 Quote by straycat
I've been pondering your last post. It seems you are drawing a distinction between (1) the number of dimensions of Hilbert space and (2) the number of eigenspaces of the measurement operator. I realize these are different,
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Yes, this is essential. The number of dimensions in Hilbert space is given by the physics, and by physics alone, of the system, and might be very well infinite-dimensional. I think making assumptions on the finiteness of this dimensionality is dangerous. After all, you do not know what degrees of freedom are hidden deep down there. So we should be somehow independent of the number of dimensions of the Hilbert space.
However, the number of eigenspaces of the measurement operator is purely determined by the measurement apparatus. It is given by the resolution by which we could, in principle, determine the quantity we're trying to measure, using the apparatus in question. You and I agree that this must be a finite number, and a rather well-determined one. This is probably where we are differing in opinion, and where you seem to claim "micromeasurements" of eventually unknown physics of which we are not aware versus "macromeasurements" which are just our own coarse-graining of these micromeasurements- while I claim that with every specific measurement goes a certain, well-defined number of outcomes (which could eventually be more fine-grained than the observed result but that this should not be dependent on "unknown physics", but that a detailled analysis of the measurement setup should reveil that to us). I would even claim that a good measurement apparatus makes the observed number of outcomes about equal to the real number of eigenspaces.
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Suppose we assume the APP. Given a particular measurement to be performed, suppose we have K total fine-grained outcomes, with k_i the number of fine-grained outcomes corresponding to the i^th coarse-grained result. eg, we have N position detector elements, i an integer in [1,N], and the sum of k_i over all i equals K. So the probability of detection at the i^th detector element is k_i / K, and we define:
E_i = k_i / K
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Yes, but I'm claiming now that for a good measurement system, k_i = 1 for all i, and even if it isn't (for instance, you measure with a precision of 1 mm, and your numerical display only displays up to 1 cm resolution), you're not free to fiddle with k_i as you like.
Also, you now have a strange outcome! You ALWAYS find probability E_i for outcome i, no matter what was the quantum state ! Even if the quantum state is entirely within the E_i eigenspace, you'd still have a fractional probability ? That would violate the rule that two measurements applied one after the other will give the same result.
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So if I claim that p_i can depend only upon E_i (ie p_i = E_i), then it seems to me that I could argue, using the same reasoning that you use above for the Born rule, that the APP is non-contextual.
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No, non-contextuality has nothing to do with the number E_i you're positioning here, it is a property of being only a function of the eigenspace (spanned by the k_i subspaces) and the quantum state, no matter how the other eigenspaces are sliced up. Of course, in a way, you're right: if the outcome is INDEPENDENT on the quantum state (as it is in your example), you are indeed performing a non-contextual measurement. In fact, the outcome has nothing to do with the system: outcome i ALWAYS appears with probability E_i. But I imagine that you only want to consider THOSE OUTCOMES i THAT HAVE A PART OF the quantum state in them, right ? And THEN you become dependent on what happens in the other eigenspaces.
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That is, Hilbert space assumes the state is represented by f, and the sum of |f|^2 over all states equals 1 (by normalization). So really it's not that surprising that |f|^2 is the only way to get a probability.
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Well, there's a difference in the following sense: if you start out with a normalized state, you will always keep a normalized state under unitary evolution, and if you change basis (change measurement), you can keep the same normalized vector. That cannot be said for the E_i and k_i construction, which needs to be redone after each evolution, and after each different measurement basis.
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Do you say "at first sight" because a careful analysis indicates that it's not all that reasonable?
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Well, it is reasonable, but it is an EXTRA assumption (and, according to Gleason, logically equivalent to postulating the Born rule). It is hence "just as" reasonable as postulating the Born rule.
What I meant with "at first sight" is that one doesn't realise the magnitude of the step taken! In unitary QM, there IS no notion of probability. There is just a state vector, evolving deterministically by a given differential equation of first order, in a hilbert space. From the moment that you require, no matter how little, a certain quality of a probability issued from that vector, you are in fact implicitly postulating an entire construction: namely that probabilities ARE going to be generated from this state vector (probabilities for what, for whom?), that only part of the state vector is going to be observed (by whom?) etc... So the mere statement of a simple property of the probabilities postulates in fact an entire machinery - which is not obvious at first sight. Now if your aim is to DEDUCE the appearance of probabilities from the unitary machinery, then implicitly postulating this machinery is NOT reasonable, because it implies that you are postulating what you were trying to deduce in one way or another.
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I actually agree with you here. The argument in my mind goes like this: consider a position measurement. If you want to come up with a continuous measurement variable, this would be it. But from a practical perspective, a position measurement is performed via an array or series of discrete measurement detectors. The continuous position measurement is then conceived as the theoretical limit as the number of detector elements becomes infinite. But from a practical, and perhaps from a theoretical, perspective, this limit cannot ever be achieved: the smallest detector element I can think of would be (say) an individual atom, for example the atoms that make up x-ray film.
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That's what I meant, too. There's a natural "resolution" to each measurement device, which is given by the physics of the apparatus. An x-ray film will NOT be in different quantum states for positions which differ much less than the size of an atom (or even a bromide xtal). This is not "unknown physics" with extra degrees of freedom. I wonder whether a CCD type camera will be sensing on a better resolution than one pixel (meaning that the quantum states would be different for hits at different positions on the same pixel). Of course, there may be - and probably there will be - some data reduction up to the display, but one cannot invent, at will, more fine-grained measurements than the apparatus is actually naturally performing. And this is what determines the slicing-up of the Hilbert space in a finite number of eigenspaces, which will each result in macroscopically potentially distinguishable "pointer states". And I think it is difficult (if not hopeless) to posit that these "micromeasurements" will arrange themselves each time in such a way that they work according to the APP, but give rise to the Born rule on the coarse-grained level. Mainly because the relationship between finegrained and coarse grained is given by the measurement apparatus itself, and not by the quantum system under study (your E_i = k_i/K is fixed by the physics of the apparatus, independent of the state you care to send onto it ; the number of atoms on the x-ray film per identified "pixel" on the scanner is fixed, and not depending on how it was irradiated).
cheers,
Patrick.
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