Juan Pablo Vigneaux

Whereas Shannon entropy is related to the growth rate of multinomial coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is connected to their q-deformation; when q is a prime power, these q-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. In particular, the q-binomial coefficients count vector subspaces of given dimension. We obtain in this way a combinatorial explanation for the nonadditivity of the quadratic entropy, which arises from a recursive counting of flags. We show that statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the q-binomial probability distribution, that generates at time n a vector subspace of \mathbb F_q^n (here \mathbb F_q is the finite field of order q). The concentration of measure on certain "typical subspaces" allows us to extend the asymptotic equipartition property to this setting. The size of the typical set is quantified by the quadratic entropy. We discuss the applications to Shannon theory, particularly to source coding, when messages correspond to vector spaces.

We define \emphstratified measures on Euclidean space as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize discrete-continuous mixtures. A stratified measure ρcan thus be represented as \sum_i=1^k q_i \rho_i, where (q_1,..,q_k) is a probability vector and each \rho_i is absolutely continuous with respect to the m_i-Hausdorff measure \mu_i on a m_i-rectifiable set E_i (e.g. a smooth m_i-manifold if m_i>0 or a countable set if m_i=0). We introduce a set of strongly typical realizations of ρ^⊗n that occur with high probability; they are supported on a finite union of strata E_i_1\times ⋯\times E_i_n whose dimensions “per factor” concentrate around the \emphmean dimension \sum_i=1^k q_i m_i. For each n, an appropriate sum of Hausdorff measures on the different strata gives a notion of reference “volume”; the exponential growth rate of the typical set’s volume is quantified by Csiszar’s \emphgeneralized entropy of ρwith respect to μ=\sum_i=1^k \mu_i. This entropy satisfies a chain rule; the conditional term is related to the volume growth of the typical realizations in each stratum. Under suitable hypotheses, our notion of mean dimension coincides with Rényi’s information dimension when applied to stratified measures, but the generalized entropy used here differs from Rényi’s dimensional entropy.

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin’s definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis α-entropy, depending on the value of the parameter.

The magnitude of finite categories is a generalization of the Euler characteristic. It is defined using the coarse incidence algebra of rational-valued functions on the given finite category, and a distinguished element in this algebra: the Dirichlet zeta function. The incidence algebra may be identified with the algebra of n\times n matrices over the rational numbers, where n is the cardinality of the underlying object set. The Moore-Penrose pseudoinverse of a matrix is a generalization of the inverse; it exists and is unique for any given matrix over the complex numbers. In this article, we derive a new method for calculating the magnitude of a finite category, using the pseudoinverse of the matrix that corresponds to the zeta function. The magnitude equals the sum of the entries of this pseudoinverse.