Truncated moment sequences and a solution to the channel separability problem

TL;DR

This paper introduces a novel method using truncated moment sequences and semidefinite programming to determine the separability of quantum channels, providing a practical algorithm and numerical insights into 2-qubit and qutrit channels.

Contribution

It presents a new approach linking Choi matrix separability to truncated moment sequences, enabling a certificate-based algorithm for quantum channel separability.

Findings

Algorithm successfully certifies separability in various quantum channels.

Numerical analysis of 2-qubit and qutrit channels demonstrates effectiveness.

Method can resolve cases where negativity criterion is inconclusive.

Abstract

We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamio{\l}kowski isomorphism. We explore three classes of separability across different cuts between systems and ancillae and we provide a solution based on the mapping of the coordinates of the Choi state (in a fixed basis) to a truncated moment sequence (tms) $y$. This results in an algorithm which gives a separability certificate using semidefinite programming. The computational complexity and the performance of it depend on the number of variables $n$ in the tms and on the size of the moment matrix $M_t(y)$ of order $t$. We exploit the algorithm to numerically investigate separability of families of 2-qubit and single-qutrit channels; in the latter case we can provide an answer for examples explored earlier through the criterion based on the negativity $N$, a criterion which remains inconclusive for Choi matrices with $N=0$.