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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. 126. ISSN 0894984 (In Press)

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Official URL: http://link.springer.com/article/10.1007/s1095901607126
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Abstract
In this work, we study a version of the general question of how well a Haardistributed 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 Haardistributed orthogonal matrix (Formula presented.). If (Formula presented.) denotes the vector formed by the first mcoordinates 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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