Given two separable Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, I am wondering: what are the necessary and sufficient conditions for there to be a completely positive trace-preserving (CPTP) map $\Phi:B(\mathcal{H}_1)\to B(\mathcal{H}_2)$?
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The map $\Phi(\rho) = \mathrm{tr}(\rho)\sigma$, for some state $\sigma\in B(\mathcal H_2)$ with $\mathrm{tr}(\sigma)=1$ is CPTP and exists for any pair of separable Hilbert spaces.
(As always for these questions, let me advertise my list of canonical examples for quantum channels.)
Norbert Schuch
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