Exact analytical relation between the entropies and the dominant eigenvalue of random reduced density matrices

TL;DR

This paper establishes an exact analytical relationship between the entropy measures and the dominant eigenvalue of random reduced density matrices, using Random Matrix Theory and simulations, with implications for quantum entanglement.

Contribution

It provides the first explicit analytical relation connecting entropy and dominant eigenvalue of random reduced density matrices, supported by RMT and numerical validation.

Findings

Analytical relation between entropy and eigenvalue derived from RMT

Validation through extensive computer simulations

Relevance to quantum entanglement generation

Abstract

In this paper, we show how the entropy (including the von Neumann entropy obtained by tracing across various sizes of subsystems, the entanglement gap, as well as different degrees of R\'{e}nyi entropy) of the random reduced density matrices are related to their dominant eigenvalue. Analytical results are deduced from Random Matrix Theory (RMT) for decentralized Wishart matrices and backed up by computer simulations. The correlation between our study and entanglement generated by quantum computing is provided with various examples.