Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras

In the foundations of quantum mechanics and quantum computing, there is often a split between research using infinite dimensional structures and research using finite dimensional structures. On the one hand, in axiomatic quantum foundations there is often a focus on finite dimensional spaces and matrix mechanics (e.g. [1, 45, 23, 46, 5, 3, 8, 9, 10, 13, 27]), and the same is true for circuit based quantum computing (e.g. [16, 31]). On the other hand, infinite dimensional spaces arise naturally in subjects such as quantum field theory [47], and moreover the register space in a scalable quantum computer arguably has an infinite dimensional aspect (see e.g. [36]), which has led some researchers to use infinite dimensional spaces in the semantics of quantum programming languages [6, 37, 38, 21]. The ‘spaces’ in quantum theory are really non-commutative, so we understand them as W*-algebras, by analogy to Gelfand duality [11, 1.4].