Euclidean Distance between Haar Orthogonal and Gaussian Matrices

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 Yn of order n and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix Un. If F m i denotes the vector formed by the first m-coordinates of the ith row of Yn − √ nUn and α = m n , our main result shows that the euclidean norm of F i converges exponentially fast to √ ( 2− 43 (1−(1−α)3/2) α ) m, up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm ǫn(m) = sup1≤i≤n,1≤j≤m |yi,j − √ nui,j | and we find a coupling that improves by a factor √ 2 the recently proved best known upper bound of ǫn(m). Applications of our results to Quantum Information Theory are also explained.