6,663 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
Research on nonlinear optical materials: an assessment. IV. Photorefractive and liquid crystal materials
This panel considered two separate subject areas: photorefractive materials used for nonlinear optics and liquid crystal materials used in light valves. Two related subjects were not considered due to lack of expertise on the panel: photorefractive materials used in light valves and liquid crystal materials used in nonlinear optics. Although the inclusion of a discussion of light valves by a panel on nonlinear optical materials at first seems odd, it is logical because light valves and photorefractive materials perform common functions
A recurrent network model of somatosensory parametric working memory in the prefrontal cortex
A parametric working memory network stores the information of an analog stimulus in the form of persistent neural activity that is monotonically tuned to the stimulus. The family of persistent firing patterns with a continuous range of firing rates must all be realizable under exactly the same external conditions (during the delay when the transient stimulus is withdrawn). How this can be accomplished by neural mechanisms remains an unresolved question. Here we present a recurrent cortical network model of irregularly spiking neurons that was designed to simulate a somatosensory working memory experiment with behaving monkeys. Our model reproduces the observed positively and negatively monotonic persistent activity, and heterogeneous tuning curves of memory activity. We show that fine-tuning mathematically corresponds to a precise alignment of cusps in the bifurcation diagram of the network. Moreover, we show that the fine-tuned network can integrate stimulus inputs over several seconds. Assuming that such time integration occurs in neural populations downstream from a tonically persistent neural population, our model is able to account for the slow ramping-up and ramping-down behaviors of neurons observed in prefrontal cortex
Martingale Models for Quantum State Reduction
Stochastic models for quantum state reduction give rise to statistical laws
that are in most respects in agreement with those of quantum measurement
theory. Here we examine the correspondence of the two theories in detail,
making a systematic use of the methods of martingale theory. An analysis is
carried out to determine the magnitude of the fluctuations experienced by the
expectation of the observable during the course of the reduction process and an
upper bound is established for the ensemble average of the greatest
fluctuations incurred. We consider the general projection postulate of L\"uders
applicable in the case of a possibly degenerate eigenvalue spectrum, and derive
this result rigorously from the underlying stochastic dynamics for state
reduction in the case of both a pure and a mixed initial state. We also analyse
the associated Lindblad equation for the evolution of the density matrix, and
obtain an exact time-dependent solution for the state reduction that explicitly
exhibits the transition from a general initial density matrix to the L\"uders
density matrix. Finally, we apply Girsanov's theorem to derive a set of simple
formulae for the dynamics of the state in terms of a family of geometric
Brownian motions, thereby constructing an explicit unravelling of the Lindblad
equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics
On the dominance of J(P)=0(+) ground states in even-even nuclei from random two-body interactions
Recent calculations using random two-body interactions showed a preponderance
of J(P)=0(+) ground states, despite the fact that there is no strong pairing
character in the force. We carry out an analysis of a system of identical
particles occupying orbits with j=1/2, 3/2 and 5/2 and discuss some general
features of the spectra derived from random two-body interactions. We show that
for random two-body interactions that are not time-reversal invariant the
dominance of 0(+) states in this case is more pronounced, indicating that
time-reversal invariance cannot be the origin of the 0(+) dominance.Comment: 8 pages, 3 tables and 3 figures. Phys. Rev. C, in pres
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