Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

Abstract

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random $n\times n$ unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) $k\times k$ submatrix converges in distribution, after multiplying by a normalization factor $\sqrt{n}$ and as $n\to \infty$, to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.

Cited by in corpus (2)

• Random-matrix models of monitored quantum circuits
• Random covariant quantum channels