Identifying the Information Gain of a Quantum Measurement

TL;DR

This paper demonstrates that quantum measurements can be simulated with classical communication equal to their quantum mutual information, extending classical and quantum information theories to more general measurement scenarios.

Contribution

It generalizes Winter's measurement compression theorem to arbitrary inputs and identifies the quantum mutual information of a measurement as the measure of information gained.

Findings

Quantum measurements can be asymptotically simulated with classical communication equal to their quantum mutual information.

The quantum mutual information of a measurement quantifies the information gained, regardless of input state.

The proof introduces a novel one-shot state merging protocol and uses the post-selection technique for quantum channels.

Abstract

We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically simulated by an amount of classical communication equal to the quantum mutual information of the measurement, if sufficient shared randomness is available. This result generalizes Winter's measurement compression theorem for fixed independent and identically distributed inputs [Winter, CMP 244 (157), 2004] to arbitrary inputs, and more importantly, it identifies the quantum mutual information of a measurement as the information gained by performing it, independent of the input state on which it is performed. Our result is a generalization of the classical reverse Shannon theorem to quantum-to-classical channels. In this sense, it can be seen as a quantum reverse Shannon theorem for quantum-to-classical channels, but with the entanglement assistance and quantum communication replaced by shared randomness and classical communication, respectively. The proof is based on a novel one-shot state merging protocol for "classically coherent states" as well as the post-selection technique for quantum channels, and it uses techniques developed for the quantum reverse Shannon theorem [Berta et al., CMP 306 (579), 2011].