26,504 research outputs found

    The asymptotic spectrum of graphs and the Shannon capacity

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    We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional clique cover number, the complement of the fractional orthogonal rank and the fractional Haemers bounds

    The asymptotic lift of a completely positive map

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    Starting with a unit-preserving normal completely positive map L: M --> M acting on a von Neumann algebra - or more generally a dual operator system - we show that there is a unique reversible system \alpha: N --> N (i.e., a complete order automorphism \alpha of a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*-dynamical system (N,\mathbb Z), whick we identify as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed point algebra of (N,\mathbb Z). In general, we show the action of the asymptotic lift is trivial iff L is {\em slowly oscillating} in the sense that limnρLn+1ρLn=0,ρM. \lim_{n\to\infty}\|\rho\circ L^{n+1}-\rho\circ L^n\|=0,\qquad \rho\in M_* . Hence \alpha is often a nontrivial automorphism of N.Comment: New section added with an applicaton to the noncommutative Poisson boundary. Clarification of Sections 3 and 4. Additional references. 23 p

    Minimizing Communication for Eigenproblems and the Singular Value Decomposition

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    Algorithms have two costs: arithmetic and communication. The latter represents the cost of moving data, either between levels of a memory hierarchy, or between processors over a network. Communication often dominates arithmetic and represents a rapidly increasing proportion of the total cost, so we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds were presented on the amount of communication required for essentially all O(n3)O(n^3)-like algorithms for linear algebra, including eigenvalue problems and the SVD. Conventional algorithms, including those currently implemented in (Sca)LAPACK, perform asymptotically more communication than these lower bounds require. In this paper we present parallel and sequential eigenvalue algorithms (for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms that do attain these lower bounds, and analyze their convergence and communication costs.Comment: 43 pages, 11 figure

    Application of Shemesh theorem to quantum channels

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    Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,AK)\mathcal{A}(A_1,\ldots A_K). By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra A(A1,AK)\mathcal{A}(A_1,\ldots A_K) one can predict the asymptotic dynamics for a class of operations
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