Comparison of quantum channels and statistical experiments

TL;DR

This paper characterizes when one quantum channel can approximate another through post-processing by comparing success probabilities across all input ensembles, extending Le Cam's deficiency to quantum channels.

Contribution

It introduces an operational criterion for quantum channel approximation based on success probabilities, extending classical statistical concepts to quantum information theory.

Findings

Provides a criterion for quantum channel approximation via success probabilities.

Extends Le Cam's deficiency concept to quantum channels.

Utilizes properties of the diamond norm and its dual for proofs.

Abstract

For a pair of quantum channels with the same input space, we show that the possibility of approximation of one channel by post-processings of the other channel can be characterized by comparing the success probabilities for the two ensembles obtained as outputs for any ensemble on the input space coupled with an ancilla. This provides an operational interpretation to a natural extension of Le Cam's deficiency to quantum channels. In particular, we obtain a version of the randomization criterion for quantum statistical experiments. The proofs are based on some properties of the diamond norm and its dual, which are of independent interest.