# Only assuming the universe evolves according to a positive trace-preserving map, is there a proof that all subsystem evolution must be CPTP?

If we only assume that the wavefunction of the universe evolves according to $e^{-iHt}$, is there any proof that all subsystems of the universe (partial traces over parts of the universe) must evolve according to a completely positive, trace-preserving (CPTP) map?

An example of a perfectly valid quantum map that is not completely positive is given in the paragraph containing Eq. 6 of the paper: Who's afraid of not completely positive maps?. This was possible because they made the system and its environment entangled at the time $t=0$. So my question is whether such a proof exists for the case where there is no initial entanglement.

• Comments are not for extended discussion; this conversation has been moved to chat. – heather May 18 '18 at 21:55
• @heather: Is there any way to recover the comments that were moved to chat? The link you gave seems to lead to a removed page. – user1271772 Jun 12 '18 at 20:12
• The room had been deleted for inactivity, which is why it wasn't showing up. I've undeleted it. – heather Jun 12 '18 at 20:27
• a paper about this came out recently on the arxiv: arxiv.org/abs/1806.02381 – glS Jun 18 '18 at 14:48
• @glS: I mentioned this paper in my comment to Neil: "Finally it seems someone has come up with an explanation for the non-positive map seen for the case where the system and bath are initially correlated: arxiv.org/pdf/1806.02381.pdf, though for me it is still too early to tell if this refute's the 2005 paper that we discussed" in a comment to his answer to: quantumcomputing.stackexchange.com/questions/2058/… – user1271772 Jun 18 '18 at 16:35

1. Let's say the universe evolves from $t=0$ to $t_f$ then the unitary evolution $U$ from $0$ to $t_f$ induces a CP evolution on the subsystem. To see this, note that the composition of CP maps is CP. Now, the reduced (system) evolution is $Tr_E U\rho_s\otimes\rho_E U^\dagger$ which is a composition of the map $\rho_s\rightarrow \rho_s\otimes\rho_E$ (which is CP), unitary evolution, and partial trace (again CP). So overall it is CP.
2. Take the dynamics between $t_i>t_0$ and $t_f$, this might not be CP (or even linear!), which is why NCP maps (and similarly non-linear maps) can be physical. However, this is a slightly tricky subject since it is unclear how you would go about constructing the map. For example, different ways of doing process tomography could lead to different results.