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I am trying to learn Zurek's theory of decoherence. Right now I am reading Decoherence, einselection and the existential interpretation (the rough guide) which seems like an easier read than his big Rev.Mod.Phys article from 2003.

In the middle of Section 3 (Decoherence and einselection), Zurek tries to motivate what constitutes a useful predictability criterion. Here is a quote from him:

The ability to retain correlations is the defining characteristic of the preferred 'pointer' basis of the apparatus. In simple models of measurement cum decoherence, the selection of the preferred basis of the apparatus can be directly tied to the form of the interaction with the environment. Thus, an observable $\hat{O}$ which commutes with the complete (i.e. self-, plus the interaction with the environment) Hamiltonian of the apparatus, $[\hat{H}_{\mathcal{A}}+\hat{H}_{\mathcal{AE}},\hat{O}]=0$ (Eq. 3.5), will be the pointer observable. This criterion can be fulfilled only in the simplest case: typically, $[\hat{H}_{\mathcal{A}},\hat{H}_{\mathcal{AE}}]\neq 0$, hence equation (3.5) cannot be satisfied exactly.

I am confused about his last sentence. Even if the individual Hamiltonians do not commute, you can still diagonalize the total Hamiltonian and use the set of the resulting eigenstates to deduce the structure of $\hat{O}$.

When Zurek says, "...cannot be satisfied exactly," is he

  1. stating a mathematical fact,

  2. just emphasizing that diagonalizing the total Hamiltonian will be very impractical even when you figure out $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{AE}}$ individually?

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  • $\begingroup$ Zurek also addresses this question in this video (a lecture given at the Perimeter Institute, pirsa.org/displayFlash.php?id=07080044), see the interval 32:13 - 33:18, with a focus on 33:00. But I can't make out what he is saying towards the end. $\endgroup$ – wcc Aug 6 '18 at 3:58
  • $\begingroup$ in this context the "environment" is generally taken to be something that we do not fully control/know, so my guess would be that the author is here simply pointing out that if the full Hamiltonian does not commute with the apparatus Hamiltonian, then to find the pointer states of the full system we would need to diagonalize the full Hamiltonian, which is unfeasible in practice $\endgroup$ – glS Aug 6 '18 at 16:35
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I would like to put an answer to my own question, as I am not satisfied with the responses so far and would like to carry the discussion in a different direction.

1 . Regarding definition of environment

As long as we agree that the trio of system, apparatus, and environment are all quantum-mechanical (we have already assigned Hilbert spaces to them at the beginning), and know which degrees of freedom to trace out at the end, I don't think there is any ambiguity on what defines the environment. The only question that arises is whether the environment splits into an "immediate" environment and "remote" environment. Zurek address this in Section III of his original paper from 1981. He acknowledges that the coupling between the these two types of the environment can be strong, but claims (without further justification) that the final conclusion does not change whether we include this coupling or not. Still hand-wavy, but this I can buy and accept.

2. Why diagonalization of Hamiltonian does not yield predictable states (pointer states)

What are pointer states? They form the basis in which the reduced density matrix of the (system+apparatus) eventually diagonalize or nearly diagonalize. An isolated (system+apparatus) duo suffers from the ambiguity of basis choice and observables. An example that Zurek likes to give is how a CNOT gate with system as the control and the apparatus as the target, in $\{|0\rangle,|1\rangle\}$ basis, can "flip" so that the apparatus now becomes the control and the system becomes the target in $\{|+\rangle,|-\rangle\} = \{\frac{|0\rangle + |1\rangle}{\sqrt{2}},\frac{|0\rangle - |1\rangle}{\sqrt{2}}\}$ basis. After decoherence we no longer have the freedom in choosing basis, and we are restricted to working with pointer state basis. The apparatus-environment coupling is crucial in determining the form of the pointer state basis.

Now, the original question was motivated by asking why we need more refined criteria like minimizing change in von Neumann entropy or purity ("predictability sieve") to look for pointer states rather than simply looking at the Hamiltonian and diagonlize it.

Let's stick to the basics and focus on the definitions. The pointer states live in the Hilbert space of the apparatus, $\mathcal{H}_{A}$. The apparatus-environmental coupling, $\hat{H}_{\mathcal{AE}}$, acts on the Hilbert space $\mathcal{H}_{A} \otimes \mathcal{H}_{E}$. If we know the form of the self-Hamiltonian of the apparatus $\hat{H}_{A}$ and the coupling $\hat{H}_{AE}$, in principle we can diagonalize the total Hamiltonian $\hat{H}_{A} + \hat{H}_{AE}$. The resulting eigenstates from the diagonalization live in the Hilbert space $\mathcal{H}_{A} \otimes \mathcal{H}_{E}$. Let's say they have the form $|\Psi_{AE}\rangle = \sum_{i} c_i |a_i\rangle_A |\varepsilon_i\rangle_E $, possibly after Schmidt decomposition.

Now let's use $|\Psi_{AE}\rangle$ to construct a candidate pointer observable. We consider the structure of $Tr_{E} |\Psi_{AE}\rangle \langle \Psi_{AE}| = \sum_i |c_i|^2 |a_i \rangle \langle a_i| $. This observable commutes with $\hat{H}_A$ and if $[\hat{H}_{A},\hat{H}_{AE}] = 0$, it commutes with $\hat{H}_{AE}$ as well so the observable is invariant under the total Hamiltonian. If $[\hat{H}_{A},\hat{H}_{AE}] \neq 0$, this observable, which we constructed from $|\Psi_{AE}\rangle$ and taking the trace over the environment, is not useful because it now changes in time under the action of the total Hamiltonian.

So my point is that even if we manage to get eigenstates of the total Hamiltonian, living in $\mathcal{H}_A \otimes \mathcal{H}_E$, they may not yield insight on what kind of observables we can construct in $\mathcal{H}_A$ that happen to stay invariant under time evolution.

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