A Singular Value Inequality for Heinz Means

We prove a matrix inequality for matrix monotone functions, and apply it to prove a singular value inequality for Heinz means recently conjectured by X. Zhan.Comment: 4 pages; Mistake in proof of Theorem 1 correcte

Trace inequalities for completely monotone functions and Bernstein functions

We prove a matrix trace inequality for completely monotone functions and for Bernstein functions. As special cases we obtain non-trivial trace inequalities for the power function x->x^q, which for certain values of q complement McCarthy's trace inequality and for others strenghten it.Comment: 16 page

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all $n\times n$ non-negative matrices $A$ and $B$, $\rho(A\circ B)\le\rho((A\circ A)(B\circ B))^{1/2}\le\rho(AB)$, where $\circ$ represents the Hadamard product.Comment: 4.1 page

On the asymmetry of the relative entropy

The quantum relative entropy $S(\rho||\sigma)$ is a widely used dissimilarity measure between quantum states, but it has the peculiarity of being asymmetric in its arguments. We quantify the amount of asymmetry by providing a sharp upper bound in terms of two parameters: the trace norm distance between the two states, and the smallest of the smallest eigenvalues of both states. The bound is essentially the asymmetry between two binary distributions governed by these two parameters.Comment: 9 pages, 3 figure