The incompatibility relation between visibility of interference and distinguishability of paths

A model of the Young double-slit experiment is formulated in a fully quantum theoretical setting. The state and dynamics of a wall which has the double slits in it, as well as the state of a particle incoming to the double slits, are described in quantum theoretical terms. Incompatibility between producing the interference pattern and distinguishing the particle path is studied and their quantitative relation is established. It is argued that the uncertainty relation involved in the double-slit experiment is not the Ozawa-type uncertainty relation but the Kennard-type uncertainty relation of the position and the momentum of the double-slit wall. A possible experiment to test the incompatibility relation is suggested. It is also argued that various phenomena which occur at the interface of a quantum system and a classical system, including measurement, decoherence, interference and distinguishability, can be understood as different aspects of entanglement.