We study here the elementary properties of the relative entropyH(A,B) = Tr[φ(A)−φ(B)−φ(B)(A−B)] for φ a convex function and A,B bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if φ is operator monotone. We use this to appropriately define H(A,B) in infinite dimension. In this paper we introduce a relative entropy H(A,B) of two bounded o...

We introduce an axiomatic approach to entropies and relative that relies only on minimal information-theoretic axioms, namely monotonicity under mixing data-processing as well additivity for product distributions. find these axioms induce sufficient structure establish continuity in the interior of probability simplex meaningful upper lower bounds, e.g., we every entropy satisfying must lie bet...

Entropy games and matrix multiplication games have been recently introduced by Asarin et al. They model the situation in which one player (Despot) wishes to minimize the growth rate of a matrix product, whereas the other player (Tribune) wishes to maximize it. We develop an operator approach to entropy games. This allows us to show that entropy games can be cast as stochastic mean payoff games ...

We establish the entangling power of a unitary operator on a general finite-dimensional bipartite quantum system with and without ancillas, and give relations between the entangling power based on the von Neumann entropy and the entangling power based on the linear entropy. Significantly, we demonstrate that the entangling power of a general controlled unitary operator acting on two equal-dimen...