The typicality of symmetry-induced entanglement

TL;DR

This paper investigates the symmetric separability problem under charge conservation, showing most symmetric separable states are far from symmetric separability, with implications for quantum tasks and entanglement complexity.

Contribution

It introduces the use of number entanglement as a witness to quantify the distance from symmetric separability in charge-conserving quantum states.

Findings

Most symmetric and separable states are far from being symmetrically separable.

Number entanglement concentrates Gaussian around a positive mean.

Almost all states violate symmetric separability in high dimensions.

Abstract

In the presence of a globally conserved charge $N$, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all $N$. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by $N$. We also touch upon the question of the complexity of SSP, and multiparty entanglement.