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Physical Review A

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

Xin-Wei Zha, Kuan Fang Ren

Published Jan 31, 2008• 42 citations
Publisher

Abstract

In this paper, we introduce a transformation operator expression to give a criterion for faithful teleportation of an arbitrary two-qubit state via a four-qubit entangled state. The general relation between the transformation operator and the SLOCC invariant of quantum channel is obtained and its applicability is illuminated by some known examples for the measurement in the Bell basis.

DOI
10.1103/physreva.77.014306

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

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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.

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It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.

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

Xin-Wei Zha, Kuan Fang Ren

We agree with the correction of the index made by Chen et al. [previous Comment, Phys. Rev. A 79, 036301 (2009)]. We reconfirm that our protocol can be applied to multipartite quantum channels and nonsymmetric quantum channels and that Rogolin's protocol is only a special case of ours.

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The relation of genuine multiqubit entanglement and controlled teleportation

Xin-Wei Zha, Haiyang Song

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.

arXiv:0911.3949•Nov 20, 2009•0 citations
In this paper, we introduce a transformation operator expression to give a criterion for faithful teleportation of an arbitrary two-qubit state via a four-qubit entangled state. The general relation between the transformation operator and the SLOCC invariant of quantum channel is obtained and its applicability is illuminated by some known examples for the measurement in the Bell basis.
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.
A criterion for he tquantum teleportation of an arbitrary N -particle state via a 2 N -particle quantum channel is presented by introducing a term of the "judgment operator". Using the criterion, not only the qualitative judgment of the possibility of successful teleportation can be made but also the quantitative calculation of the probability of successful teleportation can be explicitly given. In addition, a new genuine four-qubit entangled state is proposed, which could not belong to the category of previously known states under stochastic local operations and classical communication.
We find an invariant for nnn qubits and propose residual entanglement for nnn qubits by means of the invariant. Thus, we establish a relation between stochastic local operations and classical communication (SLOCC) entanglement and residual entanglement. Invariant and residual entanglement can be used for SLOCC entanglement classification of nnn qubits.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
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.