By hypothesis, Sensor-LEFT is a different state from Sensor-RIGHT—otherwise it wouldn’t be a very sensitive Sensor. So the final state doesn’t factorize any further; it’s entangled.
But this entanglement is not likely to manifest in difficulties of calculation. Suppose the Sensor has a little LCD screen that’s flashing “LEFT” or “RIGHT”. This may seem like a relatively small difference to a human, but it involves avogadros of particles—photons, electrons, entire molecules—occupying different positions.
So, since the states Sensor-LEFT and Sensor-RIGHT are widely separated in the configuration space, the volumes (Sensor-LEFT Atom-LEFT) and (Sensor-RIGHT Atom-RIGHT) are even more widely separated.
The question still left in my mind is what is meant by “widely separated”, and why states that are widely separated have volumes that are widely separated.
For example, take a chaotic system evolving from an initial state. (Perhaps an energetic particle in a varying potential field.) After evolving, the probability that was concentrated at that initial state flows out to encompass a rather large region of configuration space. Presumably in this case the end states can be widely separated, but the probability volumes are not.
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.
It seems like the relevant section is
The question still left in my mind is what is meant by “widely separated”, and why states that are widely separated have volumes that are widely separated.
For example, take a chaotic system evolving from an initial state. (Perhaps an energetic particle in a varying potential field.) After evolving, the probability that was concentrated at that initial state flows out to encompass a rather large region of configuration space. Presumably in this case the end states can be widely separated, but the probability volumes are not.
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.