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posted by Fnord666 on Sunday January 19 2020, @09:55PM   Printer-friendly
from the merely-impossible dept.

'Remarkable' Mathematical Proof Describes How to Solve Seemingly Impossible Computing Problem:

You enter a cave. At the end of a dark corridor, you encounter a pair of sealed chambers. Inside each chamber is an all-knowing wizard. The prophecy says that with these oracles' help, you can learn the answers to unanswerable problems. But there's a catch: The oracles don't always tell the truth. And though they cannot communicate with each other, their seemingly random responses to your questions are actually connected by the very fabric of the universe. To get the answer you seek, you must first devise... the questions.

Computer scientists are buzzing about a new mathematical proof that proposes a quantum-entangled system sort of like the one described above. It seems to disprove a 44-year-old conjecture and details a theoretical machine capable of solving the famous halting problem, which says a computer cannot determine whether it will ever be able to solve a problem it's currently trying to solve.


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  • (Score: 3, Informative) by maxwell demon on Monday January 20 2020, @04:58PM (15 children)

    by maxwell demon (1608) on Monday January 20 2020, @04:58PM (#945887) Journal

    Experiments about the fundamentals of quantum mechanics are not done in particle accelerators. They are done with lasers, with electrons, with atoms, and a lot of other systems. There are countless laboratories in the world that do quantum experiments, and all of those find that quantum mechanics works.

    --
    The Tao of math: The numbers you can count are not the real numbers.
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  • (Score: 2) by JoeMerchant on Monday January 20 2020, @09:11PM (14 children)

    by JoeMerchant (3937) on Monday January 20 2020, @09:11PM (#945975)

    And yet, when I try to get physicists (real PhD quantum and nuclear physicists, with plenty of time on their hands) to explain something fundamental like: how Bell's inequality really proves that a hidden variable doesn't exist? They fail, they even fail to convince themselves except with statements like "they've collected lots of statistics and the evidence is overwhelming..." and yet still seem to sound unconvinced. Follow:

    there are "a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions." The two particles travel away from each other to two distant locations, at which measurements of spin are performed, along axes that are independently chosen. Each measurement yields a result of either spin-up (+) or spin-down (−); it means, spin in the positive or negative direction of the chosen axis.

    The probability of the same result being obtained at the two locations depends on the relative angles at which the two spin measurements are made, and is strictly between zero and one for all relative angles other than perfectly parallel or antiparallel alignments (0° or 180°). Since total angular momentum is conserved, and since the total spin is zero in the singlet state, the probability of the same result with parallel (antiparallel) alignment is 0 (1). This last prediction is true classically as well as quantum mechanically.

    Bell's theorem is concerned with correlations defined in terms of averages taken over very many trials of the experiment. The correlation of two binary variables is usually defined in quantum physics as the average of the products of the pairs of measurements....

    Measuring the spin of these entangled particles along anti-parallel directions (i.e., facing in precisely opposite directions, perhaps offset by some arbitrary distance) the set of all results is perfectly correlated. On the other hand, if measurements are performed along parallel directions (i.e., facing in precisely the same direction, perhaps offset by some arbitrary distance) they always yield opposite results, and the set of measurements shows perfect anti-correlation. This is in accord with the above stated probabilities of measuring the same result in these two cases. Finally, measurement at perpendicular directions has a 50% chance of matching, and the total set of measurements is uncorrelated....

    To date, Bell's theorem is generally regarded as supported by a substantial body of evidence and there are few supporters of local hidden variables, though the theorem is continually the subject of study, criticism, and refinement, and the popularity of non-local hidden variable theories has been on the rise.

    And, quite simply, if the measurement done at point A in some way affects the results of measurement of entangled particles done at point B, that would constitute non-causal / FTL communication, which everyone (in the field) swears is not possible... So, if non-causal communication is not happening, it would seem clear that hidden variables could explain whatever is observed.

    --
    🌻🌻 [google.com]
    • (Score: 0) by Anonymous Coward on Monday January 20 2020, @10:00PM

      by Anonymous Coward on Monday January 20 2020, @10:00PM (#946014)

      I hate idiots like this who think THEY are the smart one in the room.

    • (Score: 0) by Anonymous Coward on Tuesday January 21 2020, @01:40AM (1 child)

      by Anonymous Coward on Tuesday January 21 2020, @01:40AM (#946110)

      Most physicists know very little about Bells theorem. Most are trained to look away.

      Bells theorem + experimental results rule out a wide class of hidden variable theories. Specifically this in which the detection of an entangled photon is determined only by the properties of that photon and the polariser/detector which it encounters.You can come up with a more elaborate model, but one of the kind in bells theorem won't work.

      • (Score: 2) by JoeMerchant on Tuesday January 21 2020, @11:38AM

        by JoeMerchant (3937) on Tuesday January 21 2020, @11:38AM (#946296)

        Most are trained to look away.

        Thus, my increased suspicion of Bell's Theorem as opposed to, say, Gallileo's demonstration of gravitational attraction.

        --
        🌻🌻 [google.com]
    • (Score: 2) by FatPhil on Tuesday January 21 2020, @08:33AM (9 children)

      by FatPhil (863) <pc-soylentNO@SPAMasdf.fi> on Tuesday January 21 2020, @08:33AM (#946239) Homepage
      Nope, entanglement is not communication - you can't chose what is shared, you can only know that whatever is shared is coherent. The only way the information extracted from the second (a meaningless term under special relativity) measurement can be interpreted is by the addition of another bit of information telling you how to interpret the quantum one. So in order to communicate one bit of information, you need to transmit one bit via quantum channel and one bit from a traditional channel. The quantum channel actually told you nothing.

      If even Einstein can consider quantum mechanics "spooky" (and downright disbelieving some aspects of QM that were untestable at the time), there's no shame in still getting perturbed by some of its claims, but the core claims are all lab-proved, time and time again. Every objection to an experiment proving Bell's inequality has simply led to a trickier experimental setup that proves exactly the same result as the experiment that was objected to. I don't think there are any outstanding objections any more, many many decades later. (Under any of the conventonal interpretations of QM - there's a slightly out-there new hidden-variable interpretation that's been proposed which seems to bubble up into the discussion occasionally, but it has no traction yet, and it's different enough that I have no idea what it says regarding EPR and Bell.)
      --
      Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
      • (Score: 2) by JoeMerchant on Tuesday January 21 2020, @11:49AM (8 children)

        by JoeMerchant (3937) on Tuesday January 21 2020, @11:49AM (#946299)

        you can only know that whatever is shared is coherent... by the addition of another bit of information telling you how to interpret the quantum one... The quantum channel actually told you nothing.

        Granted, now: clearly demonstrate that whatever is shared is not simply a hidden variable which was determined at the time of entanglement?

        the core claims are all lab-proved, time and time again.

        And Bernie Madoff delivered spooky returns on investment, time and time again, until he didn't.

        a trickier experimental setup that proves exactly the same result as the experiment that was objected to.

        If you had said "a simpler experimental setup..." I'd believe we're onto something truly worthwhile.

        there's a slightly out-there new hidden-variable interpretation that's been proposed

        I've caught the occasional reference to a special-relativity hidden-variable, but the best I can come up with on short notice is this:

        For example, E. T. Jaynes[40] argued in 1989 that there are two hidden assumptions in Bell's theorem that limit its generality. According to him:

        Bell interpreted conditional probability P(X | Y) as a causal influence, i.e. Y exerted a causal influence on X in reality. This interpretation is a misunderstanding of probability theory. As Jaynes shows,[41] "one cannot even reason correctly in so simple a problem as drawing two balls from Bernoulli's Urn, if he interprets probabilities in this way."
        Bell's inequality does not apply to some possible hidden variable theories. It only applies to a certain class of local hidden variable theories. In fact, it might have just missed the kind of hidden variable theories that Einstein is most interested in.

        In any event it disturbs me that our "men of science" seem to be, in large part, taking things like on faith just as blind as those who send money to TV evangelists...

        --
        🌻🌻 [google.com]
        • (Score: 2) by FatPhil on Tuesday January 21 2020, @12:38PM (7 children)

          by FatPhil (863) <pc-soylentNO@SPAMasdf.fi> on Tuesday January 21 2020, @12:38PM (#946311) Homepage
          > clearly demonstrate that whatever is shared is not simply a hidden variable which was determined at the time of entanglement?

          There's 56 years of history behind that particular endeavour (plus the work before Bell's paper). Many are happy that their chosen interpretation supports that, and and their experimental results support their interpretation, and I think that's true for most commonly accepted interpretations (some interpretations make the question meaningless, of course, which I'd say makes it vacuously true).

          > If you had said "a simpler experimental setup..." I'd believe we're onto something truly worthwhile.

          Removal of the posibility of alternative explanations requires active prevention of them, that's just how things work. The Bell inequality is one of the hypotheses where there's been more such refinements than any other one than I can think of, because there are so many alternative "but it could be"s that need to be nullified. IIRC the wikipedia page had a good list of most of them.
          --
          Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
          • (Score: 2) by JoeMerchant on Tuesday January 21 2020, @03:55PM (6 children)

            by JoeMerchant (3937) on Tuesday January 21 2020, @03:55PM (#946373)

            Removal of the posibility of alternative explanations requires active prevention of them, that's just how things work.

            But, back to Gallileo - dropping two objects of unequal size and mass off of a high place and observing they fall at the same rate is a simple demonstration that removes the possibility of any number of alternative, incorrect explanations...

            there are so many alternative "but it could be"s that need to be nullified.

            I recognize that I'm a naive poseur, but my initial and only objection to the whole thing boils down to what is referred to in the field as "hidden variables," and as long as there are demonstrations that:

            - for particles A and B entangled at point 1 time T,

            - A moved to point 2, B moved to point 3, some significant distance apart,

            - if something about A measured at time U has some bearing on something about B measured at time U,

            - I find it logically inescapable that that bearing is accountable to either:

            -- a hidden variable established at point 1 time T, or

            -- faster than light communication of something from A at point 2 time U to B at point 3 time U.

            Explanations built up around theories of indeterminate collapsing states are all well and good, and those mathematical/theoretical constructs may, or may not, be self consistent and consistent with the physical observations, but... I fail to see how they refute the seemingly simpler logical explanation: "the states were determined at time T, we just didn't measure them until time U."

            It's all well and good to say: It's very esoteric and confusing, you wouldn't understand it unless you were a rare expert in the field. However, if Bernie Madoff gave me that explanation for the returns on his fund, I wouldn't be investing in it.

            I would also, romantically, far prefer the explanation of faster than light communication to be true - but I believe the Fermi paradox is a fairly strong piece of circumstantial evidence that FTL just doesn't happen in this universe.

            --
            🌻🌻 [google.com]
            • (Score: 3, Insightful) by FatPhil on Tuesday January 21 2020, @05:17PM (5 children)

              by FatPhil (863) <pc-soylentNO@SPAMasdf.fi> on Tuesday January 21 2020, @05:17PM (#946413) Homepage
              However, it's not just a question of "hidden variables", it's about locality and determinism too, plus the other things which cause experts into fall into different camps regarding interpretations. The experiment either needs to be interpretation neutral or needs to be able to reject interpretations (a much harder thing to do, probably a Nobel earner).
              --
              Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
              • (Score: 2) by JoeMerchant on Tuesday January 21 2020, @05:46PM (4 children)

                by JoeMerchant (3937) on Tuesday January 21 2020, @05:46PM (#946439)

                I know the horse is dead, but I just can't stop kicking it...

                it's about locality and determinism too

                Does it have to be? Forget quantum mechanics mostly altogether: put your two particles (or, rather, large sampling of particle pairs) into their "entangled state" in any of these experiments (time T). Then, measure them later (time U). Where is the outcome that says the later measurement state couldn't possibly have been determined at the time the particles were "entangled." As I understand the rules of engagement, no measurement is permitted until time U, ergo you just don't know what happened after T until U - what's the possible proof that the state did not become determinate at T which does not involve FTL travel between the measurement points?

                Yes, yes, and here I'm devolving into complexity just like the rest of the field, which I'd rather not do, but... just because there's a "masking bit" which must be communicated between the points to conduct the experiment, and therefore no new information can be communicated between B and C, doesn't have a damn thing to do with willful ignorance of a state determined at the time of entanglement - insisting that it wasn't determined at that time and place doesn't make it so.

                I realize that a rather large number of "smart guys" have put a rather large amount of thought into this problem for a rather long amount of time, but it still feels after all these decades of "progress" have passed that there's that rather basic question left unanswered while buzzing around all the details and intricacies of more highly developed theories.

                --
                🌻🌻 [google.com]
                • (Score: 2) by FatPhil on Tuesday January 21 2020, @11:30PM (3 children)

                  by FatPhil (863) <pc-soylentNO@SPAMasdf.fi> on Tuesday January 21 2020, @11:30PM (#946586) Homepage
                  I've only learnt 2 different versions of the dumbed-down-for-not-hardcore-physicists version, once in a format interactive enough environment to feel that I was convinced, and couldn't find any more holes to pick at.

                  I would hate to be responsible for mangling what I learnt even more by trying to pass it on. At times like this, I just run off to the internet, there are many reliable resources out there. Wikipedia's pretty good, and has plenty of outward pointing links.
                  --
                  Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
                  • (Score: 2) by JoeMerchant on Wednesday January 22 2020, @02:32AM (2 children)

                    by JoeMerchant (3937) on Wednesday January 22 2020, @02:32AM (#946633)

                    Thanks for your patience... usually there's a "hidden gotcha" that they don't explain up front. In college I had a few "NP hard" problems explained to me in ways that weren't NP hard, I think one had to do with factoring boolean expressions - which I thought: HA, I've already written algorithms that do that... but when I finally dug into it the ACTUAL boolean expressions in the NP hard problem are product of sums expressions, not sum of products expressions - well, of course, what f'ing idiot writes complex boolean expressions in product of sums form?!? For one thing, they're so much harder to work with like that, doh!

                    Like I've said, I've breezed through some Wikipedia articles, kind of stepped my way through the supposed proof experiments, and challenged "great physicists" to explain it to me - all to no avail, and 1) as you said: Einstein didn't seem to get it, so I guess I'm in good company, and 2) I've explained a thing or two to some PhD physicists over the years that I proved numerically (computer algorithms) with large numbers of demonstrations of equal results and had the PhD leave notes in the computer code to the effect of "I don't understand this part, see Joe for an explanation." So, I guess that point is: PhDs can be a sort of bifurcating group, some are so brilliant that academia is the only place they can be appreciated, some are so obtuse that academia is the only place they can survive, and many are both at the same time which might explain where theories of quantum superposition came from in the first place ;-)

                    --
                    🌻🌻 [google.com]
                    • (Score: 2) by FatPhil on Wednesday January 22 2020, @02:32PM (1 child)

                      by FatPhil (863) <pc-soylentNO@SPAMasdf.fi> on Wednesday January 22 2020, @02:32PM (#946827) Homepage
                      The scary thing is that behind all the verbal explanations and pretty diagrams they do actually have serious hardcore mathematical equations with complex integrations over structures in more dimensions than we can ever happily pretend to imagine (and this is just the boring bog-standard stuff, nothing like M-Theory needed). The reason the scientists never pull out those big guns is because they would kill us before they'd even explained what every symbol meant. When someone who I know has the equations at his fingertips (and performs his experiments against that mathematical model) explains it in terms of an English-language fluffy explanation responds to a request for clarification with "you're just going to have to trust me", my only response is "OK, will do". Their maths predicted why the magnetic moment of the electron is different from the naive 1 that it should be under the high-school explanation of the field to 12 decimal places, and their science measured it to 12 decimal places, and their two numbers agree down to unity in the 12th decimal place (which is within both values' margins of error). This is their territory, their maths is king, if they tell me their maths implies something, and their experiments confirm something, then I know my place, and I shut up. There are plenty of mathematicians who could destroy their work if it is wrong.
                      --
                      Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
                      • (Score: 2) by JoeMerchant on Wednesday January 22 2020, @04:03PM

                        by JoeMerchant (3937) on Wednesday January 22 2020, @04:03PM (#946856)

                        The reason the scientists never pull out those big guns is because they would kill us before they'd even explained what every symbol meant.

                        The reason 90+% of the PhD physicists I've worked with never pull out those big guns is because they know the big guns would squash them flat like a bug before they ever found the trigger...

                        Their maths predicted why the magnetic moment of the electron is different from the naive 1 that it should be under the high-school explanation of the field to 12 decimal places, and their science measured it to 12 decimal places, and their two numbers agree down to unity in the 12th decimal place (which is within both values' margins of error).

                        Which is awesome, and believable, and there's tons of great work in the field like that.

                        There are plenty of mathematicians who could destroy their work if it is wrong.

                        For big guns in the field of quantum entanglement, I think the number is actually quite small. Sure, plenty of people get degrees, publish papers, review each others' papers for publication, etc. but the "don't look there, don't question it" ethos seems very strong in some of the fundamentals of this area.

                        But, then, it's of little to no practical value in my life - so, I sit happily skeptical on the outside just abusing them for failing to communicate clearly.

                        --
                        🌻🌻 [google.com]
    • (Score: 0) by Anonymous Coward on Tuesday January 21 2020, @09:46PM

      by Anonymous Coward on Tuesday January 21 2020, @09:46PM (#946541)

      As a real Ph.D. physicist who used to design and model high energy physics detectors, I certainly can't explain Bell's inequality in any form that you would find satisfactory because I don't understand it at a base level. It is irrelevant to my work, just as it is to most physicists. I am very interested in those issues, and I'm probably better read up on those philosophical underpinning type of issues than most because of my personal interest, but that isn't my line of work. And probably like the other physicists you've asked this of, I have a high level of understanding and can explain it at a high level, but you greatly underestimate how hard it is to explain very fundamental things. Those "why" questions that children ask can be VERY tough to answer because they get very fundamental very fast. It's not that we are trained to "look away", it is that unless you are working on that kind of thing, you don't need to know the philosophical implications for QM. I would bet that outside of a very small subset (that especially includes those teaching classes on the material), most working professional mathematicians could not produce to you a satisfactory proof of just about most of the most famous or best known ideas in mathematics without considerable effort on their part. That doesn't say anything about the foundations of mathematics nor does it say anything about them, but very few of us are von Neumans or Hilberts or Feynmans.