# A Construction of Dynamical Entropy on CAR Algebras

## TL;DR

This paper develops a new definition of dynamical entropy specifically for CAR algebras, which are used to describe quantum spin systems, extending the AOW entropy framework from von Neumann algebras.

## Contribution

It formulates a novel dynamical entropy on CAR algebras based on the AOW entropy construction, enabling analysis of quantum spin systems.

## Key findings

- Entropy computed for 2x2 matrix algebra case
- Provides a tool for quantifying quantum information and uncertainty in spin systems
- Links dynamical entropy to quantum spin system models

## Abstract

The dynamical entropy on von Neumann algebras defined by Accardi, Ohya and Watanabe (AOW entropy) is a natural noncommutative extension of the classical dynamical entropy. On the other hand, quantum spin lattice systems currently used in quantum computing and communication processes are mathematically described by $C^*$-algebras called CAR algebras. Therefore, in order to obtain the average amount of quantum information and to calculate the uncertainty of the dynamics of quantum spin systems, it is necessary to define dynamical entropy on CAR algebras. In this paper, we formulate dynamical entropy on CAR algebras based on the construction of the AOW entropy. Moreover, we compute the introduced entropy for a $2 \times 2$ matrix algebra case which has relation to the quantum spin system.