Abstract
In this paper we present two families of probability logics (denoted QLP and \(QLP^{ORT}\)) suitable for reasoning about quantum observations. Assume that \(\alpha \) means “O = a”. The notion of measuring of an observable O can be expressed using formulas of the form \(\square \lozenge \alpha \) which intuitively means “if we measure O we obtain \(\alpha \)”. In that way, instead of non-distributive structures (i.e., non-distributive lattices), it is possible to relay on classical logic extended with the corresponding modal laws for the modal logic \({\textbf{B}}\). We consider probability formulas of the form \(CS_{z_{1},\rho _{1}; \ldots ; z_{m},\rho _{m}} \square \lozenge \alpha \) related to an observable O and a possible world (vector) w: if a is an eigenvalue of O, \(w_{1}\), ..., \(w_{m}\) form a base of a closed subspace of the considered Hilbert space which corresponds to eigenvalue a, and if w is a linear combination of the basis vectors such that \(w=c_{1}\cdot w_{1}+ \cdots + c_{m}\cdot w_{m}\) for some \(c_{i}\in {\mathbb {C}}\), then \(\Vert c_{1}-z_{1}\Vert \le \rho _{1}\), ..., \(\Vert c_{m}-z_{m}\Vert \le \rho _{m}\), and the probability of obtaining a while measuring O in the state w is equal to \(\Sigma _{i=1}^{m}\Vert c_{i}\Vert ^{2}\). Formulas are interpreted in reflexive and symmetric Kripke models equipped with probability distributions over families of subsets of possible worlds that are orthocomplemented lattices, while for \(QLP^{ORT}\) also satisfy ortomodularity. We give infinitary axiomatizations, prove the corresponding soundness and strong completeness theorems, and also decidability for QLP-logics.
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Notes
The actual complex numbers depend on d and D. The square modules of the chosen complex numbers correspond to the simple symmetric case, which clearly demonstrates the interference.
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Ilić Stepić, A., Ognjanović, Z. & Perović, A. Probability Logics for Reasoning About Quantum Observations. Log. Univers. 17, 175–219 (2023). https://doi.org/10.1007/s11787-023-00326-y
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DOI: https://doi.org/10.1007/s11787-023-00326-y