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Physical Review A

General form of genuine multipartite entanglement quantum channels for teleportation

Ping-Xing Chen, Shi-Yao Zhu, Guang‐Can Guo

Published Sep 19, 2006• 125 citations
Publisher

Abstract

Recently Yeo and Chua [Phys. Rev. Lett. 96, 060502 (2006)] presented an explicit protocol for faithfully teleporting an arbitrary two-qubit state via a genuine four-qubit entanglement channel. Here we generalize completely their results to teleporting an arbitrary $N$-qubit state via genuine $N$-qubit entanglement channels. And we present the general form of the genuine multipartite entanglement channels, namely, the sufficient and necessary condition the genuine $N$-qubit entanglement channels must satisfy to teleport an arbitrary $N$-qubit state.

DOI
10.1103/physreva.74.032324

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Physical Review A•Sep 14, 2005•50 citations
Recently Yeo and Chua [Phys. Rev. Lett. 96, 060502 (2006)] presented an explicit protocol for faithfully teleporting an arbitrary two-qubit state via a genuine four-qubit entanglement channel. Here we generalize completely their results to teleporting an arbitrary NNN-qubit state via genuine NNN-qubit entanglement channels. And we present the general form of the genuine multipartite entanglement channels, namely, the sufficient and necessary condition the genuine NNN-qubit entanglement channels must satisfy to teleport an arbitrary NNN-qubit state.
In this letter we present an explicit protocol calB2{\\cal B}_2calB2​ for faithfully teleporting an arbitrary 2-qubit state with a class of 4-qubit entangled states. We construct these states in a way different from that of Yeo and Chua [Phys. Rev. Lett. 94, 060502 (2006)]. We will show that the genuine states presented by Yeo and Chua are only a subset of our entangled states. The generalizations of the protocol calB2{\\cal B}_2calB2​ to general protocols calBn{\\cal B}_ncalBn​ and calDn{\\cal D}_ncalDn​, i.e., teleporting an arbitrary nnn-qubit or nnn-qudit state by using the genuine 2n2n2n-qubit or 2n2n2n-qudit entangled states we construct, are also sketched.
We construct a genuine (2N+1)(2N+1)(2N+1)-qubit entangled state to perform controlled teleportation of an arbitrary NNN-qubit state. The constructed state is a complementarity to the genuine 2N2N2N-qubit entangled state constructed by Yeo and Chua for N=2N=2N=2 [Phys. Rev. Lett. 96, 060502 (2006)] and by Chen, Zhu, and Guo for any NNN [Phys. Rev. A 74, 032324 (2006)]. We also quantify the entanglement of the state and classify it with the well-known GHZGHZGHZ and WWW states by means of the recently proposed generalized global entanglement and the associated auxiliary measures [Phys. Rev. A 74, 022314 (2006)]. Our study is of general importance with respect to exploring and exploiting the genuine multiqubit entanglement.
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.
We explicitly show a protocol in which an arbitrary two qubit state $\ensuremath{\mid}\ensuremath{\phi}⟩=a\ensuremath{\mid}00⟩+b\ensuremath{\mid}01⟩+c\ensuremath{\mid}10⟩+d\ensuremath{\mid}11⟩$ is faithfully and deterministically teleported from Alice to Bob. We construct the 16 orthogonal generalized Bell states that can be used to teleport the two qubits. The local operations Bob must perform on his qubits in order to recover the teleported state are also constructed. They are restricted only to single-qubit gates. This means that a controlled-NOT gate is not necessary to complete the protocol. A generalization where NNN qubits are teleported is also shown. We define a generalized magic basis, which possesses interesting properties. These properties help us to suggest a generalized concurrence from which we construct a measure of entanglement that has a clear physical interpretation: A multipartite state has maximum entanglement if it is a genuine quantum teleportation channel.
We propose a protocol Dn{\cal D}_nDn​ for faithfully teleporting an arbitrary nnn-qudit state with the tensor product state (TPS) of nnn generalized Bell states (GBSs) as the quantum channel. We also put forward explicit protocol Dn′{\cal D}'_nDn′​ and Dn′′{\cal D}''_nDn′′​ for faithfully teleporting an arbitrary nnn-qudit state with two classes of 2n2n2n-qudit GESs as the quantum channel, where the GESs are a kind of genuine entangled states we construct and can not be reducible to the TPS of nnn GBSs.
The multipartite state in the Rigolin's protocol [Phys. Rev. A 71, 032303 (2005)] for teleporting an arbitrary two-qubit state is just a product state of NNN Einstein-Podolsky-Rosen pairs in essence, not a genuine multipartite entangled state, and this protocol in principle is equivalent to the Yang-Guo protocol [Chin. Phys. Lett. 17, 162 (2000)].