On some class of random variables leading to violations of the Bell inequality

Volume 129, number 5,6

PHYSICS LETTERS A

30 May 1988

ON SOME CLASS OF RANDOM VARIABLES LEADING TO VIOLATIONS OF THE BELL INEQUALITY Marek CZACHOR ul. Czerwonych KosynierOw 69/1, 81-332 Gdynia, Poland Received 13 November 1987; revised manuscript received 26 January 1988; accepted for publication 25 March 1988 Communicated by J.P. Vigier

A class of probabilistic problems must be calculated by means of “type-B random variables” (BRV). The Bell inequality can be violated in theories based on local BRV. An essential feature of such theories is a symmetry breaking of a probability density of hidden variables.

1. Introduction In this Letter I would like to show in what way the method one collects data can influence interpretations of violations of the Bell inequalities (BI). The approach presented clarifies, in my opinion, problems raised in several previous papers [1—31,and an explicit example of a local, deterministic, hiddenvariables model which violates B! indicates what “dangers” can appear when one neglects undetected signals, and stresses the role of symmetry conditions imposed on probability densities of the hidden vanables and observable averages. I show that this role can be more important than it is usually thought, especially if one admits theories involving “type-B random variables” (see the end ofthe next section). But before we pass on to the details let us first concentrate on the following simple probabilistic problem.

2. A Princess and Knights-Errant Once upon a time there lived a Princess who was imprisoned in a Tower. There were four windows in the Tower and each of them was facing a different quarter of the globe. Through the windows the Princess could see: a road (N), a river (W), a sea (E) and very steep rocks (S). The Princess was everyday awaiting impatiently a knight who might release her,

so each day she was looking many times through the windows. Most often she was looking through the N window the W and E ones were equally interestingly for her, but she was almost not interested in the Southern one. One day two Knights-Errant came. Both left the road and decided to observe the Tower for a week before they would make any attempt to set the Princess free. One of them was watching it from the North-East side and the other from the North-West one. Each time any of the two noticed her opening a window he put a “plus” sign if it was in the righthand side of the Tower, and a “minus” sign in the opposite case. Any one who knew the Princess personally knew that she was looking at the road 15 times more often than at the rocks, and 1.5 times more often than at the river or the sea, so he would be able to draw the probability distribution given in fig. 1. This distribution describes an idealised case of the Princess always looking through some window, but if we changed the normalisation it could be understood as a part of some more realistic density involving the situation “the Princess is not looking through any window”. Let us now consider two cases. (A) The Knights are well informed. It means they know the Princess is always looking through some window and that she changes the window every 28 minutes. Moreover each of them knows there are not

0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

291

Volume 129, number 5,6

PHYSICS LETTERS A

30 May 1988

Table 2 Knight

Window

15

10

—~-—ó

NE NW

~

1 1

2 “36

3 4 ‘~36 ‘~6 ‘~

Fig. 1. Probability density of the hidden variables of the Tower.

only the windows they can see but also two invisible ones. So if, for example, the Princess was absent in the NE side for 56 minutes then the NE Knight knew with certainty that she was looking two times through the Western or Southern windows. After a lapse of a week the Knights calculated averages. The results could be collected in table 1 where the Knights put 0 if the Princess was looking through the invisible windows. These averages could also be calculated in a hidden-variables manner, i.e. by means of the probability distribution given in fig. 1. For example =$p(2)NE(A)dA, whereA=[0,4/36} and

100

0.2 —0.2

<

>

=

0, if A~[2/36, 4/36],

(B) The Knights have no information except that they can see. This situation is more realistic and, in Table I Window N

E

150 150

100

p(A) dA

w ere NE: [0, 2/36)—*{—l, l} NE(A)= 1, if AE [1/36, 2/36),

1,

if Ae[2/36,3/36),

The above conditional expectation values yield the correct averages because the integration is performed over the whole domains of the random van-

if Ac [0, 1 / 36).

1,

3/36)

h

_1~ if Ae[l/36,2/36).

1, if Ac [1/36, 2/36),

292

100

r

NW(A)=

NE (A) =

NE NW

150 150

Princess is sometimes looking through the visible windows, and even does not have to know about the existence of the invisible ones. After a lapse of a week the results were as shown in table 2. As we can see these averages differ from those of the previous point. In order to obtain them in the hidden-variables manner we must calculate as follows, p(A)T’~E(A)dA = . [0.2/36) (A) ~ [0. 2/36) P NW — /11/36, 3/36)p(A)NW(A) dA

NE: A—~{—l,0, l},

Knight

value

=—l, if Ae[0, 1/36), NW: [1/36, 3/36)—~{—1,l},

A

—

E

fact, is met in the majority of actually performed experiments. Now each Knight knows only that the

5

=

Average

N

Average value W

W or S

E or

110 100

110

s 5/36 5/36

ables (in general conditional expectation values are not the observable averages of observables). It is important to note that these random variables are not mapping the whole space of the hidden variables into { — 1, 1 } but only its subspaces. Note also that now the Knights cannot attach the value 0 to the invisible windows simply because they do not know how many times she looks through them. Contrary to the previous point the random variables are not three-valued ({ 1, 0, 1 }) but . two-valued ({ 1, 1 }). —

—

Amoral following from this story can be summarized in two points:

Volume 129, number 5,6

PHYSICS LETTERS A

(a) The way one collects data influences measured average values of random variables (because then the random variables are different). This result cannot be a surprise since it is well known (a conclusion from problem the Bertrand paradox) that the in each probabilistic one must determine way the data are to be collected. Otherwise the problem can be ambiguous. (b) There exist two kinds ofrandom variables. The ones described in point A let us call “type-A random variables” (ARV) and those of point B “type-B random variables” (BRV). Of course BRV are more general than ARV. From the point of view of this Letter it is important to note that in the usual theorethical approach to B! only ARV were taken into account while during experiments average values of BRV are actually measured.

30 May 1988

fv~pB(A’)B~(A’)dA’ fv~pB(2’)dA’


8>= fu,,xvpp(A,A’)ula(A)Rp(A’) 2,2’) cU. dA’ dAdA’ Let now A, A’ ~‘, J~x\’flP( B, B’ a ~ and let us consider the following expression,

> + — J~><,,,p(A, A’ ) A(A) B(A’) dAdA’

C= +
J~~~p(A, A’) cIA dA’

+ fvxv’ p(A, A’) dAdA’

+

—

~

,O(A A’) dAdA’

.IuxvP(A,A’)A’(A) B’(A’) dAdA’ Iu~vp(A,A’)dAdA’

3. A general formulation of the problem In this section I will show the way BRV can influence solutions of the question of BI violation. Let us take two probabilistic spaces (A, U, ~z)and (Li, V, i-’), and let us denote by d and ~ sets of random variables acting on A and ~, respectively. Let us then consider Borel coverings A = U Ua and ~ = U V~ jkiB

of the two spaces, and denote by and ~ sets of BRV mapping the elements Ua and V~into { — 1, 1), i.e. .‘

~={Aacd;

—

Only if there exist mappings Ui-+U, U’ i-~U, V ~, V’ i—p ~ that map simultaneously the domains U, U’, V and V’ into some U and V, such that each ~

—

—

J

expectation value can be written as ~(A, A’) A(A) 11(2’) cIA dA’ Ox 9

with the same range of integration and the same form of ~, then the Bell inequality I Cl ~ 2 holds. Otherwise the sole estimation of the range of C is the “tnangle inequality” I Cl ~ 4. As long as we consider BRV then additional symmetry conditions must be imposed on p in order to get B!.

VacIA,A,~:Ua—{—1, l}},

i={Bpe~ V/kIB, Bfl: V 8-+{— 1, l}}. If we take probability densities PA:

A—*{0}u R+

PB:

i1—+~÷ U {0},

4. Example Now I will provide an explicit example of local BRV violating B!. Let ‘A= [0, 2] =‘B~ Ua=Va= [a, a+2] and Aa(A)Ba(A)

p:Ax~—$~~u {0},

1, if Ac[a,a+l), =—l, if Aa[a+l,a+2]

we can introduce expectation values of BRV in the following OfAa and way Be!),(integration over the whole domains

4 for all AcA A=Q=[0, Pa(A)=PB(A’)=l/ and A’eL~,a4]. nd Let letp(A, A’) be defined by means of the distribution of probability density presented in

= Iu~JU,,PA(A)dA P~(2)Aa(2) cIA

fig. 2. Theassumed squares forwhichp(2, A’ )0 are, for convenience, to be closed sets. 293

Volume 129, number 5,6

PHYSICS LETTERS A

o 4 0

~

~

._.~i.. 1

0

_._~

0

1 “8

1

1

0

f~3

0

~8

“8

0

~

0

0

4

A

Fig. 2. Probability densityp(A, A’).

The respective averages are = =0 for any adA and IIEIB, and =(24a4fl+7afl)/2, for (a,/3)a [0, 1] =1

—

afl/2

x [0,

1]

,

for (a fl) (1 2] < (1 2]

<

[0, 1]

= ( —6 + 4/3+8a — 5a/J) /2,

for (a fl)a [0 1] >< (1 2]

lated (ref. [6], eq. (3.21)) but was not considered as fundamental. The above example indicates that this fact cannot be neglected as long as one admits “type-B random variables”. Finally, one important feature of models based on BRV must be stressed. Note that if a signal (for instance the photon) is in a state characterized by a hidden variable A which lies outside a domain ofsome random variable (corresponding to some setting of an experimental device) then it will never be detected. No matter how “ideal” the detector is. Hence the very idea of BRV incorporates undetected signals. There are local hidden-variables models yielding the same predictions for cascades as quantum . mechanics [3 4], but their essential feature is the assumption that the detectors are not ideal. In the model proposed by Ferrero and Santos [4], forexample, some photons after having passed a polariser are more likely to be detected than if there was no polariser on their route. Such a construction would not be possible if one considered ideal detectors, so it is commonly believed that local realism can in

=(—6+4a+8/3—5a/3)/2, for (a,/i)a (1,2]

.

Three points must be stressed now: (a) This set of random variables is analogous to the one used for a derivation of B! (experimentally there is no difference at the level of individual events), (b) These BRV are local in Bell’s sense (if the pair is in a state (2,2’) a Ax~then the result B(A’) does not depend on what Aa we choose). (c) This system of local BRV is deterministic in the following sense: the possible outcomes of the BRV are uniquely described by a point (2,2’) in the Ax~ plane. Set now A=A0, A’ ~A2, B=B0 and B’ =B2. We get

principle be tested by means of more efficient equipment. The example of the Princess shows that if we admit local theories based on BRV then this belief is groundless. Moreover, practically each expenimentally obtained average is one of some BRV. Aspect et al. [5] were aware of this fact while performing the most precise of their experiments but assumed that “the ensemble of photons actually detected is representative of the ensemble of all photons emitted”. Considerations of section 3 prove that as long as there are some “undetected signals” admitted (and experimentally there is no way of excluding them) then additional assumptions are needed. It seems that the role of the symmetry conditions should be reexamined.

C=4.

References

The secret of this “paradoxical” result lies in the symmetry braking of p (fig. 2). As has been stated in section 3 we cannot perform a suitable change of the range of integration and this is the reason why this system of BRV does not have to satisfy B!. Moreover, this symmetry braking results in a non-symmetric form of since (A0B0> ~ . The requirement of symmetry was sometimes formu-

[1] M. Kupczyñski, Phys. Lett. A 121 (1987) 205.

294

30 May 1988

[2] W. De Baere, Lett. Nuovo Cimento 40 (1984) 488. [3] T.W. Marshall, E. Santos and F. Selleri, Phys. Lett. A 98 (1983) 5. and E. Santos, Phys. Lett. A 116 (1986)356. [4] M. Ferrero [5] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49 (1982) 91. [6] J.F. Clauser and A. Shimony, Rep. Prog. Phys. 41 (1978) 1881.