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I am struggling with the following exercise, and was wondering if anybody had any good tips on how to attack the problem/where to begin:

Given a separable quantum state $$\rho_{AB'}=\sum_{i=1}^{k}p_{i}\sigma_{i}^{A}\otimes \tau_{i}^{B},$$ find a quantum channel $T=B(\mathcal{H_{B}})\rightarrow B(\mathcal{H_{B'}})$ for $\mathcal{H}_{B'}=\mathbf{C}^{k}$ such that $$\rho_{AB'}=(id_{A}\otimes T)(\sum_{i=1}^{k}p_{i}\sigma_{i}^{A}\otimes|i\rangle\langle i|).$$

Any hints/comments would be greatly appreciated.

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You can just use $T(|i\rangle\!\langle i|)=\tau_i$.

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  • $\begingroup$ But would that be a quantum channel? I feel that it is a to easy solution, shouldn't I try to define something more general/at least argue better? $\endgroup$ Commented Apr 28 at 7:29
  • $\begingroup$ @PinkElephants yes it's a channel because |i> are orthogonal by definition here. You can verify it being cptp to make sure more formally. How much you "have to argue" is about what you are doing not about the question itself $\endgroup$
    – glS
    Commented Apr 28 at 8:38

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