More specifically, this usual proof shows that every quantum channel has an environmental representation where the system $\rho$ gets coupled to a *pure* state of the environment.

**Question:** Which quantum channels have an environmental representation where the systems gets coupled to the *uniform* (identity matrix-)state of the environment? Is this known?

[edit: Oh, never mind. I see now this works by using “canonical Kraus form”. Will edit…]

[edit: Ah, it does not work generally, but for unistochastic channels…]

]]>I have spelled out (here) the proof in one direction (that every endomorphic quantum channel has an evironmental representation)

]]>added pointer to what seems to be the original proof:

Lindblad 1975 (top of p. 149 and inside the proof of Lem. 5).

]]>this is a bare sub-section — meant to be `!include`

ed into relevant entries such as at *quantum channel* and at *quantum decoherence* — towards the theorem that quantum channels are exactly the “bath-averages of bath-coupled unitary evolutions”