On Sat, Aug 30, 2025 at 2:04 AM Jesse Mazer <[email protected]> wrote:
> On Thu, Aug 28, 2025 at 8:05 PM Bruce Kellett <[email protected]> > wrote: > >> On Fri, Aug 29, 2025 at 2:47 AM Jesse Mazer <[email protected]> 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. 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) > > > >> 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. "The ingenious method of introducing a quantum-mechanical equivalent of > probabilities that Mittelstaedt follows in his approach relies on a new > operator F^N_k > whose ‘intuitive’ role is to measure the relative frequency of the outcome > a_k in a given sequence of N outcomes." > > The full details would presumably be in Mittelstaedt's book The > Interpretation of Quantum Mechanics and the Measurement Process in the > paper's bibliography. > I have no interest in looking into this further, since it clearly cannot work to give meaning to probability in an Everettian model. It was always a fringe idea that didn't work out. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAFxXSLTP9wHwPSYjLpjp6ieJ%3DBNV9DZhxReZ0LJ3Myz0hLkybg%40mail.gmail.com.
