Newton’s identities and positivity of trace class integral operators

Homa, Gábor and Balka, Richárd and Bernád, J.Z. and Károly, M. and Csordás, András (2023) Newton’s identities and positivity of trace class integral operators. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 56 (14). ISSN 1751-8113

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Abstract

We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton’s identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger’s uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.

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