16 research outputs found
Almost commuting unitary matrices related to time reversal
The behavior of fermionic systems depends on the geometry of the system and
the symmetry class of the Hamiltonian and observables. Almost commuting
matrices arise from band-projected position observables in such systems. One
expects the mathematical behavior of almost commuting Hermitian matrices to
depend on two factors. One factor will be the approximate polynomial relations
satisfied by the matrices. The other factor is what algebra the matrices are
in, either the matrices over A for A the real numbers, A the complex numbers or
A the algebra of quaternions.
There are potential obstructions keeping k-tuples of almost commuting
operators from being close to a commuting k-tuple. We consider two-dimensional
geometries and so this obstruction lives in KO_{-2}(A). This obstruction
corresponds to either the Chern number or spin Chern number in physics. We show
that if this obstruction is the trivial element in K-theory then the
approximation by commuting matrices is possible.Comment: 33 pages, 2 figures. In version 2 some formulas have been corrected
and some proofs have been rewritten to improve the expositio
Random repeated quantum interactions and random invariant states
We consider a generalized model of repeated quantum interactions, where a
system is interacting in a random way with a sequence of
independent quantum systems . Two types of randomness
are studied in detail. One is provided by considering Haar-distributed
unitaries to describe each interaction between and
. The other involves random quantum states describing each copy
. In the limit of a large number of interactions, we present
convergence results for the asymptotic state of . This is achieved
by studying spectral properties of (random) quantum channels which guarantee
the existence of unique invariant states. Finally this allows to introduce a
new physically motivated ensemble of random density matrices called the
\emph{asymptotic induced ensemble}