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Physical Review A

Comment on “General relation between the transformation operator and an invariant under stochastic local operations and classical communication in quantum teleportation”

Xiu‐Bo Chen, Qiaoyan Wen, Gang Xu, Yixian Yang, Fu-Chen Zhu

Published Mar 16, 2009• 22 citations
Publisher

Abstract

In this paper, we make a few corrections to a paper by Zha and Ren [Phys. Rev. A 77, 014306 (2008)]. The Zha-Ren protocol for teleportation in principle is equivalent to Rigolin's protocol [G. RigolinPhys. Rev. A 71, 032303 (2005)] and the associated Comment [F. G. DengPhys. Rev. A 72, 036301 (2005)]. We feel that the transformation operator is not well suited as a criterion for the faithful teleportation, but can be used as a means to transform an arbitrary four-qubit entangled state into a tensor product state of two Bell states. We give all the transformation operators.

DOI
10.1103/physreva.79.036301

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In this paper, we make a few corrections to a paper by Zha and Ren [Phys. Rev. A 77, 014306 (2008)]. The Zha-Ren protocol for teleportation in principle is equivalent to Rigolin's protocol [G. RigolinPhys. Rev. A 71, 032303 (2005)] and the associated Comment [F. G. DengPhys. Rev. A 72, 036301 (2005)]. We feel that the transformation operator is not well suited as a criterion for the faithful teleportation, but can be used as a means to transform an arbitrary four-qubit entangled state into a tensor product state of two Bell states. We give all the transformation operators.
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)].
In a recent paper [D. Li, X. Li, H. Huang, and X. Li, Phys. Rev. A 76, 032304 (2007)], Li et al. proposed the definition of the residual entanglement for general nnn qubits by means of the stochastic local operations and classical communication. Here we argue that their definition is not suitable for the case of odd-nnn qubits.
In D. Li, X. Li, H. Huang, and X. Li [Phys. Rev. A 76, 032304 (2007)], we defined the residual entanglement for odd nnn qubits. In this reply, we improve the definition by averaging the original residual entanglement with respect to each qubit, which makes the defined quantity invariant under any permutation of all the qubits.
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.