6,426 research outputs found
Generalizations of entanglement based on coherent states and convex sets
Unentangled pure states on a bipartite system are exactly the coherent states
with respect to the group of local transformations. What aspects of the study
of entanglement are applicable to generalized coherent states? Conversely, what
can be learned about entanglement from the well-studied theory of coherent
states? With these questions in mind, we characterize unentangled pure states
as extremal states when considered as linear functionals on the local Lie
algebra. As a result, a relativized notion of purity emerges, showing that
there is a close relationship between purity, coherence and (non-)entanglement.
To a large extent, these concepts can be defined and studied in the even more
general setting of convex cones of states. Based on the idea that entanglement
is relative, we suggest considering these notions in the context of partially
ordered families of Lie algebras or convex cones, such as those that arise
naturally for multipartite systems. The study of entanglement includes notions
of local operations and, for information-theoretic purposes, entanglement
measures and ways of scaling systems to enable asymptotic developments. We
propose ways in which these may be generalized to the Lie-algebraic setting,
and to a lesser extent to the convex-cones setting. One of our original
motivations for this program is to understand the role of entanglement-like
concepts in condensed matter. We discuss how our work provides tools for
analyzing the correlations involved in quantum phase transitions and other
aspects of condensed-matter systems.Comment: 37 page
Impressions of convexity - An illustration for commutator bounds
We determine the sharpest constant such that for all complex
matrices and , and for Schatten -, - and -norms the inequality
is valid. The main theoretical
tool in our investigations is complex interpolation theory.Comment: 32 pages, 88 picture
Acceleration of the PDHGM on strongly convex subspaces
We propose several variants of the primal-dual method due to Chambolle and
Pock. Without requiring full strong convexity of the objective functions, our
methods are accelerated on subspaces with strong convexity. This yields mixed
rates, with respect to initialisation and with respect to
the dual sequence, and the residual part of the primal sequence. We demonstrate
the efficacy of the proposed methods on image processing problems lacking
strong convexity, such as total generalised variation denoising and total
variation deblurring
Introduction to Cirquent Calculus and Abstract Resource Semantics
This paper introduces a refinement of the sequent calculus approach called
cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.)
sequents are disjoint sequences of formulas, in cirquent calculus they are
permitted to share elements. Explicitly allowing or disallowing shared
resources and thus taking to a more subtle level the resource-awareness
intuitions underlying substructural logics, cirquent calculus offers much
greater flexibility and power than sequent calculus does. A need for
substantially new deductive tools came with the birth of computability logic
(see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed
formal theory of computational resources, which has stubbornly resisted any
axiomatization attempts within the framework of traditional syntactic
approaches. Cirquent calculus breaks the ice. Removing contraction from the
full collection of its rules yields a sound and complete system for the basic
fragment CL5 of computability logic. Doing the same in sequent calculus, on the
other hand, throws out the baby with the bath water, resulting in the strictly
weaker affine logic. An implied claim of computability logic is that it is CL5
rather than affine logic that adequately materializes the resource philosophy
traditionally associated with the latter. To strengthen this claim, the paper
further introduces an abstract resource semantics and shows the soundness and
completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio
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