# What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?

In Nielsen and Chuang, 10th Anniversary Edition, there is a definition of the operator sum representation of a quantum operation: $$\mathcal{E}(\rho)=\sum_{k}\langle e_k|U[\rho\otimes|e_0\rangle\langle e_0|]U^\dagger|e_k\rangle$$ and then the operation element is defined as $$E_k\equiv \langle e_k|U|e_0 \rangle$$. In both cases the dimensions of $$U$$ on the one hand, and $$\langle e_k|$$, and $$|e_0\rangle$$ are different ($$U$$ is an operator acting on composite system, whereas $$|e\rangle$$ is a state of the component system).

How can the inner product be calculated in this case? Is this some sort of notational convention or is there a different explanation? It's a trivial question, but I can't find an explicit answer anywhere. It comes down to the identity $$I\otimes |e_k\rangle = |e_k\rangle$$ the meaning of which I'm struggling to understand.

• – glS
Feb 22 at 10:12
• There is an implicit $I\otimes \langle e_k|$ so that nothing happens on the system and the environment is traced out. Feb 22 at 10:53

TL;DR: We can understand the object $$E_k=\langle e_k|U|e_0\rangle$$ rigorously in two steps. First, think of $$\langle e_k|$$ and $$|e_0\rangle$$ as linear functions. Next, treat the implicit operation in $$\langle e_k|U|e_0\rangle$$ as function composition.

## $$\langle e_k|$$ and $$|e_0\rangle$$ are functions

Usually, we think of elements of a set $$A$$ and functions to $$A$$ as different types of objects. However, we can view elements of $$A$$ as just a special type of function to $$A$$: namely, one whose domain is some singleton set $$\{\star\}$$. Then we identify $$a\in A$$ with $$a:\{\star\}\to A$$ defined by $$a(\star)=a$$ where I allowed myself a slight abuse of notation by denoting both objects as $$a$$.

Similarly, we typically think of an object like $$|e_0\rangle$$ as an element of some complex Hilbert space $$\mathcal{H}_E$$. However, we can also think of $$|e_0\rangle$$ as the linear function $$|e_0\rangle:\mathbb{C}\to\mathcal{H}_E$$ defined by $$|e_0\rangle(z):=z|e_0\rangle$$ for all $$z\in\mathbb{C}$$. In fact, this function is just the linear extension of $$|e_0\rangle:\{\star\}\to\mathcal{H}_E$$ defined by $$|e_0\rangle(\star)=|e_0\rangle$$ which we saw earlier. Physically, the function $$|e_0\rangle$$ can be understood as state preparation.

Next, an object such as $$\langle e_k|$$ is a dual of $$|e_k\rangle$$. In other words, $$\langle e_k|$$ is a linear function $$\langle e_k|:\mathcal{H}_E\to\mathbb{C}$$ defined by $$\langle e_k|(|e_j\rangle)=\langle e_k|e_j\rangle=\delta^j_k$$ and extended by linearity to all of $$\mathcal{H}_E$$. Physically, the function $$\langle e_k|$$ can be thought of as an "effect" that is a possible outcome in a measurement. It evaluates to the amplitude between the input and the fixed state $$|e_k\rangle$$ it is the dual of.

## Function composition

Thus, the object $$E_k=\langle e_k|U|e_0\rangle$$ is built from three functions: \begin{align} |e_0\rangle&:\mathbb{C}\to\mathcal{H}_E\\ U&:\mathcal{H}_S\otimes\mathcal{H}_E\to\mathcal{H}_S\otimes\mathcal{H}_E\\ \langle e_k|&:\mathcal{H}_E\to\mathbb{C}. \end{align}\tag1 We would like to compose them together, but unfortunately their domains and codomains disagree. We can fix this by extending any function that doesn't explicitly act on $$\mathcal{H}_S$$ to all of $$\mathcal{H}_S\otimes\mathcal{H}_E$$ by assuming that it acts on $$\mathcal{H}_S$$ as identity. Then $$(1)$$ becomes \begin{align} |e_0\rangle&:\mathcal{H}_S\otimes\mathbb{C}\to\mathcal{H}_S\otimes\mathcal{H}_E\\ U&:\mathcal{H}_S\otimes\mathcal{H}_E\to\mathcal{H}_S\otimes\mathcal{H}_E\\ \langle e_k|&:\mathcal{H}_S\otimes\mathcal{H}_E\to\mathcal{H}_S\otimes\mathbb{C} \end{align}\tag2 where I continue to abuse notation by not giving new names to new objects made of old ones. We can further simplify $$(2)$$ by noting that $$\mathcal{H}\otimes\mathbb{C}=\mathcal{H}$$ \begin{align} |e_0\rangle&:\mathcal{H}_S\to\mathcal{H}_S\otimes\mathcal{H}_E\\ U&:\mathcal{H}_S\otimes\mathcal{H}_E\to\mathcal{H}_S\otimes\mathcal{H}_E\\ \langle e_k|&:\mathcal{H}_S\otimes\mathcal{H}_E\to\mathcal{H}_S. \end{align}\tag3 Finally, we can compose the three functions to obtain $$E_k=\langle e_k|U|e_0\rangle:\mathcal{H}_S\to\mathcal{H}_S.\tag4$$

## Physical meaning

Mathematically, this is just the composition of the three functions. Physically, we can understand the action of $$E_k=\langle e_k|U|e_0\rangle$$ as consisting of three steps. First, $$E_k$$ sends any $$|\psi\rangle\in\mathcal{H}_S$$ from the smaller Hilbert space of system $$S$$ to $$|\psi\rangle\otimes|e_0\rangle$$ in the larger Hilbert space of both the system $$S$$ and the environment $$E$$, then it applies unitary $$U$$ on the larger Hilbert space, and finally it measures the environment obtaining whatever (non-degenerate) outcome is associated with $$|e_k\rangle$$.