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I am reading through Nielsen & Chuang, and I am on the section about operator sum representation. They performed this derivation.

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Why is it important and useful for us to bundle together the unitary transformation and environment state into a new operator, instead of keeping it separate?

Furthermore, how is E_k an operator? Is it not just an expectation value so it is a single scalar?

Also, let me know if that is actually what is going on here.

Thank you!

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To add to Daniele's answer, $E_k$ is an operator—and not a scalar—because the notation in Nielsen & Chuang is sloppy. What is meant is that $E_k=\iota_k^\dagger U\iota_0$ where $\iota_k:\mathbb C^d\to\mathbb C^d\otimes\mathbb C^{d'}$ is defined via $|x\rangle\mapsto|x\rangle\otimes|e_k\rangle$. Thus $E_k$ is an operator that maps $$ \mathbb C^d\overset{\iota_0}\to\mathbb C^d\otimes\mathbb C^{d'}\overset{U}\to\mathbb C^d\otimes\mathbb C^{d'}\overset{\iota_k^\dagger}\to\mathbb C^d\,, $$ that is, $E_k\in\mathbb C^{d\times d}$. And Nielsen & Chuang write $|e_k\rangle=\iota_k$ in abuse of notation, presumably to keep notation more clean.

As an example—to give you a better feeling how these $\iota$ operators work—if $U$ were a product operator $U=U_1\otimes U_2$, then \begin{align*} E_k|x\rangle&=\iota_k^\dagger (U_1\otimes U_2)|x\rangle\otimes|e_0\rangle\\ &=\iota_k^\dagger (U_1|x\rangle\otimes U_2|e_0\rangle)\\ &=U_1|x\rangle\langle e_k|U_2|e_0\rangle \end{align*} for all $x$ meaning $E_k=\langle e_k|U_2|e_0\rangle U_1$. In general, of course, $U$ is not of product form so the relation between the $E_k$ and the system-environment $U$ is not as simple.

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The formalism is useful to describe a noisy transformation $\mathcal{E}$ only in the domain of a system of interest $\mathcal{H}$.

According to the book, you can select an orthogonal bases $\{e_i\}_i$ such that the environment is in the basis state $|e_0\rangle$.

At this point you have the Kraus operators, which embed the action of the environment into a mapping $\mathcal{H} \rightarrow \mathcal{H}$

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To add to Daniele's and Frederik's answers: the operator sum representation is even more useful than the Nielson and Chuang derivation might suggest - though, the derivation is very helpful from a pedagogical viewpoint.

In the book's derivation, the operator sum representation is used to describe how a quantum system changes under the influence of some environment. So, in this case, $\rho$ describes the quantum system before the interaction and $\mathcal{E}(\rho)$ describes the very same quantum system after the interaction. The size of the system has not changed.

But in general, the operator sum representation can do more than that. Here are two examples of quantum channels where the size of the quantum system changes:

  1. Adding a qubit in state $|0\rangle$ to your system. In this case you have one single Kraus operator $E_0 = |0\rangle$.
  2. "Throwing away" a qubit from your system. This is the same as tracing out the qubit. The Kraus operators are $E_0 = \langle 0|$ and $E_1 = \langle 1|$

Of course, one could think of much more complicated examples. I only wanted to highlight the fact that the operator sum representation can be used to describe any physical process that a quantum system can undergo even if the size of the system changes.

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