9 research outputs found
Irreversible Quantum Baker Map
We propose a generalization of the model of classical baker map on the torus,
in which the images of two parts of the phase space do overlap. This
transformation is irreversible and cannot be quantized by means of a unitary
Floquet operator. A corresponding quantum system is constructed as a completely
positive map acting in the space of density matrices. We investigate spectral
properties of this super-operator and their link with the increase of the
entropy of initially pure states.Comment: 4 pages, 3 figures include
From quantum graphs to quantum random walks
We give a short overview over recent developments on quantum graphs and
outline the connection between general quantum graphs and so-called quantum
random walks.Comment: 14 pages, 6 figure
Quantum ergodicity for graphs related to interval maps
We prove quantum ergodicity for a family of graphs that are obtained from
ergodic one-dimensional maps of an interval using a procedure introduced by
Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take
the L^2 functions on the interval. The proof is based on the periodic orbit
expansion of a majorant of the quantum variance. Specifically, given a
one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an
increasingly refined sequence of partitions of the interval. To this sequence
we associate a sequence of graphs, whose directed edges correspond to elements
of the partitions and on which the classical dynamics approximates the
Perron-Frobenius operator corresponding to the map. We show that, except
possibly for subsequences of density 0, the eigenstates of the quantum graphs
equidistribute in the limit of large graphs. For a smaller class of observables
we also show that the Egorov property, a correspondence between classical and
quantum evolution in the semiclassical limit, holds for the quantum graphs in
question.Comment: 20 pages, 1 figur
Quantum Iterated Function Systems
Iterated functions system (IFS) is defined by specifying a set of functions
in a classical phase space, which act randomly on an initial point. In an
analogous way, we define a quantum iterated functions system (QIFS), where
functions act randomly with prescribed probabilities in the Hilbert space. In a
more general setting a QIFS consists of completely positive maps acting in the
space of density operators. We present exemplary classical IFSs, the invariant
measure of which exhibits fractal structure, and study properties of the
corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include
Entropic bounds on semiclassical measures for quantized one-dimensional maps
Quantum ergodicity asserts that almost all infinite sequences of eigenstates
of a quantized ergodic system are equidistributed in the phase space. On the
other hand, there are might exist exceptional sequences which converge to
different (non-Liouville) classical invariant measures. By the remarkable
result of N. Anantharaman and S. Nonnenmacher math-ph/0610019, arXiv:0704.1564
(with H. Koch), for Anosov geodesic flows the metric entropy of any
semiclassical measure must be bounded from below. The result seems to be
optimal for uniformly expanding systems, but not in general case, where it
might become even trivial if the curvature of the Riemannian manifold is
strongly non-uniform. It has been conjectured by the same authors, that in
fact, a stronger bound (valid in general case) should hold.
In the present work we consider such entropic bounds using the model of
quantized one-dimensional maps. For a certain class of non-uniformly expanding
maps we prove Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps
we are able to construct some explicit sequences of eigenstates which saturate
the bound. This demonstrates that the conjectured bound is actually optimal in
that case.Comment: 38 pages, 4 figure
Locality for quantum systems on graphs depends on the number field
Adapting a definition of Aaronson and Ambainis [Theory Comput. 1 (2005),
47--79], we call a quantum dynamics on a digraph "saturated Z-local" if the
nonzero transition amplitudes specifying the unitary evolution are in exact
correspondence with the directed edges (including loops) of the digraph. This
idea appears recurrently in a variety of contexts including angular momentum,
quantum chaos, and combinatorial matrix theory. Complete characterization of
the digraph properties that allow such a process to exist is a long-standing
open question that can also be formulated in terms of minimum rank problems. We
prove that saturated Z-local dynamics involving complex amplitudes occur on a
proper superset of the digraphs that allow restriction to the real numbers or,
even further, the rationals. Consequently, among these fields, complex numbers
guarantee the largest possible choice of topologies supporting a discrete
quantum evolution. A similar construction separates complex numbers from the
skew field of quaternions. The result proposes a concrete ground for
distinguishing between complex and quaternionic quantum mechanics.Comment: 9 page