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.)
answered Apr 13, 2020 at 10:34