Categories and Types in Logic, Language, and Physics

Volume 8222 of the series Lecture Notes in Computer Science pp 90-107

A Logical Basis for Quantum Evolution and Entanglement

  • Richard F. BluteAffiliated withDepartment of Mathematics and Statistics, University of Ottawa
  • , Alessio GuglielmiAffiliated withDepartment of Computer Science, University of Bath
  • , Ivan T. IvanovAffiliated withDepartment of Mathematics, Vanier College
  • , Prakash PanangadenAffiliated withSchool of Computer Science, McGill University
  • , Lutz StraßburgerAffiliated withINRIA Saclay, École Polytechnique

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We reconsider discrete quantum causal dynamics where quantum systems are viewed as discrete structures, namely directed acyclic graphs. In such a graph, events are considered as vertices and edges depict propagation between events. Evolution is described as happening between a special family of spacelike slices, which were referred to as locative slices. Such slices are not so large as to result in acausal influences, but large enough to capture nonlocal correlations.

In our logical interpretation, edges are assigned logical formulas in a special logical system, called BV, an instance of a deep inference system. We demonstrate that BV, with its mix of commutative and noncommutative connectives, is precisely the right logic for such analysis. We show that the commutative tensor encodes (possible) entanglement, and the noncommutative seq encodes causal precedence. With this interpretation, the locative slices are precisely the derivable strings of formulas. Several new technical results about BV are developed as part of this analysis.