A refinement of quantum mechanics by algorithmic randomness

Abstract: The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics 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. In this paper, based on the toolkit of algorithmic randomness, we present a refinement of the Born rule, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. Algorithmic randomness is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We then present an operational refinement of the Born rule for mixed states, as an alternative rule to it, based on algorithmic randomness. In particular, we give a precise definition for the notion of mixed state. We then show that all of the refined rules of the Born rule for both pure states and mixed states can be derived from a single postulate, called the principle of typicality, in a unified manner. We do this from the point of view of the many-worlds interpretation of quantum mechanics. Finally, we make an application of our framework to the BB84 quantum key distribution protocol in order to demonstrate how properly our framework works in practical problems in quantum mechanics, based on the principle of typicality.