210 research outputs found
Homogeneous Structures: Model Theory meets Universal Algebra (online meeting)
The workshop "Homogeneous Structures: Model Theory meets Universal
Algebra'' was centred around transferring recently obtained advances
in universal algebra from the finite to the infinite. As it turns out,
the notion of homogeneity together with other model-theoretic concepts
like -categoricity and the Ramsey property play an
indispensable role in this endeavour
C*-algebras associated to coverings of k-graphs
A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an
embedding of universal C*-algebras. We show how to build a (k+1)-graph whose
universal algebra encodes this embedding. More generally we show how to realise
a direct limit of k-graph algebras under embeddings induced from coverings as
the universal algebra of a (k+1)-graph. Our main focus is on computing the
K-theory of the (k+1)-graph algebra from that of the component k-graph
algebras.
Examples of our construction include a realisation of the Kirchberg algebra
\mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class
of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens
algebras.Comment: 44 pages, 2 figures, some diagrams drawn using picTeX. v2. A number
of typos corrected, some references updated. The statements of Theorem 6.7(2)
and Corollary 6.8 slightly reworded for clarity. v3. Some references updated;
in particular, theorem numbering of references to Evans updated to match
published versio
The lowest discriminant ideal of a Cayley-Hamilton Hopf algebra
Discriminant ideals of noncommutative algebras , which are module finite
over a central sublagebra , are key invariants that carry important
information about , such as the sum of the squares of the dimensions of its
irreducible modules with a given central character. There has been substantial
research on the computation of discriminants, but very little is known about
the computation of discriminant ideals. In this paper we carry out a detailed
investigation of the lowest discriminant ideals of Cayley-Hamilton Hopf
algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose
identity fiber algebras are basic. The lowest discriminant ideals are the most
complicated ones, because they capture the most degenerate behaviour of the
fibers in the exact opposite spectrum of the picture from the Azumaya locus. We
provide a description of the zero sets of the lowest discriminant ideals of
Cayley-Hamilton Hopf algebras in terms of maximally stable modules of Hopf
algebras, irreducible modules that are stable under tensoring with the maximal
possible number of irreducible modules with trivial central character. In
important situations, this is shown to be governed by the actions of the
winding automorphism groups. The results are illustrated with applications to
the group algebras of central extensions of abelian groups, big quantum Borel
subalgebras at roots of unity and quantum coordinate rings at roots of unity.Comment: 33 page
On the nature and decay of quantum relative entropy
Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features.
An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty.
We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity.
Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels
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