Relative entropy between quantum ensembles

Huaijing Huang, Zhaoqi Wu, Chuanxi Zhu +1 more

Mary Beth Ruskai

This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb.

Subhash Kak

Айрат Мидхатович Бикчентаев, Fuad Kıttaneh, Mohammad Sal Moslehian +1 more

Максим Евгеньевич Широков

It is proved that the decrease of quantum relative entropy under the action of a quantum operation is a lower semicontinuous function of the pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under the action of quantum operations. It also implies the lower semicontinuity of the modulus of the joint convexity of quantum relative entropy (as a function of ensembles of quantum states). Various corollaries and applications of these results are considered. Bibliography: 42 titles.

C. S. Sharma