It is said in a lecture note[1] by John Preskill that,
Equivalently, we may imagine measuring system $B$ in the basis $\{|a\rangle\}$, but failing to record the measurement outcome, so we are forced to average over all the possible post-measurement states, weighted by their probabilities. The result is that the initial density operator $\boldsymbol{\rho} = |\psi\rangle\langle \psi|$ is subjected to a linear map $\mathcal{E}$, which acts as
$$\mathcal{E}(\boldsymbol {\rho}) = \sum_a M_a\boldsymbol{\rho} M^{\dagger}_a, \tag{3.32}$$
where the operators $\{M_a\}$ obey the completeness relation eq.(3.25).
The justification for this name will emerge shortly. Eq.(3.32) is said to be an operator-sum representation of the quantum channel, and the operators $\{M_a\}$ are called the Kraus operators or operation elements of the channel.
It seems that Kraus operators and measurement operators are the same thing. Is that right?
[1]: Lecture Notes for Ph219/CS219: Quantum Information Chapter 3 (John Preskill, 2018)
