What does ‘widely separated’ mean? I suspect that this can be defined without recourse to a full detailed treatment of decoherence. Let’s give that a try. (I’m going to feel free to edit this until someone responds, since I’m kind of thinking out loud).
The obvious but wrong answer is that given two initial components |a> and |b>, the measurement process produces consequences such that U|a> is orthogonal to U|b>.. of course, that’s trivially true since = = 0. Even if they were overlapping everywhere, the unitary process of time evolution would make their overlap integral keep canceling out. And meanwhile they would be interfering with each other—not independent at all.
What we need is for the random phase approximation to become applicable. If we are able to consider a local system, it can be by exchanging a particle with the outside. The applicability of this approach to a single universal wavefunction is not clear. We will need to be able to speak of information loss and dissipation in unitary language.
I had another flawed but more promising notion, that one could get somewhere by considering a second split after a first. You have two potentially decoherent vectors |a> and |b>, with = 0; then you split |b> = |c> + |d> such that = 0. The idea was that |a> and |b> are ‘widely separated’ if any choice of |c> and |d> will have = = 0… except that you can always choose some crazy superposed |c> that explicitly overlaps |a> and |b>.
Based on this, I thought instead about an operator that takes macro-scale measurements, like ‘is that screen reading X’. Then you can require that |c> and |d> each are in the same kernel of each these operators as |b> is. That might be sufficient even without splitting |b> - as long as you can construct a macro-scale measurement that indicates |b> instead of |a>, then they’re going to be distinguishable by that, so they won’t interfere. But that in itself doesn’t prove that you can’t smash the computer and get them to interfere again.
Of course, all that puts it backwards, focusing on how you could possibly establish a perfectly ordinary decoherent state, rather than focusing on how you maintain an utterly abnormal coherent state (this is the approach the sequence suggests).
You need to be able to split of a subspace of the Hilbert space such that the ‘outside’ is completely independent of the ‘inside’. Nearly completely causally independent, at least on some time domain. For example, in an interferometer, all the rest of the universe depends on is that the inside of the interferometer is only doing interferometry, not, say, exploding. If there were such a dependence (and it was a true dependence such that the various outcomes actually produced a different effect), then the joint configurations rule would kick in, and the subspace could not interfere because of the different effects on the outside.
The problem here is, I do not know of any mathematical language for expressing causal dependence in quantum mechanics. If there is one, this is a very brief statement in it.
What does ‘widely separated’ mean? I suspect that this can be defined without recourse to a full detailed treatment of decoherence. Let’s give that a try. (I’m going to feel free to edit this until someone responds, since I’m kind of thinking out loud).
The obvious but wrong answer is that given two initial components |a> and |b>, the measurement process produces consequences such that U|a> is orthogonal to U|b>.. of course, that’s trivially true since = = 0. Even if they were overlapping everywhere, the unitary process of time evolution would make their overlap integral keep canceling out. And meanwhile they would be interfering with each other—not independent at all.
What we need is for the random phase approximation to become applicable. If we are able to consider a local system, it can be by exchanging a particle with the outside. The applicability of this approach to a single universal wavefunction is not clear. We will need to be able to speak of information loss and dissipation in unitary language.
I had another flawed but more promising notion, that one could get somewhere by considering a second split after a first. You have two potentially decoherent vectors |a> and |b>, with = 0; then you split |b> = |c> + |d> such that = 0. The idea was that |a> and |b> are ‘widely separated’ if any choice of |c> and |d> will have = = 0… except that you can always choose some crazy superposed |c> that explicitly overlaps |a> and |b>.
Based on this, I thought instead about an operator that takes macro-scale measurements, like ‘is that screen reading X’. Then you can require that |c> and |d> each are in the same kernel of each these operators as |b> is. That might be sufficient even without splitting |b> - as long as you can construct a macro-scale measurement that indicates |b> instead of |a>, then they’re going to be distinguishable by that, so they won’t interfere. But that in itself doesn’t prove that you can’t smash the computer and get them to interfere again.
Of course, all that puts it backwards, focusing on how you could possibly establish a perfectly ordinary decoherent state, rather than focusing on how you maintain an utterly abnormal coherent state (this is the approach the sequence suggests).
You need to be able to split of a subspace of the Hilbert space such that the ‘outside’ is completely independent of the ‘inside’. Nearly completely causally independent, at least on some time domain. For example, in an interferometer, all the rest of the universe depends on is that the inside of the interferometer is only doing interferometry, not, say, exploding. If there were such a dependence (and it was a true dependence such that the various outcomes actually produced a different effect), then the joint configurations rule would kick in, and the subspace could not interfere because of the different effects on the outside.
The problem here is, I do not know of any mathematical language for expressing causal dependence in quantum mechanics. If there is one, this is a very brief statement in it.