Quantum channels and representation theory

In the study of d -dimensional quantum channels ( d ⩾ 2 ) , an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the intera...

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Bibliographic Details
Published in:Journal of mathematical physics 2005-08, Vol.46 (8), p.082103-082103-22
Main Author: Ritter, William Gordon
Format: Article
Language:English
Subjects:
ALGEBRA
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
CLIFFORD ALGEBRA
DENSITY MATRIX
Exact sciences and technology
HAMILTONIANS
Mathematical methods in physics
Mathematics
Physics
POLYNOMIALS
QUANTUM MECHANICS
Sciences and techniques of general use
SPHERES
SU-2 GROUPS
SYMMETRY
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Summary:In the study of d -dimensional quantum channels ( d ⩾ 2 ) , an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su ( n ) is a depolarizing channel for all n , but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su ( 2 ) , we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g 2 and the Clifford algebra.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.1945768