Probability Logics for Reasoning About Quantum Observations

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.