10 research outputs found
Extensions and degenerations of spectral triples
For a unital C*-algebra A, which is equipped with a spectral triple and an
extension T of A by the compacts, we construct a family of spectral triples
associated to T and depending on the two positive parameters (s,t).
Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact
quantum metric spaces it is possible to define a metric on this family of
spectral triples, and we show that the distance between a pair of spectral
triples varies continuously with respect to the parameters. It turns out that a
spectral triple associated to the unitarization of the algebra of compact
operators is obtained under the limit - in this metric - for (s,1) -> (0, 1),
while the basic spectral triple, associated to A, is obtained from this family
under a sort of a dual limiting process for (1, t) -> (1, 0).
We show that our constructions will provide families of spectral triples for
the unitarized compacts and for the Podles sphere. In the case of the compacts
we investigate to which extent our proposed spectral triple satisfies Connes' 7
axioms for noncommutative geometry.Comment: 40 pages. Addedd in ver. 2: Examples for the compacts and the Podle`s
sphere plus comments on the relations to matricial quantum metrics. In ver.3
the word "deformations" in the original title has changed to "degenerations"
and some illustrative remarks on this aspect are adde
Matrix Compactification On Orientifolds
Generalizing previous results for orbifolds, in this paper we describe the
compactification of Matrix model on an orientifold which is a quotient space as
a Yang-Mills theory living on a quantum space. The information of the
compactification is encoded in the action of the discrete symmetry group G on
Euclidean space and a projective representation U of G. The choice of Hilbert
space on which the algebra of U is realized as an operator algebra corresponds
to the choice of a physical background for the compactification. All these data
are summarized in the spectral triple of the quantum space.Comment: 28 pages, late
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa