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

Recently, an explicit protocol 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 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 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 G[Xi]>1/2. In addition, we conjecture that when G[Xi]< 1/4, then no entanglement could be teleported. It is shown that to determine the usefulness of Xi for E-0, it is necessary to analyze G[Xi].