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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.


  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?


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?

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  • $\begingroup$ What exactly is your question? Would "just sum over $i$" be an answer, if in addition you consider the post-measurement state of the POVM? $\endgroup$ – Norbert Schuch Sep 3 at 4:39
  • $\begingroup$ @NorbertSchuch my questions are - Can one see Naimark's theorem as a special case of the Stinespring dilation of a channel? If yes, can one write the POVM as a channel and show that its Stinespring representation is equivalent to a projective measurement in the larger space? $\endgroup$ – user1936752 Sep 3 at 18:48
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    $\begingroup$ I would say it is the opposite: Channels are special cases of measurements (where you forget the outcome). And yes, Stinespring uses a partial trace, which can be seen as measuring the ancilla (Naimark!) and then forgetting the result. $\endgroup$ – Norbert Schuch Sep 3 at 23:24
  • $\begingroup$ @NorbertSchuch that's a great comment! I understand the link now. I'm happy to accept your comment as an answer if you wish. $\endgroup$ – user1936752 Sep 4 at 18:56
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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.

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