Roeland, Mattia Walschaers, Frédéric Grosshans, Valentina Parigi, Damian Markham, and Nicolas Treps, "Certification of Non-Gaussian States with Operational Measurements", PRX Quantum 2 2, 020333 (2021).

[2] Benjamin Morris, Lukas J. Fiderer, Ben Lang, and Daniel Goldwater, "Witnessing Bell violations through probabilistic negativity", arXiv:2105.01685.

[3] Mattia Walschaers, "Non-Gaussian Quantum States and Where to Find Them", arXiv:2104.12596.

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