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Quantum Information and Computation

Construction of entanglement witnesses based on concurrence for multi-qubit states

Hoshang Heydari

Published Sep 1, 2008• 1 citation
Publisher

Abstract

We establish a relation between concurrence and entanglement witnesses. In particular, we construct entanglement witnesses for three-qubit W and GHZ states in terms of concurrence. We also generalize our construction for multi-qubit states. This result gives the finest entanglement witnesses for a given class of state without any optimization.

DOI
10.26421/qic8.8-9-8

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We establish a relation between concurrence and entanglement witnesses. In particular, we construct entanglement witnesses for three-qubit W and GHZ states in terms of concurrence. We also generalize our construction for multi-qubit states. This result gives the finest entanglement witnesses for a given class of state without any optimization.
We construct quantum gate entanglers for different classes of multipartite states based on definition of W and GHZ concurrence classes. First, we review the basic construction of concurrence classes based on orthogonal complement of a positive operator valued measure (POVM) on quantum phase. Then, we construct quantum gates entanglers for different classes of multi-qubit states. In particular, we show that these operators can entangle multipartite state if they satisfy some conditions for W and GHZ classes of states. Finally, we explicitly give the W class and GHZ classes of quantum gate entanglers for four-qubit states.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.
We construct an important missing piece in the entanglement theory of pure three-qubit states, which is a polynomial measure of W-entanglement. This works in parallel to the three-tangle, which is a polynomial measure of GHZ-entanglement, and to the bipartite concurrence, which is a polynomial measure of bipartite entanglement. We also show that these entanglement measures are ordered, the bipartite measure is larger than the W measure, which is larger than the GHZ measure. It is meaningful then to consider these three types of three-qubit entanglement, which are also ordered, bipartite is weaker than W, which is weaker than GHZ, in parallel to the order of the three kinds of equivalence classes of entangled three-qubit states.