Quantum logic is undecidable

TL;DR

This paper proves that the first-order theory of closed subspaces in complex Hilbert spaces, involving orthogonality, is undecidable, revealing fundamental limits on algorithmic reasoning in quantum logic.

Contribution

It demonstrates the undecidability of quasi-identities in quantum logic, connecting group theory, hypergraph approaches, and quantum satisfiability problems.

Findings

Undecidability of quasi-identities in quantum logic.

Connection to Slofstra's group theory results.

Implication for quantum satisfiability problems.

Abstract

We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature $(\lor,\perp,0,1)$, where `$\perp$' is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent result of Slofstra in combinatorial group theory. It follows upon reinterpreting that result in terms of the hypergraph approach to quantum contextuality, for which it constitutes a proof of the inverse sandwich conjecture. It can also be interpreted as stating that a certain quantum satisfiability problem is undecidable.