Upper entropy axioms and lower entropy axioms

The paper suggests the concepts of an upper entropy and a lower entropy. We propose a new axiomatic definition, namely, upper entropy axioms, inspired by axioms of metric spaces, and also formulate lower entropy axioms. We also develop weak upper entropy axioms and weak lower entropy axioms. Their conditions are weaker than those of Shannon–Khinchin axioms and Tsallis axioms, while these conditions are stronger than those of the axiomatics based on the first three Shannon–Khinchin axioms. The subadditivity and strong subadditivity of entropy are obtained in the new axiomatics. Tsallis statistics is a special case of satisfying our axioms. Moreover, different forms of information measures, such as Shannon entropy, Daroczy entropy, Tsallis entropy and other entropies, can be unified under the same axiomatics.     

Extracting deep information from limited observations on an evolved social network

We provide empirical evidence that in a social network which evolves over time, it is possible to extract deep information about the system from limited observations. In this paper, we consider a simple piece of readily available evidence on access to financial services by individuals in the UK. Detailed statistical analysis has shown that the decisions of agents on whether or not to have a basic financial account such as a bank account is heavily influenced by other individuals on their social network. We consider a small amount of straightforward and readily accessible information. We deduce from this, using an agent-based model, the type of social network across which information and influence on behaviour flows between agents in this context. Specifically, we show that information appears to flow across a small world network.     

Small accessible quantum information does not imply security

The security of quantum key distribution is typically defined in terms of the mutual information between the distributed key S and the outcome of an optimal measurement applied to the adversary's system. We show that even if this so-called accessible information is small, the key S might not be secure enough to be used in applications such as one-time pad encryption. This flaw is due to a locking property of the accessible information: one additional (physical) bit of information can increase the accessible information by more than one bit.     

Axiomatic approach to the cosmological constant

A theory of the cosmological constant Λ is currently out of reach. Still, one can start from a set of axioms that describe the most desirable properties a cosmological constant should have. This can be seen in certain analogy to the Khinchin axioms in information theory, which fix the most desirable properties an information measure should have and that ultimately lead to the Shannon entropy as the fundamental information measure on which statistical mechanics is based. Here we formulate a set of axioms for the cosmological constant in close analogy to the Khinchin axioms, formally replacing the dependence of the information measure on probabilities of events by a dependence of the cosmological constant on the fundamental constants of nature. Evaluating this set of axioms one finally arrives at a formula for the cosmological constant given by , where G is the gravitational constant, m_e the electron mass, and α_el the low-energy limit of the fine structure constant. This formula is in perfect agreement with current WMAP data. Our approach gives physical meaning to the Eddington-Dirac large-number hypothesis and suggests that the observed value of the cosmological constant is not at all unnatural.     

Unknowability of Quantum State forbids perfectly quantum operations

We analyze the connection between quantum operations and accessible information. And we find that the accessible information decreases under quantum operations. We show that it is impossible to perfectly manipulate an unknown state in an open quantum system. That the accessible information decreases under quantum operations gives a fundamental limitation in the microscopic world.     

Accessible Information for Equally-Distant Partially-Entangled Alphabet State Resource

We have proposed a quantum system with equally-distant partially-entangled alphabet states which has the minimal mutual overlap and the highly distinguishability,these quantum states are used as the “signal states” of the quantum communication.We have also constructed the positive operator-valued measure for these “signal states” and discussed their entanglement properties and measurement of cntanglement.We calculate the accessible information for these alphabet states and show that the accessible information is closely related to the entanglement of the “signal states”:the higher the entanglement of the “signal states”,the better the accessible information of the quantum system,and the accessible information reaches its maximal value when the alphabet states have their maximal entanglement.     

Towards information theory for q-nonextensive statistics without q-deformed distributions

In this paper we extend our recent results [P. Jizba, T. Arimitsu Physica A 340 (2004) 110] on q-nonextensive statistics with non-Tsallis entropies. In particular, we combine an axiomatics of Rényi with the q-deformed version of Khinchin axioms to obtain the entropy which accounts both for systems with embedded self-similarity and q-nonextensivity. We find that this entropy can be uniquely solved in terms of a one-parameter family of information measures. The corresponding entropy maximizer is expressible via a special function known under the name of the Lambert W-function. We analyze the corresponding “high” and “low-temperature” asymptotics and make some remarks on the possible applications.     

Gödel's theorem and information

Gödel's theorem may be demonstrated using arguments having an informationtheoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.     

Spectral Triples of Holonomy Loops

The machinery of noncommutative geometry is applied to a space of connections. A noncommutative function algebra of loops closely related to holonomy loops is investigated. The space of connections is identified as a projective limit of Lie-groups composed of copies of the gauge group. A spectral triple over the space of connections is obtained by factoring out the diffeomorphism group. The triple consist of equivalence classes of loops acting on a hilbert space of sections in an infinite dimensional Clifford bundle. We find that the Dirac operator acting on this hilbert space does not fully comply with the axioms of a spectral triple.     

Equivalence of Euclidean and Wightman field theories

A new inversion formula for the Laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions implies that some of the limits needed for the inversion formula exist for any Euclidean Green's function with an even number of variables. We then prove that the initial Osterwalder-Schrader axioms [1] and the weak spectral condition are equivalent with the Wightman axioms.The research described in this publication was made possible in part by Grant No. 93-011-147 from the Russian Foundation for Basic Research     

Quantum phase transition of ultracold bosons in double-well optical lattices

We study the superfluid-to-Mott insulator transition of bosons in a two-legged ladder optical lattice of a type accessible in current experiments on double-well optical lattices. The zero-temperature phase diagram is mapped out, with a focus on its dependence upon interchain hopping and the tilt between double wells. We find that the unit-filling Mott phase exhibits a nonmonotonic behavior as a function of the tilt parameter, producing a reentrant phase transition between the Mott insulator and superfluid phases.     

On the definition of information

In this note a new system of axioms for the information is given without using the concept of probability. Furthermore, it is shown that the information determines the probability uniquely, except for some degenerate cases.     

The differential and cone structures of spacetime

We propose to model spacetime by a differential space rather than by a differential manifold. A differential space is the pair (M, C), where M is any set, and C a family of real functions on M, satisfying certain axioms; C is called a differential structure of a corresponding differential space. This concept suitably generalizes the manifold concept. We show that C can be chosen in such a way that it contains all information about the causal structure of spacetime. This information can be read out of C with the help of only one postulate, namely that physical signals travel along piecewise smooth curves in (M, C). We effectively construct the Minkowski spacetime, with its cone structure, in this way. Some comments are made.     

Truncation of power law behavior in "scale-free" network models due to information filtering

We formulate a general model for the growth of scale-free networks under filtering information conditions-that is, when the nodes can process information about only a subset of the existing nodes in the network. We find that the distribution of the number of incoming links to a node follows a universal scaling form, i.e., that it decays as a power law with an exponential truncation controlled not only by the system size but also by a feature not previously considered, the subset of the network "accessible" to the node. We test our model with empirical data for the World Wide Web and find agreement.     

Resonant Forcing of Chaotic Dynamics

We study resonances of multidimensional chaotic map dynamics. We use the calculus of variations to determine the additive forcing function that induces the largest response, that is, the greatest deviation from the unperturbed dynamics. We include the additional constraint that only select degrees of freedom be forced, corresponding to a very general class of problems in which not all of the degrees of freedom in an experimental system are accessible to forcing. We find that certain Lagrange multipliers take on a fundamental physical role as the efficiency of the forcing function and the effective forcing experienced by the degrees of freedom which are not forced directly. Furthermore, we find that the product of the displacement of nearby trajectories and the effective total forcing function is a conserved quantity. We demonstrate the efficacy of this methodology with several examples.     

Spin Axioms in Different Geometries of Relativistic Continuum Physics

The 24 components of the relativistic spin tensor consist of 3 + 3 basic spin fields and 9 + 9 constitutive fields. Empirically only three basic spin fields and nine constitutive fields are known. This empirem can be expressed by two spin axioms, one of them denying purely relativistic spin fields, and the other one relating the three additional basic fields and the nine additional constitutive fields to the known (and measurable) ones. This identification by the spin axioms is material-independent and does not mix basic spin fields with constitutive properties. The approaches to the Weyssenhoff fluid and the Dirac-electron fluid found in literature are discussed with regard to these spin axioms. The conjecture is formulated, that another reduction from six to three basic spin fields which does not obey the spin axioms introduces special material properties by not allowed mixing of constitutive and basic fields.     

A Lie group structure underlying the triplectic geometry

We consider the pair of degenerate compatible antibrackets satisfying a generalization of the axioms imposed in the triplectic quantization of gauge theories. We show that these data encode a Lie group structure such that the antibrackets are related to the left- and right- invariant vector fields on the group. Conversely, every Lie group admits a “triplectic bundle.” The standard triplectic quantization axioms then correspond to Abelian Lie groups.     

Fluorescence lifetime of a single molecule as an observable of meta-basin dynamics in fluids near the glass transition

Using single molecule spectroscopy, we show that the fluorescence lifetime trajectories of single probe molecules embedded in a glass-forming polymer melt exhibit strong fluctuations of a hopping character. Using molecular dynamics simulations targeted to explain these experimental observations, we show that the lifetime fluctuations correlate strongly with the average square displacement function of the matrix particles. The latter observable is a direct probe of the meta-basin transitions in the potential energy landscape of glass-forming liquids. We thus show here that single molecule experiments can provide detailed microscopic information on system properties that hitherto have been accessible via computer simulations only.     

A note on Aerts' description of separated entities

The theoretical scheme proposed by Aerts for describing two separated entities as a whole within a question-state structure is considered. The quoted author claims that two relevant axioms characterizing quantum physics cannot hold for a quantum, nonclassical entity consisting of two quantum separate entities. We suggest that Aerts' theory is not adequate, from the empirical point of view, to describe this situation.