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 $$Y_n$$ Y n of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix $$U_n$$ U n . If $$F_i^m$$ F i m denotes the vector formed by the first m-coordinates of the ith row of $$Y_n-\sqrt{n}U_n$$ Y n - n U n and $$\alpha \,=\,\frac{m}{n}$$ α = m n , our main result shows that the Euclidean norm of $$F_i^m$$ F i m converges exponentially fast to $$\sqrt{ \big (2-\frac{4}{3} \frac{(1-(1 -\alpha )^{3/2})}{\alpha }\big )m}$$ ( 2 - 4 3 ( 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 $$\epsilon _n(m)\,=\,\sup _{1\le i \le n, 1\le j \le m} |y_{i,j}- \sqrt{n}u_{i,j}|$$ ϵ n ( m ) = sup 1 ≤ i ≤ n , 1 ≤ j ≤ m | y i , j - n u i , j | and we find a coupling that improves by a factor $$\sqrt{2}$$ 2 the recently proved best known upper bound on $$\epsilon _n(m)$$ ϵ n ( m ) . Our main result also has applications in Quantum Information Theory.