EPR Phenomenon

By T.Sakaguchi

From the time when Einstein, Podolsky and Rosen[1] argued the incompleteness of quantum mechanics by showing so-called EPR thought experiment, there has been heated discussions on the nonlocal nature of quantum mechanics[2]. Among them, Bell[3] proposed an inequality which has to be satisfied if the nature has a locality, and many experimental tests have subsequently been performed[4-9].

According to the experiments, the inequality is really violated, and so we have to conclude that the nature possesses a nonlocality. But how can this be possible by local interaction? One of the answers could be that is quantum mechanics''. But if one takes Everett's formulation of quantum mechanics, it can be understood from a more fundamental level. This is shown below in detail. I will not introduce any new interpretation to explain the phenomenon. The only assumption the formulation is based upon is that the physical law which governs microscopic phenomena is also applicable to macroscopic human beings. Those who are not familiar with Everett's formulation of quantum mechanics should read his original paper[10].

We use the same notation as the previous paper for describing observer state: An observer state in which an observed value X^i, one of the eigenvalues of an observable X, is recorded, is written as |[X^i]>. When an object system is in a superposition |psi> = sum_i C_i |X^i>, the measurement process can be described as follows:

U(t)|psi>|[]> = sum_i C_i |X^i>|[X^i]>,                       (1)

where U(t) is a time evolution operator obtained from a Hamiltonian which includes the interaction between the object and observer systems. It means that when the observer observes the object system in the state |psi>, the observer state |[]> branches into a number of different observer states |[X^i]>, each of which describes independent observer. This is a direct consequence of (A) the linearity of the time evolution operator U(t) and of (B) the relation which should be satisfied if the object system is prepared in an eigenstate |X^i> and the observer system is prepared to observe the observable X:

U(t)|X^i>|[]> = |X^i>|[X^i]>.                                 (2)

When the observer system comes into interaction with the object system for a second time, we can apply the time evolution operator to the branched states again and we get a superposition of states |X^i>|[X^i X^i]>, where the same value with the one observed first is recorded in each observer's memory. Therefore, it will appear to each observer, who is described by each observer state in the superposition, that the object system in the state |psi> has collapsed into one of the eigenstates of the observable X. The collapse-of-wavefunction phenomenon of the object system described above is nothing but the branching process of the observer state, which can never happen in classical physics. It plays an essential role in the EPR phenomenon. That is, the nonlocal nature comes from the fact that a state of a pair of observers branches into a number of states of mutually correlated pairs by the local interaction between the object and observer systems:

We consider a two particle system in a state:

sum_{ij} C_{ij} |X^i>_1 |X^j>_2.                              (3)

The two particles can be separated at any distance from each other. When the observer "1" observes the observable X of the particle "1" prepared in the state |X^i>_1, the measurement process will be described as:

U_1 |X^i>_1 |>_2 |[]>_1 |>_2 = |X^i>_1 |[X^i]>_1 |>_2 |[]>_2. (4)

It should be noted that the measurement process is local and it does not affect the state of the system "2". When the observer "2" observes an observable Y, which does not commute with X, of the particle "2" prepared in the state |X^j>_2, the measurement process will be described as:

U_2 |>_1 |X^j>_2 |[]>_1 |[]>_2
= |>_1 |[]>_1 sum_j' _2<Y^j'|X^j>_2 |Y^j'>_2 |[Y^j']>_2.    (5)

That is, the state of the system "1" remains unaffected, while the state of the observer "2" branches into the states |[Y^j']>_2. The state vector |Psi> of the whole system after the two observations performed therefore becomes

|Psi> = U_1 U_2 sum_{ij} C_{ij} |X^i>_1 |X^j>_2 |[]>_1 |[]>_2
= sum_{ij'} |X^i>_1|[X^i]>_1 sum_j C_{ij}
_2<Y^j'|X^j>_2 |Y^j'>_2 |[Y^j']>_2
= sum_{ij'} K_{ij'} |X^i Y^j'> |[(X^i Y^j')]>           (6)

where

def
K_{ij'} = sum_j C_{ij} _2<Y^j'|X^j>_2,                        (7)

def
|X^i Y^j'> = |X^i>_1 |Y^j'>_2                                 (8)

and
def
|[(X^i Y^j')]> = |[X^i]>_1 |[Y^j']>_2.                        (9)

Preparing N identical pairs of particles in the same state described in Eq.(3) and performing the above observation sequentially, we will get a superposition of branched states of the form

K_{ii'} K_{jj'}...K_{kk'} |X^i Y^i'>|X^j Y^j'>...|X^k Y^k'>
\otimes |[(X^i Y^i')(X^j Y^j')...(X^k Y^k')]>              (10)

where

def
|[(X^i Y^i')(X^j Y^j')...(X^k Y^k')]> = |[X^i X^j...X^k]>|[Y^i' Y^j'...Y^k']>
(11)

In the limit N -> infinity, we can show that (X^p Y^p') in [...] appears |K_{pp'}|^2 N times in almost all branched states in the superposition. Therefore, each pair of observers in the superposition will conclude that the pair (X^p Y^p') can be obtained with probability

P(X^p,Y^p') = |K_{pp'}|^2
= sum_{qr} C^*_{pq} C_{pr} <X^q|Y^p'><Y^p'|X^r>. (12)

If the state Eq.(3) is the spin singlet state of spin-1/2 particles;

(C_{ij}) = 1/sqrt{2} [ 0, 1]
[-1, 0]                                 (13)

and

(<S^i_z|S^j_z'> = [    cos(theta/2),  -i sin(theta/2)]
[- i sin(theta/2),     cos(theta/2)]

where theta is the angle between the z-axis and z'-axis, the probability becomes

(P(S^i_z, S^j_z')) = 1/2 [sin^2(theta/2), cos^2(theta/2)]
[cos^2(theta/2), sin^2(theta/2)]    (14)

The branching process caused by the local interaction between the object and observer systems is local and does not affect the observation performed by the other observer at a remote site as shown in Eqs.(4) and (5). This is why superluminal communication is impossible. The probability derived above, which is shown to violate Bell's inequality, can be recognized only when the two observers meet each other to compare the data in their world.

References:
[1]  A.Einstein, B.Podolsky and N.Rosen: Phys.Rev. 47 (1935) 777.
[2]  For a review, see Quantum Theory and Measurement, Edited by
J.A.Wheeler and W.H.Zurek, Princeton University Press (1983).
[3]  J.S.Bell: Physics 1 (1965) 195.
[4]  J.F.Clauser and A.Shimony: Rep.Prog.Phys. 41 (1971) 1881.
[5]  S.J.Freedman and J.F.Clauser: Phys.Rev.Lett. 28 (1972) 938.
[6]  J.F.Clauser: Phys.Rev.Lett. 36 (1976) 1223.
[7]  E.S.Fry and R.C.Thompson: Phys.Rev.Lett. 37 (1976) 465.
[8]  A.Aspect, P.Grangier and G.Roger: Phys.Rev.Lett. 47 (1981) 460; 49 (1982) 91.
[9]  A.Aspect, J.Dalibard and G.Roger: Phys.Rev.Lett. 49 (1982) 1804.
[10] H.Everett III: Rev.Mod.Phys. 29 (1957) 454.