At 3:06 PM 10/19/4, Keith Nagel wrote: >I may be being boneheaded here, help me out. I thought that I showed >by adding extra dimensions it was possible to do exactly what you >describe above, changing the outcome probabilities for the three visible >axis of measurement. If I didn't, show me where I blew it. Perhaps >I'm not understanding all the constraints on the results required by >experiment? Here's the 4D table again. I'll add the constraint ( if >I understand your argument ) that we only choose the visible axis >columns to calculate our final probability. > >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 >BE 8/16 >BF 16/16 >BG 8/16 >CE 8/16 >CF 8/16 >CG 16/16 > >96/144 = .666... > >full table >160/256 = .625
OK, just to check that I understand what you are saying I'll attempt to rephrase it. You are saying that nature uses the above table but only columns A, B, and C are applied to Alice's sensors and columns E, F and G are applied to Bob's sensors. We thus can take the above table and covert it to the following form: A B C E F G 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 All I did to obtain this table was to cut and paste the initial 4 dimensional table and delete columns D and H. It seems this is what you inrend bcasue you get the tabulation: AE 16/16 AF 8/16 AG 8/16 BE 8/16 BF 16/16 BG 8/16 CE 8/16 CF 8/16 CG 16/16 Do I have this all correct? If so, the following is my response. We could also, for convenience and consistency with prior 3 dimensional tables rename E, F, and G to D, E and F. This gives: A B C E F G 0 0 0 1 1 1 0 0 0 1 1 1 * 0 0 1 1 1 0 0 0 1 1 1 0 * 0 1 0 1 0 1 0 1 0 1 0 1 * 0 1 1 1 0 0 0 1 1 1 0 0 * 1 0 0 0 1 1 1 0 0 0 1 1 * 1 0 1 0 1 0 1 0 1 0 1 0 * 1 1 0 0 0 1 1 1 0 0 0 1 * 1 1 1 0 0 0 1 1 1 0 0 0 * This is just the original 3 dimensional table, Table 1, but with some rows duplicated. For convenience I have flagged the rows which are handily each duplicates of the row preceeding them. My point was that duplicating entries merely has the effect of weighting those entries. To see a similar table with weights consider 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 We can eliminate the *'ed rows by assigning weights g=h=i=j=k=m=n=p=2 in Table 5. Since, in the above table all the weights are exactly equal to 2, we can normalize them to 1, i.e. g=h=i=j=k=m=n=p=1, so we are then right back to: 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 The same process applies no matter how many dimensions you use. You always end up with Table 2. That is because under the experiment protocol, Alice and Bob only have three possible ways to observe. You can get much fancier, say by setting weights g and p to zero, and plaing around with the weights by any imaginary way. However, as I showed, there is no way to get better than 5/9 overall matches without knowlege of which axes Bob and Alice have chosen at least some of the time. Regards, Horace Heffner
