The issues are most simply addressed by using a simple schematic quantum model of measurement and preparation that goes back to J. von Neumann (1932: sec. VI.3). Let \(\HC_s\) be the Hilbert space of the system to be measured, henceforth referred to as a particle, and \(\HC_m\) that of the measuring device or apparatus. For example, \(\HC_s\) could be the 2-dimensional Hilbert space of the spin of a spin-half particle, whereas the quantum description of its position might be included in \(\HC_m\), along with all the many degrees of freedom needed to describe the apparatus itself. Let \(\{\ket{s^j}\}\) be an orthonormal basis for \(\HC_s\), with states labeled by a superscript so subscripts can label time. Let