Title: Refining Ky Fan's majorization relation with linear programming
 
Abstract: Quantum information theory is the study of the quantum-physical limits on rates for faithful storage, transmission, and interconversion of information. As such, it lies in the intersection of matrix analysis and information theory. After a brief introduction to the basics of the theory, I will discuss the so-called regularized formulas problem, which refers to the fact that known formulas for optimal information processing rates, such as quantum channel capacities, are generically computationally intractable. Next, I will explain the spin alignment conjecture and how an affirmative resolution of it implies that the quantum capacities of certain quantum channels are readily computable. In one form, the conjecture is that a strong refinement of Ky Fan's majorization relation holds where the summands are restricted to have some specific tensor product structure. The main contribution of this work is in this vein: a proof of a separable version of Ky Fan’s majorization relation for a sum of two operators that are each a tensor product of two positive semi-definite operators.  To prove it, upper bounds are established on 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, that I call 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.

Zoom Information:

https://unm.zoom.us/j/98912959181

Meeting ID: 989 1295 9181
Passcode: 811422