14,050 research outputs found
Categorical formulation of quantum algebras
We describe how dagger-Frobenius monoids give the correct categorical
description of certain kinds of finite-dimensional 'quantum algebras'. We
develop the concept of an involution monoid, and use it to construct a
correspondence between finite-dimensional C*-algebras and certain types of
dagger-Frobenius monoids in the category of Hilbert spaces. Using this
technology, we recast the spectral theorems for commutative C*-algebras and for
normal operators into an explicitly categorical language, and we examine the
case that the results of measurements do not form finite sets, but rather
objects in a finite Boolean topos. We describe the relevance of these results
for topological quantum field theory.Comment: 34 pages, to appear in Communications in Mathematical Physic
Eleven Dimensional Origin of String/String Duality: A One Loop Test
Membrane/fivebrane duality in D=11 implies Type IIA string/Type IIA fivebrane
duality in D=10, which in turn implies Type IIA string/heterotic string duality
in D=6. To test the conjecture, we reproduce the corrections to the 3-form
field equations of the D=10 Type IIA string (a mixture of tree-level and
one-loop effects) starting from the Chern-Simons corrections to the 7-form
Bianchi identities of the D=11 fivebrane (a purely tree-level effect). K3
compactification of the latter then yields the familiar gauge and Lorentz
Chern-Simons corrections to 3-form Bianchi identities of the heterotic string.
We note that the absence of a dilaton in the D=11 theory allows us to fix both
the gravitational constant and the fivebrane tension in terms of the membrane
tension. We also comment on an apparent conflict between fundamental and
solitonic heterotic strings and on the puzzle of a fivebrane origin of
S-duality.Comment: 30 pages (including 5 postscript figures included), LaTeX, Footnote 8
has been removed; the apparent disagreement with Townsend is only one of
semantics, not substanc
Geometry of abstraction in quantum computation
Quantum algorithms are sequences of abstract operations, performed on
non-existent computers. They are in obvious need of categorical semantics. We
present some steps in this direction, following earlier contributions of
Abramsky, Coecke and Selinger. In particular, we analyze function abstraction
in quantum computation, which turns out to characterize its classical
interfaces. Some quantum algorithms provide feasible solutions of important
hard problems, such as factoring and discrete log (which are the building
blocks of modern cryptography). It is of a great practical interest to
precisely characterize the computational resources needed to execute such
quantum algorithms. There are many ideas how to build a quantum computer. Can
we prove some necessary conditions? Categorical semantics help with such
questions. We show how to implement an important family of quantum algorithms
using just abelian groups and relations.Comment: 29 pages, 42 figures; Clifford Lectures 2008 (main speaker Samson
Abramsky); this version fixes a pstricks problem in a diagra
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