State discrimination with post-measurement information and incompatibility of quantum measurements

Abstract: We discuss the following variant of the standard minimum error state discrimination problem: Alice picks the state she sends to Bob among one of several disjoint state ensembles, and she communicates him the chosen ensemble only at a later time. Two different scenarios then arise: either Bob is allowed to arrange his measurement set-up after Alice has announced him the chosen ensemble, or he is forced to perform the measurement before of Alice's announcement. In the latter case, he can only post-process his measurement outcome when Alice's extra information becomes available. We compare the optimal guessing probabilities in the two scenarios, and we prove that they are the same if and only if there exist compatible optimal measurements for all of Alice's state ensembles. When this is the case, post-processing any of the corresponding joint measurements is Bob's optimal strategy in the post-measurement information scenario. Furthermore, we establish a connection between discrimination with post-measurement information and the standard state discrimination. By means of this connection and exploiting the presence of symmetries, we are able to compute the various guessing probabilities in many concrete examples.