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MiS Preprint
50/2016

Quantum logic is undecidable

Tobias Fritz

Abstract

We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature $(\lor,\perp,0)$, where '$\perp$' is orthogonality. Our main result is that already its purely implicational fragment is undecidable: there is no algorithm to decide whether an implication between equations in the language of orthomodular lattices is valid in all complex Hilbert spaces. This is a simple corollary of a recent result of Slofstra in combinatorial group theory, and follows upon reinterpreting that result in terms of the hypergraph approach to contextuality.

Received:
Jul 20, 2016
Published:
Aug 8, 2016
MSC Codes:
03G12, 03B25, 46L99, 81P13
Keywords:
Quantum logic, orthomodular lattices, Hilbert lattices, decidability, restricted word problem, residually finite-dimensional C*-algebras, quantum contextuality

Related publications

inJournal
2021 Repository Open Access
Tobias Fritz

Quantum logic is undecidable

In: Archive for mathematical logic, 60 (2021) 3/4, pp. 329-341