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 $\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon$ 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.

翻译：我们针对量子态集合上的Schur凹函数$f$，构造了有限$f(ρ)$的量子态$ρ$与任意$m$-部分偏序于$ρ$（按文献[1]所述意义）的量子态$σ$之差的$f(ρ)-f(σ)$的紧上界。在附加条件$\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon$下，我们还获得了该差的紧上界，并找到了当$\,\min\{\varepsilon,1/m\}\to0\,$时该界消失的简单充分条件。所得结果应用于冯·诺依曼熵。引入了有限熵量子态的$\varepsilon$-充分偏序秩概念，推导了该量的紧上界，并将其应用于量子振荡器的吉布斯态。我们还展示了如何将所得结果重新表述为具有有限或可数结果集概率分布集合上的Schur凹函数。