# Quantum operator entropies under unitary evolution

## TL;DR

This paper investigates how quantum operator entropies evolve under unitary dynamics, showing that positional entropy increases over time in a way consistent with classical thermodynamics, despite the constancy of von Neumann entropy.

## Contribution

It demonstrates that quantum operator entropies can increase during unitary evolution in a simple model, aligning quantum and classical entropy behaviors.

## Key findings

- Positional entropy increases over time during unitary evolution.
- Von Neumann entropy remains constant for pure states.
- Quantum operator entropy reflects information loss about specific observables.

## Abstract

For a quantum state undergoing unitary Schr\"odinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.