What's the difference between Kraus operators and measurement operators?

Question

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).

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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?

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^{[1]: Lecture Notes for Ph219/CS219: Quantum Information Chapter 3 (John Preskill, 2018)}

Answer 1

Quantum measurement (without results recording) is just a special case of quantum operation (quantum channel). So, yes, measurement operators (as in general measurement formalism) are indeed Kraus operators. But Kraus operators are more general. For example, they can be "rectangular", while measurement operators can't.