Heralded enhancement in quantum state discrimination

Abstract The discrimination of quantum states is a central problem in quantum information science and technology. Meanwhile, partial post-selection has emerged as a valuable tool for quantum state engineering. In this work, we bring these two areas together and ask whether partial measurements can enhance the discrimination performance between two unknown and non-orthogonal pure states. Our framework is general: the two unknown states interact with the same environment—set in a pure state—via an arbitrary unitary transformation. A measurement is then performed on one of the output modes (i.e. a partial measurement), modeled by an arbitrary positive operator-valued measure (POVM). We then allow classical communication to inform the unmeasured mode of the outcome of the partial measurement, which is subsequently measured by a POVM that is optimal in the sense that the discrimination probability of error is minimized. The two POVMs act locally and classical information is exchanged between the two modes, representing a single-round (feed-forward) form of local operations with classical communication. Under these considerations, we first show that, as expected, the minimum error probability, averaged over all possible conditional states, cannot be reduced below the minimum error probability of discriminating the original input states. Then, we devise a generic setup produces specific examples where the conditional discrimination can achieve strictly lower error probabilities than the original optimal measurement, illustrating that while post-selection does not improve the average performance, it can enable better discrimination in certain post-selected ensembles.