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Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

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González Guillén, Carlos Eduardo and Palazuelos Cabezón, Carlos and Villanueva, Ignacio (2016) Euclidean Distance Between Haar Orthogonal and Gaussian Matrices. Journal of Theoretical Probability . pp. 1-26. ISSN 0894984 (In Press)

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Official URL: http://link.springer.com/article/10.1007/s10959-016-0712-6


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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 (Formula presented.) of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix (Formula presented.). If (Formula presented.) denotes the vector formed by the first m-coordinates of the ith row of (Formula presented.) and (Formula presented.), our main result shows that the Euclidean norm of (Formula presented.) converges exponentially fast to (Formula presented.), up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm (Formula presented.) and we find a coupling that improves by a factor (Formula presented.) the recently proved best known upper bound on (Formula presented.). Our main result also has applications in Quantum Information Theory.


Item Type:Article
Uncontrolled Keywords:Random matrix theory; Gaussian measure; Haar measure
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:39895
Deposited On:05 Nov 2016 16:53
Last Modified:07 Nov 2016 15:02

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