Defining quantum divergences via convex optimization

Hamza Fawzi1 and Omar Fawzi2

1DAMTP, University of Cambridge, United Kingdom
2Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France

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We introduce a new quantum Rényi divergence $D^{\#}_{\alpha}$ for $\alpha \in (1,\infty)$ defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched $\alpha$-Rényi divergence between quantum channels for $\alpha > 1$. Second it allows us to prove a chain rule property for the sandwiched $\alpha$-Rényi divergence for $\alpha > 1$ which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.

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[1] WN Anderson, Jr and GE Trapp. Shorted operators. II. SIAM J. Appl. Math., 28 (1): 60–71, 1975. 10.1137/​0128007.

[2] Viacheslav P Belavkin and P Staszewski. $C^*$-algebraic generalization of relative entropy and entropy. In Annales de l'IHP Physique Théorique, volume 37, pages 51–58, 1982.

[3] Charles H Bennett, Aram Wettroth Harrow, Debbie W Leung, and John A Smolin. On the capacities of bipartite hamiltonians and unitary gates. IEEE Trans. Inform. Theory, 49 (8): 1895–1911, 2003. 10.1109/​TIT.2003.814935.

[4] Mario Berta and Mark M Wilde. Amortization does not enhance the max-Rains information of a quantum channel. New J. Phys., 20 (5): 053044, may 2018. 10.1088/​1367-2630/​aac153.

[5] Mario Berta, Omar Fawzi, and Marco Tomamichel. On variational expressions for quantum relative entropies. Lett. Math. Phys., Sep 2017. ISSN 1573-0530. 10.1007/​s11005-017-0990-7.

[6] Rajendra Bhatia. Positive definite matrices, volume 24. Princeton University Press, 2009. 10.1515/​9781400827787.

[7] Peter Brown, Hamza Fawzi, and Omar Fawzi. Computing conditional entropies for quantum correlations. 2020. arXiv:2007.12575.

[8] Eric Chitambar and Gilad Gour. Quantum resource theories. Reviews of Modern Physics, 91 (2): 025001, 2019. 10.1103/​RevModPhys.91.025001.

[9] Matthias Christandl and Alexander Müller-Hermes. Relative entropy bounds on quantum, private and repeater capacities. Comm. Math. Phys., 353 (2): 821–852, 2017. 10.1007/​s00220-017-2885-y.

[10] Tom Cooney, Milán Mosonyi, and Mark M. Wilde. Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication. Comm. Math. Phys., 344 (3): 797–829, Jun 2016. ISSN 1432-0916. 10.1007/​s00220-016-2645-4.

[11] Matthew J. Donald. On the Relative Entropy. Comm. Math. Phys., 105 (1): 13–34, mar 1986. ISSN 0010-3616. 10.1007/​BF01212339.

[12] Kun Fang and Hamza Fawzi. Geometric Rényi Divergence and its Applications in Quantum Channel Capacities. 2019. arXiv:1909.05758.

[13] Kun Fang, Omar Fawzi, Renato Renner, and David Sutter. Chain Rule for the Quantum Relative Entropy. Phys. Rev. Lett., 124: 100501, 2020. 10.1103/​PhysRevLett.124.100501.

[14] Hamza Fawzi and James Saunderson. Lieb's concavity theorem, matrix geometric means, and semidefinite optimization. Linear Algebra Appl., 513: 240–263, 2017. 10.1016/​j.laa.2016.10.012.

[15] Hamza Fawzi, James Saunderson, and Pablo A. Parrilo. Semidefinite approximations of the matrix logarithm. Foundations of Computational Mathematics, 2018. 10.1007/​s10208-018-9385-0. Package cvxquad at https:/​/​​hfawzi/​cvxquad.

[16] A.W. Harrow. Applications of Coherent Classical Communication and the Schur Transform to Quantum Information Theory. PhD thesis, Massachusetts Institute of Technology, 2005. arXiv:quant-ph/​0512255.

[17] Masahito Hayashi. Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing. Journal of Physics A: Mathematical and General, 35 (50): 10759, 2002. 10.1088/​0305-4470/​35/​50/​307.

[18] F. Hiai and D. Petz. The proper formula for relative entropy and its asymptotics in quantum probability. Comm. Math. Phys., 143 (1): 99–114, 1991. 10.1007/​BF02100287.

[19] Fumio Hiai and Milán Mosonyi. Different quantum $f$-divergences and the reversibility of quantum operations. Reviews in Mathematical Physics, 2017. 10.1142/​S0129055X17500234.

[20] Alexander S Holevo and Reinhard F Werner. Evaluating capacities of bosonic gaussian channels. Phys. Rev. A, 63 (3): 032312, 2001. 10.1103/​PhysRevA.63.032312.

[21] Anna Jenčová. Rényi Relative Entropies and Noncommutative $L_p$-Spaces. Annales Henri Poincaré, 19 (8): 2513–2542, 2018. 10.1007/​s00023-018-0683-5.

[22] Vishal Katariya and Mark M Wilde. Geometric distinguishability measures limit quantum channel estimation and discrimination. 2020. arXiv:2004.10708.

[23] Eneet Kaur and Mark M Wilde. Amortized entanglement of a quantum channel and approximately teleportation-simulable channels. J. Phys. A - Math. Theor., 51 (3): 035303, 2017. 10.1088/​1751-8121/​aa9da7.

[24] Fumio Kubo and Tsuyoshi Ando. Means of positive linear operators. Mathematische Annalen, 246 (3): 205–224, 1980. 10.1007/​BF01371042.

[25] Felix Leditzky, Eneet Kaur, Nilanjana Datta, and Mark M Wilde. Approaches for approximate additivity of the Holevo information of quantum channels. Phys. Rev. A, 97 (1): 012332, 2018. 10.1103/​PhysRevA.97.012332.

[26] Keiji Matsumoto. A new quantum version of $f$-divergence. 2013. arXiv:1311.4722.

[27] Milán Mosonyi and Tomohiro Ogawa. Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Comm. Math. Phys., 334 (3): 1617–1648, 2015. 10.1007/​s00220-014-2248-x.

[28] Milán Mosonyi and Tomohiro Ogawa. Strong converse exponent for classical-quantum channel coding. Communications in Mathematical Physics, 355 (1): 373–426, 2017. 10.1007/​s00220-017-2928-4.

[29] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel. On quantum Rényi entropies: A new generalization and some properties. J. Math. Phys., 54 (12): 122203, 2013. 10.1063/​1.4838856.

[30] Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani, and Leonardo Banchi. Fundamental limits of repeaterless quantum communications. Nat. Comm., 8 (1): 1–15, 2017. 10.1038/​ncomms15043.

[31] Eric M Rains. A semidefinite program for distillable entanglement. IEEE Trans. Inform. Theory, 47 (7): 2921–2933, 2001. 10.1109/​18.959270.

[32] R Tyrrell Rockafellar. Convex analysis. Number 28. Princeton University Press, 1970. 10.1515/​9781400873173.

[33] Guillaume Sagnol. On the semidefinite representation of real functions applied to symmetric matrices. Linear Algebra Appl., 439 (10): 2829–2843, 2013. 10.1016/​j.laa.2013.08.021.

[34] Barry Simon. Operator means, II: Kubo–Ando theorem. In Loewner's Theorem on Monotone Matrix Functions, pages 379–384. Springer, 2019. 10.1007/​978-3-030-22422-6_37.

[35] Marco Tomamichel. Quantum Information Processing with Finite Resources: Mathematical Foundations, volume 5. Springer, 2015. 10.1007/​978-3-319-21891-5. arXiv:1504.00233.

[36] Hisaharu Umegaki. Conditional expectation in an operator algebra, IV (entropy and information). In Kodai Mathematical Seminar Reports, volume 14, pages 59–85. Department of Mathematics, Tokyo Institute of Technology, 1962. 10.2996/​kmj/​1138844604.

[37] Tim van Erven and Peter Harremos. Rényi divergence and Kullback-Leibler divergence. IEEE Trans. Inform. Theory, 60 (7): 3797–3820, 2014. 10.1109/​TIT.2014.2320500.

[38] Xin Wang and Runyao Duan. Improved semidefinite programming upper bound on distillable entanglement. Phys. Rev. A, 94 (5): 050301, 2016. 10.1103/​PhysRevA.94.050301.

[39] Xin Wang and Mark M. Wilde. Resource theory of asymmetric distinguishability for quantum channels. Phys. Rev. Research, 1: 033170, Dec 2019. 10.1103/​PhysRevResearch.1.033169. arXiv:1907.06306.

[40] Xin Wang, Wei Xie, and Runyao Duan. Semidefinite programming strong converse bounds for classical capacity. IEEE Trans. Inform. Theory, 64 (1): 640–653, 2017. 10.1109/​TIT.2017.2741101.

[41] Xin Wang, Kun Fang, and Marco Tomamichel. On converse bounds for classical communication over quantum channels. IEEE Trans. Inform. Theory, 65 (7): 4609–4619, 2019. 10.1109/​TIT.2019.2898656.

[42] Mark M Wilde, Marco Tomamichel, and Mario Berta. Converse bounds for private communication over quantum channels. IEEE Trans. Inform. Theory, 63 (3): 1792–1817, 2017. 10.1109/​TIT.2017.2648825.

[43] Mark M. Wilde, Mario Berta, Christoph Hirche, and Eneet Kaur. Amortized channel divergence for asymptotic quantum channel discrimination. Lett. Math. Phys., 2020. 10.1007/​s11005-020-01297-7. arXiv:1808.01498.

[44] M.M. Wilde, A. Winter, and D. Yang. Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. Comm. Math. Phys., 331 (2): 593–622, 2014. ISSN 0010-3616. 10.1007/​s00220-014-2122-x. arXiv:1306.1586.

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[1] Omar Fawzi, Ala Shayeghi, and Hoang Ta, 2021 IEEE International Symposium on Information Theory (ISIT) 272 (2021) ISBN:978-1-5386-8209-8.

[2] Bjarne Bergh, Robert Salzmann, and Nilanjana Datta, "The α → 1 limit of the sharp quantum Rényi divergence", Journal of Mathematical Physics 62 9, 092205 (2021).

[3] Samuel O. Scalet, Álvaro M. Alhambra, Georgios Styliaris, and J. Ignacio Cirac, "Computable Rényi mutual information: Area laws and correlations", Quantum 5, 541 (2021).

[4] Peter Brown, Hamza Fawzi, and Omar Fawzi, "Computing conditional entropies for quantum correlations", Nature Communications 12 1, 575 (2021).

[5] Kun Fang and Zi-Wen Liu, "No-go theorems for quantum resource purification: new approach and channel theory", arXiv:2010.11822.

[6] Christoph Hirche, "Quantum Network Discrimination", arXiv:2103.02404.

[7] Andres F. Ducuara and Paul Skrzypczyk, "Quantum state betting, dependence measures, quantum Rényi divergences, and resource monotones: a four-way correspondence", arXiv:2106.12711.

[8] Dawei Ding, Sumeet Khatri, Yihui Quek, Peter W. Shor, Xin Wang, and Mark M. Wilde, "Bounding the forward classical capacity of bipartite quantum channels", arXiv:2010.01058.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-22 15:34:53) and SAO/NASA ADS (last updated successfully 2021-10-22 15:34:54). The list may be incomplete as not all publishers provide suitable and complete citation data.