Logic of infinite quantum systems

Gabriele Saladini

This paper addresses the Infinite Quantum Field Paradox. Using the Saladini Theorem, it proves that divergences arise only from the impossible assumption of infinite field mode processing.-Finite Capacity: Agents cannot access, process, or measure an infinite number of field modes simultaneously.-Logical Nullification: Divergences are non-operational for finite physical systems.-Systemic Collapse: The paradox is logically and physically nullified within finite comprehension.

Apostol Vourdas

Gilles Barthe, Minbo Gao, Jam Kabeer Ali Khan +7 more

We present sound and complete relational program logics for infinite-dimensional quantum and classical-quantum programs. The logics model assertions as self-adjoint unbounded linear relations, which simultaneously support quantitative and qualitative reasoning. Our main theoretical results include new convergence theorems and infinite-dimensional duality theorems for infinite-dimensional quantum states, which we use to establish completeness.

Herman Dishkant

Andre Kornell

The quantum propositional logic of Birkhoff and von Neumann is the logic of Boolean observables on a quantum system. The quantum predicate logic of Weaver extends this quantum propositional logic to a logic of Boolean observables on composite quantum systems. We investigate a variant of this quantum predicate logic in which the physical systems are modeled by hereditarily atomic von Neumann algebras. Such physical systems may be infinite-dimensional, but their superselection sectors are finite-dimensional. From the perspective of noncommutative mathematics, they are a quantum generalization of sets. We define an equality relation on each such quantum set. This definition allows us to axiomatize several classes of discrete quantum structures within quantum predicate logic, including quantum graphs, quantum groups and quantum metric spaces. We likewise characterize the quantum isomorphism of simple graphs. The proof that discrete quantum groups may be so axiomatized is due to Stefaan Vaes.

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