Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Abstract

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states ρ _S and ρ _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision ε (S,T) > 0 such that for any LOCC protocol mapping ρ _S into a probabilistic ensemble (p_α, ρ_α), the fidelity between ρ _T and any possible final state ρ_α satisfies F(ρ _T , ρ_α) = 1 - ε(S,T).

PACS: 03.65.Bz; 03.67.-a; 89.70+c.