EPR and Bell Revisited (DRAFT #2) 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. Suppose 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 2 row throw out Tabulation of Table 4 still shows Bell's inequality to be in effect. However. suppose at the formation of an entangled pair two conjugate rows of Table 4 are randomly and arbitrarily thrown out by the process that sets the hidden variables. Since Table 4 is symmetrical, we can arbitrarily throw out conjugate rows 2 and 7 to obtain Table 5. Conjugate row pairs are defined as (2,7), (3,6) and (4,5). i A B C D E F 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 Table 5 - Hidden variable table for observations by Alice and Bob Tabulation of Table 5 creates Table 6. a b matches - - ------- A D 4/4 A E 0/4 A F 1/4 B D 0/4 36 possibilites B E 4/4 18 matches B F 2/4 match probability 0.5 C D 1/4 C E 2/4 C F 4/4 Table 6 - Expected results based on Table 5 Here again the term "match" refers to spin orientations being cojugate, i.e. 1 and 0 or 0 and 1 in the columns chosen in Table 5. We now see that the imaginary experimental results shown in Table 3 can actually be obtained experimentally. Entries A D, B E, and C F in Table 6 remain at 4/4 no matter which rows are thrown out. The other rows of Table 6 will average 1/4 if the (random conjugate row pair throw-out) selection process is repeated enough. Bell's inequality no longer applies due to an unexpected limitation in the way spin might be carried by hidden variables in a given particle, and due to the rules by which the hidden variables are chosen at the time of entanglement. The spins in each of the three axes may not be independent after all, and may in fact be carried by hidden variables, at least with regard to this experiment. Bell's assumptions are destroyed if the spin values of a particle are not necessarily independent variables. There is no way to directly determine dependence of the hidden variables because spin can only be measured in one axis at a time. Futher, there could be a nearly infinite number of such variables, a set of three for each possible orientation of the orthogonal axes. This hypothesis sounds outlandish, but it certainly is no more outlandish than the all possible paths hypothesis in QED, and other quantum mysteries. Regards, Horace Heffner
