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I'm studying weak measurement and I would like to ask about the interaction Hamiltonian required for weak measurement.

The interaction unitary operator of weak measurement is defind as $\hat{U} = e^\hat{-iH} = e^{-ig\hat{\Pi}_s\hat{Y}_p}$ where $g$ is the interaction strength, $\hat{\Pi}_s$ is the projection operator on the system $s$, and $\hat{Y}_p$ is the pauli $\hat{Y}$ operator on the pointer $p$.

My questions is as follows:

Should the pointer operator always be the Pauli $\hat{Y}$ operator? Can't it be Pauli $\hat{X}$ operator or others?

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As far as the $U$ is a unitary operator, it is fine. Consider this Hamiltonian of the form $H = g * (p \otimes x)$, $px$ is well known that it is Hermitian.

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  • $\begingroup$ Thank you for your answer. By the way, what if the system operator $\Pi$ would be non-Hermitian? $\endgroup$
    – Alex
    Commented Oct 11, 2022 at 15:51
  • $\begingroup$ @Alex I don't know what happens when we have non-Hermitian operator as system. But I hope this paper which I have not read gives you some answer: arxiv.org/pdf/1204.3296.pdf $\endgroup$
    – FearlessVirgo
    Commented Oct 11, 2022 at 16:03
  • $\begingroup$ Ok, I will read that paper. If we have a non-Hermitian operator in the system and a Hermitian operator in the pointer, do we need to call the entire operator as non-Hermitian? $\endgroup$
    – Alex
    Commented Oct 11, 2022 at 16:06
  • $\begingroup$ @Alex Its trivially true, take $H^{\dagger} \neq H$ $\endgroup$
    – FearlessVirgo
    Commented Oct 11, 2022 at 16:10

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