Teleportation with a Mixed State of Four Qubits and the Generalized Singlet Fraction

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. {\bf 96}, 060502 (2006)]. Here, we show that ${\mathcal{E}}_0$ with an arbitrary four-qubit mixed state resource $\Xi$ 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 $\mathcal{G}[\Xi ]$, 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 $\mathcal{G}[\Xi ]>1/2$. In addition, we conjecture that when $\mathcal{G}[\Xi ]<1/4$ then no entanglement can be teleported. It is shown that to determine the usefulness of $\Xi$ for ${\mathcal{E}}_0$, it is necessary to analyze $\mathcal{G}[\Xi ]$.

11 pages

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