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QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Published online by Cambridge University Press:  04 July 2003

David W. Kribs
David W. Kribs
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada (kribs@math.purdue.edu)
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Abstract

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We show that the representations of the Cuntz $C^*$-algebras $\mathcal{O}_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.

AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15

Keywords

operatorcompletely positive mapquantum channelorthogonal waveletisometric dilationCuntz algebra
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 2 , June 2003 , pp. 421 - 433
Copyright
Copyright © Edinburgh Mathematical Society 2003