# On modelling the measurement?

## Background

So I had the following heuristic idea to model a measurement. Let's say I have $$2$$ Hamiltonians. They are $$H_{system}$$ and $$H_{detector}$$. Now,

$$\hat H = \begin{cases} \hat H_{system} \otimes \hat I + \hat I \otimes \hat H_{detector} & t < -t_0\\ \hat H_{system} \otimes \hat I + \hat I \otimes \hat H_{detector} + H_{int} & -t_0\leq t \leq t_0\\ H_{system} \otimes \hat I + \hat I \otimes \hat H_{detector} & t > t_0\\ \end{cases}$$

$$H_{int}$$ is the interaction Hamiltonian between $$\hat H_{detector}$$ and $$\hat H_{system}$$ in the interaction of $$2t_0$$. Let the system initially be in the state $$|s\rangle$$ and the detector in the state $$|d \rangle$$.

Now, just before the measurement we have $$|s,-t_0 \rangle$$. Then, this state does not get sufficient time to change responding to the change in the Hamiltonian (sudden approximation).

$$|\Psi,-t_0 \rangle = |s,-t_0 \rangle \otimes |d,-t_0 \rangle$$

Using the unitary operator:

$$|\Psi,t_0 \rangle= \exp{(i\int_{-t_0}^{t_0} \hat H(t)dt)} |\Psi,-t_0 \rangle$$

Then post the interaction, the wavefunction $$|\Psi,t_0 \rangle$$ can be decomposed into:

$$|\Psi,t_0 \rangle = |s,t_0 \rangle \otimes |d,t_0 \rangle$$

Note, therefore it means one cannot simply think of $$H_{system}$$ as a completely isolated system:

$$|s,t_0 \rangle \neq U_{system} |s,-t_0 \rangle$$

where $$U_{system}$$ is the unitary operator of the system. (See Minimum number of ancilla qubits required to make a transformation unitary? )

So how do we model this? Recall, when $$t < -t_0$$, we can write the unitary operator as:

$$\hat U= \begin{cases} \hat U_{system} \otimes \hat U_{detector} & t < -t_0\\ \exp{(i\int_{-t_0}^{t_0} \hat H(t)dt)} & -t_0\leq t \leq t_0\\ \hat U_{system} \otimes \hat U_{detector} & t > t_0\\ \end{cases}$$

The difficult part is the time between $$-t_0$$ and $$t_0$$. Since, we do not know what the operation $$\exp{(i\int_{-t_0}^{t_0} \hat H(t)dt)}$$ is doing. We know it should not favor any state. Further, on repeating the experiment getting an interaction time of exactly $$2t_0$$ is not always possible.

Thus, we represent this by a matrix which is not unitary $$M$$:

$$|s_{final} \rangle = M |s_{initial} \rangle$$

Or M can be thought of as some kind of random matrix.

## Question

Is this a valid line of attack? Are there any papers with similar lines of attack? Where the measurement operator is obeys a (justified) probability distribution which mimics the the Born Rule?