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Positive maps for bound entangled states based on unextendible product bases

Barbara M. Terhal

Published Oct 29, 1998• 1 citation
OpenAlex

Abstract

In this note I show how to construct positive maps from any bound entangled state based on an unextendible product basis. These positive maps exhibit the inseparability of the corresponding bound entangled states. I illustrate the construction with an example.

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International Journal of Modern Physics B•Oct 14, 2022•0 citations
An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We exhibit a tripartite $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e., all three corresponding $2\ifmmode\times\else\texttimes\fi{}4$ bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local positive operator valued measurements and classical communication.
Abstract Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging from quantum nonlocality to quantum cryptography. A generalized concept called uncompletable product basis also attracts much attention. In this paper, we find some unextendible product bases that are uncompletable product bases in every bipartition, which answers a 19 year-old open question proposed by DiVincenzo et al (2003 Commun. Math. Phys. 238 379–410). As a consequence, we connect such unextendible product bases to local hiding of information, positive-partial-transpose entangled states and genuinely entangled states. Furthermore, we give a sufficient condition for the existence of an unextendible product basis that is still unextendible in every bipartition, and the existence of such a UPB is another open question proposed by Demianowic et al (2018 Phys. Rev. A 98 012313). Our results advance the understanding of the geometry of unextendible product bases.
The multiqubit bound entangled states that are uniformly mixed over the subspaces complementary to unextendible product bases (UPBs) are interesting members from the peculiar family of bound entangled states. Their entanglement is delocalized and they look separable with respect to any partition into a single party and all remaining parties. Only one example of quantifying the entanglement of this type of states in the case of three-qubit states has been studied [C. Branciard, H. Zhu, L. Chen, and V. Scarani, Evaluation of two different entanglement measures on a bound entangled state, Phys. Rev. A 82, 012327 (2010)]. We provide a systematic procedure for generating decompositions of such states in terms of minimally entangled states. This procedure enables us to measure entanglement in terms of the concurrence for a wide range of families of these bound entangled (BE) states with each family containing an infinite number of states. In particular, we study one family of three-qubit states in addition to six inequivalent families of four-qubit bound entangled states. These studied families are examples that represent all the known 1446 families of four-qubit UPB-based bound entangled states, which means that our method is applicable to all the 1446 families. In addition, a discussion on the matter of applicability of the proposed strategy to additional families is provided. Specifically, the proposed method looks promising for measuring entanglement of other families of UPB-based BE states in addition to some families of symmetric multipartite positive under partial transposition BE states.