Can a nonlinear evolution be linear at the level of reduced states?

Question

Consider a (possibly unphysical) non-linear transformations of bi-partite quantum states,

$$\mathcal{N} (a A + b B) \neq a \mathcal{N}(A) + b \mathcal{N} (B)$$ for some density matrices $A,B \in \mathcal{S}(\mathcal{H} \otimes \mathcal{H}^{\prime})$ and some $a+b=1$. Suppose that

$$Tr_{\mathcal{H}^{\prime}} \left[ \mathcal{N} (a A + b B) \right] = a Tr_{\mathcal{H}^{\prime}} [\mathcal{N}(A)] + b Tr_{\mathcal{H}^{\prime}} [\mathcal{N} (B)] $$ hence the reduced dynamics on $\mathcal{H}$ is linear.

Can we also have that the reduced dynamics is linear on $\mathcal{H}^{\prime}$? $$Tr_{\mathcal{H}} \left[ \mathcal{N} (a A + b B) \right] = a Tr_{\mathcal{H}} [\mathcal{N}(A)] + b Tr_{\mathcal{H}} [\mathcal{N} (B)] $$ or is this just nonsense and, if linear on $\mathcal{H}$, it must be nonlinear on $\mathcal{H}^{\prime}$?

Answer 1

Sure this is possible. Just put the non-linearity on the matrix elements of the output which do not appear in either reduced density matrix.

For instance, for two qubits, the entries on the anti-diagonal do not appear in the reduced density matrix. If the non-linearity lives in those entries, and the other entries are linear, you get an examples of such a map which is linear in either reduced state, but not as a whole.