201 research outputs found
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.
Realistic simulations of single-spin nondemolition measurement by magnetic resonance force microscopy
A requirement for many quantum computation schemes is the ability to measure
single spins. This paper examines one proposed scheme: magnetic resonance force
microscopy, including the effects of thermal noise and back-action from
monitoring. We derive a simplified equation using the adiabatic approximation,
and produce a stochastic pure state unraveling which is useful for numerical
simulations.Comment: 33 pages LaTeX, 9 figure files in EPS format. Submitted to Physical
Review
Qubit-Qutrit Separability-Probability Ratios
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for
the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to
high numerical accuracy, the formulas of Sommers and Zyczkowski
(quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional
hyperarea of the (separable and nonseparable) N x N density matrices, based on
the Bures (minimal monotone) metric -- and also their analogous formulas
(quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same
seven billion well-distributed (``low-discrepancy'') sample points, we estimate
the unknown volumes and hyperareas based on five additional (monotone) metrics
of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate
all of these seven volume and seven hyperarea (unknown) quantities when
restricted to the separable density matrices. The ratios of separable volumes
(hyperareas) to separable plus nonseparable volumes (hyperareas) yield
estimates of the separability probabilities of generically rank-six (rank-five)
density matrices. The (rank-six) separability probabilities obtained based on
the 35-dimensional volumes appear to be -- independently of the metric (each of
the seven inducing Haar measure) employed -- twice as large as those (rank-five
ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we
fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable
volumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of
the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite
clearly close to integral too.) The doubling relationship also appears to hold
for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit
exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.Comment: 36 pages, 15 figures, 11 tables, final PRA version, new last
paragraph presenting qubit-qutrit probability ratios disaggregated by the two
distinct forms of partial transpositio
Conditional evolution in single-atom cavity QED
We consider a typical setup of cavity QED consisting of a two-level atom
interacting strongly with a single resonant electromagnetic field mode inside a
cavity. The cavity is resonantly driven and the output undergoes continuous
homodyne measurements. We derive an explicit expression for the state of the
system conditional on a discrete photocount record. This expression takes a
particularly simple form if the system is initially in the steady state. As a
byproduct, we derive a general formula for the steady state that had been
conjectured before in the strong driving limit.Comment: 15 pages, 1 postscript figure, added discussion of mode
Quantum trajectories for Brownian motion
We present the stochastic Schroedinger equation for the dynamics of a quantum
particle coupled to a high temperature environment and apply it the dynamics of
a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on
the environmental memory time scale, in the mean, our result recovers the
solution of the known non-Lindblad quantum Brownian motion master equation. A
remarkable feature of our approach is its localization property: individual
quantum trajectories remain localized wave packets for all times, even for the
classically chaotic system considered here, the localization being stronger the
smaller .Comment: 4 pages, 3 eps figure
Entanglement production in Quantized Chaotic Systems
Quantum chaos is a subject whose major goal is to identify and to investigate
different quantum signatures of classical chaos. Here we study entanglement
production in coupled chaotic systems as a possible quantum indicator of
classical chaos. We use coupled kicked tops as a model for our extensive
numerical studies. We find that, in general, presence of chaos in the system
produces more entanglement. However, coupling strength between two subsystems
is also very important parameter for the entanglement production. Here we show
how chaos can lead to large entanglement which is universal and describable by
random matrix theory (RMT). We also explain entanglement production in coupled
strongly chaotic systems by deriving a formula based on RMT. This formula is
valid for arbitrary coupling strengths, as well as for sufficiently long time.
Here we investigate also the effect of chaos on the entanglement production for
the mixed initial state. We find that many properties of the mixed state
entanglement production are qualitatively similar to the pure state
entanglement production. We however still lack an analytical understanding of
the mixed state entanglement production in chaotic systems.Comment: 16 pages, 5 figures. To appear in Pramana:Journal of Physic
Entanglement purification of unknown quantum states
A concern has been expressed that ``the Jaynes principle can produce fake
entanglement'' [R. Horodecki et al., Phys. Rev. A {\bf 59}, 1799 (1999)]. In
this paper we discuss the general problem of distilling maximally entangled
states from copies of a bipartite quantum system about which only partial
information is known, for instance in the form of a given expectation value. We
point out that there is indeed a problem with applying the Jaynes principle of
maximum entropy to more than one copy of a system, but the nature of this
problem is classical and was discussed extensively by Jaynes. Under the
additional assumption that the state of the copies of the
quantum system is exchangeable, one can write down a simple general expression
for . We show how to modify two standard entanglement purification
protocols, one-way hashing and recurrence, so that they can be applied to
exchangeable states. We thus give an explicit algorithm for distilling
entanglement from an unknown or partially known quantum state.Comment: 20 pages RevTeX 3.0 + 1 figure (encapsulated Postscript) Submitted to
Physical Review
Quantum nonlinear dynamics of continuously measured systems
Classical dynamics is formulated as a Hamiltonian flow on phase space, while
quantum mechanics is formulated as a unitary dynamics in Hilbert space. These
different formulations have made it difficult to directly compare quantum and
classical nonlinear dynamics. Previous solutions have focussed on computing
quantities associated with a statistical ensemble such as variance or entropy.
However a more direct comparison would compare classical predictions to the
quantum for continuous simultaneous measurement of position and momentum of a
single system. In this paper we give a theory of such measurement and show that
chaotic behaviour in classical systems can be reproduced by continuously
measured quantum systems.Comment: 11 pages, REVTEX, 3 figure
Classical model for bulk-ensemble NMR quantum computation
We present a classical model for bulk-ensemble NMR quantum computation: the
quantum state of the NMR sample is described by a probability distribution over
the orientations of classical tops, and quantum gates are described by
classical transition probabilities. All NMR quantum computing experiments
performed so far with three quantum bits can be accounted for in this classical
model. After a few entangling gates, the classical model suffers an exponential
decrease of the measured signal, whereas there is no corresponding decrease in
the quantum description. We suggest that for small numbers of quantum bits, the
quantum nature of NMR quantum computation lies in the ability to avoid an
exponential signal decrease.Comment: 14 pages, no figures, revte
Effects of noise on quantum error correction algorithms
It has recently been shown that there are efficient algorithms for quantum
computers to solve certain problems, such as prime factorization, which are
intractable to date on classical computers. The chances for practical
implementation, however, are limited by decoherence, in which the effect of an
external environment causes random errors in the quantum calculation. To combat
this problem, quantum error correction schemes have been proposed, in which a
single quantum bit (qubit) is ``encoded'' as a state of some larger number of
qubits, chosen to resist particular types of errors. Most such schemes are
vulnerable, however, to errors in the encoding and decoding itself. We examine
two such schemes, in which a single qubit is encoded in a state of qubits
while subject to dephasing or to arbitrary isotropic noise. Using both
analytical and numerical calculations, we argue that error correction remains
beneficial in the presence of weak noise, and that there is an optimal time
between error correction steps, determined by the strength of the interaction
with the environment and the parameters set by the encoding.Comment: 26 pages, LaTeX, 4 PS figures embedded. Reprints available from the
authors or http://eve.physics.ox.ac.uk/QChome.htm
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