Exponential Modalities and Complementarity (extended abstract)

Robin Cockett
(University of Calgary)
Priyaa Varshinee Srinivasan
(University of Calgary)

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics.

The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.

In Kohei Kishida: Proceedings of the Fourth International Conference on Applied Category Theory (ACT 2021), Cambridge, United Kingdom, 12-16th July 2021, Electronic Proceedings in Theoretical Computer Science 372, pp. 207–220.
A full version of this paper, containing all proofs, appears at https://arxiv.org/abs/2103:05191
Published: 3rd November 2022.

ArXived at: http://dx.doi.org/10.4204/EPTCS.372.15 bibtex PDF
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