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

Abstract Recently, an explicit protocol 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 with an arbitrary four-qubit mixed state resource Ξ 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[Ξ], 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[Ξ] > 1/2. In addition, we conjecture that when G[Ξ] < 1/4 then no entanglement can be teleported. It is shown that to determine the usefulness of Ξ for E0, it is necessary to analyze G[Ξ].