Truncated su(2) moment problem for spin and polarization states

TL;DR

This paper investigates the compatibility of expectation values with spin operator moments in quantum systems, linking it to bipartite qubit state extension problems, and provides approximate solutions applicable to large spins.

Contribution

It introduces a novel connection between the truncated su(2) moment problem and bipartite qubit state extension, offering operational solutions for large spin systems.

Findings

Operational approximate solutions for large spin numbers

Exact solutions in the limit of infinite spin

Efficient semidefinite programming methods for low spins

Abstract

We address the problem whether a given set of expectation values is compatible with the first and second moments of the generic spin operators of a system with total spin j. Those operators appear as the Stokes operator in quantum optics, as well as the total angular momentum operators in the atomic ensemble literature. We link this problem to a particular extension problem for bipartite qubit states; this problem is closely related to the symmetric extension problem that has recently drawn much attention in different contexts of the quantum information literature. We are able to provide operational, approximate solutions for every large spin numbers, and in fact the solution becomes exact in the limiting case of infinite spin numbers. Solutions for low spin numbers are formulated in terms of a hyperplane characterization, similar to entanglement witnesses, that can be efficiently solved with semidefinite programming.