Positivity of state, trace, and moment polynomials, and applications in quantum information

TL;DR

This paper surveys the positivity and optimization of state, trace, and moment polynomials within multivariate operator theory, highlighting their applications in quantum information through semidefinite programming hierarchies.

Contribution

It provides a unified framework for understanding positivity of these polynomials, introduces sum of squares certificates, and develops a hierarchy of semidefinite programs for quantum information applications.

Findings

Positivity certificates via sums of squares are established.

A convergent hierarchy of semidefinite programs is developed.

Applications in quantum information theory are demonstrated.

Abstract

State, trace, and moment polynomials are polynomial expressions in several operator or random variables and positive functionals on their products (states, traces or expectations). While these concepts, and in particular their positivity and optimization, arose from problems in quantum information theory, yet they naturally fit under the umbrella of multivariate operator theory. This survey presents state, trace, and moment polynomials in a concise and unified way, and highlights their similarities and differences. The focal point is their positivity and optimization. Sums of squares certificates for unconstrained and constrained positivity (Positivstellens\"atze) are given, and parallels with their commutative and freely noncommutative analogs are discussed. They are used to design a convergent hierarchy of semidefinite programs for optimization of state, trace, and moment polynomials. Finally, circling back to the original motivation behind the derived theory, multiple applications in quantum information theory are outlined.