  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.
