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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.

Sanchayan Dutta
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user1271772 No more free time
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2 Answers2

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There are two possible answers.

  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.

    See theorem 5.4 in John Watrous's lecture notes

    Note that this extends to multiple systems since each would just involve a different partial trace.

  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.

Aharon Brodutch
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So my question is whether such a proof exists for the case where there is no initial entanglement.

The answer to this is no. As mentioned in the paper, if the subsystem is separable from the rest of the universe then a global unitary evolution will act as a CPTP map on the subsystem. You can verify this explicitly by writing down an unentagled pure state, evolving it by a unitary and then tracing out one part. This calculation is done explicitly in Chapter 8 of Nielsen and Chuang.

biryani
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