Geodesic
Quantifying non-Gaussianity via the Hellinger distance
Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 2, pp. 242-257 Cet article a éte moissonné depuis la source Math-Net.Ru

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Non-Gaussianity is an important resource for quantum information processing with continuous variables. We introduce a measure of the non-Gaussianity of bosonic field states based on the Hellinger distance and present its basic features. This measure has some natural properties and is easy to compute. We illustrate this measure with typical examples of bosonic field states and compare it with various measures of non-Gaussianity. In particular, we highlight its similarity to and difference from the measure based on the Bures distance (or, equivalently, fidelity).
Keywords: bosonic field, Gaussian state, non-Gaussianity, Hellinger distance
Mots-clés : Bures distance. 
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Yue Zhang; Shunlong Luo. Quantifying non-Gaussianity via the Hellinger distance. Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 2, pp. 242-257. http://geodesic.mathdoc.fr/item/TMF_2020_204_2_a5/

[1] G. Giedke, J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states”, Phys. Rev. A, 66:3 (2002), 032316, 7 pp. | DOI

[2] J. Eisert, M. B. Plenio, “Introduction to the basics of entanglement theory in continuous-variable systems”, Internat. J. Quantum Inf., 1:4 (2003), 479–506 | DOI

[3] A. Ferraro, S. Olivares, M. G. A. Paris, Gaussian States in Quantum Information, Bibliopolis, Napoli, 2005

[4] S. L. Braunstein, P. van Loock, “Quantum information with continuous variables”, Rev. Modern Phys., 77:2 (2005), 513–577, arXiv: quant-ph/0410100 | DOI | MR

[5] X.-B. Wang, T. Hiroshima, A. Tomita, M. Hayashi, “Quantum information with Gaussian states”, Phys. Rep., 448:1–4 (2007), 1–111 | DOI | MR

[6] C. Weedbrook, S. Pirandola, R. G. Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, S. Lloyd, “Gaussian quantum information”, Rev. Modern Phys., 84:2 (2012), 621–669, arXiv: 1110.3234 | DOI

[7] G. Adesso, S. Ragy, A. R. Lee, “Continuous variable quantum information: Gaussian states and beyond”, Open Syst. Inf. Dyn., 21:2 (2014), 1440001, 47 pp. | DOI | MR

[8] I. E. Segal, “Foundations of the theory of dynamical systems of infinitely many degrees of freedom. II”, Canadian J. Math., 13 (1961), 1–18 | DOI | MR

[9] A. S. Kholevo, Veroyatnostnye i statisticheskie aspekty kvantovoi teorii, MTsNMO, M., 2020 | MR

[10] R. Simon, “Peres–Horodecki separability criterion for continuous variable systems”, Phys. Rev. Lett., 84:12 (2000), 2726–2729, arXiv: quant-ph/9909044 | DOI

[11] R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics”, Phys. Rev. A, 36:8 (1987), 3868–3880 | DOI | MR

[12] S. Fu, S. Luo, Y. Zhang, “Gaussian states as minimum uncertainty states”, Phys. Lett. A, 384:1 (2020), 126037, 5 pp. | DOI | MR

[13] F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, P. Grangier, “Quantum key distribution using Gaussian-modulated coherent states”, Nature, 421:6920 (2003), 238–241, arXiv: quant-ph/0312016 | DOI

[14] V. D'Auria, S. Fornaro, A. Porzio, S. Solimeno, S. Olivares, M. G. A. Paris, “Full characterization of Gaussian bipartite entangled states by a single homodyne detector”, Phys. Rev. Lett., 102:2 (2009), 020502, 3 pp. | DOI

[15] D. Daems, F. Bernard, N. J. Cerf, M. I. Kolobov, “Tripartite entanglement in parametric down-conversion with spatially structured pump”, J. Opt. Soc. Am. B, 27:3 (2010), 447–451, arXiv: 1002.1798 | DOI

[16] L.-M. Duan, G. Giedke, J. I. Cirac, P. Zoller, “Inseparability criterion for continuous variable systems”, Phys. Rev. Lett., 84:12 (2000), 2722–2725, arXiv: quant-ph/9908056 | DOI

[17] G. Adesso, A. Serafini, F. Illuminati, “Quantification and scaling of multipartite entanglement in continuous variable systems”, Phys. Rev. Lett., 93:22 (2004), 220504, arXiv: quant-ph/0406053 | DOI

[18] P. Marian, T. A. Marian, “Entanglement of formation for an arbitrary two-mode Gaussian state”, Phys. Rev. Lett., 101:22 (2008), 220403, 4 pp., arXiv: 0809.0321 | DOI

[19] G. Adesso, A. Datta, “Quantum versus classical correlations in Gaussian states”, Phys. Rev. Lett., 105:3 (2010), 030501, 4 pp., arXiv: 1003.4979 | DOI

[20] P. Giorda, M. G. A. Paris, “Gaussian quantum discord”, Phys. Rev. Lett., 105:2 (2010), 020503, 4 pp., arXiv: 1003.3207 | DOI

[21] T. Opatrný, G. Kurizki, D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement”, Phys. Rev. A, 61:3 (2000), 032302, 7 pp. | DOI

[22] P. T. Cochrane, T. C. Ralph, G. J. Milburn, “Teleportation improvement by conditional measurements on the two-mode squeezed vacuum”, Phys. Rev. A, 65:6 (2002), 062306, 6 pp. | DOI

[23] S. Olivares, M. G. A. Paris, R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction”, Phys. Rev. A, 67:3 (2003), 032314, 5 pp., arXiv: quant-ph/0209140 | DOI

[24] N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal”, Phys. Rev. Lett., 95:7 (2005), 070501, 4 pp., arXiv: quant-ph/0410058 | DOI

[25] F. Dell'Anno, S. De Siena, L. Albano, F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources”, Phys. Rev. A, 76:2 (2007), 022301, 11 pp., arXiv: 0706.3701 | DOI

[26] F. Dell'Anno, S. De Siena, F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources”, Phys. Rev. A, 81:1 (2010), 012333, 12 pp., arXiv: 0910.2713 | DOI

[27] R. Takagi, Q. Zhuang, “Convex resource theory of non-Gaussianity”, Phys. Rev. A, 97:6 (2018), 062337, 14 pp., arXiv: 1804.04669 | DOI

[28] L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, G. Adesso, “Gaussian quantum resource theories”, Phys. Rev. A, 98:2 (2018), 022335, 13 pp., arXiv: 1801.0545 | DOI

[29] F. Albarelli, M. G. Genoni, M. G. A. Paris, A. Ferraro, “Resource theory of quantum non-Gaussianity and Wigner negativity”, Phys. Rev. A, 98:5 (2018), 052350, 17 pp., arXiv: 1804.05763 | DOI

[30] R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, S. P. Walborn, “Quantum entanglement beyond Gaussian criteria”, Proc. Natl. Acad. Sci. USA, 106:51 (2009), 21517–21520 | DOI

[31] M. Allegra, P. Giorda, M. G. A. Paris, “Role of initial entanglement and non-Gaussianity in the decoherence of photon-number entangled states evolving in a noisy channel”, Phys. Rev. Lett., 105:10 (2010), 100503, 4 pp. | DOI

[32] R. Tatham, L. Mišta, Jr., G. Adesso, N. Korolkova, “Nonclassical correlations in continuous-variable non-Gaussian Werner states”, Phys. Rev. A, 85:2 (2012), 022326, 12 pp., arXiv: 1111.1101 | DOI

[33] C. Navarrete-Benlloch, R. García-Patrón, J. H. Shapiro, N. J. Cerf, “Enhancing quantum entanglement by photon addition and subtraction”, Phys. Rev. A, 86:1 (2012), 012328, 9 pp., arXiv: 1201.4120 | DOI

[34] K. K. Sabapathy, A. Winter, “Non-Gaussian operations on bosonic modes of light: Photon-added Gaussian channels”, Phys. Rev. A, 95:6 (2017), 062309, 17 pp., arXiv: 1604.07859 | DOI

[35] M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, R. Tualle-Brouri, “Non-Gaussianity of quantum states: An experimental test on single-photon-added coherent states”, Phys. Rev. A, 82:6 (2010), 063833, 5 pp., arXiv: 1012.0466 | DOI

[36] A. Allevi, S. Olivares, M. Bondani, “Manipulating the non-Gaussianity of phase-randomized coherent states”, Opt. Exp., 20:22 (2012), 24850–24855, arXiv: 1210.0303 | DOI

[37] C. Baune, A. Schönbeck, A. Samblowski, J. Fiurášek, R. Schnabel, “Quantum non-Gaussianity of frequency up-converted single photons”, Opt. Exp., 22:19 (2014), 22808–22816, arXiv: 1406.1602 | DOI

[38] C. Hughes, M. G. Genoni, T. Tufarelli, M. G. A. Paris, M. S. Kim, “Quantum non-Gaussianity witnesses in phase space”, Phys. Rev. A, 90:1 (2014), 013810, 9 pp., arXiv: 1403.6264 | DOI

[39] R. Filip, L. Mišta, Jr., “Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states”, Phys. Rev. Lett., 106:20 (2011), 200401, 4 pp. | DOI

[40] M. G. Genoni, M. L. Palma, T. Tufarelli, S. Olivares, M. S. Kim, M. G. A. Paris, “Detecting quantum non-Gaussianity via the Wigner function”, Phys. Rev. A, 87:6 (2013), 062104, 9 pp., arXiv: 1304.3340 | DOI

[41] I. Straka, A. Predojević, T. Huber, L. Lachman, L. Butschek, M. Miková, M. Mičuda, G. S. Solomon, G. Weihs, M. Ježek, R. Filip, “Quantum non-Gaussian depth of single-photon states”, Phys. Rev. Lett., 113:22 (2014), 223603, 5 pp., arXiv: 1403.4194 | DOI

[42] M. G. Genoni, M. G. A. Paris, K. Banaszek, “Measure of the non-Gaussian character of a quantum state”, Phys. Rev. A, 76:4 (2007), 042327, 6 pp., arXiv: 0704.0639 | DOI

[43] M. G. Genoni, M. G. A. Paris, K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy”, Phys. Rev. A, 78:6 (2008), 060303, 4 pp., arXiv: 0805.1645 | DOI | MR

[44] M. G. Genoni, M. G. A. Paris, “Quantifying non-Gaussianity for quantum information”, Phys. Rev. A, 82:5 (2010), 052341, 19 pp., arXiv: 1008.4243 | DOI

[45] P. Marian, T. A. Marian, “Relative entropy is an exact measure of non-Gaussianity”, Phys. Rev. A, 88:1 (2013), 012322, 6 pp., arXiv: 1308.2939 | DOI

[46] J. S. Ivan, M. S. Kumar, R. Simon, “A measure of non-Gaussianity for quantum states”, Quantum Inf. Process., 11:3 (2012), 853–872 | DOI | MR

[47] I. Ghiu, P. Marian, T. A. Marian, “Measures of non-Gaussianity for one-mode field states”, Phys. Scr., T153 (2013), 014028, arXiv: 1210.1929 | DOI

[48] P. Marian, I. Ghiu, T. A. Marian, “Gaussification through decoherence”, Phys. Rev. A, 88:1 (2013), 012316, 11 pp., arXiv: 1211.1701 | DOI

[49] W. Son, “Role of quantum non-Gaussian distance in entropic uncertainty relations”, Phys. Rev. A, 92:1 (2015), 012114, arXiv: 1504.05725 | DOI

[50] K. Baek, H. Nha, “Non-Gaussianity and entropy-bounded uncertainty relations: Application to detection of non-Gaussian entangled states”, Phys. Rev. A, 98:4 (2018), 042314, 7 pp., arXiv: 1811.10207 | DOI

[51] S. Fu, S. Luo, Y. Zhang, “Quantifying non-Gaussianity of bosonic fields via an uncertainty relation”, Phys. Rev. A, 101:1 (2020), 012125, 8 pp. | DOI

[52] P. Marian, T. A. Marian, “Hellinger distance as a measure of Gaussian discord”, J. Phys. A: Math. Theor., 48:11 (2015), 115301, 21 pp., arXiv: 1408.4477 | DOI | MR

[53] S. Luo, Q. Zhang, “Informational distance on quantum-state space”, Phys. Rev. A, 69:3 (2004), 032106, 8 pp. | DOI | MR

[54] P. Marian, T. A. Marian, “Squeezed states with thermal noise. I. Photon-number statistics”, Phys. Rev. A, 47:5 (1993), 4474–4486 ; “Squeezed states with thermal noise. II. Damping and photon counting”, 4487–4495 | DOI | DOI

[55] D. Bures, “An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite $W^*$-algebras”, Trans. Amer. Math. Soc., 135 (1969), 199–212 | DOI | MR

[56] M. Nilsen, I. Chang, Kvantovye vychisleniya i kvantovaya informatsiya, Mir, M., 2006 | MR | Zbl

[57] G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, 122, Princeton Univ. Press, Princeton, NJ, 1989 | DOI | MR

[58] S. Olivares, M. G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise”, J. Opt. B, 7:10 (2005), S392–S397 | DOI

[59] K. K. Sabapathy, C. Weedbrook, “ON states as resource units for universal quantum computation with photonic architectures”, Phys. Rev. A, 97:6 (2018), 062315, 12 pp., arXiv: 1802.05220 | DOI

[60] V. V. Dodonov, O. V. Man'ko, V. I. Man'ko, A. Wünsche, “Hilbert–Schmidt distance and non-classicality of states in quantum optics”, J. Modern Opt., 47:4 (2000), 633–654 | DOI | MR

[61] M. Ozawa, “Entanglement measures and the Hilbert–Schmidt distance”, Phys. Lett. A, 268:3 (2000), 158–160 | DOI | MR

[62] J. Lee, M. S. Kim, Č. Brukner, “Operationally invariant measure of the distance between quantum states by complementary measurements”, Phys. Rev. Lett., 91:8 (2003), 087902, 4 pp., arXiv: quant-ph/0303111 | DOI

[63] V. Vedral, “The role of relative entropy in quantum information theory”, Rev. Modern Phys., 74:1 (2002), 197–234 | DOI | MR

[64] A. Wehrl, “General properties of entropy”, Rev. Modern Phys., 50:2 (1978), 221–260 | DOI | MR

[65] A. Wehrl, “On the relation between classical and quantum mechanical entropy”, Rep. Math. Phys., 16:3 (1979), 353–358 | DOI | MR

[66] E. H. Lieb, “Proof of an entropy conjecture of Wehrl”, Commun. Math. Phys., 62:1 (1978), 35–41 | DOI | MR

[67] A. Orłowski, “Classical entropy of quantum states of light”, Phys. Rev. A, 48:1 (1993), 727–731 | DOI

[68] S. Luo, “A simple proof of Wehrl's conjecture on entropy”, J. Phys. A: Math. Gen., 33:16 (2000), 3093–3096 | DOI | MR

[69] F. Mintert, K. .{Z}yczkowski, “Wehrl entropy, Lieb conjecture, and entanglement monotones”, Phys. Rev. A, 69:2 (2004), 022317, 11 pp., arXiv: quant-ph/0307169 | DOI | MR