From the non-negativity of classical relative entropy, we have The second part of the claim follows from the fact that, since -log is strictly convex, equality is achieved in if and only if (Pi j) is a permutation matrix, which implies ρ = σ, after a suitable labeling of the eigenvectors {vi} and {wi}. Joint convexity of relative entropy The relative entropy is jointly convex. For and states we have Monotonicity of relative entropy The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations on density matrices, . This inequality is called Monotonicity of quantum relative entropy and was first proved by Lindblad. An entanglement measure Let a composite quantum system have state space and ρ be a density matrix acting on H. The relative entropy of entanglement of ρ is defined by where the minimum is taken over the family of separable states. A physical interpretation of the quantity is the optimal distinguishability of the state ρ from separable states. Clearly, when ρ is not entangled by Klein's inequality. Relation to other quantum information quantities One reason the quantum relative entropy is useful is that several other important quantum information quantities are special cases of it. Often, theorems are stated in terms of the quantum relative entropy, which lead to immediate corollaries concerning the other quantities. Below, we list some of these relations. Let ρAB be the joint state of a bipartite system with subsystem A of dimension nA and B of dimension nB. Let ρA, ρB be the respective reduced states, and IA, IB the respective identities. The maximally mixed states are IA/nA and IB/nB. Then it is possible to show with direct computation that where I(A:B) is the quantum mutual information and S(B|A) is the quantum conditional entropy. References Michael A. Nielsen, Isaac L. Chuang, "Quantum Computation and Quantum Information" Marco Tomamichel, "Quantum Information Processing with Finite Resources -- Mathematical Foundations". arXiv:1504.00233 Quantum mechanical entropy Quantum information theory