Quantum logical entropy: fundamentals and general properties

TL;DR

This paper introduces quantum logical entropy, extending classical logical entropy to quantum systems, and explores its fundamental properties and applications to density matrices and post-selected systems.

Contribution

It presents the first comprehensive analysis of quantum logical entropy, establishing its properties and extending it to post-selected quantum systems.

Findings

Quantum logical entropy is well-defined for generic density matrices.

Several fundamental properties of quantum logical entropy are proven.

Extension of quantum logical entropy to post-selected systems is demonstrated.

Abstract

Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how fundamental concepts of this entropy and other related definitions can be applied to the study of quantum systems, leading to the introduction of the quantum logical entropy. Moreover, we prove several properties of this entropy for generic density matrices that may be relevant to various areas of quantum mechanics and quantum information. Furthermore, we extend the notion of quantum logical entropy to post-selected systems.