Distinguishability measures between ensembles of quantum states

A quantum ensemble $\left\{(p_x,{\rho }_x)\right\}$ is a set of quantum states each occurring randomly with a given probability. Quantum ensembles are necessary to describe situations with incomplete a priori information, such as the output of a stochastic quantum channel (generalized measurement), and play a central role in quantum communication. In this paper, we propose measures of distance and fidelity between two quantum ensembles. We consider two approaches: the first one is based on the ability to mimic one ensemble given the other one as a resource and is closely related to the Monge-Kantorovich optimal transportation problem, while the second one uses the idea of extended-Hilbert-space (EHS) representations which introduce auxiliary pointer (or flag) states. Both types of measures enjoy a number of desirable properties. The Kantorovich measures, albeit monotonic under deterministic quantum operations, are not monotonic under generalized measurements. In contrast, the EHS measures are. This property can be regarded as a generalization of the monotonicity under deterministic maps of the trace distance and the fidelity between states. The EHS measures are equivalent to convex optimization problems and are bounded by the Kantorovich measures which are equivalent to linear programs. We present operational interpretations for both types of measures. We also show that the EHS fidelity between ensembles provides an interpretation of the fidelity between mixed states as the fidelity between all pure-state ensembles whose averages are equal to the mixed states being compared. We finally use the measures to define distance and fidelity for stochastic quantum channels and positive operator-valued measures. These quantities may be useful in the context of tomography of stochastic quantum channels and quantum detectors.