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

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 $\frac{1}{2}\mathrm{\parallel }\rho -\sigma {\mathrm{\parallel }}_1\le \epsilon$ and find simple sufficient conditions for vanishing this bound with $\min \{\epsilon ,1\mathrm{/}m\}\to 0$. 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.