# Generalized Wigner-von Neumann entropy and its typicality

## TL;DR

This paper introduces a generalized Wigner-von Neumann entropy applicable to pure and mixed quantum states, demonstrating its typicality, basis independence, and connection to classical thermodynamic entropies.

## Contribution

It proposes a new generalized quantum entropy that unifies and extends previous definitions, with properties that emerge in large Hilbert spaces and relate to thermodynamic concepts.

## Key findings

- GWvN entropy becomes basis-independent and approaches ln N for large N.
- The entropy's evolution is typical and aligns with the quantum H theorem.
- GWvN entropy matches Boltzmann and von Neumann entropies in specific ensembles.

## Abstract

We propose a generalization of the quantum entropy introduced by Wigner and von Neumann in 1929 [Zeitschrift f\"ur Physik 57, 30 (1929)]. Our generalization is applicable to both quantum pure states and mixed states. When the dimension $N$ of the Hilbert space is large, as a result of typicality, this generalized Wigner-von Neumann (GWvN) entropy becomes independent of choices of basis and is asymptotically equal to $\ln N$. The dynamic evolution of our entropy is also typical, reminiscent of quantum H theorem proved by von Neumann. For the microcanonical ensemble, the GWvN entropy is equivalent to the Boltzmann entropy; for a subsystem in a mixed state, the GWvN entropy is equivalent to the familiar von Neumann entropy, which is zero for pure states. The GWvN entropy can be used to derive the Gibbs ensemble.