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Journal of Logic, Language and Information
Published in: Journal of Logic, Language and Information 2/2018

27-11-2017

On the Modal Logic of the Non-orthogonality Relation Between Quantum States

Author: Shengyang Zhong

Published in: Journal of Logic, Language and Information | Issue 2/2018

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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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Footnotes
1
Throughout this paper we make the convention that \(\{ 0 , \dots , n-1 \} = \emptyset \) when \(n = 0\).
 
2
I adopt this name from Moore (1995).
 
3
I adopt this name from Baltag and Smets (2005).
 
4
I adopt this name from Moore (1995).
 
5
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).
 
6
For the notion of a formula defining a property of Kripke frames, please refer to Section 3.1 in Blackburn et al. (2001).
 
7
The idea of this construction is inspired by Exercise 4.5.1 in Blackburn et al. (2001).
 
8
For a proof, please refer to textbooks in modal logic, for example, Section 4.3 of Blackburn et al. (2001).
 
9
For the notion of a formula corresponding to a property of Kripke frames, please refer to Definition 3.5 in Blackburn et al. (2001).
 
10
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.
 
11
For a set A, card(A) denotes the cardinality of A.
 
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Metadata
Title
On the Modal Logic of the Non-orthogonality Relation Between Quantum States
Author
Shengyang Zhong
Publication date
27-11-2017
Publisher
Springer Netherlands
Published in
Journal of Logic, Language and Information / Issue 2/2018
Print ISSN: 0925-8531
Electronic ISSN: 1572-9583
DOI
https://doi.org/10.1007/s10849-017-9262-2

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