Quantum Collections

TL;DR

This paper introduces a new categorical perspective on W*-algebras, constructing free exponentials that generalize function collections, and connects completely positive maps to states on these exponentials.

Contribution

It develops a novel analogy between W*-algebras and sets, constructing free exponentials and relating positive maps to states, advancing the categorical understanding of operator algebras.

Findings

Construction of free exponential for W*-algebras

Every unital normal completely positive map arises from a state

Establishment of an analogy between W*-algebras and sets

Abstract

We develop the viewpoint that the opposite of the category of W*-algebras and unital normal *-homomorphisms is analogous to the category of sets and functions. For each pair of W*-algebras, we construct their free exponential, which in the context of this analogy corresponds to the collection of functions from one of these W*-algebras to the other. We also show that every unital normal completely positive map between W*-algebras arises naturally from a normal state on their free exponential.