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arXiv:quant-ph/0607108 · Physical Review A

Teleportation with a mixed state of four qubits and the generalized singlet fraction

Ye Yeo

Published Nov 6, 2006• 36 citations
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Abstract

Recently, an explicit protocol ${\mathcal{E}}_{0}$ for faithfully teleporting arbitrary two-qubit states using genuine four-qubit entangled states was presented by us [Phys. Rev. Lett. 96, 060502 (2006)]. Here, we show that ${\mathcal{E}}_{0}$ with an arbitrary four-qubit mixed-state resource $\ensuremath{\Xi}$ is equivalent to a generalized depolarizing bichannel with probabilities given by the maximally entangled components of the resource. These are defined in terms of our four-qubit entangled states. We define the generalized singlet fraction $\mathcal{G}[\ensuremath{\Xi}]$ and illustrate its physical significance with several examples. We argue that in order to teleport arbitrary two-qubit states with average fidelity better than is classically possible, we have to demand that $\mathcal{G}[\ensuremath{\Xi}]>1∕2$. In addition, we conjecture that when $\mathcal{G}[\ensuremath{\Xi}]<1∕4$, then no entanglement could be teleported. It is shown that to determine the usefulness of $\ensuremath{\Xi}$ for ${\mathcal{E}}_{0}$, it is necessary to analyze $\mathcal{G}[\ensuremath{\Xi}]$.

DOI
10.1103/physreva.74.052305

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Recently, an explicit protocol E0{\mathcal{E}}_{0}E0​ for faithfully teleporting arbitrary two-qubit states using genuine four-qubit entangled states was presented by us [Phys. Rev. Lett. 96, 060502 (2006)]. Here, we show that E0{\mathcal{E}}_{0}E0​ with an arbitrary four-qubit mixed-state resource \ensuremathΞ\ensuremath{\Xi}\ensuremathΞ is equivalent to a generalized depolarizing bichannel with probabilities given by the maximally entangled components of the resource. These are defined in terms of our four-qubit entangled states. We define the generalized singlet fraction G[\ensuremathΞ]\mathcal{G}[\ensuremath{\Xi}]G[\ensuremathΞ] and illustrate its physical significance with several examples. We argue that in order to teleport arbitrary two-qubit states with average fidelity better than is classically possible, we have to demand that G[\ensuremathΞ]>1∕2\mathcal{G}[\ensuremath{\Xi}]>1∕2G[\ensuremathΞ]>1∕2. In addition, we conjecture that when G[\ensuremathΞ]<1∕4\mathcal{G}[\ensuremath{\Xi}]<1∕4G[\ensuremathΞ]<1∕4, then no entanglement could be teleported. It is shown that to determine the usefulness of \ensuremathΞ\ensuremath{\Xi}\ensuremathΞ for E0{\mathcal{E}}_{0}E0​, it is necessary to analyze G[\ensuremathΞ]\mathcal{G}[\ensuremath{\Xi}]G[\ensuremathΞ].
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.
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.
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 present an explicit protocol E0 for faithfully teleporting an arbitrary two-qubit state via a genuine four-qubit entangled state. By construction, our four-partite state is not reducible to a pair of Bell states. Its properties are compared and contrasted with those of the four-party Greenberger-Horne-Zeilinger and W states. We also give a dense coding scheme D0 involving our state as a shared resource of entanglement. Both D0 and E0 indicate that our four-qubit state is a likely candidate for the genuine four-partite analogue to a Bell state.
We show that standard teleportation with an arbitrary mixed state resource is equivalent to a generalized depolarizing channel with probabilities given by the maximally entangled components of the resource. This enables the usage of any quantum channel as a generalized depolarizing channel without additional twirling operations. It also provides a nontrivial upper bound on the entanglement of a class of mixed states. Our result allows a consistent and statistically motivated quantification of teleportation success in terms of the relative entropy and this quantification can be related to a classical capacity.
Teleportation for pure states, mixed states with standard and optimal protocols are introduced and investigated systematically. An explicit equation governing the teleportation of finite dimensional quantum pure states by a generally given non-local entangled state is presented. For the teleportation of a mixed state with an arbitrary mixed state resource, an explicit expression is obtained for the quantum channel associated with the standard teleportation protocol. The corresponding transmission fidelity is calculated. It is shown that the standard teleportation protocol is not optimal. The optimal quantum teleportation is further studied, its fidelity is given and shown to be related to the fully entangled fraction of the quantum resource, rather than the single fraction as in the standard teleportation protocol.