Sums of hermitian squares and the BMV conjecture

TL;DR

This paper proves that the BMV conjecture holds for words of length up to 13, extending previous results, by connecting it to sums of hermitian squares and semidefinite programming.

Contribution

It establishes the validity of the BMV conjecture for longer words and introduces a novel approach using sums of hermitian squares and semidefinite programming.

Findings

BMV conjecture verified for words of length ≤ 13

Connection made between hermitian squares and the conjecture

Example of a polynomial with nonnegative trace but not sum of hermitian squares

Abstract

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.