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arXiv:2601.08268

Optimization of maximal quantum f-divergences between unitary orbits

Hoang Minh Nguyen, Hoang Hieu Nguyen, Công Trình Lê

Published Jan 13, 2026• 0 citations
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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 $$ U \longmapsto \widehat S_f(ρ\|U^*σU) $$ 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 $ρ$ and $σ$, while the maximum corresponds to pairing the decreasing eigenvalues of $ρ$ with the increasing eigenvalues of $σ$. 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, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum $f$-divergences.

DOI
10.48550/arxiv.2601.08268

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Maximal quantum fff-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 fff-divergences along unitary orbits of two quantum states. For any operator convex function f:(0,+∞)→Rf:(0,+\infty)\to\mathbb{R}f:(0,+∞)→R, we determine the exact minimum and maximum of U⟼S^f(ρ∥U∗σU)U \longmapsto \widehat S_f(ρ\|U^*σU)U⟼Sf​(ρ∥U∗σU) 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 ρρρ and σσσ, while the maximum corresponds to pairing the decreasing eigenvalues of ρρρ with the increasing eigenvalues of σσσ. Hence, the range of the maximal quantum fff-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 fff-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum fff-divergences.
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Information Geometry of the Geodesic Quantum fff-Divergences
We study the differential, statistical, and geometrical consequences generated by the geodesic quantum fff-divergences introduced in [14], which are constructed by interpolating the relative modular operator and the commutant Radon-Nikodym derivative using a geodesic with parameter t∈[0,1]t\in [0,1]t∈[0,1]. For an invertible state ρρρ and an operator convex function fff, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric gρ,t(f)g_{ρ,t}^{(f)}gρ,t(f)​. Furthermore, we also compare these metrics with the Petz-Hasegawa metric and find the meaning of the interpolation parameter ttt in this new geometry. We next show that the interpolation of relative modular operators ΓtΓ_tΓt​ in the reference purification of a state σσσ defines a canonical finite binary experiment (pt,qt)(p_t,q_t)(pt​,qt​), and introduce a log-likelihood cumulant function Ψρ,σ(t,s)Ψ_{ρ,σ}(t,s)Ψρ,σ​(t,s), recovering the Nussbaum-Szkoła distributions at t=0t=0t=0 and the Matsumoto construction at t=1t=1t=1. Finally, using Busemann functions, we endow this statistical framework with a geometric meaning in the cone of positive operators.
While many bounds have been proved for partial trace inequalities over the last decades for a large variety of quantities, recent problems in quantum information theory demand sharper bounds. In this work, we study optimal bounds for partial trace quantities in terms of the spectrum; equivalently, we determine the best bounds attainable over unitary orbits of matrices. We solve this question for Schur-convex functionals acting on a single partial trace in terms of eigenvalues for self-adjoint matrices and then we extend these results to singular values of general matrices. We subsequently extend the study to Schur-convex functionals that act on several partial traces simultaneously and present sufficient conditions for sharpness. In cases where closed-form maximizers cannot be identified, we present quadratic programs that yield new computable upper bounds for any Schur-convex functional. We additionally present examples demonstrating improvements over previously known bounds. Finally, we conclude with the study of optimal bounds for an nnn-qubit system and its subsystems of dimension 222.