A certified exact-arithmetic counterexample shows the stationary quantum mechanical bootstrap at level three is not tight in two dimensions, while eigenstate constraints seem to close the gap.
Sums of hermitian squares and the BMV conjecture
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
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.
citation-role summary
background 1
citation-polarity summary
fields
cs.SC 1years
2026 1verdicts
ACCEPT 1roles
background 1polarities
background 1representative citing papers
citing papers explorer
-
Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone
A certified exact-arithmetic counterexample shows the stationary quantum mechanical bootstrap at level three is not tight in two dimensions, while eigenstate constraints seem to close the gap.