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Quantum logic is undecidable

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Abstract

We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature \((\vee ,\perp ,0,1)\), 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.

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Notes

  1. See also the survey [10] for a more recent exposition from a geometrical perspective.

  2. The formulation of [19] asks for the existence of a quantum model on H, but this is clearly equivalent: the existence of a quantum model requires the existence of a quantum representation to begin with; conversely, one can use a quantum representation and an arbitrary state in its underlying Hilbert space to obtain a quantum model.

  3. See [31] for an introduction to von Neumann algebras from the perspective of quantum logic, including a treatment of their lattices of projections (Section 6.2).

  4. Where as usual in von Neumann algebra theory, normal means continuous with respect to the ultraweak topologies.

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Acknowledgements

We thank John Harding, Christian Herrmann, Ravi Kunjwal, Anthony Leverrier, Mladen Pavičić, Stefan Schmidt, William Slofstra, Rob Spekkens, Karl Svozil, Andreas Thom and Moritz Weber for discussions. Special thanks go to Christian Herrmann for copious help with the literature and useful feedback, and to William Slofstra for pointing out that Theorem 14 actually follows from the arguments used in an earlier version of this paper to prove a weaker result. Most of this paper was written while the author was with the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany.

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    Tobias Fritz

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Fritz, T. Quantum logic is undecidable. Arch. Math. Logic 60, 329–341 (2021). https://doi.org/10.1007/s00153-020-00749-0

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