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arXiv:1501.00768 · Journal of Physics A Mathematical and Theoretical

Three-qubit entanglement witnesses with the full spanning properties

Seung-Hyeok Kye

Published May 19, 2015• 21 citations
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Abstract

We construct three-qubit entanglement witnesses with the full spanning property, by which we mean that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes (PPT) states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect fully bi-separable states as well as all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to positive multi-linear maps.

DOI
10.1088/1751-8113/48/23/235303

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We construct three-qubit entanglement witnesses with the full spanning property, by which we mean that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes (PPT) states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect fully bi-separable states as well as all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to positive multi-linear maps.
Recently, Qi and Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated with indecomposable positive linear maps, but whether partial transposes of these witnesses are also optimal has not been evaluated. We show that partial transposes of these entanglement witnesses have the spanning property and that these witnesses are indeed optimal witnesses detecting positive-partial- transposed entangled states (nondecomposable optimal entanglement witnesses).
We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decompositions, and show that this is true in most cases if the number of product vectors is sufficiently small. In the three-qubit case, generic five-dimensional spaces give rise to faces of the convex set consisting of all separable states, which are affinely isomorphic to the five-dimensional simplex with six vertices. We construct, as a byproduct, three-qubit entangled positive partial transpose edge states of rank 4 with explicit formulas. This covers those entanglements which cannot be constructed from an unextendible product basis.
Multiqubit positive-partial-transpose (PPT) entangled states play an important role in quantum information theory. We characterize such states of minimum rank in three-qubit system, namely rank four. Depending on whether the Lorentz invariant is zero, we classify such states into two types. The PPT entangled states constructed by unextendible product bases (UPB) have nonzero invariants, which belong to type I. We provide a method to effectively determine whether a state can be constructed from UPB. For states with zero invariant, which belong to type II, we provide an explicit expression up to equivalence of stochastic local operations and classical communications (SLOCC). It turns out that we can represent them with only one complex parameter. We further study SLOCC-equivalence relation within the expression. We also investigate the Lorentz invariants of multiqubit states with rank less than three and analyze their range.
We propose separability criteria for three-qubit states in terms of diagonal and anti-diagonal entries to detect entanglement with positive partial transposes. We report that the phases, that is, the angular parts of anti-diagonal entries, play a crucial role in determining whether a given three-qubit state is separable or entangled, and they must obey even an identity for separability in some cases. These criteria are strong enough to detect PPT (positive partial transpose) entanglement with nonzero volume. In several cases when all the entries are zero except for diagonal and anti-diagonal entries, we characterize separability using phases. These include the cases when anti-diagonal entries of such states share a common magnitude, and when ranks are less than or equal to six. We also compute the lengths of rank six cases and find three-qubit separable states with lengths 8 whose maximum ranks of partial transposes are 7.
Abstract We interpret multi-partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them. We apply this result to multi-qubit X -shaped Hermitian matrices, and characterize the conditions for them to be genuine entanglement witnesses, in terms of entries. Furthermore, we find all optimal entries among them. They turn out to have spanning properties, and so they detect a non-zero volume set of multi-qubit genuine entanglement. We also characterize decomposability for X -shaped entanglement witnesses.