Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach
Thea Li, Vladimir Zamdzhiev
TL;DR
This work builds a new semantic model for MALL by combining finite-dimensional operator spaces with von Neumann (co)algebras, where proofs of positive polarity interpret as CPTP maps (Schrödinger picture) and proofs of negative polarity as CPU maps (Heisenberg picture). Central to the construction is the Haagerup tensor, which allows vN-algebras and vN-coalgebras to be internal monoid/comonoid objects in $ extbf{FdOS}$, enabling a duality that mirrors quantum theory. The authors establish a duality between the Schrödinger and Heisenberg pictures at the categorical level, and they implement Hyland–Schalk gluing to carve a category $ extbf{Q}$ that faithfully contains the required quantum operations while preserving MALL structure. The framework supports both mixed-state and pure-state quantum computation (via density operators and unitary evolutions) and points to future directions in type systems, recursion, and connections with state-and-effect theories. Overall, the paper provides a rigorous, algebraically rich model that unifies quantum operations with linear logic semantics through a self-dual Haagerup-tensor setting.
Abstract
We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.