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

TL;DR

This paper derives optimal bounds for Schur-convex functionals of partial traces over unitary orbits, improving quantum information bounds and providing computational methods for complex systems.

Contribution

It introduces new sharp bounds for partial trace functionals in quantum systems, extending to multiple traces and offering computational tools for non-closed form cases.

Findings

Derived optimal bounds for spectral and singular value functionals

Extended bounds to multiple partial traces with conditions for sharpness

Provided quadratic programs for computable upper bounds in complex cases

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 $2$.