How can I understand these two equations about the probability of measurement?

Question

I have trouble understanding these two equations in the nielsen-and-chuang textbook. Suppose we perform a measurement described by the operator $M_m$, if the initial state is $|\psi_i\rangle$, then the probability of getting result m is:

$$ p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle $$

The form of this equation looks like the overlap between two states, but I'm not exactly sure what does $ M_m|\psi_i\rangle$ mean? Is this relevant to the projection operator?

Also, given the initial state $|\psi_i\rangle$, the state after obtaining the result m is

$$ |\psi_i^m\rangle = \frac{M_m|\psi_i\rangle}{\sqrt{\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle}} $$

Why that's the case? Thanks!!

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Cross-posted on quantumcomputing.SE

Answer 1

The measurement is a projection so does not preserve the norm. Thus, if $M_m$ takes $\vert \psi_i\rangle $ to the unnormalized state $\vert\phi\rangle := M_m\vert \psi_i\rangle$, its normalized version is \begin{align} \vert\psi_i^m\rangle &= \frac{\vert \phi\rangle}{\sqrt{\langle \phi \vert \phi\rangle}}=\frac{M_m\vert \psi_i\rangle}{\sqrt{\langle \psi_i\vert M_m^\dagger M_m\vert \psi_i\rangle}} \end{align}