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

• URL: http://arxiv.org/abs/2604.13033v1
• Date: Tue, 14 Apr 2026 17:59:12 GMT
• Title: Partial majorization and Schur concave functions on the sets of quantum and classical states
• Abstract summary: We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f()-f()$ for a quantum state $$ with finite $f()$ and any quantum state $$ $m$-partially majorized by the state $$.<n>We 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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon$ 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.

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