The channel causal-cut criterion

The schedule statement has a factorization form. This is the channel interface compared with Cheung's schedule and no-signalling source records CheungVIIJ; the proof here does not import a physical cone.

For a channel $\Theta :L({\mathcal{H}}_A\otimes {\mathcal{H}}_C)\to L({\mathcal{H}}_B)$, the following are equivalent:

1. $\Theta (\rho )=\Theta (\sigma )$ whenever ${\mathrm{Tr}}_A\rho ={\mathrm{Tr}}_A\sigma$;

2. there is a channel $\theta :L({\mathcal{H}}_C)\to L({\mathcal{H}}_B)$ with

   $$\Theta =\theta \circ {\mathrm{Tr}}_A;$$
3. $${\Theta }^{\ast }(L({\mathcal{H}}_B))\subseteq I_A\otimes L({\mathcal{H}}_C).$$

These conditions imply invariance of the joint $B$–reference output under every local trace-preserving intervention on $A$, with the reference untouched.