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Take a qubit initialized to $|0\rangle$. Apply a Hadamard transform to it. Measure it with an apparatus along the $|0\rangle,\, |1\rangle$ basis. If zero, spare a living cat. If 1, kill the cat. Construct a box surrounding the Earth. Construct a larger box surrounding the solar system. According to Zurek, pointer states and decoherence only happens through interaction with the environment, but how do we time that moment?

If the environment is chosen to be outside the earth box, it takes time for any signal to reach the box boundaries due to the speed of light. The environment does not even register anything about the cat till then. So does decoherence only happen when cat signals reach the box boundaries? Then it's sensitive to the choice of system/environment split. What about the solar system box then?

Maybe it takes time for the environment to register anything at all about the cat status, but we can retrodate the moment based upon what the environment registers? If so, when to retrodate? Certainly not before the Hadamard transform. Pointer states evolve in time, but surely it's not proper to evolve backward till before the Hadamard transform? We can evolve backward the pointer state to the period between Hadamard and apparatus measurement, although it's not proper to retrodate moment of decoherence before then, because counterfactually, it's still possible to apply an inverse Hadamard before measurement? Do we invoke irreversible internal decoherence? Counterfactually too, in principle, we can reflect all information back before reaching the box walls and unevolve the apparatus? These are all counterfactuals, you know.

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Take two volumes of some gas at different temperatures. Put them into contact. They will evolve to a thermal equilibrium. But at what time exacly does thermal equilibrium happens? –  Kostya Jul 27 '12 at 13:25
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First of all, welcome to the measurement problem, which is one of the central interpretational issues of quantum mechanics.

There are not, as yet, quantitiative answers which can allow you to rule out the possibility that the entire Earth and various objects in low-Earth orbit are in a massive entangled state — if you believe that consciousness is a (poorly understood!) function of the brain as a physical system, that is. The reason why is related to the question of how you could ever hope to prove that the state that the Earth is in is an entangled one, rather than a decoherent mixture.

There are two hallmarks of an entangled state, as distinct from a mixed one.

  1. One is that, if you have sufficient control over the system, an entangled state can be transformed into a pure one. Even in the case of the cat alone, let alone the Earth entire, the system is so massively complicated that you cannot hope to reverse the process which put its status as alive in doubt; to say nothing of the person who killed the cat (assuming that the cat was killed in the open rather than in an unpierceable box).

  2. Another is that an entangled state is correlated not just with respect to one basis of measurement, but with respect to multiple bases of measurement — as in the EPR thought experiment, and which implicitly plays a role in protocols of quantum communication (such as quantum teleportation) which rely on highly entangled states. However, if the correlation is spread across very large numbers of particles (such as a system comprised of a cat, someone who may or may not kill them, and the rest of the Earth), the joint correlation would be hopeless to actually measure. This is a manifestation of the monogamy of entanglement, the principle that what distinguishes entanglement from 'classical' correlation is diluted when distributed across larger systems, and that maximal entanglement can only be manifested between a single particle with one other.

So while the entire Earth may in principle be in a massive entangled state — and indeed, an advocate of the Many Worlds Interpretation of quantum mechanics would say that of course it's in a massive entangled state; it always is, and so is the matter that makes up your body and your brain — there is no practical way that you could discover whether it is.

If you took the practical approach of saying that the state decoheres at the point that you could never hope to prove that it is entangled, then it is a question of technological limitations; and even if you could demonstrate entanglement in a system, there would be a question of the complexity of proving it, which I imagine would increase dramatically with the size of the system which you encapsulate into a box. So the proper question isn't really "when has the system decohered?", but "how quickly does the complexity of stopping decoherence grow?", where 'decoherence' is just the process of catching up more matter into the entangled state rather than the status of having lost entanglement. Concievably there would be a point where the resources of the entire universe would not suffice to distill the entanglement into a handful of particles where it could be observed, just as (and for precisely the same reason as) there would come a point where we lack the resources to reduce entropy in a sufficiently complicated system.

But in short, it seems to me that the problem is not a yes/no problem, but a how hard is it to do X problem. In that way, at least, we could hope to get a quantitative answer in the domain of our current theories.

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Nice try about trying to pick out the preferred basis from "delayed" environmental coupling from a huge box surrounding us, but it doesn't quite work out.

Say you pick a solar system box, and only wait for light signals and slower matter to carry information about the interior dynamics. The problem is, the interior scales as the volume while the box boundary area scales as the area, which is a channel bottleneck. The environment outside the solar system is limited in what it can probe about the solar system because of this bottleneck. This isn't quite quantum gravity holography because we're not taking about densities large enough to produce black holes. This is something else. Any "pointer states" picked out in this manner can only have a coarse grained resolution because of this. They are more like projection operators.

Try to analyze this using consistent histories. We have some projectors in the environment, delayed by light cones. Now, you wish to include some further projectors here on Earth a few hours earlier at a finer grained resolution. A signal, say a radio wave encoding of a photo of the live/dead cat is transmitted out of the solar system to be picked up by the "environment". We can easily set up coarse grained projection operators outside the solar system a few hours later corresponding to "a photo of a live cat" or "a photo of a dead cat". If we set up coarse grained projection operators here on earth a few hours earlier, to "living cat" and "dead cat", we satisfy the approximate decoherence consistency conditions with nearly perfect correlation. So far, so good. However, trouble comes in when you try to pick out finer grained projectors than what can be picked up by the environment. This has been pointed out by Kent and Dowker. Even fixing the quasiclassical environmental operators, there are still many finer grained "decohered in the sense of satisfying approximate consistency conditions" choices of projectors here on earth hours earlier which aren't remotely quasiclassical. They have to commute with the life/death status of the cat and the photo information, but otherwise, within say "living cat as captured by photo", we can have smeared cat projectors, pardon the expression.

The environment can only pick out "pointer projectors" up to a certain resolution which can be far larger than what one would expect from "internal decoherence". Beyond that, even the environment can't pick out the "correct" quasiclassical preferred basis.

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Well if the information about the inside state will reach the outside of the box ever, even in the future the decoherence will happen. This is due to non-locality.

The only way to avoid decoherence is to isolate the system as much as possible in the time interval before you return the system back to a basis state.

You cannot say "look, I will return the system to the basis state before the light reaches the box wall so there wil be no decoherence".

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