Probability Logics for Reasoning About Quantum Observations

Abstract

In this paper we present two families of probability logics (denoted QLP and QLPORT) suitable for reasoning about quantum observations. Assume that α means “O = a”. The notion of measuring of an observable O can be expressed using formulas of the form □ ◊α which intuitively means “if we measure O we obtain α”. 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 B. We consider probability formulas of the form CSz1,ρ1;…;zm,ρm□◊α related to an observable O and a possible world (vector) w: if a is an eigenvalue of O, w1,.., wm 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= c1· w1+ ⋯ + cm· wm for some ci∈ C, then ‖ c1- z1‖ ≤ ρ1,.., ‖ cm- zm‖ ≤ ρm, and the probability of obtaining a while measuring O in the state w is equal to Σi=1m‖ci‖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 QLPORT also satisfy ortomodularity. We give infinitary axiomatizations, prove the corresponding soundness and strong completeness theorems, and also decidability for QLP-logics.