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Consider a channel $\mathcal{E}: A\rightarrow B$ that maps quantum states $\rho_A\in A$ to

$$\mathcal{E}(\rho_A) = \sum_i \langle i\vert \rho_A\vert i\rangle\tau_i\ ,\tag{1}$$

where $\tau_i\in B$ is a quantum state and $\langle i\vert \rho_A\vert i\rangle$ is the probability of outcome $i$ when $\rho_A$ is measured in the standard basis. I have two questions

1) According to this answer on another question I asked, the quantity $\langle i\vert \rho_A\vert i\rangle\tau_i$ is not well-defined due to dimensional inconsistency. What is the mathematically correct way to express the action of this channel?

2) How do I write the Kraus operators of this channel?

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  • $\begingroup$ Can you elaborate how those Kraus operators map $A\to B$ (the Hilbert spaces)? $\endgroup$ – Norbert Schuch Oct 19 at 16:32
  • $\begingroup$ @NorbertSchuch, perhaps I am using incorrect/confusing notation - sorry for this but maybe it is clearer after the edit? The idea is that the channel measures the state in $A$ and obtains some outcome $i$ with probability $p_i$ (at this point the register $A$ ceases to exist). Then the channel outputs the state $\tau_i\in B$ with this probability. So the Kraus operators must take us from $A$ to $B$. $\endgroup$ – user1936752 Oct 19 at 17:45
  • $\begingroup$ Yes. But the new channel you write takes A+B as an input and maps it onto B. And on the joint A+B system (with input $\rho_A\otimes I$) the map is not trace preserving. On the other hand, if the $I_B$ is not part of the input to the channel, then the form you write is not the Kraus form. $\endgroup$ – Norbert Schuch Oct 19 at 21:02
  • $\begingroup$ @NorbertSchuch, I see your point - thank you for this comment. I realize I do not know how to correctly express the action of this channel. I have edited the question to reflect this as well along with my original goal to write our the Kraus operators. $\endgroup$ – user1936752 Oct 19 at 22:25
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  1. Your way of writing the channel [Equation (1)] is completely fine. (Why wouldn't it be?)

  2. As always, there are many ways of writing the Kraus operators. E.g., you could decompose $$ \tau_i = \sum_j \vert a_{ij}\rangle\langle a_{ij}\vert $$ (e.g., an eigenvalue decomposition with the eigenvalues absorbed in the $\vert a_{ij}\rangle$, but any such decomposition will be fine), and choose Kraus operators $$ K_{ij} = \vert a_{ij}\rangle\langle i|\ . $$ That they satisfy the completeness relation is left as an exercise to the reader ;) (We know they have to satisfy them, since they are Kraus operators of a trace preserving map.)

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