17,219 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
-algebraic drawings of dendroidal sets
In recent years the theory of dendroidal sets has emerged as an important
framework for higher algebra. In this article we introduce the concept of a
-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an
object in the category of presheaves on -algebras. We show that the
construction is functorial and, in fact, it is the left adjoint of a Quillen
adjunction between combinatorial model categories. We use this construction to
produce a bridge between the two prominent paradigms of noncommutative geometry
via adjunctions of presentable -categories, which is the primary
motivation behind this article. As a consequence we obtain a single mechanism
to construct bivariant homology theories in both paradigms. We propose a
(conjectural) roadmap to harmonize algebraic and analytic (or topological)
bivariant K-theory. Finally, a method to analyse graph algebras in terms of
trees is sketched.Comment: 28 pages; v2 expanded version with some improvements; v3 revised and
added references; v4 some changes according to the suggestions of the
referees (to appear in Algebr. Geom. Topol.
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