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Unextendible product bases and locally unconvertible bound entangled states

Sergey Bravyi
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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 $ρ_T$ can not be converted to each other by LOCC, unless $S$ and $T$ coincide up to local unitaries. More specifically, there exists a finite precision $ε>0$ such that for any LOCC protocol mapping $ρ_S$ into a probabilistic ensemble $(p_j,ρ_j)$, the fidelity between $ρ_T$ and any possible final state $ρ_j$ is smaller than $1-ε$.

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Sergey Bravyi. 2003-10-29. Unextendible product bases and locally unconvertible bound entangled states. https://arxiv.org/abs/quant-ph/0310172

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