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

Abstract

We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f(\ddot{I})-f(\ddot{I}ƒ)$ for a quantum state $\ddot{I}$ with finite $f(\ddot{I})$ and any quantum state $\ddot{I}ƒ$ $m$-partially majorized by the state $\ddot{I}$ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition $\frac{1}{2}\Vert \ddot{I}-\ddot{I}ƒ{\Vert }_1\le \epsilon$ and find simple sufficient conditions for vanishing this bound with $\min \{\epsilon ,1/m\}\to 0$. The obtained results are applied to the von Neumann entropy. The concept of $\epsilon$-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.

20 pages, 3 figures, any comments are welcome