After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience [2], it occurred to me that this entropy might have applications to classical mechanics as it appears to be generally useful for dimensionality reduction.

## Motivation:

Let's suppose we have an ergodic dynamical system with $N$ observables $x_i(t) \in \mathbb{R}$ which are sampled using a sequence of $n$ measurements so we have a dataset $X \in \mathbb{R}^{N \times n}$. Given $X$, we may compute the statistics $X_i = x_i - \langle x_i \rangle$ and $\sigma_i^2 = \langle X_i^2 \rangle$ which allows us to define the Pearson correlation matrix with entries:

\begin{equation} R_{i,j} = \frac{X_i \cdot X_j}{\sigma_i \cdot \sigma_j} \tag{1} \end{equation}

Given $R \in \mathbb{R}^{N \times N}$, we may define the density matrix $\rho = \frac{R}{N}$ which is positive semi-definite, Hermitian and has unit-trace. Thus, we may calculate the entropy of $R$ using the von Neumann entropy:

\begin{equation} S(\rho) = -\text{tr}(\rho \cdot \ln \rho) = - \sum_{i=1}^N \lambda_i \cdot \ln \lambda_i \tag{2} \end{equation}

where $\lambda_i$ are the eigenvalues of $\rho$.

If $N$ is large, in order to compress the dataset $X$ so that we keep the dimensions that contain $95\%$ of the statistical information it is sufficient to find the discrete subset $S \subset [1,N]$ that maximises:

\begin{equation} - \sum_{i \in S} \lambda_i \cdot \ln \lambda_i \tag{3} \end{equation}

subject to the constraint $\frac{- \sum_{i\in S} \lambda_i \cdot \ln \lambda_i}{- \sum_{i=1}^N \lambda_i \cdot \ln \lambda_i} \leq \frac{95}{100}$ which may be done using sorting algorithms such as Quick Sort. Moreover, the cardinality $\lvert S \rvert$ provides us with an approximate upper-bound on the intrinsic dimension of an ergodic dynamical system.

## Question:

Has the von Neumann entropy ever been used in classical mechanics?

## Note:

It is worth noting that the idea of viewing the Pearson correlation matrix as a density matrix is quite natural and not new [4].

## References:

von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik (Mathematical foundations of quantum mechanics) Princeton University Press., . ISBN 978-0-691-02893-4.

H. Felippe et al. The von Neumann entropy for the Pearson correlation matrix: A test of the entropic brain hypothesis. Arxiv. 2021.

E.T. Jaynes. Information theory and statistical mechanics. The Physical Review. 1957.

linello (https://physics.stackexchange.com/users/10941/linello), Can a correlation matrix be regarded as a quantum density matrix?, URL (version: 2016-09-28): https://physics.stackexchange.com/q/282904