Unambiguous comparison of quantum measurements

The goal of comparison is to reveal the difference of compared objects as fast and reliably as possible. In this paper we formulate and investigate the unambiguous comparison of unknown quantum measurements represented by nondegenerate sharp positive operator valued measures. We distinguish between measurement devices with a priori labeled and unlabeled outcomes. In both cases we can unambiguously conclude only that the measurements are different. For the labeled case it is sufficient to use each unknown measurement only once and the average conditional success probability decreases with the Hilbert space dimension as 1/d. If the outcomes of the apparatuses are not labeled, then the problem is more complicated. We analyze the case of two-dimensional Hilbert space. In this case single shot comparison is impossible and each measurement device must be used (at least) twice. The optimal test state in the two-shot scenario gives the average conditional success probability 4/9. Interestingly, the optimal experiment detects unambiguously the difference with nonvanishing probability for any pair of observables.