In this paper we will study a function of simultaneus measurements for quantum events (s-map) which will be compared with the conditional states on an orthomodular lattice as a basic structure for quantum logic.
By a “quantum logic” we mean a pair F, P where P is a set and F is a set of functions from P to the closed real unit interval satisfying three postulates which we describe in intuitive terms here. Cf. [2], [4], [7]. P may be interpreted as the set of events and F the set of states of a “physical system”, and f(x) then becomes the probability of occurrence of the event x in the state f. Since the outcome of an experiment is an estimate for some f(x), or a collection of such estimates, it is natural to identify events which cannot be distinguished by experiment.
The theory of orthomodular lattice and its relationships that depend on s-map and condi-tional states definitions has shown in several types of probabilistic relations. An orthomodularlattice with a conditional states can be applied and defined as a model for non-compatible events.Studying of s-maps or conditional states on an orthomodular lattice helps us to describe suchproperties of random events, which are difficult to be described by Boolean algebra. In a par-ticularly observed quantum or classical system the information, that we cannot access results inour description of the system be coming mixed even if we have perfect initial knowledge.That is, if the system is quantum the conditional states will be given by a state matrix and ifclassical, the conditional states will be given by a probability distribution. We show conditionswhen a quantum system has the same properties as the classical probability space.Acknowledgement. This work was supported by Science and Technology Assistance AgencyunderthecontractNo. APVT-20-003204, VEGA-1/3014/06, VEGA-1/3321/06, VEGA-1/3006/06