PhD Math Defence: Complex Matrix Scalings, Extremal Permanents, and the Geometric Measure of Entanglement

CANDIDATE:   GEORGE HUTCHINSON

ABSTRACT:

An n x n matrix with complex entries is said to be doubly quasi-stochastic (DQS) if all row and column sums are equal to one. Given a positive definite matrix A and a diagonal matrix D, we say that D*AD is a (complex matrix) scaling of A if D*AD is doubly quasi-stochastic. Motivated by a result of Pereira and Boneng concerning the application of complex matrix scalings to the geometric measure of entanglement of certain symmetric states, we embark upon an investigation of these scalings and their properties.

We begin with an existence theorem that unifies complex matrix scalings together with other classical notions of scaling (Sinkhorn scaling, Marshall and Olkin's scaling of copositive real matrices, etc.). We then discuss the notion of double quasi-stochasticity as it pertains to tensors (ie. hypermatrices) and we extend some classical scaling results to these higher-order objects.

Returning to the study of complex matrix scalings of positive definite matrices, we disprove a conjecture of Pereira and Boneng concerning the cardinality of the set of complex matrix scalings for a given positive definite matrix A. In particular, we show that a given 3 x 3 real matrix has at most 6 scalings, and that when n > 3 there exist n x n matrices with infinitely many scalings. Using the application to quantum entanglement as motivation, we consider complex matrix scalings of extremal permanent; that is, given positive definite A, we investigate the scaling(s) of A that have maximal (minimal) permanent. We prove that scalings with maximal permanent satisfy a certain optimization condition and use this condition to derive topological properties of this set as well as a lower bound on the permanent of these "maximal scalings". We also arrive at an upper bound on the permanent of certain "minimal scalings" and use these results to bound the geometric measure of entanglement of symmetric states that satisfy certain conditions.

We close with a discussion of possible future work and open problems in the study of matrix scalings, including: a scaling algorithm that builds on the work of O'Leary; the permanent conjecture of Chollet and Drury; the problem of maximizing the permanent of an n x n matrix with prescribed eigenvalues; and mutually unbiased bases.