Quantum Symmetries, Cartan
Decompositions and Quantum System
Identification in Arbitrary
Dimensions
Professor Domenico DÕAlessandro
Systems and Control Theory
Department of Mathematics
Iowa State University
Decompositions of Lie groups have been extensively used in control
theory to design control algorithms for bilinear, right invariant, systems with state varying on a Lie
group. They also play an important role in quantum information theory as they
are used to analyze quantum dynamics. Motivated by recent results on
entanglement of quantum systems,
we clarify the relation between quantum symmetries and Cartan Lie group
decompositions. As a consequence, we obtain a novel method to construct a
decomposition for unitary operators on a multipartite quantum system starting
from decompositions concerning the single subsystems. The resulting
decomposition, which we call of the 'odd-even type', contains, as a special case, the concurrence
canonical decomposition (CCD) presented in entanglement theory. The
generalization consists of allowing any possible dimension and different types
of Cartan decompositions for the single subsystems. The results are applied to
a system theoretic problem of interest in spin dynamics and in particular in
nuclear magnetic resonance. The problem is that of characterizing models of
networks of particles with spin, driven by an electro-magnetic field, which are
input-output equivalent. These are models which give the same value of the total
magnetization for every input field. A complete classification of equivalent
models can be obtained in terms of the introduced Cartan decompositions
.Friday, December 2, 2005
3:30 – 4:30 p.m.