Positive state polynomials

CS/Math seminar - Igor Klep, University of Ljubljana

The talk will discuss state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. The motivation behind this theory arises from the study of correlations in quantum networks. We will give a state analog of Artin's solution to Hilbert's 17th problem showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Further, archimedean Positivstellensätze in the spirit of Putinar and Helton-McCullough are presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués-Pironio-Acín (NPA) scheme for optimization of noncommutative polynomials. 

Based on joint work with Victor Magron, Jurij Volčič, Jie Wang.

QNC 4104 and Zoom

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