Abstract
It is well known that the non-orthogonality relation between the (pure) states of a quantum system is reflexive and symmetric, and the modal logic \(\mathbf {KTB}\) is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies each one of these two properties and both, respectively. We also show that the modal logics involved are decidable.
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Notes
Throughout this paper we make the convention that \(\{ 0 , \dots , n-1 \} = \emptyset \) when \(n = 0\).
I adopt this name from Moore (1995).
I adopt this name from Baltag and Smets (2005).
I adopt this name from Moore (1995).
For the notion of a formula being canonical for a property of Kripke frames, please refer to Definition 4.31 in Blackburn et al. (2001).
For the notion of a formula defining a property of Kripke frames, please refer to Section 3.1 in Blackburn et al. (2001).
The idea of this construction is inspired by Exercise 4.5.1 in Blackburn et al. (2001).
For a proof, please refer to textbooks in modal logic, for example, Section 4.3 of Blackburn et al. (2001).
For the notion of a formula corresponding to a property of Kripke frames, please refer to Definition 3.5 in Blackburn et al. (2001).
x, y, \(u_i\) and \(v_j\) are used to denote points in a Kripke frame in this paper, except that in this first-order formula they serve as individual variables.
For a set A, card(A) denotes the cardinality of A.
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Acknowledgements
The axiomatizations and their decidability of the modal logics of state spaces and state spaces satisfying Superposition are obtained in Section 4.3 of my PhD thesis (Zhong 2015) finished at the Institute for Logic, Language and Computation (ILLC), University of Amsterdam, in 2015. I am very grateful to my supervisors, Prof. Johan van Benthem, Dr. Alexandru Baltag and Prof. Sonja Smets, for their supervision and the detailed comments on my thesis which are invaluable in the writing of this paper. I am also very grateful to the members in my PhD committee, Dr. Nick Bezhanishvili, Prof. Robert Goldblatt, Prof. John Harding, Prof. Yde Venema, Prof. Ronald de Wolf and Prof. Mingsheng Ying, for their comments on my thesis which are very helpful in writing this paper. The results were also presented in Tsinghua Logic Colloquium on Information Flow in a Social World, and I am very grateful to the audience for the interesting discussion. I thank very much the two reviewers of this paper for their detailed and helpful comments.
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My PhD research was funded by China Scholarship Council (CSC), and the writing of this paper is supported by NSSFC Grant 14ZDB015 and the International Postdoctoral Exchange Fellowship Program run by the China Postdoctoral Council.
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Zhong, S. On the Modal Logic of the Non-orthogonality Relation Between Quantum States. J of Log Lang and Inf 27, 157–173 (2018). https://doi.org/10.1007/s10849-017-9262-2
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DOI: https://doi.org/10.1007/s10849-017-9262-2