7,472 research outputs found
Statistical Geometry in Quantum Mechanics
A statistical model M is a family of probability distributions, characterised
by a set of continuous parameters known as the parameter space. This possesses
natural geometrical properties induced by the embedding of the family of
probability distributions into the Hilbert space H. By consideration of the
square-root density function we can regard M as a submanifold of the unit
sphere in H. Therefore, H embodies the `state space' of the probability
distributions, and the geometry of M can be described in terms of the embedding
of in H. The geometry in question is characterised by a natural Riemannian
metric (the Fisher-Rao metric), thus allowing us to formulate the principles of
classical statistical inference in a natural geometric setting. In particular,
we focus attention on the variance lower bounds for statistical estimation, and
establish generalisations of the classical Cramer-Rao and Bhattacharyya
inequalities. The statistical model M is then specialised to the case of a
submanifold of the state space of a quantum mechanical system. This is pursued
by introducing a compatible complex structure on the underlying real Hilbert
space, which allows the operations of ordinary quantum mechanics to be
reinterpreted in the language of real Hilbert space geometry. The application
of generalised variance bounds in the case of quantum statistical estimation
leads to a set of higher order corrections to the Heisenberg uncertainty
relations for canonically conjugate observables.Comment: 32 pages, LaTex file, Extended version to include quantum measurement
theor
Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces
We investigate general differential relations connecting the respective
behavior s of the phase and modulo of probability amplitudes of the form
\amp{\psi_f}{\psi}, where is a fixed state in Hilbert space
and is a section of a holomorphic line bundle over some complex
parameter space. Amplitude functions on such bundles, while not strictly
holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions
involving the U(1) Berry-Simon connection on the parameter space. These
conditions entail invertible relations between the gradients of the phase and
modulo, therefore allowing for the reconstruction of the phase from the modulo
(or vice-versa) and other conditions on the behavior of either polar component
of the amplitude. As a special case, we consider amplitude functions valued on
the space of pure states, the ray space , where
transition probabilities have a geometric interpretation in terms of geodesic
distances as measured with the Fubini-Study metric. In conjunction with the
generalized Cauchy-Riemann conditions, this geodesic interpretation leads to
additional relations, in particular a novel connection between the modulus of
the amplitude and the phase gradient, somewhat reminiscent of the WKB formula.
Finally, a connection with geometric phases is established.Comment: 11 pages, 1 figure, revtex
Testing statistical bounds on entanglement using quantum chaos
Previous results indicate that while chaos can lead to substantial entropy
production, thereby maximizing dynamical entanglement, this still falls short
of maximality. Random Matrix Theory (RMT) modeling of composite quantum
systems, investigated recently, entails an universal distribution of the
eigenvalues of the reduced density matrices. We demonstrate that these
distributions are realized in quantized chaotic systems by using a model of two
coupled and kicked tops. We derive an explicit statistical universal bound on
entanglement, that is also valid for the case of unequal dimensionality of the
Hilbert spaces involved, and show that this describes well the bounds observed
using composite quantized chaotic systems such as coupled tops.Comment: 5 pages, 3 figures, to appear in PRL. New title. Revised abstract and
some changes in the body of the pape
Hidden variable interpretation of spontaneous localization theory
The spontaneous localization theory of Ghirardi, Rimini, and Weber (GRW) is a
theory in which wavepacket reduction is treated as a genuine physical process.
Here it is shown that the mathematical formalism of GRW can be given an
interpretation in terms of an evolving distribution of particles on
configuration space similar to Bohmian mechanics (BM). The GRW wavefunction
acts as a pilot wave for the set of particles. In addition, a continuous stream
of noisy information concerning the precise whereabouts of the particles must
be specified. Nonlinear filtering techniques are used to determine the dynamics
of the distribution of particles conditional on this noisy information and
consistency with the GRW wavefunction dynamics is demonstrated. Viewing this
development as a hybrid BM-GRW theory, it is argued that, besides helping to
clarify the relationship between the GRW theory and BM, its merits make it
worth considering in its own right.Comment: 13 page
Nuclear Structure Calculations with Low-Momentum Potentials in a Model Space Truncation Approach
We have calculated the ground-state energy of the doubly magic nuclei 4He,
16O and 40Ca within the framework of the Goldstone expansion starting from
various modern nucleon-nucleon potentials. The short-range repulsion of these
potentials has been renormalized by constructing a low-momentum potential
V-low-k. We have studied the connection between the cutoff momemtum Lambda and
the size of the harmonic oscillator space employed in the calculations. We have
found a fast convergence of the results with a limited number of oscillator
quanta.Comment: 6 pages, 8 figures, to be published on Physical Review
Analgesic prescribing trends in a national sample of older veterans with osteoarthritis: 2012-2017
Few investigations examine patterns of opioid and nonopioid analgesic prescribing and concurrent pain intensity ratings before and after institution of safer prescribing programs such as the October 2013 Veterans Health Administration system-wide Opioid Safety Initiative (OSI) implementation. We conducted a quasi-experimental pre–post observational study of all older U.S. veterans (≥50 years old) with osteoarthritis of the knee or hip. All associated outpatient analgesic prescriptions and outpatient pain intensity ratings from January 1, 2012 to December 31, 2016, were analyzed with segmented regression of interrupted time series. Standardized monthly rates for each analgesic class (total, opioid, nonsteroidal anti-inflammatory drug, acetaminophen, and other study analgesics) were analyzed with segmented negative binomial regression models with overall slope, step, and slope change. Similarly, segmented linear regression was used to analyze pain intensity ratings and percentage of those reporting pain. All models were additionally adjusted for age, sex, and race. Before OSI implementation, total analgesic prescriptions showed a steady rise, abruptly decreasing to a flat trajectory after OSI implementation. This trend was primarily due to a decrease in opioid prescribing after OSI. Total prescribing after OSI implementation was partially compensated by continuing increased prescribing of other study analgesics as well as a significant rise in acetaminophen prescriptions (post-OSI). No changes in nonsteroidal anti-inflammatory drug prescribing were seen. A small rise in the percentage of those reporting pain but not mean pain intensity ratings continued over the study period with no changes associated with OSI. Changes in analgesic prescribing trends were not paralleled by changes in reported pain intensity for older veterans with osteoarthritis
Dynamical state reduction in an EPR experiment
A model is developed to describe state reduction in an EPR experiment as a
continuous, relativistically-invariant, dynamical process. The system under
consideration consists of two entangled isospin particles each of which undergo
isospin measurements at spacelike separated locations. The equations of motion
take the form of stochastic differential equations. These equations are solved
explicitly in terms of random variables with a priori known probability
distribution in the physical probability measure. In the course of solving
these equations a correspondence is made between the state reduction process
and the problem of classical nonlinear filtering. It is shown that the solution
is covariant, violates Bell inequalities, and does not permit superluminal
signaling. It is demonstrated that the model is not governed by the Free Will
Theorem and it is argued that the claims of Conway and Kochen, that there can
be no relativistic theory providing a mechanism for state reduction, are false.Comment: 19 pages, 3 figure
Intruder States and their Local Effect on Spectral Statistics
The effect on spectral statistics and on the revival probability of intruder
states in a random background is analysed numerically and with perturbative
methods. For random coupling the intruder does not affect the GOE spectral
statistics of the background significantly, while a constant coupling causes
very strong correlations at short range with a fourth power dependence of the
spectral two-point function at the origin.The revival probability is
significantly depressed for constant coupling as compared to random coupling.Comment: 18 pages, 10 Postscript figure
Inferring Networks of Substitutable and Complementary Products
In a modern recommender system, it is important to understand how products
relate to each other. For example, while a user is looking for mobile phones,
it might make sense to recommend other phones, but once they buy a phone, we
might instead want to recommend batteries, cases, or chargers. These two types
of recommendations are referred to as substitutes and complements: substitutes
are products that can be purchased instead of each other, while complements are
products that can be purchased in addition to each other.
Here we develop a method to infer networks of substitutable and complementary
products. We formulate this as a supervised link prediction task, where we
learn the semantics of substitutes and complements from data associated with
products. The primary source of data we use is the text of product reviews,
though our method also makes use of features such as ratings, specifications,
prices, and brands. Methodologically, we build topic models that are trained to
automatically discover topics from text that are successful at predicting and
explaining such relationships. Experimentally, we evaluate our system on the
Amazon product catalog, a large dataset consisting of 9 million products, 237
million links, and 144 million reviews.Comment: 12 pages, 6 figure
- …