Year: 2005
Type: article
Abstract: 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 g2 and the Clifford algebra.
Source: Journal of Mathematical Physics
DOI: 10.1063/1.1945768
Author: William Gordon Ritter
Institution: Harvard University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2055066733