1,067 research outputs found
Quantum channels as a categorical completion
We propose a categorical foundation for the connection between pure and mixed
states in quantum information and quantum computation. The foundation is based
on distributive monoidal categories.
First, we prove that the category of all quantum channels is a canonical
completion of the category of pure quantum operations (with ancilla
preparations). More precisely, we prove that the category of completely
positive trace-preserving maps between finite-dimensional C*-algebras is a
canonical completion of the category of finite-dimensional vector spaces and
isometries.
Second, we extend our result to give a foundation to the topological
relationships between quantum channels. We do this by generalizing our
categorical foundation to the topologically-enriched setting. In particular, we
show that the operator norm topology on quantum channels is the canonical
topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201
Semantics for a Quantum Programming Language by Operator Algebras
This paper presents a novel semantics for a quantum programming language by
operator algebras, which are known to give a formulation for quantum theory
that is alternative to the one by Hilbert spaces. We show that the opposite
category of the category of W*-algebras and normal completely positive
subunital maps is an elementary quantum flow chart category in the sense of
Selinger. As a consequence, it gives a denotational semantics for Selinger's
first-order functional quantum programming language QPL. The use of operator
algebras allows us to accommodate infinite structures and to handle classical
and quantum computations in a unified way.Comment: In Proceedings QPL 2014, arXiv:1412.810
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Poisson boundaries of monoidal categories
Given a rigid C*-tensor category C with simple unit and a probability measure
on the set of isomorphism classes of its simple objects, we define the
Poisson boundary of . This is a new C*-tensor category P, generally
with nonsimple unit, together with a unitary tensor functor . Our
main result is that if P has simple unit (which is a condition on some
classical random walk), then is a universal unitary tensor functor
defining the amenable dimension function on C. Corollaries of this theorem
unify various results in the literature on amenability of C*-tensor categories,
quantum groups, and subfactors.Comment: v2: 37 pages, minor changes, to appear in Ann. Sci. Ecole Norm. Sup.;
v1: 37 page
Reduced operator algebras of trace-preserving quantum automorphism groups
Let be a finite dimensional C-algebra equipped with its canonical
trace induced by the regular representation of on itself. In this paper, we
study various properties of the trace-preserving quantum automorphism group
\G of . We prove that the discrete dual quantum group \hG has the
property of rapid decay, the reduced von Neumann algebra L^\infty(\G) has the
Haagerup property and is solid, and that L^\infty(\G) is (in most cases) a
prime type II-factor. As applications of these and other results, we deduce
the metric approximation property, exactness, simplicity and uniqueness of
trace for the reduced -algebra C_r(\G), and the existence of a
multiplier-bounded approximate identity for the convolution algebra L^1(\G).Comment: Section 6 removed and replaced by a more general solidity resul
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