Introduction to quantum probability

This thesis is a review of the quantum logic approach to quantum probability theory which was first studied by Birkhoff and von Neumann in 1936. Our study is based on the more general quantum structure which is known as the $\sigma$-orthocomplete orthomodular poset or quantum logic. We introduce the definitions of states and observables and show that the quantum probability theory is a generalization of the classical probability theory due to Kolmogorov. Discussions about joint distributions of observables are given, and we also give the general expression of the uncertainty relation in the logico-algebraic sense which generalizes the well-known Heisenberg uncertainty relation. Two examples, the classical probability structure of Kolmogorov and the Hilbert space quantum logic, are given as illustrations throughout the whole discussion.