Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

TL;DR

This paper derives the exact distribution of the minimum eigenvalue of reduced density matrices for random pure states, providing insights into entanglement properties and correlated random variables in quantum systems.

Contribution

It proves a conjecture and derives the full distribution of the minimum eigenvalue for both real and complex random states, a rare exactly solvable case.

Findings

Exact distribution for real and complex states

Validation of a conjecture on eigenvalue averages

Applicable to quantum chaos and random matrix theory

Abstract

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).