Distinguishability measures between ensembles of quantum states

TL;DR

This paper introduces new measures of distance and fidelity between quantum ensembles, based on optimal transportation and extended-Hilbert-space representations, with applications in quantum communication and channel tomography.

Contribution

It proposes two novel approaches for quantifying differences between quantum ensembles, including operational interpretations and a new perspective on mixed state fidelity.

Findings

Kantorovich measures are monotonic under deterministic operations but not under generalized measurements.

EHS measures are monotonic under all quantum operations.

EHS fidelity offers a new interpretation of mixed state fidelity.

Abstract

A quantum ensemble $\{(p_x, \rho_x)\}$ 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. We present operational interpretations for both types of measures. We also show that the EHS fidelity between ensembles provides a novel interpretation of the fidelity between mixed states--the latter is equal to the maximum of the fidelity between all pure-state ensembles whose averages are equal to the mixed states being compared. We finally use the new measures to define distance and fidelity for stochastic quantum channels and positive operator-valued measures (POVMs). These quantities may be useful in the context of tomography of stochastic quantum channels and quantum detectors.