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Decoherence is often presented as a program to solve the measurement problem using only the bare bones quantum mechanics framework of a Hilbert space and unitary Schrödinger time evolution. As often stated, there is no need to postulate a collapse or splittings into many worlds for decoherence to work.

However, I have my misgivings. What do you guys think of the following example? The system is an electron and the environment contains a Stern-Gerlach apparatus. In configuration A, the apparatus measures the spin of the electron in the z-direction, and in configuration B, the x-direction. A standard decoherence analysis will pick out the z-basis as the pointer states for configuration A, and the x-basis for B. So far, so good.

Now instead, set up a control qubit $\left( |0\rangle + |1\rangle \right)/\sqrt{2}$ using a Hadamard gate. A value of 0 sets up the apparatus in configuration A, while a value of 1 sets it up in configuration B. The crucial detail here is the environment is in a superposition. Now as far as I know, all the standard criteria for picking out system pointer states like for instance, diagonalizing the reduced density matrix, the predictability sieve, purification time, efficiency threshold, etc. , all give out crap for the pointer basis in this example. None of them can capture the fact that for a value of 0, the pointer states lie in the z-direction, but for a value of 1, they lie in the x-direction instead.

Would it be fair to claim we need something extra like a collapse of the environment into configuration A or B, or a splitting into the worlds of configuration A and configuration B with each world treated in a different manner, to actually make decoherence work? Or are there loopholes allowing us to stick to a bare bones quantum mechanical framework?

Most treatments of decoherence assume the system is in a superposition right after preparation but hardly any consider the scenario where the environment is also in a superposition during preparation. This, I think, is incomplete. And this is no idle matter either because the environment is always in a superposition. Structure formation leading to the condensation of matter into superclusters, galaxies, stars and planets have their origins in quantum fluctuations. Even major patterns here on earth like large scale weather events depend sensitively upon quantum fluctuations via the butterfly effect. The unpredictable, or at least not fully predictable, behavior of experimenters are also sensitive to quantum fluctuations in their brain. Without presupposing either collapse or a many worlds split, how can one get down to a proper decoherence analysis?

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The issue here is that the wavefunction domain for the environment is enormously high dimensional, essentially infinite dimensional, so there is no productive way to imagine a superposition of environmental degrees of freedom at the start which in time recoheres to cancel out certain branches of the superposition after measurement.

If quantum mechanics is exactly correct, it is possible to get macroscopic interference in principle, just using reversibility of the laws of physics. If you can start with an electron in a superposition of z spin, measure its spin, then take the post-measurement system including the measuring device, and time-reverse it. Then you lose the information about the measurement and recohere the electron back into its original superposition. This reversal requires that different macroscopic branches recombine into a coherent electronic state, and it is practically impossible.

It is this essentially practically unobservable character of macroscopic superpositions that lead many people to renounce the concept, and to claim that there is a separation between quantum and macroscopic realms.

In decoherence approaches, the assumption is that the enormous quantum wavefunction space can be taken to be empty for macroscopic systems, so that new branches are produced into regions where there aren't already things to interfere them out of existence. This is plausible considering how big the wavefunction space is, but it is a little annoying to have to do this sort of metaphysics.

Addendum: After reading the middle

I answered this question before reading the middle, only the beginning and the end. My answer is fine for the general question, but there is a thought experiment in the middle of this question which is completely incorrect. This thought experiment will produce an environment in a decohered macroscopic superposition of "measure A" and "measure B", which then will decohere the spin of the particle. Decohering relative to already decohered states is still decohering. There is no paradox or difficulty with first setting up the state using the superposed qubit, because setting up the experiment is an act of measurement in itself.

To see to true problem with a macroscopic superposition, read the stuff above. You can recohere decohered stuff, but it requires the same effort as any other reversal of entropy gain, and it is practically impossible.

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It is possible that the downvote is because I didn't adress the flawed thought experiment in the middle of the question. I didn't notice it, and I added something to address it. The answer above the header is the original answer, unchanged, and it is correct. –  Ron Maimon Dec 1 '11 at 2:51
I didn't downvote, but there is a conceptual problem behind something which is "practically impossible": it doesn't address the foundations of the subject. Your answer is tantamount to "the measured system decoheres because it interacts with the decohered macroscopic system; and the macroscopic system is decohered because you'll never be able to prove that it hasn't". That's not a very good answer. –  Niel de Beaudrap Dec 1 '11 at 16:17
@Niel: But I believe this is the standard answer. It is also important to note that the realm of humans must be decohered, so that we can do classical entropic irreversible computation at our scale, so there is a fundamental reason why the decoherence must happen for macroscopic systems, it must reproduce the second law at our scale. I am not satisfied with this answer 100%, because we don't have evidence of actual exponential speedup in QM yet, but I can't see how you can give a better one given the current state of knowledge. –  Ron Maimon Dec 1 '11 at 17:59
If it is not possible to formulate a satisfying (and non-circular) answer, then this fact is itself the most useful thing that you could tell the OP, however popular other partial attempts to answer the question might be in the folklore. –  Niel de Beaudrap Dec 13 '11 at 20:15
There do exist macroscopic quantized systems: superfluids, superconductors, where coherence can be kept and reversibility might be possible. Macroscopic systems where the near infinity of phases are inaccessible and incoherent go into statistical formulations and it is impossible to reverse them. The same argument holds for classical statistical mechanics and nobody is tied into knots about it. –  anna v Jan 17 '12 at 14:25
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A partial attempt to address this issue is made by invoking the idea of quantum discord. The basic idea of quantum discord is the environment needn't be in a specific state prior to interacting with the system. All that is necessary is that it factorizes and there is no correlation.

Let's start with the simple example of a qubit, and an environment which is initially in a maximally mixed state, not a pure one. Assume the pointer states are $|0\rangle$ and $|1\rangle$, and it's the same no matter what state the environment is in, and that the pointer states are exact. This is only a toy model after all. Suppose $$|0\rangle\otimes|e\rangle \to |0\rangle \otimes U |e\rangle$$ and $$|1\rangle\otimes|e\rangle \to |1\rangle \otimes V |e\rangle$$ where U and V are unitary matrices acting upon the environment.

Now you might think, if the environment is in a maximally mixed state before interacting, it will still be maximally mixed after interacting, so how can there be decoherence? It's possible, however.

In block matrix form, an initial qubit state $\alpha|0\rangle + \beta |1\rangle$ transforms as $${1\over N}\begin{pmatrix}|\alpha|^2 I & \alpha\beta^* I\\\alpha^*\beta I & |\beta|^2 I\end{pmatrix} \to {1\over N}\begin{pmatrix}|\alpha|^2 I & \alpha\beta^* UV^{-1}\\\alpha^*\beta VU^{-1} & |\beta|^2 I\end{pmatrix}$$ for the density matrix where N is the dimensionality of the state space of the environment. Taking the partial trace over the environment, we get $$\begin{pmatrix}|\alpha|^2 & \alpha\beta^* Tr[UV^{-1}]/N\\ \alpha^*\beta Tr[VU^{-1}]/N& |\beta|^2\end{pmatrix}$$. For generic unitary matrices, the two traces divided by N scales as $1/\sqrt{N}$ assuming some very mild statistical distribution properties.

How can a maximally mixed environment record any information about the qubit? It can't, but it can still decohere the qubit!

Physically, consider a molecule decohered by light shining on it and scattering off it. If most of the photons are coming from only one direction, e.g. sunlight coming only from the direction of the sun at a certain spectral distribution, and the photons are scattered off in different directions at a different spectral distribution, we can see how the scattered photons carry off information about the location of the molecule, its energy level prior to the scattering, and the difference between its energy levels (assuming it's an inelastic scattering).

However, place the molecule in a closed box filled with blackbody radiation in thermal equilibrium. The blackbody radiation can still decohere the position of the molecule and its energy levels even though the blackbody photons can't carry any information about the molecule!

The OP's question is about a different case though, where the different environmental states have different pointer states. This has also been covered by Zurek. Assume a dilute gas of environmental particles scatter off the molecule from different directions and velocities. The pointer states depend upon the direction and velocity of the scattering probe, as can be shown by an examination of the S-matrix. What happens in this case after a number of collisions is thermalization, not decoherence in the form of dephasing in a specific pointer state basis.

That's still not what the OP's question is about. The previous paragraph is for an environment in a thermal state. The OP's question is about an environment in a superposition which is nonthermal. There is also only one interaction, and not multiple scatterings. I'm afraid the question is still open as it stands.

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I think the thought experiment in the question strikes right at the heart of the Heisenberg cut. If what is measured depends on the environment, but the environment is in a superposition, then of course the pointer state will have to depend upon which classical state the environment is in. Some parts of the environment have to be treated as classical. Not all, most certainly, but some. Niels Bohr ended up with the same conclusion as well.

I guess one way out is to redefine the system to include the qubit, but I suppose this is not what the OP intended, especially if the qubit is far away from the electron.

I think the most interesting situation comes about when the Hadamard gate operation is applied after the electron is prepared. If there is indeed a collapse of the environment (or splitting as implied in the question), it can only happen after the electron has been prepared. Thus, the electron cannot have any pointer state before the environment collapses. I don't think this is all that much of an issue because presumably, the electron is shielded from external interactions in the interim and there is no reason why isolated systems must have pointer states. The more confusing issue is if we want to assign a pointer basis to the electron before it is measured, we have to collapse the environment before the electron spin is measured. Either that, or give up the idea of pointer states entirely.

Let's see what happens in a consistent histories framework suitably generalized so that the choice of projection operators at later times depend on the outcome of projection operators at earlier times. One of the main restrictions on families in consistent histories is that the chain operators have to be built up from a time ordered product of projection operators. It is crucial to include a pair of projection operators distinguishing configuration A from configuration B, but it is clear from my earlier assumption that these operators can only be dated after the Hadamard gate operation. Time ordering implies that the projection operators defining the pointer basis for the electron spin can only be dated after the Hadamard gate operation. That is the same point I made in the previous paragraph. The spin has no pointer basis before the environment collapses.

I am not so sure if my consistent histories explanation is all that satisfactory, though. The question appears to be not so much about decoherence as such as much as it is about the choice of pointer states. In that case, consistent histories has its own problems as to how to select quasi-classical frameworks and that is analogous to choosing the pointer states. At any rate, any decent quasi-classical framework will have to depend upon the specifics of the environment for these sorts of thought experiments.

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A nice tool from quantum information theory which can come in handy here is the concept of POVMs. We have a two element POVM -- actually PVM -- when the Stern-Gerlach apparatus is aligned in the z direction, and another two element POVM when it's in the x direction. If the environment is in a decohered superposition with probabilities p and 1-p, the POVM to use in this case is the four element POVM formed by taking the union of the original POVMs and rescaling the elements by p and 1-p respectively.

But back to the original question, I agree standard decoherence tools need to be sharpened to be adapted to this particular example.

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Any collapse of the environment at a finite time will be a physical modification of the dynamics of quantum mechanics, and this is hard to do in a manner which respects relativity. On the other hand, a collapse of the memory of the environment at the end of time will not lead to any such difficulty. The two state formalism of Aharonov is one way to do this. This collapse at the end of time is propagated retrocausally and contextually to the electron picking out the correct pointer states. With retrocausal influences, it is beside the point whether the environment "splits" before or after the electron was prepared.

The two state formalism is a wonderful improvement over plain old quantum mechanics and can explain so much more.

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