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?

  • $\begingroup$ Can you elaborate how those Kraus operators map $A\to B$ (the Hilbert spaces)? $\endgroup$ Oct 19 '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 $B$. $\endgroup$ Oct 19 '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. $\endgroup$ Oct 19 '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. $\endgroup$ Oct 19 '19 at 22:25
  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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.