Entropy of quantum states

TL;DR

This paper introduces an algebraic definition of entropy for quantum states that resolves ambiguity in von Neumann entropy and aligns with thermodynamic principles, generalizing quantum entropy concepts.

Contribution

It provides a purely algebraic entropy definition for quantum states, extending von Neumann entropy to more general algebraic settings and ensuring thermodynamic consistency.

Findings

The algebraic entropy satisfies thermodynamic properties.

It reduces to von Neumann entropy in quantum mechanics.

It equals the von Neumann entropy of a unique density matrix in multiplicity-free representations.

Abstract

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so defined satisfies all the desirable thermodynamic properties, and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.