Quantum categories for quantum logic


Vladimir Vasyukov


The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors $\mathsf{[E,Sets]}$ where $\mathsf{E}$ is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called $quantos$ – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative $C^*$-algebras, which in turn are studied in their quantum habitat of noncommutative $C^*$-algebras – more fundamental structures than commutative $C^*$-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors $\mathsf{[E,QSets]}$ is constructed where $\mathsf{QSets}$ is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.




Non-classical logics


Abramsky, Duncan, 2004 – Abramsky, S., Duncan, R. “Categorical quantum logic”, Proc. QPL, 2004. pp. 3–20.
Beran, 1984 – Beran, L. Orthomodular Lattices: Algebraic Approach, Prague, Academia, 1984. 394 pp.
Birkhoff, 1967 – Birkhoff, G., Lattice Theory, Providence, Rhode Island, 1967. 423 pp. Birkhoff, Neumann, 1936 – Birkhoff, G., Neumann, J. von. “The logic of quantum mechanics”, Annal. Math, 1936, Vol. 37, pp. 823–843.
Crane, 2007 – Crane, L. What is the mathematical structure of quantum space-time?, 2007. arXiv: [gr-qc/0706.4452].
Cutland, Gibbins, 1982 – Cutland, N. J., Gibbins, P. F. “A regular sequent calculus for quantum logic in which ∧ and ∨ are dual”, Log. et Anal, 1982, Vol. 25, No. 95, pp. 221–248.
Dalla Chiara, Giuntini, 2002 – Dalla Chiara, M.-L., Giuntini, R. “Quantum Logic”, Handbook of Philosophical Logic (2nd Edition), eds by D. Gabbay and F. Guenthner. Vol. 6, 2002, pp. 129–228.
Dunn et al., 2013 – Dunn, J.M., Moss, L.S. and Wang, Z. “The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing“, Journal of Philosophical Logic, 2013, Vol. 42, Issue 3, pp. 443–459.
Goldblatt, 1974 – Goldblatt, R.I. “Semantic analysis of orthologic”, J. Phil. Log, 1974, Vol. 3, No. 1–2, pp. 19–35.
Goldblatt, 1979 – Goldblatt, R.I. Topoi. The categorial analysis of logic. North-Holland, Amsterdam, N.Y., Oxford, 1979. 486 pp.
Hardegree, 1981 – Hardegree, G.M. “An axiom system for orthomodular quantum logic”, Studia Logica, 1981, Vol. 40, No. 1, pp. 1–12.
Isham, Doring, 2007 – Isham, Ch., Doring, A. A topos foundation for theories of physics (in 4 parts), 2007. arXiv: [quantph/0703060], [quant-ph/0703062], [quantph/0703064], [quant-ph/0703066].
Landsman, 2017 – Landsman, K. “Foundations of Quantum Theory. From Classical Concepts to Operator Algebras”, Fundamental Theories of Physics, 2017, Vol. 188, 881 pp.
Nishimura, 1980 – Nishimura, H. “Sequential method in quantum logic”, J.Symb.Log., 1980, Vol. 45, No. 2, pp. 335–352.
Takeuti, 1981 – Takeuti, G. “Quantum Set Theory”, in: Current Issues in Quantum Logic, Ettore Majorana International Science Series, eds by E.G. Beltrametti and B.C. van Fraassen. Plenum, New York, Vol. 8, 1981, pp. 303–322.
Vasyukov, 1989 – Vasyukov, V.L. “Kvantovaya logika v toposakh” [Quantum Logic in Topoi], in Investigations on Non-Classical Logics, ed by V.A. Smirnov. Moscow: Nauka, 1989. pp. 338–348. (In Russian) Vasyukov, 2005 – Vasyukov, V.L. Kvantovaya logika [Quantum Logic]. Moscow, 2005. 192 pp. (In Russian)