507 research outputs found
Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics
We re-examine the problem of the "Loschmidt echo", which measures the
sensitivity to perturbation of quantum chaotic dynamics. The overlap squared
of two wave packets evolving under slightly different Hamiltonians is
shown to have the double-exponential initial decay in the main part of phase space. The
coefficient is the self-averaging Lyapunov exponent. The average
decay is single exponential with a different
coefficient . The volume of phase space that contributes to
vanishes in the classical limit for times less than the
Ehrenfest time . It is only after
the Ehrenfest time that the average decay is representative for a typical
initial condition.Comment: 4 pages, 4 figures, [2017: fixed broken postscript figures
Classical predictability and coarse-grained evolution of the quantum baker's map
We investigate how classical predictability of the coarse-grained evolution
of the quantum baker's map depends on the character of the coarse-graining. Our
analysis extends earlier work by Brun and Hartle [Phys. Rev. D 60, 123503
(1999)] to the case of a chaotic map. To quantify predictability, we compare
the rate of entropy increase for a family of coarse-grainings in the decoherent
histories formalism. We find that the rate of entropy increase is dominated by
the number of scales characterising the coarse-graining.Comment: 28 pages, 1 figur
Quantum probabilities as Bayesian probabilities
In the Bayesian approach to probability theory, probability quantifies a
degree of belief for a single trial, without any a priori connection to
limiting frequencies. In this paper we show that, despite being prescribed by a
fundamental law, probabilities for individual quantum systems can be understood
within the Bayesian approach. We argue that the distinction between classical
and quantum probabilities lies not in their definition, but in the nature of
the information they encode. In the classical world, maximal information about
a physical system is complete in the sense of providing definite answers for
all possible questions that can be asked of the system. In the quantum world,
maximal information is not complete and cannot be completed. Using this
distinction, we show that any Bayesian probability assignment in quantum
mechanics must have the form of the quantum probability rule, that maximal
information about a quantum system leads to a unique quantum-state assignment,
and that quantum theory provides a stronger connection between probability and
measured frequency than can be justified classically. Finally we give a
Bayesian formulation of quantum-state tomography.Comment: 6 pages, Latex, final versio
Preparation information and optimal decompositions for mixed quantum states
Consider a joint quantum state of a system and its environment. A measurement
on the environment induces a decomposition of the system state. Using
algorithmic information theory, we define the preparation information of a pure
or mixed state in a given decomposition. We then define an optimal
decomposition as a decomposition for which the average preparation information
is minimal. The average preparation information for an optimal decomposition
characterizes the system-environment correlations. We discuss properties and
applications of the concepts introduced above and give several examples.Comment: 13 pages, latex, 2 postscript figure
Semiclassical properties and chaos degree for the quantum baker's map
We study the chaotic behaviour and the quantum-classical correspondence for
the baker's map. Correspondence between quantum and classical expectation
values is investigated and it is numerically shown that it is lost at the
logarithmic timescale. The quantum chaos degree is computed and it is
demonstrated that it describes the chaotic features of the model. The
correspondence between classical and quantum chaos degrees is considered.Comment: 30 pages, 4 figures, accepted for publication in J. Math. Phy
Classical limit in terms of symbolic dynamics for the quantum baker's map
We derive a simple closed form for the matrix elements of the quantum baker's
map that shows that the map is an approximate shift in a symbolic
representation based on discrete phase space. We use this result to give a
formal proof that the quantum baker's map approaches a classical Bernoulli
shift in the limit of a small effective Plank's constant.Comment: 12 pages, LaTex, typos correcte
C++QED: An object-oriented framework for wave-function simulations of cavity QED systems
We present a framework for efficiently performing Monte Carlo wave-function
simulations in cavity QED with moving particles. It relies heavily on the
object-oriented programming paradigm as realised in C++, and is extensible and
applicable for simulating open interacting quantum dynamics in general. The
user is provided with a number of ``elements'', eg pumped moving particles,
pumped lossy cavity modes, and various interactions to compose complex
interacting systems, which contain several particles moving in electromagnetic
fields of various configurations, and perform wave-function simulations on such
systems. A number of tools are provided to facilitate the implementation of new
elements.Comment: 31 pages, 8 figures, 3 table
Chaos for Liouville probability densities
Using the method of symbolic dynamics, we show that a large class of
classical chaotic maps exhibit exponential hypersensitivity to perturbation,
i.e., a rapid increase with time of the information needed to describe the
perturbed time evolution of the Liouville density, the information attaining
values that are exponentially larger than the entropy increase that results
from averaging over the perturbation. The exponential rate of growth of the
ratio of information to entropy is given by the Kolmogorov-Sinai entropy of the
map. These findings generalize and extend results obtained for the baker's map
[R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992)].Comment: 26 pages in REVTEX, no figures, submitted to Phys. Rev.
Complex joint probabilities as expressions of determinism in quantum mechanics
The density operator of a quantum state can be represented as a complex joint
probability of any two observables whose eigenstates have non-zero mutual
overlap. Transformations to a new basis set are then expressed in terms of
complex conditional probabilities that describe the fundamental relation
between precise statements about the three different observables. Since such
transformations merely change the representation of the quantum state, these
conditional probabilities provide a state-independent definition of the
deterministic relation between the outcomes of different quantum measurements.
In this paper, it is shown how classical reality emerges as an approximation to
the fundamental laws of quantum determinism expressed by complex conditional
probabilities. The quantum mechanical origin of phase spaces and trajectories
is identified and implications for the interpretation of quantum measurements
are considered. It is argued that the transformation laws of quantum
determinism provide a fundamental description of the measurement dependence of
empirical reality.Comment: 12 pages, including 1 figure, updated introduction includes
references to the historical background of complex joint probabilities and to
related work by Lars M. Johanse
Hypersensitivity to Perturbations in the Quantum Baker's Map
We analyze a randomly perturbed quantum version of the baker's
transformation, a prototype of an area-conserving chaotic map. By numerically
simulating the perturbed evolution, we estimate the information needed to
follow a perturbed Hilbert-space vector in time. We find that the Landauer
erasure cost associated with this information grows very rapidly and becomes
much larger than the maximum statistical entropy given by the logarithm of the
dimension of Hilbert space. The quantum baker's map thus displays a
hypersensitivity to perturbations that is analogous to behavior found earlier
in the classical case. This hypersensitivity characterizes ``quantum chaos'' in
a way that is directly relevant to statistical physics.Comment: 8 pages, LATEX, 3 Postscript figures appended as uuencoded fil
- …