An axiomatic characterization of a two-parameter extended relative entropy

The uniqueness theorem for a two-parameter extended relative entropy is proven. This result extends our previous one, the uniqueness theorem for a one-parameter extended relative entropy, to a two-parameter case. In addition, the properties of a two-parameter extended relative entropy are studied.Comment: 11 page

Synthesizing judgements: a functional equations approach

AbstractWe discuss several conditions which are reasonable requirements for functions synthesizing either ratio or measure judgements (or both) and determine all synthesizing functions satisfying either shorter or longer lists of such assumptions (yielding more general or more specific synthesizing procedures, respectively)

Measurements and Information in Spin Foam Models

We present a problem relating measurements and information theory in spin foam models. In the three dimensional case of quantum gravity we can compute probabilities of spin network graphs and study the behaviour of the Shannon entropy associated to the corresponding information. We present a general definition, compute the Shannon entropy of some examples, and find some interesting inequalities.Comment: 15 pages, 3 figures. Improved versio

Determination of all semisymmetric recursive information measures of multiplicative type on n positive discrete probability distributions

AbstractInformation measures Δm (entropies, divergences, inaccuracies, information improvements, etc.), depending upon n probability distributions which we unite into a vector distribution, are recursive of type μ if Δm(p1, p2, p3,…,pm)=Δm−1(p1+p2, p3,…,pm)+μ(p1+p2)Δ2p1p1+p2,p2p1+p2. If also a similar equation holds with three instead of two distinguished vectors, then μ has to be multiplicative, except if all Δm are identically 0. The information measure is semisymmetric if Δ3(p1, p2, p3) = Δ3(p1, p3, p2). We determine all semisymmetric (in particular, symmetric) recursive information measures of multiplicative type, allowing first only positive probabilities. Previously the cases n ⪕ 3 have been examined mainly for μ(t) = μ(τ1, τ2,…, τn) = τ1α1 τ2α2 … τnαn and some probabilities were allowed to be 0. This has made the proofs easier. But permitting certain probabilities to be 0 would exclude most information measures important for applications, so the description of appropriate domains became complicated. However, we show how the measures which we determine here can be extended to the “old” domains and to more general ones

A classification of bisymmetric polynomial functions over integral domains of characteristic zero

We describe the class of n-variable polynomial functions that satisfy Acz\'el's bisymmetry property over an arbitrary integral domain of characteristic zero with identity

On the characterization of Shannon's entropy by Shannon's inequality

1. In [2,5,6,7] a.o. several interpretations of the inequality for all such that were given and the following was prove

On the Orthogonal Stability of the Pexiderized Quadratic Equation

The Hyers--Ulam stability of the conditional quadratic functional equation of Pexider type f(x+y)+f(x-y)=2g(x)+2h(y), x\perp y is established where \perp is a symmetric orthogonality in the sense of Ratz and f is odd.Comment: 10 pages, Latex; Changed conten

Origin of Complex Quantum Amplitudes and Feynman's Rules

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. In this paper, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.Comment: v2: Clarifications, and minor corrections and modifications. Results unchanged. v3: Minor changes to introduction and conclusio

Strong superadditivity and monogamy of the Renyi measure of entanglement

Employing the quantum R\'enyi $\alpha$-entropies as a measure of entanglement, we numerically find the violation of the strong superadditivity inequality for a system composed of four qubits and $\alpha>1$. This violation gets smaller as $\alpha\rightarrow 1$ and vanishes for $\alpha=1$ when the measure corresponds to the Entanglement of Formation (EoF). We show that the R\'enyi measure aways satisfies the standard monogamy of entanglement for $\alpha = 2$, and only violates a high order monogamy inequality, in the rare cases in which the strong superadditivity is also violated. The sates numerically found where the violation occurs have special symmetries where both inequalities are equivalent. We also show that every measure satisfing monogamy for high dimensional systems also satisfies the strong superadditivity inequality. For the case of R\'enyi measure, we provide strong numerical evidences that these two properties are equivalent.Comment: replaced with final published versio

Information theoretical properties of Tsallis entropies

A chain rule and a subadditivity for the entropy of type $\beta$, which is one of the nonadditive entropies, were derived by Z.Dar\'oczy. In this paper, we study the further relations among Tsallis type entropies which are typical nonadditive entropies. The chain rule is generalized by showing it for Tsallis relative entropy and the nonadditive entropy. We show some inequalities related to Tsallis entropies, especially the strong subadditivity for Tsallis type entropies and the subadditivity for the nonadditive entropies. The subadditivity and the strong subadditivity naturally lead to define Tsallis mutual entropy and Tsallis conditional mutual entropy, respectively, and then we show again chain rules for Tsallis mutual entropies. We give properties of entropic distances in terms of Tsallis entropies. Finally we show parametrically extended results based on information theory.Comment: The subsection on data processing inequality was deleted. Some typo's were modifie