Abstract
We construct for a Schur concave function f on the set of quantum states a tight upper bound on the difference f(ρ)-f(σ) for a quantum state ρ with finite f(ρ) and any quantum state σ m-partially majorized by the state ρ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition 1/2ρ-σ₁≤ε and find simple sufficient conditions for vanishing this bound with min{ε,1/m}→0. The obtained results are applied to the von Neumann entropy. The concept of ε-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.