In Section 8.5 of the 10th anniversary edition, Nielsen and Chuang discuss the limitations of the quantum operations framework. They give an example of a qubit prepared in an unknown state $\rho$, entangled with a qubit in the environment that is $|0\rangle$ if $\rho$ is in the bottom half of the Bloch sphere, and $|1\rangle$ if $\rho$ is in the top half.
Suppose the interaction is such that a controlled-NOT is performed between the principal system and the extra qubit in the laboratory system. Thus, if the system’s Bloch vector was initially in the bottom half of the Bloch sphere it is left invariant by the process, while if it was initially in the top half of the Bloch sphere it is rotated into the bottom half of the Bloch sphere. Obviously, this process is not an affine map acting on the Bloch sphere, and therefore, by the results of Section 8.3.2, it cannot be a quantum operation.
(Section 8.3.2 shows that a single-qubit operation is an affine map on the Bloch sphere)
However, it seems to me that the correct way to model this scenario is by considering a state $$ \rho = p \rho_0 \otimes |0\rangle \langle 0| + (1-p) \rho_1 \otimes |1\rangle \langle 1|. $$ Then, the operation is not only well-described by the quantum operations formulation, but is in fact simply the unitary $CNOT$, with the control on the second qubit. Why do they claim that this is not a quantum operation?
Further, if this is in fact not a quantum operation, what would be a formalism that can describe it? In every other source I've encountered, an arbitrary physical process has been assumed to be modellable as a quantum operation.