Entropy of Quantum Measurements

TL;DR

This paper introduces a new measure called $ ho$-entropy for quantum effects and observables, providing bounds, properties, and applications to quantum measurement models, enhancing understanding of information gain in quantum systems.

Contribution

It defines and analyzes the properties of $ ho$-entropy for quantum effects and observables, offering new bounds, characterizations, and simplified proofs in quantum measurement theory.

Findings

Bounds on $ ho$-entropy for effects and observables

Inequalities relating $ ho$-entropy of effects and their sums

Characterizations of when bounds are tight

Abstract

If $a$ is a quantum effect and $\rho$ is a state, we define the $\rho$-entropy $S_a(\rho )$ which gives the amount of uncertainty that a measurement of $a$ provides about $\rho$. The smaller $S_a(\rho )$ is, the more information a measurement of $a$ gives about $\rho$. In Section~2, we provide bounds on $S_a(\rho )$ and show that if $a+b$ is an effect, then $S_{a+b}(\rho )\ge S_a(\rho )+S_b(\rho )$. We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their $\rho$-entropies. In Section~3, we employ $S_a(\rho )$ to define the $\rho$-entropy $S_A(\rho )$ for an observable $A$. We show that $S_A(\rho )$ directly provides the $\rho$-entropy $S_\iscript (\rho )$ for an instrument $\iscript$. We establish bounds for $S_A(\rho )$ and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider $\rho$-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.