Refining Ky Fan's majorization relation with linear programming

TL;DR

This paper introduces a linear programming approach to refine Ky Fan's majorization relation for tensor products of positive semi-definite operators, leading to new bounds and an application to quantum information theory.

Contribution

It develops a separable version of Ky Fan's relation using linear programming and geometric bounds, and applies it to resolve the spin alignment conjecture at the 2-letter level.

Findings

Established tight upper bounds on eigenvalue overlaps.

Proved the spin alignment conjecture at the 2-letter level.

Showed additivity of coherent information for platypus channels.

Abstract

A separable version of Ky Fan's majorization relation is proven for a sum of two operators that are each a tensor product of two positive semi-definite operators. In order to prove it, upper bounds are established for the relevant largest eigenvalue sums in terms of the optimal values of certain linear programs. The objective function of these linear programs is the dual of the direct sum of the spectra of the summands. The feasible sets are bounded polyhedra determined by positive numbers, called alignment terms, that quantify the overlaps between pairs of largest eigenvalue spaces of the summands. By appealing to geometric considerations, tight upper bounds are established on the alignment terms of tensor products of positive semi-definite operators. As an application, the spin alignment conjecture in quantum information theory is affirmatively resolved to the 2-letter level. Consequently, the coherent information of platypus channels is additive to the 2-letter level.