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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?

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