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arXiv:1006.1152 · Physical Review A

Evaluation of two different entanglement measures on a bound entangled state

Cyril Branciard, Huangjun Zhu, Lin Chen, Valerio Scarani

Published Jul 23, 2010• 9 citations
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Abstract

We consider the mixed three-qubit bound entangled state defined as the normalized projector on the subspace that is complementary to an unextendible product basis [C. H. Bennett et al., Phys. Rev. Lett. 82, 5385 (1999)]. Using the fact that no product state lies in the support of that state, we compute its entanglement by providing a basis of its subspace formed by ``minimally entangled'' states. The approach is in principle applicable to any entanglement measure; here we provide explicit values for both the geometric measure of entanglement and a generalized concurrence.

DOI
10.1103/physreva.82.012327

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We consider the mixed three-qubit bound entangled state defined as the normalized projector on the subspace that is complementary to an unextendible product basis [C. H. Bennett et al., Phys. Rev. Lett. 82, 5385 (1999)]. Using the fact that no product state lies in the support of that state, we compute its entanglement by providing a basis of its subspace formed by ``minimally entangled'' states. The approach is in principle applicable to any entanglement measure; here we provide explicit values for both the geometric measure of entanglement and a generalized concurrence.
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.
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.
We give an explicit expression for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space. Each expression of the measure has its own geometrically meaningful interpretation. Such an interpretation allows oneself to take one step toward a complete understanding for the general properties of the entanglement measure. The states that lie on joint surfaces separating different ranges of definition, designated as shared states, seem to have particularly interesting features. The properties of the shared states are fully discussed.
We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general unextendible product bases, nevertheless, they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's theorem from algebraic geometry.