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arXiv · 2601.14158

Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits

Pablo Costa Rico·Pavel Shteyner
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Abstract

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 $n$-qubit system and its subsystems of dimension $2$.

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Pablo Costa Rico, Pavel Shteyner. 2026-08-25. Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits. https://arxiv.org/abs/2601.14158

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