Linearly independent pure-state decomposition and quantum state discrimination

We put the pure-state decomposition mathematical property of a mixed state to a physical test. We present a protocol for preparing two known nonorthogonal quantum states with well-defined a priori probabilities. Hence we characterize all the possible decompositions of a rank-two mixed state by means of the complex overlap between the two involved states. The physical test proposes a scheme for quantum state recognition of one of the two linearly independent states that arise from the decomposition. We find that the two states associated with the balanced pure-state decomposition have the smaller overlap modulus and therefore the smallest probability of being discriminated conclusively, while in the nonconclusive process they have the highest probability of having an error. In addition, we design an experimental scheme that allows discriminating conclusively and optimally two nonorthogonal states prepared with different a priori probabilities. Thus we propose a physical implementation for this linearly independent pure-state decomposition and state discrimination test by using twin photons generated in the process of spontaneous parametric down-conversion. The information state is encoded in a one-photon polarization state, whereas the second single photon is used for heralded detection.