Quantum Formalism in Lower-Dimensional Real and Complex Hilbert Spaces

C. S. Sharma

Anuj Dawar, Nihil Shah

The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiablity for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiablity problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.

Gianluca Stefanucci

Anuj Dawar, Nihil Shah

The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiability for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiability problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.

Paul Bracken

Shubhayan Sarkar, David Trillo, Marc-Olivier Renou +1 more

Quantum information theory, the formalism for representing information contained in quantum systems, is based on complex Hilbert spaces. It was recently shown that in quantum networks involving three parties with nontrivial locality constraints, this formalism predicts correlations that cannot be explained by real quantum theory, a variant of quantum mechanics based on real Hilbert spaces. In this work, we study a scenario withparties sharing quantum systems in a star network. We then construct a multipartite Bell inequality that exhibits a gap between quantum theory and its variant based on real numbers, which grows linearly with, and is thus arbitrarily large in the asymptotic limit. This implies, that, as the number of parties grows, Hilbert space formalism based on real numbers becomes exceedingly worse at describing complex networks of quantum systems. We also compute the tolerance of this gap to experimental errors.