The <generalized measurement postulate> specifies operators $\{M_i\}$ satisfying
$$
\sum_iM_i^\dagger M_i=I.
$$
On state $\rho$, outcome $i$ has probability
$$
\boxed{p_i=\operatorname{Tr}(M_i^\dagger M_i\rho)},
$$
and, when $p_i>0$, conditional state
$$
\boxed{\rho_i=\frac{M_i\rho M_i^\dagger}{p_i}}.
$$
The effects $E_i=M_i^\dagger M_i$ form a <positive operator-valued measure>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 5 / a / i / Solution

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{wiki}

A positive operator-valued measure is a family of positive operators $E_i$ satisfying $\sum_iE_i=I$. It determines measurement probabilities $\Pr(i)=\operatorname{Tr}(E_i\rho)$; choosing measurement operators with $E_i=M_i^\dagger M_i$ additionally determines state updates.

= POVM
{c}
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 Incoming links: Positive operator-valued measure