1,507 research outputs found
Morphogen Gradient from a Noisy Source
We investigate the effect of time-dependent noise on the shape of a morphogen
gradient in a developing embryo. Perturbation theory is used to calculate the
deviations from deterministic behavior in a simple reaction-diffusion model of
robust gradient formation, and the results are confirmed by numerical
simulation. It is shown that such deviations can disrupt robustness for
sufficiently high noise levels, and the implications of these findings for more
complex models of gradient-shaping pathways are discussed.Comment: Four pages, three figure
Optimal quantum detectors for unambiguous detection of mixed states
We consider the problem of designing an optimal quantum detector that
distinguishes unambiguously between a collection of mixed quantum states. Using
arguments of duality in vector space optimization, we derive necessary and
sufficient conditions for an optimal measurement that maximizes the probability
of correct detection. We show that the previous optimal measurements that were
derived for certain special cases satisfy these optimality conditions. We then
consider state sets with strong symmetry properties, and show that the optimal
measurement operators for distinguishing between these states share the same
symmetries, and can be computed very efficiently by solving a reduced size
semidefinite program.Comment: Submitted to Phys. Rev.
Quantum Detection with Unknown States
We address the problem of distinguishing among a finite collection of quantum
states, when the states are not entirely known. For completely specified
states, necessary and sufficient conditions on a quantum measurement minimizing
the probability of a detection error have been derived. In this work, we assume
that each of the states in our collection is a mixture of a known state and an
unknown state. We investigate two criteria for optimality. The first is
minimization of the worst-case probability of a detection error. For the second
we assume a probability distribution on the unknown states, and minimize of the
expected probability of a detection error.
We find that under both criteria, the optimal detectors are equivalent to the
optimal detectors of an ``effective ensemble''. In the worst-case, the
effective ensemble is comprised of the known states with altered prior
probabilities, and in the average case it is made up of altered states with the
original prior probabilities.Comment: Refereed version. Improved numerical examples and figures. A few
typos fixe
Universal 2-local Hamiltonian Quantum Computing
We present a Hamiltonian quantum computation scheme universal for quantum
computation (BQP). Our Hamiltonian is a sum of a polynomial number (in the
number of gates L in the quantum circuit) of time-independent, constant-norm,
2-local qubit-qubit interaction terms. Furthermore, each qubit in the system
interacts only with a constant number of other qubits. The computer runs in
three steps - starts in a simple initial product-state, evolves it for time of
order L^2 (up to logarithmic factors) and wraps up with a two-qubit
measurement. Our model differs from the previous universal 2-local Hamiltonian
constructions in that it does not use perturbation gadgets, does not need large
energy penalties in the Hamiltonian and does not need to run slowly to ensure
adiabatic evolution.Comment: recomputed the necessary number of interactions, new geometric
layout, added reference
Undersampled Phase Retrieval with Outliers
We propose a general framework for reconstructing transform-sparse images
from undersampled (squared)-magnitude data corrupted with outliers. This
framework is implemented using a multi-layered approach, combining multiple
initializations (to address the nonconvexity of the phase retrieval problem),
repeated minimization of a convex majorizer (surrogate for a nonconvex
objective function), and iterative optimization using the alternating
directions method of multipliers. Exploiting the generality of this framework,
we investigate using a Laplace measurement noise model better adapted to
outliers present in the data than the conventional Gaussian noise model. Using
simulations, we explore the sensitivity of the method to both the
regularization and penalty parameters. We include 1D Monte Carlo and 2D image
reconstruction comparisons with alternative phase retrieval algorithms. The
results suggest the proposed method, with the Laplace noise model, both
increases the likelihood of correct support recovery and reduces the mean
squared error from measurements containing outliers. We also describe exciting
extensions made possible by the generality of the proposed framework, including
regularization using analysis-form sparsity priors that are incompatible with
many existing approaches.Comment: 11 pages, 9 figure
Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination
We consider two different optimized measurement strategies for the
discrimination of nonorthogonal quantum states. The first is conclusive
discrimination with a minimum probability of inferring an erroneous result, and
the second is unambiguous, i. e. error-free, discrimination with a minimum
probability of getting an inconclusive outcome, where the measurement fails to
give a definite answer. For distinguishing between two mixed quantum states, we
investigate the relation between the minimum error probability achievable in
conclusive discrimination, and the minimum failure probability that can be
reached in unambiguous discrimination of the same two states. The latter turns
out to be at least twice as large as the former for any two given states. As an
example, we treat the case that the state of the quantum system is known to be,
with arbitrary prior probability, either a given pure state, or a uniform
statistical mixture of any number of mutually orthogonal states. For this case
we derive an analytical result for the minimum probability of error and perform
a quantitative comparison to the minimum failure probability.Comment: Replaced by final version, accepted for publication in Phys. Rev. A.
Revtex4, 6 pages, 3 figure
Mixed quantum state detection with inconclusive results
We consider the problem of designing an optimal quantum detector with a fixed
rate of inconclusive results that maximizes the probability of correct
detection, when distinguishing between a collection of mixed quantum states. We
develop a sufficient condition for the scaled inverse measurement to maximize
the probability of correct detection for the case in which the rate of
inconclusive results exceeds a certain threshold. Using this condition we
derive the optimal measurement for linearly independent pure-state sets, and
for mixed-state sets with a broad class of symmetries. Specifically, we
consider geometrically uniform (GU) state sets and compound geometrically
uniform (CGU) state sets with generators that satisfy a certain constraint.
We then show that the optimal measurements corresponding to GU and CGU state
sets with arbitrary generators are also GU and CGU respectively, with
generators that can be computed very efficiently in polynomial time within any
desired accuracy by solving a semidefinite programming problem.Comment: Submitted to Phys. Rev.
Nonlinear Peculiar-Velocity Analysis and PCA
We allow for nonlinear effects in the likelihood analysis of peculiar
velocities, and obtain ~35%-lower values for the cosmological density parameter
and for the amplitude of mass-density fluctuations. The power spectrum in the
linear regime is assumed to be of the flat LCDM model (h=0.65, n=1) with only
Om_m free. Since the likelihood is driven by the nonlinear regime, we "break"
the power spectrum at k_b=0.2 h/Mpc and fit a two-parameter power-law at k>k_b.
This allows for an unbiased fit in the linear regime. Tests using improved mock
catalogs demonstrate a reduced bias and a better fit. We find for the Mark III
and SFI data Om_m=0.35+-0.09$ with sigma_8*Om_m^0.6=0.55+-0.10 (90% errors).
When allowing deviations from \lcdm, we find an indication for a wiggle in the
power spectrum in the form of an excess near k~0.05 and a deficiency at k~0.1
h/Mpc --- a "cold flow" which may be related to a feature indicated from
redshift surveys and the second peak in the CMB anisotropy. A chi^2 test
applied to principal modes demonstrates that the nonlinear procedure improves
the goodness of fit. The Principal Component Analysis (PCA) helps identifying
spatial features of the data and fine-tuning the theoretical and error models.
We address the potential for optimal data compression using PCA.Comment: 15 pages, LaTex, in Mining the Sky, July 31 - August 4, 2000,
Garching, German
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