= Generalized measurement postulate
= Generalized measurement
{synonym}
A generalized <quantum measurement> can be described by <Kraus operators> $M_{i\alpha}$, where $i$ labels the recorded outcome and $\alpha$ labels alternatives unresolved by that record. Completeness requires
$$
\sum_{i,\alpha}M_{i\alpha}^\dagger M_{i\alpha}=I.
$$
For an input <density operator> $\rho$,
$$
p_i=\operatorname{Tr}\!\left(\sum_\alpha M_{i\alpha}^\dagger M_{i\alpha}\rho\right),\qquad
\rho_i=\frac{\sum_\alpha M_{i\alpha}\rho M_{i\alpha}^\dagger}{p_i}
$$
when $p_i>0$. The one-operator-per-outcome form is the special case with one $\alpha$ for each $i$.
The <POVM> effect $E_i=\sum_\alpha M_{i\alpha}^\dagger M_{i\alpha}$ determines the outcome <probability>, while the whole outcome map determines the <post-measurement state>. Keeping only $i$ can mix states that a refined record $(i,\alpha)$ would distinguish. Forgetting every outcome gives the trace-preserving <quantum channel> $\rho\mapsto\sum_{i,\alpha}M_{i\alpha}\rho M_{i\alpha}^\dagger$. This is the distinction between a <positive operator-valued measure> and a <quantum instrument>. <The Theory of Quantum Information by John Watrous/Basic notions of quantum information>[Section 2.3, Nondestructive measurements and instruments, equations 2.255–2.262, printed pp. 111–113]{ref}
For example, two unresolved operators $M_{00}=|0\rangle\langle0|$ and $M_{01}=|0\rangle\langle1|$ give the sole recorded outcome with certainty and reset every input <qubit> to $|0\rangle$. A single operator with effect $I$ would instead act as an <linear isometry>; it cannot describe that reset on the same two-dimensional output.
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