Euclidean distance between Haar orthogonal and gaussian matrices

Abstract

In this work we study a version of the general question of how well a Haar distributed orthogonal matrix can be approximated by a random gaussian matrix. Here, we consider a gaussian random matrix $Y_n$ of order $n$ and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix $U_n$. If $F_i^m$ denotes the vector formed by the first $m$-coordinates of the $i$th row of $Y_n-\sqrt{n}{U}_n$ and $\alpha =\frac{m}{n}$, our main result shows that the euclidean norm of $F_i^m$ converges exponentially fast to $\sqrt{\left(2-\frac{4}{3}\frac{(1-(1-\alpha {)}^{3/2})}{\alpha }\right)m}$, up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm ${\epsilon }_n(m)={\sup }_{1\le i\le n,1\le j\le m}|y_{i,j}-\sqrt{n}{u}_{i,j}|$ and we find a coupling that improves by a factor $\sqrt{2}$ the recently proved best known upper bound of ${\epsilon }_n(m)$. Applications of our results to Quantum Information Theory are also explained.

v2: minor modifications to match journal version, 26 pages, 0 figures, J Theor Probab (2016)

References in corpus (2)

• Random matrices: Universal properties of eigenvectors
• Sampling quantum nonlocal correlations with high probability

Cited by in corpus (2)

• Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations
• Random constructions in Bell inequalities: A survey