264 research outputs found
Non-additivity of Renyi entropy and Dvoretzky's Theorem
The goal of this note is to show that the analysis of the minimum output
p-Renyi entropy of a typical quantum channel essentially amounts to applying
Milman's version of Dvoretzky's Theorem about almost Euclidean sections of
high-dimensional convex bodies. This conceptually simplifies the
(nonconstructive) argument by Hayden-Winter disproving the additivity
conjecture for the minimal output p-Renyi entropy (for p>1).Comment: 8 pages, LaTeX; v2: added and updated references, minor editorial
changes, no content change
Hastings' additivity counterexample via Dvoretzky's theorem
The goal of this note is to show that Hastings' counterexample to the
additivity of minimal output von Neumann entropy can be readily deduced from a
sharp version of Dvoretzky's theorem on almost spherical sections of convex
bodies.Comment: 12 pages; v.2: added references, Appendix A expanded to make the
paper essentially self-containe
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
Lineability of non-differentiable Pettis primitives
Let X be an infinite-dimensional Banach space. In 1995, settling a long
outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued
Pettis integrable function on [0; 1] whose primitive is nowhere weakly
differentiable. Using their technique and some new ideas we show that ND, the
set of strongly measurable Pettis integrable functions with nowhere weakly
differentiable primitives, is lineable, i.e., there is an infinite dimensional
vector space whose nonzero vectors belong to ND
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