Complex Scalings, Extremal Permanents, and the Geometric Measure of Entanglement

by

George Hutchinson

A Thesis presented to The University of Guelph

In partial fulfilment of requirements for the degree of Doctor of Philosophy in Mathematics

Guelph, Ontario, Canada

c George Hutchinson, April, 2018 ABSTRACT

COMPLEX MATRIX SCALINGS, EXTREMAL PERMANENTS, AND THE

GEOMETRIC MEASURE OF ENTANGLEMENT

George Hutchinson Advisor:

University of Guelph, 2018 Dr. Rajesh Pereira

An n×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

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 dis-

prove 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

3×3 real matrix has at most 6 scalings, and that when n ≥ 4 there exist n×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 con- dition 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 × n matrix with prescribed eigenvalues; and mutually unbiased bases. Dedication

This thesis is dedicated to my family. Without their love, support, guidance, and friendship, this work would not have been possible.

iv Acknowledgements

I would like to thank my advisor, Rajesh Pereira, for his guidance and support throughout my doctoral studies. He has been wonderful to work with, and I consider myself extremely lucky to have had the opportunity to learn from him over the past three and a half years.

I would also like to thank the other members of of my advisory committee, David Kribs and Laurent Marcoux, for their support. Thank you to Allan Willms for his aid improving this thesis, and thank you to my external examiner, Fuzhen Zhang, for his thoughtful com- ments and his thought-provoking questions. As well, I would like to express my gratitude to

Susan McCormick and Carrie Tanti for their aid with administrative matters, and to Larry

Banks for his assistance with all IT-related issues.

I am extremely fortunate to have a wonderful family that has always been there for me.

Thank you Mom