10,221 research outputs found
On a tracial version of Haemers bound
We extend upper bounds on the quantum independence number and the quantum
Shannon capacity of graphs to their counterparts in the commuting operator
model. We introduce a von Neumann algebraic generalization of the fractional
Haemers bound (over ) and prove that the generalization upper
bounds the commuting quantum independence number. We call our bound the tracial
Haemers bound, and we prove that it is multiplicative with respect to the
strong product. In particular, this makes it an upper bound on the Shannon
capacity. The tracial Haemers bound is incomparable with the Lov\'asz theta
function, another well-known upper bound on the Shannon capacity. We show that
separating the tracial and fractional Haemers bounds would refute Connes'
embedding conjecture.
Along the way, we prove that the tracial rank and tracial Haemers bound are
elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam,
Combinatorica, 2019). We also show that the inertia bound (an upper bound on
the quantum independence number) upper bounds the commuting quantum
independence number.Comment: 39 pages, comments are welcom
Investigation of commuting Hamiltonian in quantum Markov network
Graphical Models have various applications in science and engineering which
include physics, bioinformatics, telecommunication and etc. Usage of graphical
models needs complex computations in order to evaluation of marginal
functions,so there are some powerful methods including mean field
approximation, belief propagation algorithm and etc. Quantum graphical models
have been recently developed in context of quantum information and computation,
and quantum statistical physics, which is possible by generalization of
classical probability theory to quantum theory. The main goal of this paper is
preparing a primary generalization of Markov network, as a type of graphical
models, to quantum case and applying in quantum statistical physics.We have
investigated the Markov network and the role of commuting Hamiltonian terms in
conditional independence with simple examples of quantum statistical physics.Comment: 11 pages, 8 figure
Signals on graphs : transforms and tomograms
Development of efficient tools for the representation of large datasets is a precondition for the study of dynamics on networks. Generalizations of the Fourier transform on graphs have been constructed through projections on the eigenvectors of graph matrices. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the preseninfo:eu-repo/semantics/publishedVersio
Towers for commuting endomorphisms, and combinatorial applications
We give an elementary proof of a generalization of Rokhlin's lemma for
commuting non-invertible measure-preserving transformations, and we present
several combinatorial applications.Comment: 13 pages. Referee's comments incorporated. To appear in Annales de
l'Institut Fourie
Modeling commuting systems through a complex network analysis: a study of the Italian islands of Sardinia and Sicily
This study analyzes the inter-municipal commuting systems of the Italian islands of Sardinia and Sicily, employing weighted network analysis technique. Based on the results obtained for the Sardinian commuting network, the network analysis is used to identify similarities and dissimilarities between the two systems
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