Why is the operation in Nielsen and Chuang's Section 8.5 not a quantum operation?

Question

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.

Answer 1

This is exactly the point that N&C are making. Yes, this is a real operation that can happen and it is simply described by the controlled-NOT. However, if you wanted to describe the net effect as a quantum operation on just the qubit, you cannot because the action when viewed from that perspective is not affine. So the point is that in those circumstances, the quantum operations formalism is inadequate to describe something that actually happens.

Answer 2

After some further reading, I think I understand the problem a bit better now. My hidden assumption was that, after writing down the unitary on the larger Hilbert space, we can restrict our attention to its action on the Hilbert space of the system under study, and that this action can always be expressed in terms of Kraus operators.

However, this is not so. The recipe Nielsen and Chuang give in their Equations 8.9-8.10, for an environment in state $|e_0\rangle$ and a basis for the environment $\{ |e_k\rangle \}$, is

$$ \begin{align} \mathcal{E}(\rho) &= \mathrm{Tr}_{\mathrm{env}} \left[ U \left( \rho \otimes |e_0\rangle \langle e_0| \right) U^{\dagger} \right] \\ &= \sum_k \langle e_k| U \left( \rho \otimes |e_0\rangle \langle e_0| \right) U^{\dagger} |e_k\rangle \\ &= \sum_k (\langle e_k| U |e_0 \rangle) \rho (\langle e_k| U |e_0 \rangle)^{\dagger} \\ &= \sum_k E_k \rho E_k^{\dagger}, \end{align} $$

which assumes that the system is in a tensor product with the environment. This assumption turns out to be critical, which is the point that Section 8.5 was trying to make.

I'm not sure what to make of my initial impression that other sources claimed that quantum operations were always sufficient; perhaps I confused the direction of the fact that any physical process can be represented by a unitary in a sufficiently large Hilbert space, or I mixed this up with the fact that any physical process must be completely positive. The connection between quantum operations being insufficient as a framework and state correlation with the environment was made e.g. in these notes and in this informatively named paper, "Dynamics of open quantum systems initially entangled with environment: Beyond the Kraus representation" (see also the correction).

As the name of the paper above suggests, the generalisation to study such systems falls in the area of open quantum systems. I do not know enough to recommend any resources, but this seems like a good open-access starting point.

EDIT: This paper clarifies that, while the map $\mathcal{E}$ representing a physical process on a smaller Hilbert space $\mathcal{H}_S$ may not be completely positive on certain elements of $D(\mathcal{H}_S)$, the space of density matrices on $\mathcal{H}_S$, those are the states that are not allowed ex hypothesi by the assumptions of the problem. For example, if we assume the subsystem $S$ is entangled with the environment $R$, then the reduced state in $D(\mathcal{H}_S)$ cannot be pure, so $\mathcal{E}$ is allowed to "behave badly" on such states. If $\mathcal{E}$ was completely positive on the whole of $D(\mathcal{H}_S)$, then by the Stinespring dilation theorem, it should be expressible in terms of Kraus operators.