Probability Logics for Reasoning About Quantum Observations

In this paper we present two families of probability logics (denoted QLP and QLP^ORT ) 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 CS_z_1,ρ _1; … ; z_m,ρ _m□◊α 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· w_1+ ⋯ + c_m· w_m for some c_i∈ℂ , then ‖ c_1-z_1‖≤ρ _1 , ..., ‖ c_m-z_m‖≤ρ _m , and the probability of obtaining a while measuring O in the state w is equal to Σ _i=1^m‖ c_i‖ ^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.