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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.22353 |
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| _version_ | 1866911538567184384 |
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| author | Gunji, Yukio-Pegio Ohzawa, Yoshihiko Tokuyama, Yuki Huang, Yu Nakamura, Kyoko |
| author_facet | Gunji, Yukio-Pegio Ohzawa, Yoshihiko Tokuyama, Yuki Huang, Yu Nakamura, Kyoko |
| contents | Contextuality is widely regarded as a hallmark of quantum information, yet its structural origin is often obscured by probabilistic or operational formulations. In this work, we show that non-distributive orthomodular structure need not be postulated, but arises canonically as a left adjoint from classical Boolean contexts. We introduce a gluing functor that takes pairs of Boolean algebras and identifies only their minimal and maximal elements via a categorical pushout. The resulting lattice is orthomodular but generically non-distributive. We prove that this construction is left adjoint to a forgetful functor selecting Boolean subalgebras, thereby providing a free but constrained generation of quantum-logical structure from classical contexts. Furthermore, we demonstrate that the failure of this pushout to remain Boolean is equivalent to the absence of global sections in the sheaf-theoretic framework of Abramsky and Brandenburger. This establishes a precise correspondence between contextuality as a sheaf obstruction and non-distributivity as a colimit failure. Our results offer a categorical and lattice-theoretic reconstruction of contextuality that precedes probabilistic notions and clarifies the structural necessity of quantum logic in information-theoretic settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22353 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure Gunji, Yukio-Pegio Ohzawa, Yoshihiko Tokuyama, Yuki Huang, Yu Nakamura, Kyoko Quantum Physics Contextuality is widely regarded as a hallmark of quantum information, yet its structural origin is often obscured by probabilistic or operational formulations. In this work, we show that non-distributive orthomodular structure need not be postulated, but arises canonically as a left adjoint from classical Boolean contexts. We introduce a gluing functor that takes pairs of Boolean algebras and identifies only their minimal and maximal elements via a categorical pushout. The resulting lattice is orthomodular but generically non-distributive. We prove that this construction is left adjoint to a forgetful functor selecting Boolean subalgebras, thereby providing a free but constrained generation of quantum-logical structure from classical contexts. Furthermore, we demonstrate that the failure of this pushout to remain Boolean is equivalent to the absence of global sections in the sheaf-theoretic framework of Abramsky and Brandenburger. This establishes a precise correspondence between contextuality as a sheaf obstruction and non-distributivity as a colimit failure. Our results offer a categorical and lattice-theoretic reconstruction of contextuality that precedes probabilistic notions and clarifies the structural necessity of quantum logic in information-theoretic settings. |
| title | Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2603.22353 |