Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h_s(φ,A) of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ, we prove a few results on that, define the entropy of a dynamical system h_s(Φ), and show its invariance. The concept of sufficient families is also given and we establish that h_s(Φ) comes out to be equal to the supremum of h_s(φ,A), where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,φ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B, s₀,φ), where B is a Boolean algebra and s₀ is a state on B.