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 $\phi(X)=AXA^*$ or $\phi(X)=AX^TA^*$ is an exposed point of \fP. We also show that if a map $\phi$ is an exposed point of \fP then either $\phi$ 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 $\mathbb Q$-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