Newton's identities and positivity of trace class integral operators

TL;DR

This paper introduces a set of necessary and sufficient conditions for the positivity of trace class integral operators using elementary symmetric polynomials, along with an efficient algorithm based on Newton's identities, improving sensitivity over traditional tests.

Contribution

It provides a novel, complete characterization of positivity for trace class integral operators and an efficient computational method, advancing quantum phase-space analysis.

Findings

Conditions are necessary and sufficient for positivity.

The new algorithm is more sensitive than existing tests.

The first condition aligns with the non-negativity of linear entropy.

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-Schr\"odinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.