Brascamp-Lieb Inequality and Its Reverse: An Information Theoretic View

We generalize a result by Carlen and Cordero-Erausquin on the equivalence between the Brascamp-Lieb inequality and the subadditivity of relative entropy by allowing for random transformations (a broadcast channel). This leads to a unified perspective on several functional inequalities that have been gaining popularity in the context of proving impossibility results. We demonstrate that the information theoretic dual of the Brascamp-Lieb inequality is a convenient setting for proving properties such as data processing, tensorization, convexity and Gaussian optimality. Consequences of the latter include an extension of the Brascamp-Lieb inequality allowing for Gaussian random transformations, the determination of the multivariate Wyner common information for Gaussian sources, and a multivariate version of Nelson's hypercontractivity theorem. Finally we present an information theoretic characterization of a reverse Brascamp-Lieb inequality involving a random transformation (a multiple access channel).Comment: 5 pages; to be presented at ISIT 201

Transfer Entropy as a Log-likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic chi-squared distribution is established for the transfer entropy estimator. The result generalises the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense

Why the Tsirelson Bound? Bub's Question and Fuchs' Desideratum

To answer Wheeler's question "Why the quantum?" via quantum information theory according to Bub, one must explain both why the world is quantum rather than classical and why the world is quantum rather than superquantum, i.e., "Why the Tsirelson bound?" We show that the quantum correlations and quantum states corresponding to the Bell basis states, which uniquely produce the Tsirelson bound for the Clauser-Horne-Shimony-Holt quantity, can be derived from conservation per no preferred reference frame (NPRF). A reference frame in this context is defined by a measurement configuration, just as with the light postulate of special relativity. We therefore argue that the Tsirelson bound is ultimately based on NPRF just as the postulates of special relativity. This constraint-based/principle answer to Bub's question addresses Fuchs' desideratum that we "take the structure of quantum theory and change it from this very overt mathematical speak ... into something like [special relativity]." Thus, the answer to Bub's question per Fuchs' desideratum is, "the Tsirelson bound obtains due to conservation per NPRF."Comment: Contains corrections to the published versio

Higher Theory and the Three Problems of Physics

According to the Butterfield--Isham proposal, to understand quantum gravity we must revise the way we view the universe of mathematics. However, this paper demonstrates that the current elaborations of this programme neglect quantum interactions. The paper then introduces the Faddeev--Mickelsson anomaly which obstructs the renormalization of Yang--Mills theory, suggesting that to theorise on many-particle systems requires a many-topos view of mathematics itself: higher theory. As our main contribution, the topos theoretic framework is used to conceptualise the fact that there are principally three different quantisation problems, the differences of which have been ignored not just by topos physicists but by most philosophers of science. We further argue that if higher theory proves out to be necessary for understanding quantum gravity, its implications to philosophy will be foundational: higher theory challenges the propositional concept of truth and thus the very meaning of theorising in science.Comment: 23 pages, 1 table

New Class of Generalized Extensive Entropies for Studying Dynamical Systems in Highly Anisotropic Phase Space

Starting from the geometrical interpretation of the R\'enyi entropy, we introduce further extensive generalizations and study their properties. In particular, we found the probability distribution function obtained by the MaxEnt principle with generalized entropies. We prove that for a large class of dynamical systems subject to random perturbations, including particle transport in random media, these entropies play the role of Liapunov functionals. Some physical examples, which can be treated by the generalized R\'enyi entropies are also illustrated.Comment: 13 pages, 0 figure