 Generalized measurement postulate

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 requiresFor an input density operator $\rho$,when $p_i>0$. The one-operator-per-outcome form is the special case with one $\alpha$ for each $i$.

The POVM effect $E_i={\sum }_{\alpha }{M}_{i\alpha }^{†}{M}_{i\alpha }$ 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 $\rho \mapsto {\sum }_{i,\alpha }{M}_{i\alpha }\rho M_{i\alpha }^{†}$. This is the distinction between a positive operator-valued measure and a quantum instrument. ^[ref]

For example, two unresolved operators $M_{00}=|0⟩⟨0|$ and $M_{01}=|0⟩⟨1|$ give the sole recorded outcome with certainty and reset every input qubit to $|0⟩$. A single operator with effect $I$ would instead act as an linear isometry; it cannot describe that reset on the same two-dimensional output.