Partial majorization and Schur concave functions on the sets of quantum and classical states

TL;DR

This paper derives tight bounds on the differences of Schur concave functions between quantum states under partial majorization and trace distance constraints, with applications to entropy and Gibbs states.

Contribution

It introduces new bounds for Schur concave functions on quantum states under partial majorization and trace distance conditions, extending to classical probability distributions.

Findings

Derived tight upper bounds for Schur concave functions differences.

Applied bounds to von Neumann entropy and Gibbs states.

Introduced the concept of ε-sufficient majorization rank.

Abstract

We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f(\rho)-f(\sigma)$ for a quantum state $\rho$ with finite $f(\rho)$ and any quantum state $\sigma$ $m$-partially majorized by the state $\rho$ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition $\frac{1}{2}\|\rho-\sigma\|_1\leq\varepsilon$ and find simple sufficient conditions for vanishing this bound with $\,\min\{\varepsilon,1/m\}\to0\,$. The obtained results are applied to the von Neumann entropy. The concept of $\varepsilon$-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator. We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.