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Concurrence classes for a general multipartite state

Hoshang Heydari

Published May 14, 2005• 0 citations
OpenAlex

Abstract

In this paper, we will define concurrence classes for a general multipartite state based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we will define concurrence for an arbitrary bipartite state and concurrence classes for the three- and four-partite states. And finally, we will define concurrence for $W^{m}$ and $GHZ^{m}$ classes of a general multipartite state.

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In this paper, we will define concurrence classes for a general multipartite state based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we will define concurrence for an arbitrary bipartite state and concurrence classes for the three- and four-partite states. And finally, we will define concurrence for WmW^{m}Wm and GHZmGHZ^{m}GHZm classes of a general multipartite state.
In this paper, we propose concurrence classes for an arbitrary multi-qubit state based on orthogonal complement of a positive operator valued measure, or POVM for short, on quantum phase. In particular, we construct concurrence for an arbitrary two-qubit state and concurrence classes for the three- and four-qubit states. Finally, we construct W m and GHZ m class concurrences for multi-qubit states. The unique structure of our POVM enables us to distinguish different concurrence classes for multi-qubit states.
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.
We propose a entanglement generating set for a general multipartite state based on the of concurrence. In particular, we show that concurrence for general multipartite states can be constructed by different classes of local operators which are defined by complement of positive operator valued measure on quantum phases. The entanglement generating set consists of different classes of entanglement that are detected by these classes of operators and contributes to the degree of entanglement for a general multipartite state.
We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any NNN-partite systems consisting of NNN qubits.