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

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 $\mathcal{G}[\ensuremath{\Xi}]g1∕2$. In addition, we conjecture that when $\mathcal{G}[\ensuremath{\Xi}]l1∕4$, 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 $\mathcal{G}[\ensuremath{\Xi}]$.