Entropy and singular-value moments of products of truncated random unitary matrices

TL;DR

This paper investigates how the von Neumann entropy of products of truncated random unitary matrices behaves in the large-size limit, revealing a crossover from linear to logarithmic entropy reduction depending on the product length and matrix parameters.

Contribution

It provides a new analytical expression for the singular-value moments of matrix products using the Erlang function, advancing understanding of entropy dynamics in quantum circuit models.

Findings

Entropy decreases with product length, depending on matrix size and truncation.

A crossover from linear to logarithmic entropy decay occurs at a critical parameter value.

Derived a formula connecting singular-value moments to queueing theory functions.

Abstract

Products of truncated unitary matrices, independently and uniformly drawn from the unitary group, can be used to study universal aspects of monitored quantum circuits. The von Neumann entropy of the corresponding density matrix decreases with increasing length $L$ of the product chain, in a way that depends on the matrix dimension $N$ and the truncation depth $\delta N$. Here we study that dependence in the double-scaling limit $L,N\rightarrow\infty$, at fixed ratio $\tau=L\delta N/N$. The entropy reduction crosses over from a linear to a logarithmic dependence on $\tau$ when this parameter crosses unity. The central technical result is an expression for the singular-value moments of the matrix product in terms of the Erlang function from queueing theory.