I assume the factors that mitigate/negate the no-cloning theorem are that the bath is not a qBit, but a collection, that the state's are initially entangled. It could also be that the initial state of the cQbit is known, instead of unknown.

the no-broadcast-theorem is what covers mixed states instead of pure states. https://en.wikipedia.org/wiki/No-broadcasting_theorem

``` The theorem[1] also includes a converse: if two quantum states do commute, there is a method for broadcasting them: they must have a common basis of eigenstates diagonalizing them simultaneously, and the map that clones every state of this basis is a legitimate quantum operation, requiring only physical resources independent of the input state to implement—a completely positive map. A corollary is that there is a physical process capable of broadcasting every state in some set of quantum states if, and only if, every pair of states in the set commutes. This broadcasting map, which works in the commuting case, produces an overall state in which the two copies are perfectly correlated in their eigenbasis. ```

So it seems that there is some wiggle room, and specifically when you start working with collections instead of single qbits, things get weird.

But I'm a layman, and that was just a walk down wikipedia.