Sums of hermitian squares and the BMV conjecture

Details Sums of hermitian squares and the BMV conjecture Sums of hermitian squares and the BMV conjecture

2007-10-04

arxiv.org/abs/0710.1074v4

J. Stat. Phys. 133 (2008), no. 4, 739-760

Mathematics

Operator Algebras

21 pages; minor changes; a companion Mathematica notebook is now available in the source file

Scientific paper

10.1007/s10955-008-9632-x

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.

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