Optimization of maximal quantum f-divergences between unitary orbits

TL;DR

This paper characterizes the extremal values of maximal quantum f-divergences between quantum states along unitary orbits, providing explicit spectral formulas and revealing structural differences from other divergence measures.

Contribution

It determines the exact minimum and maximum of maximal quantum f-divergences over unitary orbits, with explicit spectral characterizations and a reduction to spectral permutation problems.

Findings

Extremal values are achieved by pairing eigenvalues in decreasing order.

The range of divergence is a closed interval between these extremal configurations.

Results extend previous extremal findings for various quantum divergences.

Abstract

Maximal quantum $f$-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum $f$-divergences along unitary orbits of two quantum states. For any operator convex function $f:(0,+\infty)\to\mathbb{R}$, we determine the exact minimum and maximum of $$ U \longmapsto \widehat S_f(\rho\|U^*\sigma U) $$ over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of $\rho$ and $\sigma$, while the maximum corresponds to pairing the decreasing eigenvalues of $\rho$ with the increasing eigenvalues of $\sigma$. Hence, the range of the maximal quantum $f$-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum $f$-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, R\'enyi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum $f$-divergences.