262 research outputs found
Primes in short intervals
Contrary to what would be predicted on the basis of Cram\'er's model
concerning the distribution of prime numbers, we develop evidence that the
distribution of , for , is approximately
normal with mean and variance , when .Comment: 29 page
Chains of large gaps between primes
Let denote the -th prime, and for any and sufficiently
large , define the quantity which measures the occurrence of
chains of consecutive large gaps of primes. Recently, with Green and
Konyagin, the authors showed that for sufficiently large . In this
note, we combine the arguments in that paper with the Maier matrix method to
show that for any fixed and sufficiently large . The
implied constant is effective and independent of .Comment: 16 pages, no figure
Single shot parameter estimation via continuous quantum measurement
We present filtering equations for single shot parameter estimation using
continuous quantum measurement. By embedding parameter estimation in the
standard quantum filtering formalism, we derive the optimal Bayesian filter for
cases when the parameter takes on a finite range of values. Leveraging recent
convergence results [van Handel, arXiv:0709.2216 (2008)], we give a condition
which determines the asymptotic convergence of the estimator. For cases when
the parameter is continuous valued, we develop quantum particle filters as a
practical computational method for quantum parameter estimation.Comment: 9 pages, 5 image
Iterative maximum-likelihood reconstruction in quantum homodyne tomography
I propose an iterative expectation maximization algorithm for reconstructing
a quantum optical ensemble from a set of balanced homodyne measurements
performed on an optical state. The algorithm applies directly to the acquired
data, bypassing the intermediate step of calculating marginal distributions.
The advantages of the new method are made manifest by comparing it with the
traditional inverse Radon transformation technique
On the ratio of consecutive gaps between primes
In the present work we prove a common generalization of Maynard-Tao's recent
result about consecutive bounded gaps between primes and on the
Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work
answers in a strong form a 60 years old problem of Erd\"os, which asked whether
the ratio of two consecutive primegaps can be infinitely often arbitrarily
small, and arbitrarily large, respectively
Parameters estimation in quantum optics
We address several estimation problems in quantum optics by means of the
maximum-likelihood principle. We consider Gaussian state estimation and the
determination of the coupling parameters of quadratic Hamiltonians. Moreover,
we analyze different schemes of phase-shift estimation. Finally, the absolute
estimation of the quantum efficiency of both linear and avalanche
photodetectors is studied. In all the considered applications, the Gaussian
bound on statistical errors is attained with a few thousand data.Comment: 11 pages. 6 figures. Accepted on Phys. Rev.
Invariant information and quantum state estimation
The invariant information introduced by Brukner and Zeilinger, Phys. Rev.
Lett. 83, 3354 (1999), is reconsidered from the point of view of quantum state
estimation. We show that it is directly related to the mean error of the
standard reconstruction from the measurement of a complete set of mutually
complementary observables. We give its generalization in terms of the Fisher
information. Provided that the optimum reconstruction is adopted, the
corresponding quantity loses its invariant character.Comment: 4 pages, no figure
Entanglement-assisted tomography of a quantum target
We study the efficiency of quantum tomographic reconstruction where the
system under investigation (quantum target) is indirectly monitored by looking
at the state of a quantum probe that has been scattered off the target. In
particular we focus on the state tomography of a qubit through a
one-dimensional scattering of a probe qubit, with a Heisenberg-type
interaction. Via direct evaluation of the associated quantum Cram\'{e}r-Rao
bounds, we compare the accuracy efficiency that one can get by adopting
entanglement-assisted strategies with that achievable when entanglement
resources are not available. Even though sub-shot noise accuracy levels are not
attainable, we show that quantum correlations play a significant role in the
estimation. A comparison with the accuracy levels obtainable by direct
estimation (not through a probe) of the quantum target is also performed.Comment: 22 pages, 9 figure
Data analysis of gravitational-wave signals from spinning neutron stars. III. Detection statistics and computational requirements
We develop the analytic and numerical tools for data analysis of the
gravitational-wave signals from spinning neutron stars for ground-based laser
interferometric detectors. We study in detail the statistical properties of the
optimum functional that need to be calculated in order to detect the
gravitational-wave signal from a spinning neutron star and estimate its
parameters. We derive formulae for false alarm and detection probabilities both
for the optimal and the suboptimal filters. We assess the computational
requirements needed to do the signal search. We compare a number of criteria to
build sufficiently accurate templates for our data analysis scheme. We verify
the validity of our concepts and formulae by means of the Monte Carlo
simulations. We present algorithms by which one can estimate the parameters of
the continuous signals accurately.Comment: LaTeX, 45 pages, 13 figures, submitted to Phys. Rev.
Ballistic transport in random magnetic fields with anisotropic long-ranged correlations
We present exact theoretical results about energetic and dynamic properties
of a spinless charged quantum particle on the Euclidean plane subjected to a
perpendicular random magnetic field of Gaussian type with non-zero mean. Our
results refer to the simplifying but remarkably illuminating limiting case of
an infinite correlation length along one direction and a finite but strictly
positive correlation length along the perpendicular direction in the plane.
They are therefore ``random analogs'' of results first obtained by A. Iwatsuka
in 1985 and by J. E. M\"uller in 1992, which are greatly esteemed, in
particular for providing a basic understanding of transport properties in
certain quasi-two-dimensional semiconductor heterostructures subjected to
non-random inhomogeneous magnetic fields
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