Elsevier

Physics Letters A

Volume 314, Issue 4, 4 August 2003, Pages 267-271
Physics Letters A

Can the states of the W-class be suitable for teleportation?

https://doi.org/10.1016/S0375-9601(03)00906-XGet rights and content

Abstract

Entangled states of the W-class are considered as a quantum channel for teleportation of an entangled state and as well the state to be teleported via a multiparticle quantum channel. Using an introduced unitary transformation in the teleportation schemes based on the multiparticle Greenberger–Horne–Zeilinger channel it is found a set of protocols main feature of which is a collection of non-local recovering operators.

Introduction

Recently along with Einstein–Podolsky–Rosen (EPR) and Greenberger–Horne–Zeilinger (GHZ) states a collection of new entangled states, particularly such as the W-class [1], the entangled states called zero sum amplitude (ZSA) [2] and a set of four-particle entangled states [3] has been proposed. Although the features of these entangled states are quite various each of them is attractive for the quantum information processing. For example, it is shown, that ZSA-states are useful in the quantum web for preparing entanglement of an unknown qubit with two types of reference states [2]. Now, as result we have a set of multiparticle entangled states and one of the main question is what informational tasks could be achieved using a given entangled state. Recently Lee et al. [4] has considered W-states for secure communication. Teleportation an entangled state of EPR-type via a multiparticle GHZ quantum channel is discussed in [5]. As to the state of the W-class and GHZ-class as a quantum channel, it is well known that these states are rather different with respect to loss of qubits and the W-channel looks more attractive because of robustness. The aim of the Letter is the investigation of teleportation of states W-class via a multipartical quantum channel and detailing of a protocol for teleportation of entangled state via the W-state quantum channel. In fact, it can be done by different ways, particularly using the GHZ channel, shared with a sender and receivers [5], [6], [7]. A start point of this investigation is the GHZ protocol introduced in [5] which, as it is shown here, can be transformed to the desired case by unitary operations. The transformation involves both a channel and a state to be teleported. It is well known that the states of the GHZ- and W-class cannot be converted from one to another by local operations and classical communication (LOCC) [1]. This fact it results in the main feature of the W-class channel, when receivers can recover an unknown teleported state by non-local operations only. Because these operations do not involve the unknown state the task is accomplished. Recently, considering teleportation of a two-qubit state via four-particle entanglement, it was found by Lee et al. [8] that the recovering operations can be non-local. It is worth paying attention that not any state from the GHZ-class is suitable as the quantum channel for teleportation of the W states. The reason, it needs the states which can be converted from the GHZ triplet by two-particle unitary operation only. But it is impossible for all members of the GHZ-class. Also it is true for the W-class and one finds a particular set of the states to be useful.

The Letter is organized as follows. First, we introduce two transformations of the GHZ protocols that result in a new channel and a new measurement. Then a set of states from the W-class are considered as the quantum channels, next we establish new type of entangled states which can be teleported using the GHZ channel and finally, discuss an optical implementation of the W-states.

Section snippets

Two types of transformations

We start from the GHZ channel represented by the three-qubit entangled state of the form |GHZ〉=12|000〉+|111〉ABC. The teleportation protocol using (1) as a quantum channel allows transmitting a two-qubit entangled state like EPR pair |A〉=α|01〉+β|10〉12, where |α|2+|β|2=1. The scheme includes five particles in the total state, that is a product |A12⊗|GHZ〉ABC, where particles A, B and C of the channel are shared with Alice and two receivers Bob and Claire, spatially separated. If Alice decided

The channel of the W-class

Consider a particular case of the transformation T, given by (5), that converts the GHZ state into a state of the W-class. It follows from the result of Cirac et al. [1], that the operator T is non-local. Now we are interested in two states from the W-class only, say of the form |W〉=13|100〉+|010〉+|001〉,|W̃〉=12|100〉+|0Ψ+.

Introduce the unitary non-local operator V V=|Ψ+〉〈00|+|11〉〈01|+|Ψ〉〈10|+|00〉〈11|. It can be represented by a network, including the conditional gates V=C12(CH)21σx2C21, where C

The states teleported via the GHZ channel

In accordance with (7) and (8) a unitary transformation R allows to investigate what kind of states could be teleported using the GHZ channel.

Let R=V, for example. Then V|A〉=α|11〉+β|Ψ〉. To teleport this entangled state, it needs a new three-particle measurement in the basis, obtained from (3), when +Φ±〉↔1/2(|010〉±|Φ±〉|1〉) and so on. The found set is represented by the states of the W-class.

Consider the GHZ channel of four qubits |GHZ〉=12|0000〉+|1111〉ABCD. Let particles A, B, C and D share a

Conclusion

Being attractive because of its robustness with respect to losses of particles it is shown in the Letter the W-states can be used for the teleportation of GHZ states as the quantum channel and to be teleported itself via the GHZ channel. In the same time the states from the W-class are different under the unitary two-particle transformations and only some of them can be converted from one to another by this way. We find also the desired collection of states using unitary transformation of the

Acknowledgements

We are grateful to Alexander Yu. Vlasov for discussion. This work was supported in part by the Delzell Foundation Inc. and INTAS Grant 00-479.

References (10)

  • L. Marinatto et al.

    Found. Phys. Lett.

    (2000)
  • W. Dur et al.

    Phys. Rev. A

    (2000)
  • A.K. Pati
  • F. Verstraete et al.
  • J. Joo et al.
There are more references available in the full text version of this article.

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Present address: Simon Fraser University, Surrey, BC Canada.

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