Fully undistillable quantum states are separable

TL;DR

This paper characterizes fully undistillable quantum states as separable and explores their properties, showing that such states have separable reductions and linking entanglement distillability to state rank and PPT conditions.

Contribution

It provides a complete characterization of undistillable states in tripartite systems and establishes new bounds on distillable entanglement for low rank bipartite states.

Findings

Undistillable states have separable reductions on subsystems.

NPT bipartite states are either distillable or have a distillable complement.

Random low rank states are almost surely 1-way distillable.

Abstract

Assume that Alice, Bob, and Charlie share a tripartite pure state $|\psi_{ABC}\rangle$. We prove that if Alice cannot distill entanglement with either Bob or Charlie using $|\psi_{ABC}\rangle$ and local operations with any one of the following configurations for classical communication: $(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C),$ and $(A\leftrightarrow B, A\leftrightarrow C)$, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems $AB$ and $AC$ are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. To prove these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.