Information capacity of quantum observable

TL;DR

This paper investigates the classical capacities of quantum measurement channels, especially for continuous observables, providing formulas, examples, and duality relations that deepen understanding of information transfer in quantum measurements.

Contribution

It derives formulas for classical capacities of quantum measurement channels, explores explicit examples, and establishes a duality between capacities and quantum ensemble information measures.

Findings

Formulas for unassisted and entanglement-assisted capacities $C$ and $C_{ea}$.

Explicit examples of entanglement-breaking channels with $C<C_{ea}$.

Duality between $C_{ea}$ and the $ ext{chi}$-quantity for dual ensembles.

Abstract

In this paper we consider the classical capacities of quantum-classical channels corresponding to measurement of observables. Special attention is paid to the case of continuous observables. We give the formulas for unassisted and entanglement-assisted classical capacities $C,C_{ea}$ and consider some explicitly solvable cases which give simple examples of entanglement-breaking channels with $C<C_{ea}.$ We also elaborate on the ensemble-observable duality to show that $C_{ea}$ for the measurement channel is related to the $\chi$-quantity for the dual ensemble in the same way as $C$ is related to the accessible information. This provides both accessible information and the $\chi$-quantity for the quantum ensembles dual to our examples.