A note on the realignment criterion

TL;DR

This paper investigates the realignment criterion for quantum state separability, providing bounds on singular values, addressing open problems, and showing limitations of existing testing schemes for equal-dimension subsystems.

Contribution

It derives bounds for symmetric functions of singular values of realignment matrices, resolving open questions and demonstrating the failure of certain separability tests in specific cases.

Findings

Bounds for elementary symmetric functions of singular values are established.

The proposed separability testing scheme fails when subsystems have equal dimensions.

Answers to open problems regarding the realignment criterion are provided.

Abstract

For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension.