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Bell's Theorem
Exercise 6.2 Show that if a state e is prepared in such a way that Aw = a
and Bo = B%, then f(A, B) = f(a, 8)e. Note that this result is valid even if
A and B do not commute, but merely happen to have a common eigenvector ».
Three particle model
A similar contradiction was derived by Mermin' for a three particle system, without the use of any debatable extension of the EPR criteria. (Mermin's argument is a simplified version of another one, with four particles, due to Greenberger, Horne, and Zeilinger.®)
The three spin ½ particles are prepared in an entangled state
4 = f(r1, F2, [3) (u, U2 U3 - VI V2 V3),
(6.3)
where the coordinate space wave function f(ri, r2, rs) has a form ensuring that the particles are widely separated, and where the spin states u and v are eigenstates of 0z, satisfying
0:٧ = U
o,u = iv
0, V = -20
0,u = u,
02V = -v.
(6.4)
It is easily seen that ‹ is an eigenfunction of several operators:
O1= O2y O3y1 = 01y 0220301= Oly 0yy 03=1=1,
(6.5)
and
(6.6)
Exercise 6.3 Verify the above equations and show that these four operators commute. Moreover show that
(012 02y 030) (01, 027 030) (010 02003x) = - (012022 03x).
(6.7)
The minus sign in (6.7) is crucial, as will soon be seen.
The EPR argument now runs as follows. We may measure, on each particle, either o, of Oy, without disturbing the other particles. The results of these measurements will be called me or my, respectively. From (6.6), we can predict with certainty that, if the three o, are measured, the results satisty
(6.8)
TN. D. Mermin, Physies Today 43 (June 1900) 9; Am. J. Phys. 58 (1990) 731.
*. M. Greenberger, M. Horne, and A. Zeilinget, in Bell& Theorem, Quantum Theory, and Conceptions of the Universe, ed. by M. Kafaice, Kluwer, Dordrecht (1989) p. 69.
The dilemma of Einstein, Podolsky, and Rosen
153
Therefore each one of the operators Ole, Ozz, and o3x corresponds to an EPR element of reality, because its value can be predicted with certainty by performing measurements on the two other, distant particles.
However, it also follows from (6.5) that we can predict with certainty the value of is by measuring Ozy and osy rather than ozz and O3z. We have
M1z Mzy M3y = 1,
(6.9)
and likewise, by cyclic permutation,
miy M2z M3y = 1,
(6.10)
and
Miy M2y M3x = 1.
(6.11)
The product of the last four equations immediately gives a contradiction.
There is a tacit assumption in the above argument, that mir in Eq. (6.8) is the same as mis in Eg. (6.9), in spite of the fact that these two ways of obtaining mix involve mutually exclusive experiments measuring ozz and oar, or measuring ow and osy. This tacit assumption is of counterfactual nature, and cannot be experimentally verified. It obviously adheres to the spirit of the EPR article it is almost forced upon us by the intuitive meaning of the word "reality"— but it is open to the same criticism that Bohr expressed in his response to Einstein, Podolsky, and Rosen.
Einstein locality and other relativistic considerations
The paradoxes or algebraic contradictions resulting from the apparently reasonable criteria proposed by EPR, prompt us to reexamine more carefully their arcument: If... we con predict with certaty... Who are "we"?
In Bohm's singlet model, the observer who measures Siz and finds +h/2 knows that if the other observer measures (or has measured, or will measure) Sz, she must find the opposite result, -h/2. However, this knowledge is uncless (it is devoid of operational meaning) because the two observers are far apart. The only thing that the first observer can do is to send a message to the second one, telling her that she can verify that Sz is -h/2, provided that she has not yet disturbed her particle by measuring another component of spin, before she received that message.
Now assume that, unbeknownst to the first observer, she measures Say and finds the result +h/2, say. Can there be any paradox here? Conceptual difficul ties may indeed arise if you demand that every physical system, such as our pair of particles, has, at every instant, a well defined quantum state (some authors
*When two observers are involved, I shall call the second one "she" rather than "he" (see footnote on page 12).
