1,298 research outputs found
Involutive Categories and Monoids, with a GNS-correspondence
This paper develops the basics of the theory of involutive categories and
shows that such categories provide the natural setting in which to describe
involutive monoids. It is shown how categories of Eilenberg-Moore algebras of
involutive monads are involutive, with conjugation for modules and vector
spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS)
construction is identified as a bijective correspondence between states on
involutive monoids and inner products. This correspondence exists in arbritrary
involutive categories
The classification problem for automorphisms of C*-algebras
We present an overview of the recent developments in the study of the
classification problem for automorphisms of C*-algebras from the perspective of
Borel complexity theory.Comment: 21 page
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
Relating Operator Spaces via Adjunctions
This chapter uses categorical techniques to describe relations between
various sets of operators on a Hilbert space, such as self-adjoint, positive,
density, effect and projection operators. These relations, including various
Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual
adjunctions, and maps between them. Of particular interest is the connection
with quantum structures, via a dual adjunction between convex sets and effect
modules. The approach systematically uses categories of modules, via their
description as Eilenberg-Moore algebras of a monad
Interactions in noncommutative dynamics
A mathematical notion of interaction is introduced for noncommutative
dynamical systems, i.e., for one parameter groups of *-automorphisms of \Cal
B(H) endowed with a certain causal structure. With any interaction there is a
well-defined "state of the past" and a well-defined "state of the future". We
describe the construction of many interactions involving cocycle perturbations
of the CAR/CCR flows and show that they are nontrivial. The proof of
nontriviality is based on a new inequality, relating the eigenvalue lists of
the "past" and "future" states to the norm of a linear functional on a certain
C^*-algebra.Comment: 22 pages. Replacement corrects misnumbering of formulas in section 4.
No change in mathematical conten
- …