Information-theoretic significance of the Wigner distribution

A coarse grained Wigner distribution p_{W}(x,u) obeying positivity derives out of information-theoretic considerations. Let p(x,u) be the unknown joint PDF (probability density function) on position- and momentum fluctuations x,u for a pure state particle. Suppose that the phase part Psi(x,z) of its Fourier transform F.T.[p(x,u)]=|Z(x,z)|exp[iPsi(x,z)] is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function Psi(x,z). The extremum solution gives an output PDF p(x,u) that is the convolution of the Wigner p_{W}(x,u) with an instrument function defining uncertainty in either position x or momentum u. The convolution arises naturally out of the approach, and is one-dimensional, in comparison with the two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes a(x) are the same at their boundaries (Examples: states a(x) with positive parity; with periodic boundary conditions; free particle trapped in a box).Comment: pdf version has 16 pages. No figures. Accepted for publ. in PR

Fisher's arrow of `time' in cosmological coherent phase space

Fisher's arrow of `time' in a cosmological phase space defined as in quantum optics (i.e., whose points are coherent states) is introduced as follows. Assuming that the phase space evolution of the universe starts from an initial squeezed cosmological state towards a final thermal one, a Fokker-Planck equation for the time-dependent, cosmological Q phase space probability distribution can be written down. Next, using some recent results in the literature, we derive an information arrow of time for the Fisher phase space cosmological entropy based on the Q function. We also mention the application of Fisher's arrow of time to stochastic inflation modelsComment: 10 pages, LaTex, Honorable Mention at GRF-199

Fisher information, Wehrl entropy, and Landau Diamagnetism

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. It is shown that such a problem can be "translated" into that of the thermal harmonic oscillator. We discover a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.Comment: no figures. Physical Review B (2005) in pres

Power laws of complex systems from Extreme physical information

Many complex systems obey allometric, or power, laws y=Yx^{a}. Here y is the measured value of some system attribute a, Y is a constant, and x is a stochastic variable. Remarkably, for many living systems the exponent a is limited to values +or- n/4, n=0,1,2... Here x is the mass of a randomly selected creature in the population. These quarter-power laws hold for many attributes, such as pulse rate (n=-1). Allometry has, in the past, been theoretically justified on a case-by-case basis. An ultimate goal is to find a common cause for allometry of all types and for both living and nonliving systems. The principle I - J = extrem. of Extreme physical information (EPI) is found to provide such a cause. It describes the flow of Fisher information J => I from an attribute value a on the cell level to its exterior observation y. Data y are formed via a system channel function y = f(x,a), with f(x,a) to be found. Extremizing the difference I - J through variation of f(x,a) results in a general allometric law f(x,a)= y = Yx^{a}. Darwinian evolution is presumed to cause a second extremization of I - J, now with respect to the choice of a. The solution is a=+or-n/4, n=0,1,2..., defining the particular powers of biological allometry. Under special circumstances, the model predicts that such biological systems are controlled by but two distinct intracellular information sources. These sources are conjectured to be cellular DNA and cellular transmembrane ion gradient

Scaling in a continuous time model for biological aging

In this paper we consider a generalization to the asexual version of the Penna model for biological aging, where we take a continuous time limit. The genotype associated to each individual is an interval of real numbers over which Dirac $\delta$--functions are defined, representing genetically programmed diseases to be switched on at defined ages of the individual life. We discuss two different continuous limits for the evolution equation and two different mutation protocols, to be implemented during reproduction. Exact stationary solutions are obtained and scaling properties are discussed.Comment: 10 pages, 6 figure

Quantum theory of incompatible observations

Maximum likelihood principle is shown to be the best measure for relating the experimental data with the predictions of quantum theory.Comment: 3 page

Quantum limits of super-resolution in reconstruction of optical objects

We investigate analytically and numerically the role of quantum fluctuations in reconstruction of optical objects from diffraction-limited images. Taking as example of an input object two closely spaced Gaussian peaks we demonstrate that one can improve the resolution in the reconstructed object over the classical Rayleigh limit. We show that the ultimate quantum limit of resolution in such reconstruction procedure is determined not by diffraction but by the signal-to-noise ratio in the input object. We formulate a quantitative measure of super-resolution in terms of the optical point-spread function of the system.Comment: 23 pages, 7 figures. Submitted to Physical Review A e-mail: [email protected]

Wigner-Yanase skew information as tests for quantum entanglement

A Bell-type inequality is proposed in terms of Wigner-Yanase skew information, which is quadratic and involves only one local spin observable at each site. This inequality presents a hierarchic classification of all states of multipartite quantum systems from separable to fully entangled states, which is more powerful than the one presented by quadratic Bell inequalities from two-entangled to fully entangled states. In particular, it is proved that the inequality provides an exact test to distinguish entangled from nonentangled pure states of two qubits. Our inequality sheds considerable light on relationships between quantum entanglement and information theory.Comment: 5 page

Value of dual testing for identifying tuberculous infection

Setting: A rural community in Chingleput district in Tamil Nadu state in south India. Objective: To determine the value of dual testing with PPD-S and PPD-B for identifying subjects with a tuberculous infection. Design: About 240,000 subjects in rural south India, all of whom were tested initially with PPD-S and PPD-B, were followed up for 15 years, mainly by total population survey once in every 212 years. The incidence of culture-positive tuberculosis was estimated using life-table technique. Results: Among 17,530 subjects with an intermediate reaction (8–11 mm) to PPD-S at intake, 285 with an induration to PPD-S exceeding the induration to PPD-B by at least 2 mm, had a significantly higher incidence of culture-positive tuberculosis than the remaining (154 and 93 per 100,000), and similarly 481 who had an induration of o10mm to PPD-B compared to those with X10mm (131 and 93 per 100,000). These subjects may be regarded as having a tuberculous infection. Infection with non-tuberculous mycobacteria conferred protection of about 30% against the development of tuberculosis over a 15-year period. Conclusion: In subjects with an intermediate reaction (8–11 mm) to PPD-S, dual testing with PPD-B enabled identification of those with a tuberculous infection. Most of the reactions were due to non-tuberculous mycobacteria