Integrable Hierarchies and Information Measures

In this paper we investigate integrable models from the perspective of information theory, exhibiting various connections. We begin by showing that compressible hydrodynamics for a one-dimesional isentropic fluid, with an appropriately motivated information theoretic extension, is described by a general nonlinear Schrodinger (NLS) equation. Depending on the choice of the enthalpy function, one obtains the cubic NLS or other modified NLS equations that have applications in various fields. Next, by considering the integrable hierarchy associated with the NLS model, we propose higher order information measures which include the Fisher measure as their first member. The lowest members of the hiearchy are shown to be included in the expansion of a regularized Kullback-Leibler measure while, on the other hand, a suitable combination of the NLS hierarchy leads to a Wootters type measure related to a NLS equation with a relativistic dispersion relation. Finally, through our approach, we are led to construct an integrable semi-relativistic NLS equation.Comment: 11 page

Information measures and classicality in quantum mechanics

We study information measures in quantu mechanics, with particular emphasis on providing a quantification of the notions of classicality and predictability. Our primary tool is the Shannon - Wehrl entropy I. We give a precise criterion for phase space classicality and argue that in view of this a) I provides a measure of the degree of deviation from classicality for closed system b) I - S (S the von Neumann entropy) plays the same role in open systems We examine particular examples in non-relativistic quantum mechanics. Finally, (this being one of our main motivations) we comment on field classicalisation on early universe cosmology.Comment: 35 pages, LATE

R\'enyi generalizations of quantum information measures

Quantum information measures such as the entropy and the mutual information find applications in physics, e.g., as correlation measures. Generalizing such measures based on the R\'enyi entropies is expected to enhance their scope in applications. We prescribe R\'enyi generalizations for any quantum information measure which consists of a linear combination of von Neumann entropies with coefficients chosen from the set {-1,0,1}. As examples, we describe R\'enyi generalizations of the conditional quantum mutual information, some quantum multipartite information measures, and the topological entanglement entropy. Among these, we discuss the various properties of the R\'enyi conditional quantum mutual information and sketch some potential applications. We conjecture that the proposed R\'enyi conditional quantum mutual informations are monotone increasing in the R\'enyi parameter, and we have proofs of this conjecture for some special cases.Comment: 9 pages, related to and extends the results from arXiv:1403.610

Escort density operators and generalized quantum information measures

Parametrized families of density operators are studied. A generalization of the lower bound of Cramer and Rao is formulated. It involves escort density operators. The notion of phi-exponential family is introduced. This family, together with its escort, optimizes the generalized lower bound. It also satisfies a maximum entropy principle and exhibits a thermodynamic structure in which entropy and free energy are related by Legendre transform.Comment: 10 page

Information measures and cognitive limits in multilayer navigation

Cities and their transportation systems become increasingly complex and multimodal as they grow, and it is natural to wonder if it is possible to quantitatively characterize our difficulty to navigate in them and whether such navigation exceeds our cognitive limits. A transition between different searching strategies for navigating in metropolitan maps has been observed for large, complex metropolitan networks. This evidence suggests the existence of another limit associated to the cognitive overload and caused by large amounts of information to process. In this light, we analyzed the world's 15 largest metropolitan networks and estimated the information limit for determining a trip in a transportation system to be on the order of 8 bits. Similar to the "Dunbar number," which represents a limit to the size of an individual's friendship circle, our cognitive limit suggests that maps should not consist of more than about $250$ connections points to be easily readable. We also show that including connections with other transportation modes dramatically increases the information needed to navigate in multilayer transportation networks: in large cities such as New York, Paris, and Tokyo, more than $80\%$ of trips are above the 8-bit limit. Multimodal transportation systems in large cities have thus already exceeded human cognitive limits and consequently the traditional view of navigation in cities has to be revised substantially.Comment: 16 pages+9 pages of supplementary materia