Yet Another Proof of the Joint Convexity of Relative Entropy

The joint convexity of the map (X,A) 7→ XAX , an integral representation of convex operator functions, and an observation of Ando are used to obtain a simple proof of both the joint convexity of relative entropy and a trace convexity result of Lieb. The latter was the key ingredient in the original proof of the strong subadditivity of quantum entropy.

[1]  Derek W. Robinson,et al.  Mean Entropy of States in Quantum‐Statistical Mechanics , 1968 .

[2]  T. Andô.  Concavity of certain maps on positive definite matrices and applications to Hadamard products , 1979 .

[3]  E. Lieb.  Convex trace functions and the Wigner-Yanase-Dyson conjecture , 1973 .

[4]  W. Näther.  Optimum experimental designs , 1994 .

[5]  Anna Vershynina,et al.  Strong Subadditivity of Quantum Entropy , 2013, Scholarpedia.

[6]  F. Hiai,et al.  Contraction coefficients for noisy quantum channels ∗ , 2015 .

[7]  M. Ruskai,et al.  A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY , 2009, 0903.2895.

[8]  Isaac H. Kim.  Operator extension of strong subadditivity of entropy , 2012, 1210.5190.

[9]  E. Lieb,et al.  Some operator inequalities of the schwarz type , 1974 .

[10]  M. Ruskai.  Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy , 2006, quant-ph/0604206.

[11]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[12]  Elliott H. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[13]  E. Effros.  A matrix convexity approach to some celebrated quantum inequalities , 2008, Proceedings of the National Academy of Sciences.

[14]  Barry Simon,et al.  Loewner's Theorem on Monotone Matrix Functions , 2019, Grundlehren der mathematischen Wissenschaften.

[15]  A. Wehrl.  General properties of entropy , 1978 .

[16]  M. Ruskai.  Inequalities for quantum entropy: A review with conditions for equality , 2002, quant-ph/0205064.

[17]  Robert König,et al.  Quantum entropy and its use , 2017 .

[18]  M. Ruskai.  Remarks on Kim's strong subadditivity matrix inequality: Extensions and equality conditionsa) , 2012, 1211.0049.

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

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

[21]  Mary Beth Ruskai.  Lieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities , 2004 .