Partial majorization and Schur concave functions on the sets of quantum and classical states
This work provides theoretical tools for comparing quantum states under partial majorization, which is relevant for quantum information theory and statistical mechanics.
The authors derive tight upper bounds on the difference of a Schur concave function between two quantum states under partial majorization, with applications to von Neumann entropy and Gibbs states. They also introduce the concept of ε-sufficient majorization rank and provide a tight bound for it.
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.