Strengthened monotonicity of relative entropy via pinched Petz recovery map

The quantum relative entropy between two states satisfies a monotonicity property, meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a “recovery map” that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved in [Fawzi and Renner, Commun. Math. Phys., 2015]. Our proof only relies on elementary properties of pinching maps and the operator logarithm.

[1]  D. Reeb,et al.  Monotonicity of the Quantum Relative Entropy Under Positive Maps , 2015, 1512.06117.

[2]  D. Petz.  Sufficient subalgebras and the relative entropy of states of a von Neumann algebra , 1986 .

[3]  R. Renner,et al.  Quantum Conditional Mutual Information and Approximate Markov Chains , 2014, Communications in Mathematical Physics.

[4]  Josip Pecaric,et al.  Jensen's operator inequality and its converses , 2006 .

[5]  A. Uhlmann.  Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory , 1977 .

[6]  M. Hayashi.  Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing , 2002, quant-ph/0208020.

[7]  Andreas Winter,et al.  Squashed Entanglement, k-Extendibility, Quantum Markov Chains, and Recovery Maps , 2018 .

[8]  C. Fuchs.  Distinguishability and Accessible Information in Quantum Theory , 1996, quant-ph/9601020.

[9]  R. Bhatia.  Positive Definite Matrices , 2007 .

[10]  Aram W. Harrow,et al.  Strengthened monotonicity of relative entropy via pinched Petz recovery map , 2015, 2016 IEEE International Symposium on Information Theory (ISIT).

[11]  Mario Berta,et al.  Monotonicity of quantum relative entropy and recoverability , 2014, Quantum Inf. Comput..

[12]  E. Carlen.  TRACE INEQUALITIES AND QUANTUM ENTROPY: An introductory course , 2009 .

[13]  M. Wolf,et al.  Positivity of linear maps under tensor powers , 2015, 1502.05630.

[14]  Mark M. Wilde,et al.  Fidelity of recovery, geometric squashed entanglement, and measurement recoverability , 2014, 1410.1441.

[15]  P. Malliavin.  Infinite dimensional analysis , 1993 .

[16]  Fernando G S L Brandão,et al.  Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution. , 2014, Physical review letters.

[17]  Mario Berta,et al.  Multivariate Trace Inequalities , 2016, ArXiv.

[18]  Mario Berta,et al.  On variational expressions for quantum relative entropies , 2015, ArXiv.

[19]  Serge Fehr,et al.  On quantum Rényi entropies: A new generalization and some properties , 2013, 1306.3142.

[20]  M. Hayashi.  Asymptotics of quantum relative entropy from a representation theoretical viewpoint , 1997, quant-ph/9704040.

[21]  Nilanjana Datta,et al.  Min- and Max-Relative Entropies and a New Entanglement Monotone , 2008, IEEE Transactions on Information Theory.

[22]  Mark M. Wilde,et al.  Recoverability in quantum information theory , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  Tetsunao Matsuta,et al.  国際会議開催報告:2013 IEEE International Symposium on Information Theory , 2013 .

[24]  E. Lieb,et al.  Proof of the strong subadditivity of quantum‐mechanical entropy , 1973 .

[25]  M. Sion.  On general minimax theorems , 1958 .

[26]  Dénes Petz,et al.  A variational expression for the relative entropy , 1988 .

[27]  林 正人.  Quantum information : an introduction , 2006 .

[28]  G. Lindblad.  Completely positive maps and entropy inequalities , 1975 .

[29]  Andreas Winter,et al.  Squashed Entanglement, $$\mathbf {k}$$k-Extendibility, Quantum Markov Chains, and Recovery Maps , 2014, 1410.4184.

[30]  Mario Berta,et al.  The Fidelity of Recovery Is Multiplicative , 2015, IEEE Transactions on Information Theory.

[31]  F. Hansen,et al.  Jensen's Operator Inequality , 2002, math/0204049.

[32]  Chandler Davis,et al.  A Schwarz inequality for convex operator functions , 1957 .

[33]  E. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[34]  A. Lichnerowicz.  Proof of the Strong Subadditivity of Quantum-Mechanical Entropy , 2018 .

[35]  Man-Duen Choi.  A schwarz inequality for positive linear maps on $C^{\ast}$-algebras , 1974 .

[36]  D. Petz.  SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS , 1988 .

[37]  Omar Fawzi,et al.  Universal recovery map for approximate Markov chains , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[38]  M. Piani.  Relative entropy of entanglement and restricted measurements. , 2009, Physical review letters.

[39]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[40]  D. Petz.  Quantum Information Theory and Quantum Statistics , 2007 .

[41]  D. Petz.  Monotonicity of quantum relative entropy revisited , 2002, quant-ph/0209053.