Tez
Graduate Poster
- Joined
- Nov 29, 2001
- Messages
- 1,104
slimshady2357 said:
Hey Tez, can you elaborate a little on this for me, or point me to a good link?
One thing I'm wondering is why it works on subsequent trials, after the first set of measurments, the entanglement is still there?
It's not effected by the measuring in some way?
How would you feel about giving a long winded explanation of exactly how the solution works?
Seriously, I'm interested.
Adam
Yeah Adam - the entanglement is destroyed by the measurement.
Heres the solution I posted back then - its exact nature was somewhat affected by the questions and quibbles that people had gone through, but the essence is there. Apologies if its not long winded enough
THE SOLUTION:
Ok heres the solution to the "million dollar challenge" logic-type problem, and I hope it amazes you as much as does me. It turns out that Jandi's reasoning about why they can win in only 75% of cases is correct as it stands, but is fundamentally flawed in that it entails a hidden assumption. The hidden incorrect assumption is that the optimal strategy can be found as a set of "locally constructible answers" for each of the psychics. By "locally constructible answers" I mean a solution strategy which says something like: "If A is asked the X question she should give such and such a answer computed as a function of all the information she has (such as previous questions, previous answers, the other psychics strategies, perhaps some random choices), but if she is asked the Y question she should give this answer...." etc. All such attempts will fail.
Another common but incorrect assumption is this. It seems, to anyone who has sweated over the problem hard enough, that SOME form of information transfer must take place between the psychics if they really are to succeed with 100% probability, since it seems impossible to win 100% of the time without at least one psychic knowing the question that one of the others has been asked. However the problem clearly states (I hope) that no communication (other than telepathic, and we all know what we think of that!) can take place, and so the solution must only take the form of a set of "local physical actions" that each psychic takes, based on whether they have been asked the X or Y question, and these actions can in no way depend on the question another psychic has been asked.
With this in mind, it is important to note that what is being tested are CORRELATIONS between the psychics answers, and not their absolute answers in any sense. For example, the equation XXX=-1 has several solutions, to satisy it you need only a certain correlation between the answers and not any one particular solution. Of course they are being asked for correlated answers based on which one of the four possible sets of correlated questions has been asked, and this is of course where it gets tricky.
All that said, the solution to the problem could be simply stated to a physicist as follows:
1. Before proposing the challenge, the psychics prepare 1000 triplets of particles (say atoms) in the GHZ (or spin-triplet state). Each psychic takes one member of each triplet. (If they're worried about being repeatedly tested by a dastardly and unbelieving Jandi, they can take as many as 10^22 particles - which of course may weigh them down a gram or two, but will allow them to keep passing the test until the end of the universe.)
2. When asked the X question they take a magnet and measure the spin of their particle in the X direction, giving the answer +1 to their captor if the spin is up and -1 if it is down. If they are asked the Y question they measure the spin of their particle in the Y direction, giving the answer +1 to their captor if the spin is up and -1 if it is down.
If they carry out this procedure they are certain to win every time. This is known in Quantum Shmantum circles as the "GHZ nonlocality paradox". It was discovered in 1988 and is much easier to understand than Bell's theorem, the first and most famous such paradox, because it exhibits what might be called "a much stronger nonlocal correlation". The term nonlocal is used beacause it doesn't matter where in the universe the other psychic is, they could be light years away (or next door), the trick will work.
For those with absolutely no physics training let me try and clarify a few things. The spin of the particle is just a physical property of a physical system, no more mysterious than its mass, position, momentum, electric charge, polarization etc. Of course if those words also mean nothing to you then you could still have problems. For this puzzle it doesnt really matter what physical property you choose, as long as it can take at least two values. Spin is physical property which has a value that changes depending on which direction you look at it from. This is not in itself mysterious - imagine a pencil pointing in some direcion, and the physical property you are measuring is whether it points "up" or "down". If you stand on your head, this value will have changed! In the particular setup I described, the magnets being oriented in the X and Y directions just amount to looking at, or measuring, the spin from two different orientations. From each such orientation the spin may appear "up" or "down".
What would actually be observed in the 1000 experiments is the following. Lets focus on the roughly 250 times that all three measure in the X direction. Then each psychic will individually see the spin to be "up" roughly half of the time and "down" the other times. The magic occurs in the correlations. For example when psychic A measures "up" and announces +1 and psychic B measures "down" and announces "-1" then psychic C will be guaranteed to have measured "up" and announce "+1" so that the product of their answers is -1. Just as likely however, is for A to measure "down", B to measure "down" and C to measure "down", which still gives a product -1. In fact all the possible combinations leading to a product of -1 are equally likely when they all measure in the X direction. Similarly when one psychic measures in the X direction and the other two in the Y direction, then all possible combinations leading to a product of +1 are equally likely.
All this kind of upsets physicists, who are used to physical properties of systems behaving in the same inherently logical fashion which may have led you to believe that there really was no way for the psychics to pass this test without being psychic. The trouble you have solving the puzzle is exactly the trouble we have. We want to believe that the physical properties of systems are inherent to, and somehow carried by, those systems, but the GHZ paradox shows that this cannot be so.
Of course I think there are interesting implications for JREF. Most double-blind testing involves looking for correlations between sets of random data. As this puzzle shows there are correlations which may appear impossible, but which are not in fact "unnatural". We dont yet know the full extent of "quantum correlation space", and so someone could presumably get JREF to agree to at least a "preliminary test" which in fact demonstrated not psychic ability but physics ability!
If youre interested in seeing the detailed calculations the original paper is D. Greenberger, M. Horne and A. Zeilinger, in Kafatos, M. "Bell Theorem, Quantum Theory and Conceptions of the Universe", Kluwer, Dordrecht, p.69 (1989), although it wont make much sense unless you have a fairly advanced understanding of quantum mechanics.
The experiment is published in Physical Review Letters, vol. 82, pages 1345-1349 (1999).
So there it is. Of course its really unfair to call this a logic puzzle (I called it a logic-type puzzle!), since the answer is to a certain extent illogical. However I spend many hours of my days (daze?) wondering to what extent we could have dreamt up quantum mechanics if we didn't have had the experimental evidence to follow - this has implications for those of us looking for deeper theories and for deeper understanding of quantum mechanics, since at present there are no unexplained experiments to guide us.
And to anyone who (quite rightly) complains that the answer is not reasonably solved by a non-physicist, I reply that puzzles about baseball trivia are far more obscure to me! I guess I just want people to be aware that quantum mechanics can be interesting even without a training in physics, and that the conceptual problems of the theory go far beyond the "particles are sometimes waves" drivel you read in a lot of "for the layman" accounts.
ADDED STUFF:
Q: Why can the psychics not measure the spins before they are asked the questions?
A: The reason you cant measure the spins beforehand is because of something else that pisses us off about quantum mechanics - when you make a measurement of one quantity, there are often other physical quantities whose results get randomized by your measurement. We say these quantities "don't commute". For example if you do measurement of spin in the X direction followed by a measurement of spin in the Y direction you do not necessarily observe the same thing as if you measure in the Y direction followed by a measurement in the X direction. Thus the psychics cannot pre-measure here - they need to know whether they should look in the X or in the Y direction - if they do premeasure and it happens that they premeasured in the Y direction but were actually asked the X question, they'll have buggered things up and the magic correlations will no longer be present...
Q: Can I see the calculations?
ok basically the spin triplet state (or GHZ state) is
|GHZ>=(|+z>|+z>|+z> - |-z>|-z>|-z>)/sqrt(2)
where these are the spins in the Z basis (spin along the Z direction). So |+z> means spin "up" in the z direction, |-z> means "down". Its an entangled state of three particles.
Lets imagine psychic A is asked the X question, the other two the Y question. To see what possible outcomes they will get, we rewrite this state using the X basis for the first particle, defined by:
|+x>=(|+z> + |-z>)/sqrt(2), |-x>=(|+z> - |-z>)/sqrt(2),
and the Y basis for the other two particles, it is defined by:
|+y>= (|+z> + i*|-z>)/sqrt(2), |-y>=(|+z> - i*|-z>)/sqrt(2),
then you get that the state |GHZ> is also equivalent to:
|GHZ>=(|+x>|+y>|+y> + |+x>|-y>|-y> + |-x>|+y>|-y> + |-x>|-y>|+y>)/2
Thus after the measurements the state will collapse to one of the 4 possibilities in the above superposition with equal likelihood, and as you can see if they answer "+1" or "-1" based on their outcome the product of their answers will always be "+1".
Similarly you can rewrite the state using the X basis for all three psychics, and you get this:
|GHZ>=(|+x>|+x>|-x> + |+x>|-x>|+x> + |-x>|+x>|+x> + |-x>|-x>|-x>)/2,
and thus the product of their answers will be "-1"...
There is no bridge. Reality is qualia. Mathematics simply can be used to accurately describe the systematic patterns in our qualia. The notion of the ontologically self-subsistent unperceived is metaphysical garbage.