24 research outputs found
Rank properties of exposed positive maps
Let \cK and \cH be finite dimensional Hilbert spaces and let \fP denote
the cone of all positive linear maps acting from \fB(\cK) into \fB(\cH). We
show that each map of the form or is an
exposed point of \fP. We also show that if a map is an exposed point
of \fP then either is rank 1 non-increasing or \rank\phi(P)>1 for
any one-dimensional projection P\in\fB(\cK).Comment: 6 pages, last section removed - it will be a part of another pape
The cone of pseudo-effective divisors of log varieties after Batyrev
In these notes we investigate the cone of nef curves of projective varieties,
which is the dual cone to the cone of pseudo-effective divisors. We prove a
structure theorem for the cone of nef curves of projective -factorial klt pairs of arbitrary dimension from the point of view of the
Minimal Model Program. This is a generalization of Batyrev's structure theorem
for the cone of nef curves of projective terminal threefolds.Comment: 15 pages. v2: Completely rewritten paper. Structure theorem for the
cone of nef curves proved in arbitrary dimension using results of Birkar,
Cascini, Hacon and McKernan. To appear in Mathematische Zeitschrif
The maximum modulus of a trigonometric trinomial
Let Lambda be a set of three integers and let C_Lambda be the space of
2pi-periodic functions with spectrum in Lambda endowed with the maximum modulus
norm. We isolate the maximum modulus points x of trigonometric trinomials T in
C_Lambda and prove that x is unique unless |T| has an axis of symmetry. This
permits to compute the exposed and the extreme points of the unit ball of
C_Lambda, to describe how the maximum modulus of T varies with respect to the
arguments of its Fourier coefficients and to compute the norm of unimodular
relative Fourier multipliers on C_Lambda. We obtain in particular the Sidon
constant of Lambda
Information-theoretic postulates for quantum theory
Why are the laws of physics formulated in terms of complex Hilbert spaces?
Are there natural and consistent modifications of quantum theory that could be
tested experimentally? This book chapter gives a self-contained and accessible
summary of our paper [New J. Phys. 13, 063001, 2011] addressing these
questions, presenting the main ideas, but dropping many technical details. We
show that the formalism of quantum theory can be reconstructed from four
natural postulates, which do not refer to the mathematical formalism, but only
to the information-theoretic content of the physical theory. Our starting point
is to assume that there exist physical events (such as measurement outcomes)
that happen probabilistically, yielding the mathematical framework of "convex
state spaces". Then, quantum theory can be reconstructed by assuming that (i)
global states are determined by correlations between local measurements, (ii)
systems that carry the same amount of information have equivalent state spaces,
(iii) reversible time evolution can map every pure state to every other, and
(iv) positivity of probabilities is the only restriction on the possible
measurements.Comment: 17 pages, 3 figures. v3: some typos corrected and references updated.
Summarizes the argumentation and results of arXiv:1004.1483. Contribution to
the book "Quantum Theory: Informational Foundations and Foils", Springer
Verlag (http://www.springer.com/us/book/9789401773027), 201
A derivation of quantum theory from physical requirements
Quantum theory is usually formulated in terms of abstract mathematical
postulates, involving Hilbert spaces, state vectors, and unitary operators. In
this work, we show that the full formalism of quantum theory can instead be
derived from five simple physical requirements, based on elementary assumptions
about preparation, transformations and measurements. This is more similar to
the usual formulation of special relativity, where two simple physical
requirements -- the principles of relativity and light speed invariance -- are
used to derive the mathematical structure of Minkowski space-time. Our
derivation provides insights into the physical origin of the structure of
quantum state spaces (including a group-theoretic explanation of the Bloch ball
and its three-dimensionality), and it suggests several natural possibilities to
construct consistent modifications of quantum theory.Comment: 16 pages, 2 figures. V3: added alternative formulation of Requirement
5, extended abstract, some minor modification
Characterizing the universal rigidity of generic frameworks
A framework is a graph and a map from its vertices to E^d (for some d). A
framework is universally rigid if any framework in any dimension with the same
graph and edge lengths is a Euclidean image of it. We show that a generic
universally rigid framework has a positive semi-definite stress matrix of
maximal rank. Connelly showed that the existence of such a positive
semi-definite stress matrix is sufficient for universal rigidity, so this
provides a characterization of universal rigidity for generic frameworks. We
also extend our argument to give a new result on the genericity of strict
complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio
O podstawowych twierdzeniach trygonometrii Łobaczewskiego
The article contains no abstrac