Extension of Information Geometry to Non-statistical Systems: Some Examples

Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.Comment: 8 pages, to be published in the proceedings of GSI2015, Lecture Notes in Computer Science, Springe

Quantum and Fisher Information from the Husimi and Related Distributions

The two principal/immediate influences -- which we seek to interrelate here -- upon the undertaking of this study are papers of Zyczkowski and Slomczy\'nski (J. Phys. A 34, 6689 [2001]) and of Petz and Sudar (J. Math. Phys. 37, 2262 [1996]). In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudar work (completing a program of Chentsov), the quantum analogue of the (classically unique) Fisher information (montone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically-defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudar? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys' prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions -- recently studied by Srednicki (Phys. Rev. A 71, 052107 [2005]) -- formed by normalizing the volume elements of the various information metrics, is also discussed in our context.Comment: 27 pages, 10 figures, slight revisions, to appear in J. Math. Phy

Hilbert-Schmidt Separability Probabilities and Noninformativity of Priors

The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (\sigma_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_{\alpha}). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},\sigma_{1}^2,\alpha) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < \alpha < \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featured. Our model can be contrasted with the three-parameter (b_{q}, \sigma_{q}^2,q) one of Abe and Rajagopal, which employs a q(Tsallis)-parameter rather than $\alpha$, and has simply q-invariant HS separability probabilities of 1/2. Our results emerge in a study initially focused on embedding certain information metrics over the two-level quantum systems into a q-framework. We find evidence that Srednicki's recently-stated biasedness criterion for noninformative priors yields rankings of priors fully consistent with an information-theoretic test of Clarke, previously applied to quantum systems by Slater.Comment: 26 pages, 12 figure

A large synoptic dataset of coastal in situ observations

Since the beginning of 2004 MUMM has performed a series of moorings in the Belgian Coastal Zone with a tripod or benthic lander, equipped with a series of oceanographic sensors. Current meters such as an Acous1c Doppler Profiler (ADP) and an Acoustic Doppler Velocimeter (ADV), a CT-­sensor capable of measuring temperature and salinity, op1cal (OBS) and acous1c backsca\er sensors (ADP and ADV) to measure suspended sediment concentra1on, a LISST 100 suspended par1cle sizer, a passive Cetacean monitoring device (C-­PoD) and a passive sampling device for chemical monitoring have ever since been moored for more than 1750 days. From autumn 2009 a permanent coastal observatory has been installed at MOW1, located to the west of the entrance to the Zeebrugge harbor. Other mooring sites include more offshore loca1ons, such as the Kwintebank, Gootebank, MOW0 and the offshore windmill farms and a very nearshore loca1on (Blankenberge)

Regularization as quantization in reducible representations of CCR

A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincar\'e covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to R\'enyi's $\alpha$. It is shown that the ``Shannon limit" $\alpha\to 1$ plays here a role of correspondence principle with the standard regularized formalism.Comment: minor extensions, version submitted for publicatio

Stretched Exponential Relaxation in the Biased Random Voter Model

We study the relaxation properties of the voter model with i.i.d. random bias. We prove under mild condions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent $d/(d+\alpha)$, where $0<\alpha\le 2$ depends on the transition rates of the non-biased voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe

Two early-stage inverse power-law dyamics in nonlinear complex systems far-from equilibrium

We consider the dynamics of the charge carriers in a tunneling-enhanced percolation network, named as a Random Resistor cum Tunneling-bond Network (RRTN), where we allow tunneling in the gap between two randomly thrown nearest neighbour metallic bonds only. Our earlier studies involve the dc and the ac nonlinear response, the percolative aspects, dielectric breakdown, low-temperature variable range hopping (VRH) conduction, etc. in the RRTN. Here we study the non-equilibrium dynamics of the carriers. With two far-from- equilibrium, initial inverse power-law relaxations extending over several decades, the dynamics has a lot of similarities with a wide variety of naturally occuring avalance-like, run-away phenomena in driven, disordered systems with statistically correlated randomness. In the power-law regime, the RRTN violates the Boltzmann's (or Debye) relaxation time approximation strongly. Beyond this regime, the response relaxes exponentially fast (acquires one time-scale) to a steady-state, and thus the relaxation approximation becomes exact.Comment: RevTex4, 6 pages, 4 figure

Cerebral spectroscopic and oxidative stress studies in patients with schizophrenia who have dangerously violently offended

<p>Abstract</p> <p>Background</p> <p>The aim of this study was to bring together all the results of <it>in vivo </it>studies of ethane excretion and cerebral spectroscopy in patients with schizophrenia who have dangerously seriously violently offended in order to determine the extent to which they shed light on the degree to which the membrane phospholipid hypothesis and the actions of free radicals and other reactive species are associated with cerebral pathophysiological mechanisms in this group of patients.</p> <p>Methods</p> <p>The patients investigated were inpatients from a medium secure unit with a DSM-IV-TR diagnosis of schizophrenia. There was no history of alcohol dependency or any other comorbid psychoactive substance misuse disorder. Expert psychiatric opinion, accepted in court, was that all these patients had violently offended directly as a result of schizophrenia prior to admission. These offences consisted of homicide, attempted murder or wounding with intent to cause grievous bodily harm. Excreted ethane was analyzed and quantified by gas chromatography and mass spectrometry (<it>m</it>/<it>z </it>= 30). 31-phosphorus magnetic resonance spectroscopy data were obtained at a magnetic field strength of 1.5 T using an image-selected <it>in vivo </it>spectroscopy sequence (TR = 10 s; 64 signal averages localized on a 70 × 70 × 70 mm³voxel).</p> <p>Results</p> <p>Compared with age- and sex-matched controls, in the patient group the mean alveolar ethane level was higher (<it>p </it>< 0.0005), the mean cerebral beta-nucleotide triphosphate was lower (<it>p </it>< 0.04) and the mean gamma-nucleotide triphosphate was higher (<it>p </it>< 0.04). There was no significant difference between the two groups in respect of phosphomonoesters, phosphodiesters or broad resonances.</p> <p>Conclusion</p> <p>Our results are not necessarily inconsistent with the membrane phospholipid hypothesis, given that the patients studied suffered predominantly from positive symptoms of schizophrenia. The results suggest that there is increased cerebral mitochondrial oxidative phosphorylation in patients with schizophrenia who have dangerously and seriously violently offended, with an associated increase in oxygen flux and subsequent electron 'leakage' from the electron transport chain leading to the formation of superoxide radicals and other reactive oxygen species. In turn, these reactive species might cause increased lipid peroxidation in neuroglial membranes, thereby accounting for the observation of increased ethane excretion.</p