Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits
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 -qubit system and its subsystems of dimension .
Cite
@article{arxiv.2601.14158,
title = {Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits},
author = {Pablo Costa Rico and Pavel Shteyner},
journal= {arXiv preprint arXiv:2601.14158},
year = {2026}
}