Hi Horace. Your posts are like a fine French cheese, they need to age a bit before reaching the peak of flavour... Are we there yet?
Anyway, as you seem to be taking some liberties with your models, let me try this notion on you. Let's question the assumption of 3 quantum variables. String theory suggests that more than our 3 visible dimensions exist; the number varying depending on the theory, time of day, etc. Consider for a moment the case of four dimensions... A B C D E F G H 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 AE 16/16 AF 8/16 AG 8/16 AH 8/16 BE 8/16 BF 16/16 BG 8/16 BH 8/16 CE 8/16 CF 8/16 CG 16/16 CH 8/16 DE 8/16 DF 8/16 DG 8/16 DH 16/16 4 matches over 16 combinations 4*16 + 12*8 160/256 = .625 Getting closer to our magic .5, but not there yet... Let's try 6 dimensions. -snip huge table- 16 full matches over 256 combinations 16*256 + 240*128 4096 + 30720 34816/65536 = .53125 Closer and closer... Now 8 dimensions 256 full matches over 65536 combinations -snip ungodly huge table- .502 The more dimensions we add, the closer we get. What do you think? If you think this is cheating ( and maybe it is ) where is the assumption of only 3 quantum variables coming from ( other than our much beleagured common sense ). K. -----Original Message----- From: Horace Heffner [mailto:[EMAIL PROTECTED] Sent: Sunday, October 17, 2004 2:08 PM To: [EMAIL PROTECTED] Subject: Re: EPR and Bell Revisited (DRAFT #6) EPR and Bell Revisited (DRAFT #6) Assume, as did Einstein, Podolski, and Rosen (EPR), the state of conjugate entangled particles is set at the time of the creation of the conjugates, at the moment of entanglement. EPR maintained that entangled particles in effect carry hidden variables, or an equivalent of a computer program, that determines how they will act when observed. Assume there are, as in Alain Aspect's experiment designed to examine this assumption, three independent quantum values involved. That is to say there are three axes of spin observation, in which a particle is in either a clockwise or counterclockwise spin state upon observation. Unfortunately, spin can only be observed in one axis, not all three at the same time. However, Bell figured out how to see if the quantum variables were set before measurement, i.e. how to see if a hidden variable was involved. The situation is shown in Table 1, below. i A B C D E F 1 0 0 0 1 1 1 Key: 2 0 0 1 1 1 0 3 0 1 0 1 0 1 i - possible combination (row) number 4 0 1 1 1 0 0 A, B, C - Alice's possible observations 5 1 0 0 0 1 1 D, E, F - Bob's corresponding observations 6 1 0 1 0 1 0 7 1 1 0 0 0 1 8 1 1 1 0 0 0 Table 1 - Possible observations by Alice and Bob Table 1 assumes that when an entangled particle pair is created that all three quantum variables, i.e. spins, are set at that time and carried as "hidden variables". Columns A, B and C are possible spins observed by a sender Alice in orthogonal axes A, B and C, and are denoted "o" for clockwise spin and "1" for counterclockwise spin. Columns D, E, and F are the corresponding spins observed by receiver Bob in the axes A, B and C. It is assumed there is no error in the detection of the spins or the transmission of the hidden variables. As the variables are independent, and it is well known from observation of single particles that the spin probability of clockwise spin being observed in any axis is 0.5, we see that there are exactly 8 equally probable combinations, possibilities denoted 1 - 8 in column i. Bell suggested that sender Alice and receiver Bob, for each particle pair, select a column at random and observe the spin. That's all there is to the experiment. To see the expected results, look at Table 2. a b matches - - ------- A D 8/8 A E 4/8 A F 4/8 B D 4/8 B E 8/8 B F 4/8 C D 4/8 C E 4/8 C F 8/8 Table 2 - Expected results In Table 2 column a indicates the axis Alice chooses to observe. Column b indicates the axis Bob chooses to observe at the same time. We can determine the probability of a match by comparing the two columns of equally probable outcomes shown in table in Table 1. By "match" here we mean the observation of opposed, i.e. conjugate, spins. For example, the first row of Table 2 has the entries, A, D, and 8/8. This means that when Alice chooses axis A, and Bob coincidentally also chooses axis A, i.e column D, then both will always observe complimentary spins. We get 8 out of 8 matches. This is the principle of, the definition of in this case, entanglement. When we look at row 2 of Table 2, we have the entries A, E, 4/8. This is because there are only 4 possible ways out of 8 outcomes, each equally probable, that a match occurs. Summing up the entries in Table 2, we see that there are 9*8 = 72 possible outcomes to the observation of a single entangled pair, and there 3*(8+4+4) = 48 possible matches. There is thus a 2/3 probability of a match for a given particle pair. That is all there is to it! If there are hidden variables, then there will be a 2/3 probability of a match. The Aspect experiment actually yields a 1/2 probability of a match. It was deduced from this there is no hidden variable involved. This is quite amazing. If the (thought) experiment data for 7200 trials is tabulated in the format of Table 2, we might expect it to look something like the idealization shown in Table 3. a b matches - - ------- A D 800/800 A E 200/800 A F 200/800 Total matches 3600 B D 200/800 Total trials 7200 B E 800/800 Match probability 0.5 B F 200/800 C D 200/800 C E 200/800 C F 800/800 Table 3 - Idealized experimental results The amazing thing that has happened is a reduction of the probability of a match when Alice and Bob have chosen differing axes to observe. We know they get a match 100 percent of the time when choosing the same axes, i.e in the combinations A D, B E, and C F. The matches in the differing axes have in effect been "discorrelated", to coin a term, reduced in matching by 50 percent from what they should be if programmed by hidden variables. The computer programs inside the particle pairs appear to have no means to accomplish this discorrelation without knowing what choice of axis was made by both Alice and Bob. Since Alice and Bob can chose the axis to observe the last moment, it appears the computer programs would have to communicate faster than light to do their work. However, perhaps Table 1 can be modified to restrict possible combinations. After all, the spin of one particle can not be measured in all three axes at once. It may be that spin direction in orthogonal axes are not independent variables. Suppose, for example, that spin in all three axes can not be the same at one time. We then have Table 4. i A B C D E F 2 0 0 1 1 1 0 3 0 1 0 1 0 1 i - possible combination (row) number 4 0 1 1 1 0 0 A, B, C - Alice's possible observations 5 1 0 0 0 1 1 D, E, F - Bob's corresponding observations 6 1 0 1 0 1 0 7 1 1 0 0 0 1 Table 4 - Hidden variable table prior to weighting considerations Tabulation of Table 4 still shows Bell's inequality to be in effect. The expected percent of total matches exceeds 50 percent. However, suppose we now simply decide to weight each row's probability. This results in Table 5. w A B C D E F g 0 0 0 1 1 1 Key: h 0 0 1 1 1 0 i 0 1 0 1 0 1 w - weight for given row j 0 1 1 1 0 0 A, B, C - Alice's possible observations k 1 0 0 0 1 1 D, E, F - Bob's corresponding observations m 1 0 1 0 1 0 n 1 1 0 0 0 1 Let T = (g+h+i+j+k+m+n+p) p 1 1 1 0 0 0 Table 5 - Prospective hidden variable table for observations by Alice and Bob Table 6, below, is a tabulation of the entries in Table 5. a b matches - - ------- A D T/T A E (g+h+n+p)/T A F (g+i+m+p)/T B D (g+h+n+p))/T Total possibilities = 9*T B E T/T Total matches = 3*T + 6*g + 6*p + 2*(h+i+j+k+m+n) B F (g+j+k+p)/T Match probability desired .5 C D (g+i+m+p)/T C E (g+j+k+p)/T C F T/T Table 6 - Expected results based on Table 5 Here again the term "match" refers to spin orientations being conjugate, i.e. 1 and 0 or 0 and 1 in the columns chosen in Table 5. We can immediately see the justification for throwing out rows 1 and 8, as their weights g and p have factors of 9. We thus set g=p=0, which throws out rows 1 and 8. Further, even with g=p=0, to obtain the desired probabilities of 1/4 for the case where differing axes are chosen, we obtain from Figure 6 the following equations: (h + n)/T = 1/4 (i + m)/T = 1/4 (j + k)/T = 1/4 which says: 4h + 4n = T 4i + 4m = T 4j + 4k = T and adding all the above: 4(h+i+j+k+m+n) = 3T 4T = 3T T=0 so the weights all disappear in a flash! This in itself is sufficient to prove the impossibility of hidden variables. Bell's inequality, however, rests on the overall observed hits. We have from Table 5 that T = h+i+j+k+m+n and from Table 6 that total possibilities = 9*T. Total matches from Table 6 is 3*T + 2*T = 5*T. We want (matches)/(possibilities) = 0.5. However, the ratio (matches)/(possibilities) = (5*T)/(9*T) = 5/9, no matter what we chose for the weights. If g and/or p are non zero the ratio is worse than 5/9. Bell's inequality thus holds no matter how the 8 possible spin combinations are weighted. Regards, Horace Heffner
