Optimal quantum state discrimination with confidentiality

We investigate quantum state discrimination with confidentiality. $N$ observers share a given quantum state belonging to a finite set of known states. The observers want to determine the state as accurately as possible and send a discrimination result to a receiver. However, the observers are not allowed to get any information about which state was given. $N-1$ or fewer observers might try to steal the information, but if $N$ observers coexist, the honest ones will keep the dishonest ones from doing anything wrong. Assume that the state set has a certain symmetry, or more precisely, is Abelian geometrically uniform; this letter describes the case of three linearly independent cyclic pure states as a special case. We propose a protocol that realizes any optimal inconclusive measurement, which is a generalized version of a minimum-error measurement and an optimal unambiguous measurement, for such a state set and ensures that any combined state of $N-1$ or fewer observers has absolutely no information about the given state. Our protocol provides a method of performing a quantum measurement securely, which could be useful in quantum information applications.