418 research outputs found
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
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
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
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 -entropies as a measure of
entanglement, we numerically find the violation of the strong superadditivity
inequality for a system composed of four qubits and . This violation
gets smaller as and vanishes for 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
, 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 , 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
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