Unextendible Product Bases and Locally Unconvertible Bound Entangled States

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 rho(S) and rho(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 epsilon(S, T) > 0 such that for any LOCC protocol mapping rho(S) into a probabilistic ensemble (p(alpha) , rho(alpha)), the fidelity between rho(T) and any possible final state rho(alpha) satisfies F(rho(T) , rho(alpha))<= 1 - epsilon(S, T).