Entanglement groups for mixed states

TL;DR

This paper generalizes an operational approach to quantify entanglement in mixed quantum states using stabilizer groups, linking it to the entanglement of purifications and exploring properties for separable states.

Contribution

It introduces a new definition of entanglement groups for mixed states based on stabilizer invariance, extending previous pure state methods and analyzing their properties.

Findings

Entanglement groups relate to the entanglement of purifications.

Separable states can have non-trivial entanglement groups.

Entanglement groups for separable states originate from multi-party entanglement with the environment.

Abstract

We extend an operational characterization of entanglement in terms of stabilizer groups from pure states to mixed states. For a density matrix $\rho_{AB}$, a stabilizer is a factorized unitary matrix $u_A \otimes u_B$ that, under conjugation, leaves $\rho_{AB}$ invariant. The entanglement group is a quotient of the stabilizer group, in which one-party stabilizers are considered trivial. This definition relates the entanglement of a density matrix to the entanglement of its purification. We give general properties of entanglement groups for mixed states, then discuss special properties for separable states. For a separable state, the entanglement group may be non-trivial. However it can only arise from multi-party entanglement with the purifying system.