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Generalized multiplicative domains and quantum error correction

Authors: Nathaniel Johnston and David W. Kribs
Journal: Proc. Amer. Math. Soc. 139 (2011), 627-639
MSC (2010): Primary 46L05, 47L05, 46N50
Published electronically: July 26, 2010
MathSciNet review: 2736344
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Abstract: Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map, and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi's characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.

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Additional Information

Nathaniel Johnston
Affiliation: Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1

David W. Kribs
Affiliation: Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Received by editor(s): July 20, 2009
Received by editor(s) in revised form: March 17, 2010
Published electronically: July 26, 2010
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society