Dynamics of Intermediate Measurements in Quantum Systems

We study the dynamics of a quantum system in which an intermediate property $m$ is measured in between initial and final measurements of two different non-commuting properties $a$ and $b$. Since this intermediate measurement must involve an interaction, we use this case to explore in more detail the dynamics of measurement and propose that the physics of this change is described by a unitary transformation parametrised by a random parameter, with the selection of $m$ described by a superposition of these unitaries. We prove a set of conditions which must hold in order for this superposition to remain unitary and then argue that in the case where $U$ is a matrix, it must be a random unitary matrix. We outline a numerical example with matrices and make some predictions based on the conditions we have found which are capable of experimental verification. We finish with a brief discussion of the complex phases which appear in the relations between non-commuting properties and their relevance for the dynamics associated with measurement.