What is the connection between Stinespring's dilation of a channel and Naimark's theorem?

Question

I'm trying to understand the connection between the Stinespring dilation of a quantum channel and Naimark's theorem that shows that POVMs can be written as projective measurements in a larger Hilbert space. On Wikipedia, the proof of Naimark's theorem is given as a special case of the Stinespring dilation - however, I cannot follow this proof.

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1. Given any channel $N$ from $\mathcal{H}_A$ to $\mathcal{H}_B$ given in terms of Kraus operators $\{A_i\}$ with $\sum_i A_i^\dagger A_i = I_A$, we can instead consider an isometry $V_{A\rightarrow BE} = \sum_i A_i\otimes \vert i\rangle_E$. Now the channel can be described as

   $$N(\rho) = \sum_i A_i\rho_A A_i^\dagger = \text{Tr}_E(V\rho V^\dagger)$$

   This is the Stinesping dilation of the channel.

2. Consider a POVM that consists of elements $\{F_i\}$. Since we know that $\sum_i F_i = I$ and $F_i > 0$, let us rewrite it as $F_i = A^\dagger_iA_i$. Now, we can consider the isometry $V_{A\rightarrow BE} = \sum_i A_i\otimes \vert i\rangle_E$. It is then true that the probability of the $i^{th}$ outcome is

   $$p_i = \text{Tr}(A_i\rho_A A^\dagger_i) = \text{Tr}(V^\dagger(I\otimes\vert i\rangle\langle i\vert)V \rho_A)$$

   Thus, we have a projective measurement on $V\rho V^\dagger$.

The mathematical description of the two things above seem very similar except that POVMs involve the additional step of picking a specific outcome $i$. I'm not sure how to express this as a channel. Moreover, if I could write the POVM as a channel, then how exactly does Naimark's theorem become a special case of the Stinespring dilation?

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TL;DR

1) How do I write the POVM above as a quantum channel?

2) If I can do 1), what is the connection between the Stinespring dilation of this channel and Naimark's theorem?

Answer 1

I would say it is the opposite: Channels are special cases of measurements (where you forget the outcome).

In particular, in the Stinespring you have a partial trace $\mathrm{Tr}_E$, which you can think of as measuring the ancilla system $E$ -- this gives exactly the construction from Naimark's theorem -- and then forgetting the result, that is, summing over all (unnormalized) post-measurement states.