Refining Ky Fan's majorization relation with linear programming

Understand

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.