The quantum measurement, formally acting on a density matrix $\rho$ as
\[\rho \rightarrow \sum_{i} M_{i} \rho M_{i}^{\dagger}.\]is a special case of quantum operation, where $i$ indices the different outcomes . The measurement Kraus operators $M_{i}$ are completely positive and satisfy condition s.t. the trace of the density matrix is preserved:
\[\sum_{i} M_{i}^{\dagger} M_{i} = I.\]Intuitively, the measurement is a process that transforms a quantum state to a mixed state. $i$ indices the different outcomes, so that the quantum state collapses stochastically into states given by each process operator $M_i$. The measurement operators are also called POVMs (positive operator-valued measures).
