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

Recently, an explicit protocol ${\cal E}_0$ 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 ${\cal E}_0$ 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 ${\cal G}[\Xi]$, 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 ${\cal G}[\Xi] > 1/2$. In addition, we conjecture that when ${\cal G}[\Xi] < 1/4$ then no entanglement can be teleported. It is shown that to determine the usefulness of $\Xi$ for ${\cal E}_0$, it is necessary to analyze ${\cal G}[\Xi]$.