Entanglement groups for mixed states

Abstract

We extend an operational characterization of entanglement in terms of stabilizer groups from pure states to mixed states. For a density matrix $\ddot{I}{}_{AB}$, a stabilizer is a factorized unitary matrix $u_A\otimes u_B$ that, under conjugation, leaves $\ddot{I}{}_{AB}$ 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.

22 pages