Can category-theoretic semantics resolve the problem of the interpretation of the quantum state vector?

TL;DR

This paper explores whether category-theoretic semantics can clarify the interpretation of quantum state vectors, arguing that computational complexity considerations challenge the idea of a direct isomorphism with physical reality.

Contribution

It introduces a novel integration of category theory and computational complexity to analyze the quantum state interpretation problem.

Findings

Category theory alone is insufficient to resolve the quantum state interpretation.

Computational complexity considerations suggest the non-existence of a universal isomorphism.

The hypothesis of a one-to-one correspondence is likely unsuitable for generic quantum systems.

Abstract

Do correctness and completeness of quantum mechanics jointly imply that quantum state vectors are necessarily in one-to-one correspondence with elements of the physical reality? In terms of category theory, such a correspondence would stand for an isomorphism, so the problem of the status of the quantum state vector could be turned into the question of whether state vectors are necessarily isomorphic to elements of the reality. As it is argued in the present paper, in order to tackle this question, one needs to complement the category-theoretic approach to quantum mechanics with the computational-complexity-theoretic considerations. Based on such considerations, it is demonstrated in the paper that the hypothesis of the isomorphism existing between state vectors and elements of the reality is expected to be unsuitable for a generic quantum system.