Optimal Estimate of a Pure Qubit State from Uhlmann–Josza Fidelity

Abstract

In the framework of collective measurements, efforts have been made to reconstruct one-qubit states. Such schemes find an obstacle in the no-cloning theorem, which prevents full reconstruction of a quantum state. Quantum Mechanics thus restricts us to obtaining estimates of the reconstruction of a pure qubit. We discuss the optimal estimate on the basis of the Uhlmann–Josza fidelity, respecting the limitations imposed by the no-cloning theorem. We derive a realistic optimal expression for the average fidelity. Our formalism also introduces an optimization parameter L. Values close to zero imply full reconstruction of the qubit (i. e., the classical limit), while larger L’s represent good quantum optimization of the qubit estimate. The parameter L is interpreted as the degree of quantumness of the average fidelity associated with the reconstruction.