On Fri, Aug 29, 2025 at 7:54 PM Bruce Kellett <bhkellet...@gmail.com> wrote:
> On Sat, Aug 30, 2025 at 2:04 AM Jesse Mazer <laserma...@gmail.com> wrote: > >> On Thu, Aug 28, 2025 at 8:05 PM Bruce Kellett <bhkellet...@gmail.com> >> wrote: >> >>> On Fri, Aug 29, 2025 at 2:47 AM Jesse Mazer <laserma...@gmail.com> >>> wrote: >>> >>>> >>>> You were discussing a case of this form: "This is easily seen if one >>>> considers a wave function with a binary outcome, |0> and |1> for example. >>>> After N repeated trials, one has 2^N strings of possible outcome sequences. >>>> One can count the number of, say, ones in each possible outcome sequence." >>>> >>>> If we are interested in statistics for N trials, let's define a >>>> "supertrial" as a sequence of N trials of the individual measurement, and >>>> say that we are repeating many supertrials and recording the results of all >>>> the individual trials in each supertrial using some kind of physical memory >>>> (persistent 'pointer states'). Each supertrial has 2^N possible outcomes, >>>> and for a given supertrial outcome O (like up, down, up, up, up, down for >>>> N=6) you can define a measurement operator on the pointer states whose >>>> eigenvalues correspond to what the records would tell you about the >>>> fraction of supertrials where the outcome was O. If I'm understanding the >>>> result in those references correctly, then if one models the interaction >>>> between quantum system, measuring apparatus, and records using only the >>>> deterministic Schrodinger equation, without any collapse assumption or Born >>>> rule, one can show that in the limit as the number of supertrials goes to >>>> infinity, all the amplitude for the whole system including the records >>>> becomes concentrated on state vectors that are parallel to the eigenvector >>>> of the measurement operator with the eigenvalue that exactly matches the >>>> frequency of outcome O that would have been predicted if you *had* used the >>>> collapse assumption and Born rule for individual measurements. And this >>>> should be true even if the probability for up vs. down on individual >>>> measurements was not 50/50 given the experimental setup. >>>> >>> >>> I haven't looked into this in any detail, but it seems to be a recasting >>> of an idea that has been around for a long time. This idea hasn't made it >>> into the mainstream because the details failed to work out. >>> >> >> Can you point to any sources that explain specific ways the details fail >> to work out? David Z Albert is very knowledgeable about results relevant to >> interpretation of QM so I'd be surprised if he missed any technical >> critique. >> > > I quote David Albert from his contribution to the book "Many Worlds? > Everett, Quantum Theory and Relativity" (Oxford,2010) > "But the business of parlaying this thought into a fully worked-out > account of probability in the Everett picture quickly runs into very > familiar and very discouraging sorts of trouble." I don't have any more > detail about this, but it seems from the fact that this is not mainstream, > that these difficulties proved insurmountable. For instance, it uses a > frequentist definition of probability, and this is known to be full of > problems. > But see my comment below on the distinction between (1) saying the result is wrong in a technical sense, vs. (2) raising the philosophical objection that it only tells us that overall frequencies in the infinite limit approach a definite answer, not that there can be a definite result to specific individual trials (or specific finite runs of trials), which may lead people to say that even though the result is valid, it doesn't resolve the measurement problem (the problem of why we only experience definite outcomes rather than superpositions when we make individual observations): > Of course there is the philosophical argument that this doesn't resolve >> the measurement problem because it doesn't lead to definite results for >> individual trials (or supertrials) but that's not taking issue with the >> technical claim about measuring frequencies of results in the limit of >> infinite trials (and David Z Albert brings up this philosophical objection >> in the last paragraph before section VI at >> https://books.google.com/books?id=_HgF3wfADJIC&lpg=PP1&pg=PA238 , and >> then in section VI he goes on to talk about why he thinks this objection >> means the fact about frequencies in the limit doesn't really resolve the >> measurement problem) >> > His essay in "Many Worlds? Everett, Quantum Theory and Relativity" states the technical result again on p. 357, saying it is one that can "easily be shown", and the part you quoted is from the second to last sentence of the paragraph where he again raises that philosophical objection to seeing this result (though valid) as a resolution to the measurement problem: "Here’s an idea: Suppose that we measure the x-spin of each of an infinite ensemble of electrons, where each of the electrons in the ensemble is initially prepared in the state α|x-up⟩ + β|x-down⟩. Then it can easily be shown that in the limit as the number of measurements already performed goes to infinity, the state of the world approaches an eigenstate of the frequency of (say) up-results, with eigenvalue |α|^2. And note that the limit we are dealing with here is a perfectly concrete and flat-footed limit of a sequence of vectors in Hilbert space, not a limit of probabilities of the sort that we are used to dealing with in applications of the probabilistic law of large numbers. And the thought has occurred to a number of investigators over the years (Sidney Coleman, and myself, and others too) that perhaps all it means to say that the probability that the outcome a measurement of the x-spin of an electron in the state α|x-up⟩ + β|x-down⟩ up is |α|^2 is that if an infinite ensemble of such experiments were to be performed, the state of the world would with certainty approach an eigenstate of the frequency of (say) up-results, with eigenvalue |α|^2. And what is particularly beautiful and seductive about that thought is the intimation that perhaps the Everett picture will turn out, at the end of the day, to be the only picture of the world on which probabilities fully and flat-footedly and not-circularly make sense. But the business of parlaying this thought into a fully worked-out account of probability in the Everett picture quickly runs into very familiar and very discouraging sorts of trouble. One doesn’t know what to say (for example) about finite runs of experiments, and one doesn’t know what to say about the fact that the world is after all very unlikely ever to be in an eigenstate of my undertaking to carry out any particular measurement of anything." > >> >> >>> There are all sorts of problems with the idea, and it doesn't appear to >>> translate well to the argument I am making. The 2^N sequences that result >>> from repeated measurements on the basic binary system do not form a >>> measurement in themselves. There is no operator for this, and no >>> eigenfunctions and there is no obvious outcome. >>> >> >> I had thought that for any measurable quantity including coarse-grained >> statistical ones, it was possible to construct a measurement operator in >> QM--doing some googling, it may be that for some coarse-grained quantities >> one has to use a "positive operator valued measure", see answer at >> https://physics.stackexchange.com/a/791442/59406 , and according to >> https://quantumcomputing.stackexchange.com/a/29326 this is not itself an >> operator though it is a function defined in terms of a collection of >> positive operators. And the page at >> https://www.damtp.cam.ac.uk/user/hsr1000/stat_phys_lectures.pdf also >> mentions that in quantum statistical mechanics, macrostates can be defined >> in terms of the density operator which is used to describe mixed states >> (ones where we don't know the precise quantum microstate and just assign >> classical probabilities to different possible microstates). I don't know if >> either was used here, but p. 13 of the paper I mentioned at >> https://www.academia.edu/6975159/Quantum_dispositions_and_the_notion_of_measurement >> indicates that some type of operator was used to derive the result about >> frequencies in the limit: >> > > There is no single outcome from a repetition of the N trials and 2^N > sequences. So it can't be an eigenvalue of some quantum operator. > Why can't there be an eigenvalue that tells you the frequency of a specific result, for example if N=6 and you want to know the frequency of the specific result "up, down, up, up, up, down" in a large collection of measurements of groups of 6 electrons? This frequency is just a type of coarse-grained macrostate that's a straightforward function of the spin microstate (the exact specification of the spin of every individual electron), so if there can be measurement operators for macrostates in QM I don't see why there'd be a problem with this one. But even assuming the result holds for such an operator when we consider the infinite limit, there would still be the same philosophical objection that this doesn't give us a definite outcome on any *specific* measurement of a group of 6 specific electrons in the infinite sequence. 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