Partial majorization and Schur concave functions on the sets of quantum and classical states
2026-04-14 • Information Theory
Information Theory
AI summaryⓘ
The authors study how certain types of functions, called Schur concave functions, change when applied to quantum states that are related by a process called partial majorization. They find precise upper limits on how much the function's value can differ between two such states, even when the states are close in a specific distance measure. They apply these results to the von Neumann entropy, a key measure of quantum uncertainty, and introduce a new concept called the epsilon-sufficient majorization rank to describe quantum states more precisely. Their work also extends to similar functions on regular probability distributions.
Schur concave functionquantum statepartial majorizationvon Neumann entropytrace distanceepsilon-sufficient majorization rankGibbs statequantum oscillatorprobability distributionmajorization
Authors
M. E. Shirokov
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.