Quantum logic is undecidable
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Mathematical Logic
Quantum logic is undecidable Tobias Fritz1 Received: 11 December 2018 / Accepted: 6 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature (∨, ⊥, 0, 1), where ‘⊥’ 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. Keywords Quantum logic · Orthomodular lattices · Hilbert lattices · Decidability · First-order theory · Restricted word problem · Finitely presented C*-algebra · Residually finite-dimensional · Quantum contextuality Mathematics Subject Classification Primary 03G12 · 03B25 · 46L99; Secondary 81P13
1 Introduction Quantum logic starts with the idea that quantum theory can be understood as a theory of physics in which standard Boolean logic gets replaced by a different form of logic, where various rules, such as the distributivity of logical and over logical or, are relaxed [1,2]. This builds on the observation that {0, 1}-valued observables behave like logical propositions: such an observable is a projection operator on Hilbert space, and it can be identified with the closed subspace that it projects onto. In this way, the conjunction (logical and) translates into the intersection of subspaces, while disjunction (logical or) is interpreted as forming the closed subspace spanned by two subspaces.
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Tobias Fritz [email protected] Perimeter Institute for Theoretical Physics, Waterloo, Canada
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Hence the closed subspaces of a complex Hilbert space H form the complex Hilbert lattice C(H), which is interpreted as the lattice of ‘quantum propositions’ and forms a particular kind of orthomodular lattice [3–5]. However, the theory of orthomodular lattices is quite rich and contains many objects other than complex Hilbert lattices. So in order to understand the laws of quantum logic, one has to find additional properties which characterize the latter kind of objects. Much effort has been devoted to this question, resulting in partial characterizations such as Piron’s theorem [6,7], Wilbur’s theorem [8] and Solèr’s theorem [9].1 However, the axioms for complex Hilbert lattices that these results suggest are quite sophisticated: atomicity, completeness or the existence of an infinite orthonormal sequence. These are conditions that cannot be expressed algebraically, i.e. as first-order properties using just a finite number of variables, algebraic operations, and quantifiers. Fortunately, there has also been a substantial amount of w
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