Title: The Principle of Typicality

Abstract:
This is a sequel to our presentation at CCR 2014. The notion of probability plays a crucial role in quantum mechanics. It appears as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. At CCR 2014 we presented an alternative rule to the Born, using the notion of Martin-Löf randomness with respect to Bernoulli measure for specifying the property of the results of quantum measurements in an operational way. This alternative rule is about pure states. In this talk we generalize this alternative rule over mixed states. We then show that all of our new rules about quantum measurements based on algorithmic randomness, including the alternative rule to the Born rule about pure states and its generalization over mixed states, can be derived from a single postulate, called the principle of typicality, in a unified way.