Noiseless Subsystems and the Structure of the
Commutant in Quantum Error Correction
John A. Holbrook,
David W. Kribs,
Received August 15, 2003; accepted January 25, 2004
The effect of noise on a quantum system can be described by a set of operators
obtained from the interaction Hamiltonian. Recently it has been shown that
generalized quantum error correcting codes can be derived by studying the algebra
of this set of operators. This led to the discovery of noiseless subsystems. They are
described by a set of operators obtained from the commutant of the noise generators.
In this paper we derive a general method to compute the structure of this commutant
in the case of unital noise.
KEY WORDS: Quantum information; quantum error correction; noiseless
PACS: 03.67.a, 03.67.Pp
Quantum mechanics promises to manipulate information for communica-
tion, cryptography and computation in a way fundamentally different from
its classical counterpart.
Although it is possible to manipulate small
quantum systems in the laboratory, the task to do so for large ones is
daunting, especially because in absence of control of noise and imperfection
of realistic devices the quantum properties of the state are destroyed.
Quantum error correction methods have recently been discovered which
protect quantum information against corruption. In particular it was shown
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada
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Perimeter Institute for Theoretical Physics, 35 King St. North, Waterloo, Ontario, Canada
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To whom correspondence should be addressed.
Quantum Information Processing, Vol. 2, No. 5, October 2003 (# 2004)
1570-0755/03/1000–0381/0 # 2004 Plenum Publishing Corporation