Optimal information extraction in probabilistic teleportation

In Li 's work on optimal probabilistic teleportation in the two-level case [W. L. Li, C. F. Li, and G. C. Guo, Phys. Rev. A 61, 034301(2000)], the authors consider the extraction of the unknown qubit \phi>=alpha(0)\0>+alpha(1)\1> from the qubit \psi>=(1/rootN)(alpha(0)beta(0)\0>+alpha(1)beta(1)\1>). In this paper, we consider the extraction in the n-level case. It is proved that, under some specific collective unitary transformation U on \psi> and auxiliary qubits as probe, the maximal probability of successfully extracting the wanted qubit is (1/N)min{beta(i)(2)}. It is also shown that the entries of such U are independent of all unknown alpha(i). The result can also be used in the purification of the entanglement via entanglement swapping.