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Chinese Physics Letters

Criterion for Genuine Multipartite Entanglement Quantum Channels

Nian-Quan Jiang, Wangyu-Jian

Published Jan 1, 2010• 9 citations
Publisher

Abstract

We introduce a character matrix for the N -qubit subsystem of a 2 N -qubit state and show the criterion for genuine entanglement channel existing between two N -qubit subsystems in the state. The criterion allows us to check conveniently whether genuine quantum channels exist or not in the 2 N -qubit state without calculating its N -qubit reduced density matrices.

DOI
10.1088/0256-307x/27/1/010302

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We introduce a character matrix for the N -qubit subsystem of a 2 N -qubit state and show the criterion for genuine entanglement channel existing between two N -qubit subsystems in the state. The criterion allows us to check conveniently whether genuine quantum channels exist or not in the 2 N -qubit state without calculating its N -qubit reduced density matrices.
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Beyond the simplest case of bipartite qubits, the composite Hilbert space of multipartite systems is largely unexplored. In order to explore such systems, it is important to derive analytic expressions for parameters which characterize the system's state space. Two such parameters are the degree of genuine multipartite entanglement and the degree of mixedness of the system's state. We explore these two parameters for an N-qubit system whose density matrix has an X form. We derive the class of states that has the maximum amount of genuine multipartite entanglement for a given amount of mixedness. We compare our results with the existing results for the N = 2 case. The critical amount of mixedness above which no N-qubit X-state possesses genuine multipartite entanglement is derived. It is found that as N increases, states with higher mixedness can still be entangled.
Recently, Paolo Facchi et al. [Phys. Rev. A. 77, 060304 (R) (2008)] introduced the notion of maximally multipartite entangled states of n qubits. Here, we give a criterion for faithful controlled teleportation of an arbitrary two-qubit state via a five-qubit entangled state and obtain the general relation between the genuine five-qubit entangled state and controlled teleportation. This criterion can be extended to teleportation of an arbitrary N-qubit state using 2N+1-qubit entangled state. Furthermore, we study the optimal match of measuring basis and quantum channel.
In this paper, an intuitive mathematical formulation is provided to generalize the residual entanglement for tripartite systems of qubits (Phys. Rev. A \textbf{61}, 052306 (2000)) to the tripartite systems in higher dimension. The spirit lies in the tensor treatment of tripartite pure states (Phys. Rev. A \textbf{72}, 022333 (2005)). A distinct characteristic of the present generalization is that the formulation for higher dimensional systems is invariant under permutation of the subsystems, hence is employed as a criterion to test the existence of genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the kronecker product approximate technique. As applications, we give the analytic approximation of the criterion for weakly mixed tripartite quantum states and consider the existence of genuine tripartite entanglement of some weakly mixed states.