Stronger-than-quantum correlations

G. Krenn
Atominstitut der Österreichischen Universitäten
Schüttelstraß e 115
A-1020 Vienna, Austria
krenn@ati.ac.at
and
K. Svozil
Institut für Theoretische Physik
Technische Universität Wien
Wiedner Hauptstraß e 8-10/136
A-1040 Vienna, Austria
svozil@tph.tuwien.ac.at

Abstract

1  Introduction

Consider Bell s inequality in the form of the Clauser-Horne-Shimony-Holt (CHSH) inequality [2,3]

-2 £ E(a,b)+E(a¢,b)+E(a,b¢) -E(a¢,b¢) £ 2     .

(1)

E is the quantum mechanical correlation function for two particle correlations which will be explained later in detail. For the moment it is only necessary to know that E may have values in the range of -1 to +1. In general the function E could be such that the four terms in (1) can take on values completely independent of each other. In such a case the maximum violation of the inequality is 4 and occurs for

E(a,b) = E(a¢,b) = E(a,b¢) = -E(a¢,b¢) = 1.

Now it is a well known fact that in quantum mechanics the maximum violation of Bell s inequality is 2Ö2 [4,5]. This implies that E is restricted to such functions which prevent a stronger violation than 2Ö2. Because the maximum possible violation 4 is not realized in quantum mechanics several questions arise. Is the limit of 2Ö2 forced by probability theory or by physics? Is a violation larger than 2Ö2 consistent with the foundations of quantum mechanics, e.g. the randomness of elementary processes? Would a stronger violation of Bell s inequality destroy the peaceful coexistence of quantum mechanics and relativity theory and enable faster-than-light communication? Related questions have been raised before by several authors [6,7,8,9,10,11].

Although we are not able to answer these questions in general, we will discuss a system in which stronger-than-quantum correlations would lead to inconsistencies with fundamental laws of physics. For this purpose we start with a detailed discussion of classical, quantum mechanical and stronger-than-quantum correlations.

2  Derivation of Bell s inequality

The assumption of local hidden variables implies the existence of a hidden classical arena. The reader may think of a mechanism determining the results of all measurements observer A (B) may perform for each individual pair of correlated particles. In the following we consider the measurements R_a, R_a¢ of observer A and R_b, R_b¢ of observer B. With the assumption of local hidden variables the results of both measurements, R_a, R_a¢ and R_b, R_b¢, respectively, are defined simultaneously for each individual pair of correlated particles. Consider a series of N such particle pairs. For each pair the values of r_a , r_a¢ (r_b , r_b¢) are determined. Writing down these values for all N particle pairs, we get four lists as shown in Fig.1. For our considerations arbitrary lists of results can be chosen. We will demonstrate that any results which can be listed in such a way have to fulfill a simple condition which is equivalent to Bell s inequality. This condition imposes a restriction on the correlation of the results and therefore on the correlation function E.

To find out the restriction for the correlation function E(a,b), we determine the number of different signs (results) in the four pairs of lists (a¢,b), (a,b), (a,b¢) and (a¢,b¢). As expressed by Eq.(2) for N particle pairs the correlation function E(a,b) is given by the number of cases n(a, b) in which different results are obtained in the measurements of R_a and R_b. Having determined the four values n(a¢,b), n(a,b), n(a,b¢) and n(a¢,b¢) (cf. Fig. 1), we make a simple observation [12].

A limit on the number n(a¢,b¢) (``outer path'' in Fig. 1) and thus on the correlation function E(a¢,b¢) (cf.Eq.(2)) is imposed by the values of n(a¢,b), n(a,b) and n(a,b¢). Along the ``inner path'' a¢®b®a®b¢ from list a¢ to list b¢ in Fig. 1 we have to change n(a¢,b) signs in the first step to get list b, n(a,b) signs in the second step to get list a, and n(a,b¢) signs in the last step to obtain list b¢. At the end of this procedure the number of different signs in lists a¢ and b¢ n(a¢,b¢) can be no greater than n(a¢,b)+n(a,b)+n(a,b¢)¹. This can be expressed by the inequality

n(a¢,b)+n(a,b)+n(a,b¢) ³ n(a¢,b¢)    .

(3)

The probability P ^¹ (a, b) for different signs (results) in measurements of R_a and R_b on N particle pairs can be approximated by the relative frequency n(a,b)/N. Analogously, the probability for equal signs P ⁼ (a, b) is approximately given by 1 - n(a,b)/N. By definition (2), the correlation function can be written as

E(a,b) = P = (a, b)-P ¹ (a,b) = 2P = (a,b)-1    .

(4)

Using these identities, Eq. (3) can easily be rewritten into the CHSH inequality [2] form

E(a,b)+E(a¢,b)+E(a,b¢) -E(a¢,b¢) £ 2    .

(5)

The bound from below

E(a,b)+E(a¢,b)+E(a,b¢) -E(a¢,b¢) ³ -2

(6)

can be derived by a similar argument, considering the number of equal signs (results) u(a,b) = N-n(a,b) instead of the number of different signs (results). u(a,b) satisfies the same inequality (3) as n(a,b). Bell's inequality in the form of Eq. (1) is given by the combination of (5) and (6).

We have seen that the value of the correlation function E(a¢,b¢) is related to the values of E(a,b), E(a¢,b) and E(a,b¢). Only results which can be represented as shown in Fig. 1 and thus are defined simultaneously and locally for all four possible experiments R_a,R_a¢ and R_b,R_b¢ (as by local realistic models) fulfill this relation and therefore also Bell's inequality.

Now let us consider a system whose correlations are such that the maximum number of sign changes along the ``inner path'' (n(a¢,b)+n(a,b)+n(a,b¢)) is smaller than the number of sign changes along the ``outer path'' (n(a¢,b¢)). Then not a single set of lists (a, b, a¢, b¢) exists, which satisfies all the correlations as defined by n(a¢,b), n(a,b), n(a,b¢) and n(a¢,b¢) (E(a¢,b), E(a,b), E(a,b¢) and E(a¢,b¢)) simultaneously. A certain fraction of the results in lists a¢ and b¢ would always be inconsistent with the values of the correlation functions. For the maximum violation of Bell s inequality permitted by quantum mechanics - 2Ö2 - this fraction is (Ö2-1)100 » 40%. For stronger-than-quantum correlations this fraction reaches 100% in the limit of a violation of Bell s inequality with the maximum value 4 (cf. section 4).

3  Classical and quantum mechanical correlations

Bell s inequality is a condition which must be fulfilled by local realistic, i.e. classical correlation functions. Quantum mechanical correlation functions violate Bell s inequality by a maximum value of 2Ö2:

| Eqm(a¢,b)+Eqm(a,b)+Eqm(a,b¢) -Eqm(a¢,b¢) | £ 2Ö2     .

In the following we will give an example for a classical as well as a quantum mechanical correlation function.

First of all we consider pairs of correlated classical particles with total angular momentum zero. [(j₁)\vec] and [(j₂)\vec] are the classical angular momenta of particle 1 and 2, respectively. Then, by measuring the angular momentum of particle 1 (2) along a direction [(a)\vec] ([(b)\vec]) defined by the angle a (b) within the plane perpendicular to the momentum of the particles the classical observable R_a = sgn ([(a)\vec] ·[(j₁)\vec]) (R_b = sgn ([(b)\vec] ·[(j₂)\vec])) can be defined. It can be shown [1] (see also [13,14]) that for such observables the classical correlation function is given by

Ec(a,b) = Ec(q) = 2 q p -1     ,

(7)

where q is the relative angle |a- b|. By comparing this function with Eq. (4) we find that

P = (q) =

q p

.

(8) This corresponds to the expectation that the probability for equal results in measurements of R_a and R_b (P ⁼ (q)) is proportional to the relative angle q. By inserting (7) into (5) one can easily see that Bell s inequality is not violated, which also implies that condition (3) is fulfilled.

To derive a quantum mechanical correlation function we now consider two particles of spin j in a singlet state. Then the correlation function is given by (cf. appendix and ref. [15])

C(q) = - j(j+1) 3  cosq    .

(9)

Again q is the relative angle |a- b| of two angles within the plane perpendicular to the momentum of the particles. To be comparable to the classical correlation function, the quantum correlation function must be normalized such that E_qm(p) = -E_qm(0) = 1 (E_qm(q) = 3/[j(j+1)] C(q)). Thus for two correlated spin-[1/2] particles in a singlet state the quantum mechanical correlation function is given by

Eqm(a,b) = - ® a   · ® b   = Eqm(q) = -cosq    ,

(10)

where the vectors [(a)\vec] and [(b)\vec] are defined by the angles a and b within the plane perpendicular to the momentum of the particles.

E_c(q) and E_qm(q) are drawn in Fig.2. One can see that for almost all angles q, the quantum mechanical correlations are stronger than the classical ones. Therefore E_qm violates Bell s inequality but the violation does not exceed 2Ö2 as one can proof by inserting (10) into (5). Results described by a quantum mechanical correlation function E_qm can in general not be represented consistently by local realistic models. As demonstrated for the angles a, a¢ and b, b¢ the results of the measurements R_a¢ and R_b¢ can not be defined in such a way as to correspond to E_qm(a¢,b¢) as well as to E_qm(a,b), E_qm(a¢,b) and E_qm(a,b¢).

4  Stronger-than-quantum correlations

We now turn our attention to -merely hypothetical- ``extremely nonclassical correlations'' and assume a stronger-than-quantum correlation function of the form

Es(a,b) = Es(q) = sgn (2q/ p-1) = sgn (Ec(q))    ,

(11)

where E_c(q) is the classical correlation function (7). E_s(q), along with E_c(q) and E_qm(q), is drawn in Fig. 2. One can clearly see that E_s(q) takes the tendency of the quantum correlation function to exceed classical correlations to an extreme. This is also expressed by the fact that, since for x = 2q/ p-1 and 0 £ q £ p

sgn (x) = ì ï í ï î -1    for  x < 0 0    for  x = 0 +1    for  0 < x = 4 p ¥ å n = 0  sin[(2n+1)x] (2n+1) = 4 p ¥ å n = 0  (-1)n cos[(2n+1)(x-p/2)] (2n+1)     , (12)

the quantum mechanical correlation function can be attributed to the first summation term in Eq. (12). By considering also terms of higher order in expansion (12) we get correlations which are stronger than the quantum correlations. Then Bell s inequality is violated by a larger value than 2Ö2.

The extreme correlation expressed by E_s(q) implies that for angles a, b with p/2 £ |a- b| £ p, the results of observers A and B are perfectly correlated (E_s(q) = 1 ,  r_a,i r_b,i = + + or - - ,  i = 1 ... N), whereas they are perfectly anticorrelated ( E_s(q) = -1 ,  r_a,i r_b,i = + - or - + ,  i = 1 ... N) for angles a, b with 0 £ | a- b | £ p/2. This cannot be accommodated by any classical theory under the assumption of local realism, nor can we think of any quantum correlation satisfying it.

The hypothetical correlation function E_s(q) gives rise to a maximum violation of Bell's inequality, since for the four angles a = p, a¢ = 6 p/ 8, b = p/ 8 and b¢ = 3 p/ 8

Es(a, b)+ Es(a¢, b) +Es(a,b¢)-Es(a¢,b¢) = 4.

A violation of Bell's inequality by the maximum value of 4 has also been studied by Popescu and Rohrlich [8] and, for a classical system, by Aerts [16]. As already mentioned in the introduction it has been shown that the maximum violation of Bell's inequality permitted by quantum mechanics is 2Ö2 [4,5].

For the angles a = p, a¢ = 6 p/ 8, b = p/ 8 and b¢ = 3 p/ 8 we now try to write down results which are correlated as defined by E_s(a,b) in the same way as shown in Fig.1 . Because n(a¢,b) = n(a,b) = n(a,b¢) = 0 (E_s(a, b) = E_s(a¢, b) = E_s(a,b¢) = 1) the results in lists a¢ and b¢ have to be identical. This demand is satisfied by the list b_in ¢ in Fig.3. At the same time these results have to be sign-reversed because n(a¢,b¢) = N (E_s(a¢,b¢) = -1), which is expressed by the list b_out ¢.

In contrast to the classical case (Fig.1) it s now no longer possible to find four lists of results which satisfy the correlations as described by E_s(a, b) (11). Therefore two different lists b¢ (b_in ¢, b_out ¢) are shown in Fig. 3. Of course the fraction of different results in these two lists may vary depending on the function E. A comparison of the correlation functions discussed in this paper (E_c, E_qm and E_s) is given in table 1. For E_s the fraction of different results in lists b_in ¢ and b_out ¢ is 100 % (cf. Fig.3). For classical correlation functions (E_c) this fraction is 0 % (b_in ¢ = b_out ¢ = b¢) and for quantum mechanical correlation functions (E_qm) it is smaller than (Ö2-1) 100 » 41.42 %. Whereas E_qm contradicts local-realistic models only on a statistical level, E_s leads to a complete contradiction. This means that out of all N particle pairs there is not a single one to which a consistent quadruple of outcomes (r_a,r_a¢, r_b and r_b¢) can be assigned. Consequently a violation of Bell s inequality by the maximum value of 4 would be a two-particle analogue to the GHZ argument [17].