Quantum information as a non-Kolmogorovian generalization of Shannon's theory

In this article we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures

Optimal Transport and Ricci Curvature: Wasserstein Space Over the Interval

In this essay, we discuss the notion of optimal transport on geodesic measure spaces and the associated (2-)Wasserstein distance. We then examine displacement convexity of the entropy functional on the space of probability measures. In particular, we give a detailed proof that the Lott-Villani-Sturm notion of generalized Ricci bounds agree with the classical notion on smooth manifolds. We also give the proof that generalized Ricci bounds are preserved under Gromov-Hausdorff convergence. In particular, we examine in detail the space of probability measures over the interval, $P(X)$ equipped with the Wasserstein metric $d^W$. We show that this metric space is isometric to a totally convex subset of a Hilbert space, $L^2[0,1]$, which allows for concrete calculations, contrary to the usual state of affairs in the theory of optimal transport. We prove explicitly that $(P(X),d^W)$ has vanishing Alexandrov curvature, and give an easy to work with expression for the entropy functional on this space. In addition, we examine finite dimensional Gromov-Hausdorff approximations to this space, and use these to construct a measure on the limit space, the entropic measure first considered by Von Renesse and Sturm. We examine properties of the measure, in particular explaining why one would expect it to have generalized Ricci lower bounds. We then show that this is in fact not true. We also discuss the possibility and consequences of finding a different measure which does admit generalized Ricci lower bounds.Comment: 47 pages, 9 figure

EPR Steering Inequalities from Entropic Uncertainty Relations

We use entropic uncertainty relations to formulate inequalities that witness Einstein-Podolsky-Rosen (EPR) steering correlations in diverse quantum systems. We then use these inequalities to formulate symmetric EPR-steering inequalities using the mutual information. We explore the differing natures of the correlations captured by one-way and symmetric steering inequalities, and examine the possibility of exclusive one-way steerability in two-qubit states. Furthermore, we show that steering inequalities can be extended to generalized positive operator valued measures (POVMs), and we also derive hybrid-steering inequalities between alternate degrees of freedom.Comment: 10 pages, 2 figure

Rigidity and flexibility of biological networks

The network approach became a widely used tool to understand the behaviour of complex systems in the last decade. We start from a short description of structural rigidity theory. A detailed account on the combinatorial rigidity analysis of protein structures, as well as local flexibility measures of proteins and their applications in explaining allostery and thermostability is given. We also briefly discuss the network aspects of cytoskeletal tensegrity. Finally, we show the importance of the balance between functional flexibility and rigidity in protein-protein interaction, metabolic, gene regulatory and neuronal networks. Our summary raises the possibility that the concepts of flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl

Decimation of the Dyson-Ising Ferromagnet

We study the decimation to a sublattice of half the sites, of the one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair interactions of the form $\frac{1}{{|i-j|}^{\alpha}}$, in the phase transition region (1< $\alpha \leq$ 2, and low temperature). We prove non-Gibbsianness of the decimated measure at low enough temperatures by exhibiting a point of essential discontinuity for the finite-volume conditional probabilities of decimated Gibbs measures. Thus result complements previous work proving conservation of Gibbsianness for fastly decaying potentials ($\alpha$ > 2) and provides an example of a "standard" non-Gibbsian result in one dimension, in the vein of similar resuts in higher dimensions for short-range models. We also discuss how these measures could fit within a generalized (almost vs. weak) Gibbsian framework. Moreover we comment on the possibility of similar results for some other transformations.Comment: 18 pages, some corrections and references added, to appear in Stoch.Proc.App

Geometrical aspects of possibility measures on finite domain MV-clans

In this paper, we study generalized possibility and necessity measures on MV-algebras of [0, 1]-valued functions (MV-clans) in the framework of idempotent mathematics, where the usual field of reals ℝ is replaced by the max-plus semiring ℝ max We prove results about extendability of partial assessments to possibility and necessity measures, and characterize the geometrical properties of the space of homogeneous possibility measures. The aim of the present paper is also to support the idea that idempotent mathematics is the natural framework to develop the theory of possibility and necessity measures, in the same way classical mathematics serves as a natural setting for probability theory. © 2012 Springer-Verlag.The authors would like to thank the anonymous referees for their relevant suggestions and helpful remarks They also acknowledge partial support from the Spanish projects TASSAT (TIN2010- 20967-C04-01), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010) and ARINF (TIN2009-14704-C03-03), as well as the ESF Eurocores-Log ICCC/MICINN project (FFI2008-03126-E/FILO). Flaminio and Marchioni acknowledge partial support from the Juan de la Cierva Program of the Spanish MICINN.Peer Reviewe

Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length $\ell$ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the R\'enyi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of $\ell^9$ for the contributions from the vacuum conformal family. The formal forms of these dissimilarity measures and the derived Fisher information metric from contributions of general operators are also given. As an application of the results, we use these dissimilarity measures to compare the excited and thermal states, and examine the eigenstate thermalization hypothesis (ETH) by showing how they behave in high temperature limit. This would help to understand how ETH in 2D CFT can be defined more precisely. We discuss the possibility that all the dissimilarity measures considered here vanish when comparing the reduced density matrices of an excited state and a generalized Gibbs ensemble thermal state. We also discuss ETH for a microcanonical ensemble thermal state in a 2D large central charge CFT, and find that it is approximately satisfied for a small subsystem and violated for a large subsystem.Comment: V1, 34 pages, 5 figures, see collection of complete results in the attached Mathematica notebook; V2, 38 pages, 5 figures, published versio