Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.     

Effect of atomic decay on purity,total correlations and entanglement of pure and mixed states

The atomic decay for a two level atom interacting with a single mode of electromagnetic field is considered. In particular for a coherent state or statistical mixture (SM) of two opposite coherent states as initial field states, the exact solution of the master equation is found. Effect of the atomic damping on the partial entropies of the atom or the field and the total entropy as a measures of the purity loss is investigated. The degree of entanglement by the negativity and the mutual information and the atomic coherence through the master equation is studied.     

Measuring information beyond communication theory—Why some generalized information measures may be useful,others not

Summary (and Keywords) Non-communication models for information theory: games and experiments. Measures of uncertainty and information: entropies, divergences, information improvements.Some useful properties of information measures, symmetry, bounds, behaviour under composition, branching, conditional measures, sources. Rényi measures, measures of higher degree.Promising and not so promising generalizations. Measures which depend not just upon the probabilities but (also) upon the subject matters.     

Further results on the relationship between μ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains

This note considers continuous-time Markov chains whose state space consists of an irreducible class, C, and an absorbing state which is accessible from C. The purpose is to provide results on μ-invariant and μ-subinvariant measures where absorption occurs with probability less than one. In particular, the well-known premise that the μ-invariant measure, m, for the transition rates be finite is replaced by the more natural premise that m be finite with respect to the absorption probabilities. The relationship between μ-invariant measures and quasi-stationary distributions is discussed.     

The idealized quantum two-slit gedanken experiment revisited—Criticism and reinterpretation

An idealized two-slit experiment is envisaged in which the hypothetical experimental set-up is constructed in such a way as to resemble a toy model giving information about the structure of quantum space–time itself. Thus starting from a very simple equation which may be interpreted as a physical realization of Gödel’s undecidability theorem, we proceed to show that space–time is very likely to be akin to a fuzzy Kähler-like manifold on the quantum level. This remarkable manifold transforms gradually into a classical space–time as we decrease the resolution in a way reversibly analogous to the processes of recovering classical space–time from the Riemannian space of general relativity.The paper’s main philosophy is to emphasize that the quintessence of the two-slit experiment as well as Feynman’s path integral could be given a different interpretation by altering our classical concept of space–time geometry and topology. In turn this would be in keeping with the development in theoretical physics since special and subsequently general relativity. In the final analysis it would seem that we have two different yet, from a positivistic philosophy viewpoint, completely equivalent alternatives to view quantum physics. Either we insist on what we see in our daily experiences, namely, a smooth four-dimensional space–time, and then accept, whether we like it or not, things such as probability waves and complex probabilities. Alternatively, we could see behind the façade of classical space–time a far more elaborate and highly complex fuzzy space–time with infinite hierarchical dimensions such as the so-called Fuzzy K3 or E–Infinity space–time and as a reward for this imaginative picture we can return to real probabilities without a phase and an almost classical picture with the concept of a particle’s path restored. We say almost classical because non-linear dynamics and deterministic chaos have long shown the central role of randomness in classical mechanics and this is reinforced once more in our model which is directly related not to Newtonian motion, but rather to a diffusion-like random walk similar to that used with great skill by Einstein and later on by Nagasawa and particularly the English-Canadian physicist Garnet Ord.     

Entropic Moments and Domains of Attraction on Countable Alphabets

Modern information theory is largely developed in connection with random elements residing in large, complex, and discrete data spaces, or alphabets. Lacking natural metrization and hence moments, the associated probability and statistics theory must rely on information measures in the form of various entropies, for example, Shannon’s entropy, mutual information and Kullback–Leibler divergence, which are functions of an entropic basis in the form of a sequence of entropic moments of varying order. The entropicmoments collectively characterize the underlying probability distribution on the alphabet, and hence provide an opportunity to develop statistical procedures for their estimation. As such statistical development becomes an increasingly important line of research in modern data science, the relationship between the underlying distribution and the asymptotic behavior of the entropic moments, as the order increases, becomes a technical issue of fundamental importance. This paper offers a general methodology to capture the relationship between the rates of divergence of the entropic moments and the types of underlying distributions, for a special class of distributions. As an application of the established results, it is demonstrated that the asymptotic normality of the remarkable Turing’s formula for missing probabilities holds under distributions with much thinner tails than those previously known.     

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C^∗-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron.Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra.Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.     

Statistical maps and generalized random walks

We continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.     

Hilbert space representations of decoherence functionals and quantum measures

We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.     

Consciousness as a state of matter

We examine the hypothesis that consciousness can be understood as a state of matter, “perceptronium”, with distinctive information processing abilities. We explore four basic principles that may distinguish conscious matter from other physical systems such as solids, liquids and gases: the information, integration, independence and dynamics principles. If such principles can identify conscious entities, then they can help solve the quantum factorization problem: why do conscious observers like us perceive the particular Hilbert space factorization corresponding to classical space (rather than Fourier space, say), and more generally, why do we perceive the world around us as a dynamic hierarchy of objects that are strongly integrated and relatively independent? Tensor factorization of matrices is found to play a central role, and our technical results include a theorem about Hamiltonian separability (defined using Hilbert–Schmidt superoperators) being maximized in the energy eigenbasis. Our approach generalizes Giulio Tononi’s integrated information framework for neural-network-based consciousness to arbitrary quantum systems, and we find interesting links to error-correcting codes, condensed matter criticality, and the Quantum Darwinism program, as well as an interesting connection between the emergence of consciousness and the emergence of time.     

Nonadiabatic quantum chaos in atom optics

Coherent dynamics of atomic matter waves in a standing-wave laser field is studied. In the dressed-state picture, wave packets of ballistic two-level atoms propagate simultaneously in two optical potentials. The probability to make a transition from one potential to another one is maximal when centroids of wave packets cross the field nodes and is given by a simple formula with the single exponent, the Landau-Zener parameter κ. If κ ? 1, the motion is essentially adiabatic. If κ ? 1, it is (almost) resonant and periodic. If κ ? 1, atom makes nonadiabatic transitions with a splitting of its wave packet at each node and strong complexification of the wave function as compared to the two other cases. This effect is referred as nonadiabatic quantum chaos. Proliferation of wave packets at κ ? 1 is shown to be connected closely with chaotic center-of-mass motion in the semiclassical theory of point-like atoms with positive values of the maximal Lyapunov exponent. The quantum-classical correspondence established is justified by the fact that the Landau-Zener parameter κ specifies the regime of the semiclassical dynamical chaos in the map simulating chaotic center-of-mass motion. Manifestations of nonadiabatic quantum chaos are found in the behavior of the momentum and position probabilities.     

Connections between generalized graph entropies and graph energy

Dehmer and Mowshowitz introduced a class of generalized graph entropies using known information‐theoretic measures. These measures rely on assigning a probability distribution to a graph. In this article, we prove some extremal properties of such generalized graph entropies by using the graph energy and the spectral moments. Moreover, we study the relationships between the generalized graph entropies and compute the values of the generalized graph entropies for special graph classes. © 2014 Wiley Periodicals, Inc. Complexity 21: 35–41, 2015     

Conditional Entropy and Information in Quantum Systems

The concepts of conditional entropy of a physical system given the state of another system and of information in a physical system about another one are generalized for quantum systems. The fundamental difference between the classical case and the quantum one is that the entropy and information in quantum systems depend on the choice of measurements performed over the systems. It is shown that some equalities of the classical information theory turn into inequalities for the generalized quantities. Specific quantum phenomena such as EPR pairs and superdense coding are described and explained in terms of the generalized conditional entropy and information.     

n-Monotone exact functionals

We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals.     

Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces

Let μ be a Gaussian measure on a separable Banach space. We prove a tight link between the logarithmic small ball probabilities of μ and certain moment generating functions. Based upon this link we provide a new lower bound for the distortion-rate function (DRF) against the small ball function. This allows us to use results of the theory of small ball probabilities to deduce lower bounds for the DRF. In particular, we obtain the correct weak asymptotics of the distortion rate function in many important cases (e.g. Brownian motion).     

Probability, geometry, and irreversibility in quantum mechanics

The main conclusion of this paper is that the Bell–Wigner–Accardi theory of quantum probabilities in spin systems may be placed within the general operator trigonometry developed independently by this author about 30 years ago. The use of the Grammian from the operator trigonometry simplifies and clarifies the analysis of Wigner. A general triangle inequality from the operator trigonometry clarifies and generalizes the analysis of Accardi. The statistical meaning of the complex numbers in quantum mechanics is seen to be that of the natural geometry of the operator trigonometry. A new connection of the operator trigonometry to CP symmetry violation is established.     

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.     

On distributionally chaotic and null systems

By a topological dynamical system, we mean a pair (X,f), where X is a compactum and f is a continuous self-map on X. A system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers. We show that there exists a null system which is distributionally chaotic. This system admits open distributionally scrambled sets, and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space. Finally such system can even exist on continua.