The question was can we describe the process of measurement in functional form?
We know hermitian operator can be used to represent measurement. In Heisenberg's picture, it even described the "motion". Once it act at a states, it instantly collapse the states and returned a real value as the result of measurement.
However, just as practical interaction has cross section time, a "measurement" must take time to happen as well. So is there any way to describe "process" of measurement as a functional process, compare to the hermitian operator's almost "instant" result description?
The important thing here is to distinguish the question into two part:
The process of measurement of an eigenvalue for each eigenvector.
The process of measurement of which collapse the probability states.
In the beginning I thought this might not be possible, but then I realized that it had nothing to do with EPR paradox, or anything that's obviously contradicted to it, as the result can still be returned as dynamics descriptions in Hilbert space based on the returned bases.
Yet, there seemed to be something intrinsic wrong about it, because(start from an explanation that explains 1 and want to explain 2) the qunatization seemed to forbid the view of a smooth transition. (start from an explanation that explains 2 and want to explain 1) If we say everything in the process of measurement was simply under the cross section time, then the collapse of the probability states would require the process of measurement to be "base insensitive", which clearly was not the case, as many of the measurement did cared about the probability weight. This would resulted such function to have return more than one result, which violates the definition of function.
Can we describe the process of measurement in functional form?