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PHYSICS LETTERS A

The Bell and GHZ theorems: a possible three-photon interference experiment and the question of nonlocality D.N. Klyshko Physics Department, MoscowState University, Moscow 119899, Russian Federation

Received 10 November 1992; accepted for publication 24 November 1992 Communicatedby V.M. Agranovich

A concrete optical experiment similar to the one proposed by Greenberger, Home and Zeilinger (GHZ) and the possibilityof obtaining experimentaldata, directly (without averaging) contradicting the Bell hidden variables theory (HVT) are discussed. The influence of accidental coincidencesis considered. The used Heisenbergrepresentation is manifestlylocal.

1. Introduction

Recently Greenberger, H o m e and Zeilinger (GhZ) [ 1,2 ] have considered three- and four-particle versions of the EPR experiments and found a contradiction of a new type between quantum theory ( Q T ) and HVT. This contradiction does not involve inequalities and in an ideal case does not need any averaging o f experimental of experimental data. This "Bell theorem without inequalities" at once received much attention [ 3 - 1 0 ] . In particular, the connection with the Kochen-Specker paradox [ 11-13 ] was discussed [4]. Zukowski [10] and Hardy [14] considered two-particle versions. In this communication a more concrete form of the optical experiment, proposed in ref. [2], is presented together with the operational procedure for measuring the Bell and G H Z observables S and Z (see section 2). In section 3 the "experimental" values Sc~p and Zexp are compared with the predictions of HVT and an elementary iteration procedure for deducing Bell inequalities [ 15 ] for N spin-1/2 particles is presented. In section. In section 4 the experiment is described in the Heisenberg representation, which makes evident that QT is local in the same degree as the Bell HVT [ 15 ] and which facilitates comparison with the HVT and classical stochastic theory of the optical field, as well as with Mermin's system [3-5 ] of three spin-I/2 particles. ElsevierSciencePublishers B.V.

In section 5 the influence of inevitable accidental coincidences is analysed. In section 6 the use of the term nonlocal in the present context is discussed.

2. E x p e r i m e n t a l procedure

Consider the experiment, depicted in fig. 1, which is a slight concretisation of the scheme, proposed in ref. [2 ] (it is a three-photon version of the experiment, proposed in ref. [ 16 ] ). We take into account only the triple events, when all three radiated photons are registered by three photon counters (in practice this is achieved by us-

Fig. 1. A three-photon six-mode interferometer for demonstration of the Bell and GHZ paradoxes. P, pump, 1 and 2, nonlinear samples, splitting the pump photon into three photons (OJo--,wa+oJ~+oJc),each belongingto two modes (k= 1, 2), the circles are the phase delays. 399

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PHYSICS LETTERS A

ing coincidence schemes). We ascribe the value + 1 or - 1 to the observable A~, when the upper (lower) detector in the channel a "clicks" ( a is the adjustable phase). Analogously we measure two other observables Bp and C r. At random moments in time t~ the detectors will click giving three random dichotomic sequences A,,, Bp~ and Cyi, where i is the number of the event. We construct the fourth observable X~,i=A,~B~,C~, which reflects the correlation between three channels and depends only on the summary phase ~0= a + fl+ 7. It is a nonlocal observable, as the three observers should somehow communicate to compare their readings and synchronize their clocks. We believe that our hypothetical experimental setup behaves in accordance with QT, so the Xo/always equal + 1 and the X.~ equal - 1, i.e. there is full correlation. At intermediate phases the X~ take the values +_ 1 randomly, but after averaging over a large number o f events we notice that 1 u

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we could have obtained eleven first observables in (3) equal t o - 1 a n d C ~ = + l .

3. The hidden variables description

According to HVT we can replace the observables A, and A~ by the deterministic functions A (2~) and A'(2~), where 2,-=2(t~) is a set of variables, which fully predetermines the instantaneous state of the whole experimental set-up, in particular, the time t~ and the values which A, A', ... would take at that moment. So we have two four-time functions S ( { L } ) = ½[A'(2, )B(2~ )C(2~ ) +... ] ,

Z( {2~})=A'(2, )A(2z)A(23)A'(24) ....

Now we average S and Z in time, take the 2~ to be ergodic stochastic variables an go over to ensemble averaging with weight p (4): (S>HvT =

( Xq, )exp =~r i~=,X~,i= c o s ( q 0 .

(1)

j d p(2)S(2),

(z)r~vT = J d2p(2)Z(2) This is an example of three-photon intensity interference [ 17,18 ]. Let us register four events ( i = 1, 2, 3, 4) with the following four phase sets,

(otfl~,) = (xyy), (yxy), (yyx), ( x x x ) ,

(2)

where x = 0 , y = n/2, so that (,01=~2=~O3=x+2y= and ~04= 3 x = 0. According to QT we should obtain

XI =A'IBICI = - I , X2=A2B'2C2=- l , X3=A3B3C'3

= -

1,

X 4 ~-A4B4C

4 ~ - -k-

1,

(3)

where A~=Ay~, A~ =Ax~ and so on. By adding eqs. (3) we get [ 9 ] the "Bell observable"

=½(A'~B,C, + . . . ) = - 2 . eqs.

(3)

Zexp -- X l X2 X 3 X 4 =A'~ B,

(4) we get

[4]

the

"GHZ

= + 1,

(7) (8)

where S(2) - ½[A'(2)B(2)C(2) +...] = + 1, z(2) =

[A(2)A'(2)B(2)B'(2)C(2)C'(2)

(9) ]2= + 1. (10)

To get eq. (10) from (6) the event numbers i should be ignored; but it is quite improbable that the four sets would coincide, so eq. (10) or (8) cannot have any relation have any relation to a real experiment. Equation (9) is easily confirmed (see also refs. [ 5,9 ] ). Consider the observable

S2(2)-½[A(B+B')+A'(B-B')]=+_I,

(ll)

we interchange the primed and unprimed quantities,

S ' 2 ( 2 ) = ½ [ A ' ( B ' + B ) + A ( B ' - B ) ] = +_I .

(ll')

Define next

C, . . . . .

1.

(5)

These are the "experimental" results; for example, 400

[ (S)HvT[ ~< 1 ,

indeed, if B ' = B , then S:=AB and i f B ' = - B , then S2=A'B'; in both cases $2= + 1. The same is true if

Se~ -- ½(X, +X~ +X~ --X~)

Multiplying observable"

(6)

S3(2) - S ( 2 ) -- ½[S2(C+C') +S'~(C-C') ] =½(A'BC+ . . . . A'B'C')= +_ 1 .

(12)

Volume 172, number 6 4. Q u a n t u m

PHYSICS LETTERSA

If we make the following designations,

description

The six-mode output field of two parametric threephoton down-converters, i.e. the input field of the three-channel interferometer, is in the following state [2,181,

Iql) = ~

1

18 January 1993

trx=m+m + , ay=i(m-m+),

trz=nl-n2,

a 2 = (nl +n2) (nl + n2 + 2 ) =3•, then the observable A~ takes the familiar form A~ =aa~ c o s ( a ) +tray s i n ( a ) = a a - n , .

(alblCi +a2b2c2) + 1 0 )

1

= ~7~ ( l a t b l C l ) + [a2b2c2 ) ) 1 - 7~ (11)+12)),

(13)

where ak, bk, Ck ( k = 1, 2) are the photon annihilation operators and, at the same time, the symbols of one-photon states of the corresponding modes. In the Heisenberg representation the 50% beamsplitter (with a phase difference a between the modes al and a2 preliminarily introduced) makes the following SU (2) transformation,

(17)

(18)

Thus our model is isomorphic with the system of three spin -1 particles an three Stern-Gerlach apparatus with orientations a, fl, y. We note by the way, that the well known conformity between photons and spins can be traced to the following property of the (classical or quantum) two-mode field: the Nth moments of the mode amplitudes have the transformation properties of a spinor with j = N / 2 [ 17]. The correlation operator takes the form

Xq, =A,~BpCy = mambmc ei~'+ h.c. ,

( 19 )

where terms like m+a mbmc are omitted as they give zero, acting on the vector ( 13 ). We find from ( 18 ) and (19) 1

1

a+ =~7~ ( +-al e ia/E + a2 e - i a / 2 ) •

(14)

Aa I~) = ~r~ (eiala2bl Cl ) + e - i a l a l b2c2 ) ) , (A,~A,~. )kl q/)

We are interested in the local operator

1

Aa -- na+ - n a _ =maeia-l- m+e -ia ,

(15)

where n = a +a and the operator m~ = a ~-at transfers the photon from mode a~ into mode a2. From these definitions we find [m + , ma] =naj --na 2 , [A,~, A,~, ] = - 2i sin ( a -

= (fT~ (e ik(a'--°t) I 1 ) + e ikta-a')] 2 ) ) ,

X,I~')= ~

1

(e-i~ll)+ei*12)) ,

( X~,X~,. )kl lg> 1 (eik(~'-~)ll)+eik(~--¢)12 >)

a ' ) (naj - n,,~).

(20)

(16)

When a photon is registered at the output mode a+, then (na+) =1 and ( n a ~ ) =0, so (A,~)=_+ 1. Analogously we define the observables Bp and C r We see that the observables ( = operators) here have the same local properties as in HVT: A,~ does not depend on fl, y and so on. The initial entangled statevector (13 ) defines the statistical properties of the source, where the photons are still "together". The time delays in the channels are lost here, as we consider only single frequency modes (the multimode theory gives the same results concerning the question of locality [ 19 ] ).

and analogously for B B and C r Thus in our subspace 2 A,~ = B p2 = C 2r =X~2 =Xo = I ,

(A,~A~,+_,~/E)2 . . . . . (X~,Xq,+_,~/2)2=X,~= - I .

(21)

Now we see why Xoi and X~ (which commute) did not fluctuate in our "experiment": I ¥ ) is their eigenvector. The negative values of some squared operators in (21 ) reflect the following property of Pauli matrices: (axay)2= (i0"z)2= - L Thus the G H Z paradox has a formal origin in the non-Abelian character of the SU (2) group. Consider again conditions (2). From (20) follows 401

Volume 172, number 6

PHYSICS LETTERS A (22)

From (27) we find

H e r e A ' = A o = a ~ , A = A ~ = a a y a n d so on. Using (21) we can rewrite this as

(X)exp = V c o s ( ~ ) ,

Z~ = X~Xo = X] Xo=A'BC. . . . .

I.

Z,~ = B B ' B B ' = I + B [ B ' , B I B ' .

(23)

If we put the c o m m u t a t o r [B', B] =2iaoz equal to zero, then we get the H V T result (8) (for any x and y = x + x / 2 ) . Once more we see that what matters is the incompatibility o f the local primed and unprimed observables. For the Bell observable we also obtain the definite eigenvalue:

SQ-r=½(A'BC+...)=½(3X,~-Xo)= - 2 I .

(24)

In case o f arbitrary x, y we obtain from (20) ( Sx,y ) = ½( 3Xx+ 2y -- X3x )

= ½[3 c o s ( x + 2y) - cos(3x) ] .

(25)

5. Influence of accidental coincidences In actual experiment the coincidence rates in eight detector triplets R + + +, R _ + +, ... are measured. Let all detectors have equal efficiencies, then from symmetry we have only two parameters, R+-R+++

= R + _ _ =... ,

R_=R ....

R_++ . . . . .

NOW the normalized correlation takes the form R+ - R +R_ "

(X)exp-- R+

(26)

There are always some "accidental" coincidences R .... which do not depend on the phases, so R+ = R~¢¢ + R ' + , where R ' are the true coincidences. As a result R'+ - R ' _ (X)exp = R'+ + R ' _ + 2 R ~ ¢ "

(27)

According to (20) ( X ) Q T =COS(~) = R~_ - R ' _ R'+ +R'__ ' ( AX2>QT =sin2(q~) .

402

(28)

18 January 1993

(29)

where V= ( I + 2 R a ~ / R ' ) -~, R v= R +v + R ' _. In an ideal case, when the photons are radiated by nonoverlapping triplets (this needs weak enough pumping) with the rate Ro, then Ra¢c/R'= (ROT) 2, where T is the coincidence window (which is taken to be much larger than the coherence time of the radiation %oh). Thus the visibility approaches unity if ROT<< 1. (Note that in the two-photon case R~¢¢/R' =ROT.) Allowing for V~ 1 we should rewrite (22) as I (S)expl = 2 V .

(30)

Thus there is a violation of the Bell inequality only when V> ½ (cf. the two-photon limit V> 0.71 [ 16 ] ). The nonunity visibility V means nonperfect correlation, i.e. Xoi and X~i would fluctuate with the means + V. The "right" answers would be observed only in some trials (consisting o f four events, see (2) ), the relation o f right to wrong trials being equal to V/( 1 - V). There seem to be logical difficulties in formulating the G H Z paradox in such unstable conditions. If we replace eqs. (3) b y ( X o ) = - V and ( X ~ ) = V , then the product ( X o ) 3 ( X ~ ) ~ - - V 4 would not produce the contradiction with HVT, as it needs nonstatistical assertions. The situation is complicated by the fact that classical stochastic models of the multiphoton interference are known [ 1 7 - 2 0 ] , which differ from the quantum ones in some case only by lesser visibility (and in other cases by the form o f the interference structure [20] ).

6. Discussion The main problem in the realization of an optical G H Z experiment is the lack of three-photon sources at present. In this connection the projects using the tow-photon sources [ 10,14 ] may be o f interest. What would be the implications if a G H Z experiment would be successfully accomplished? In the case 1 - V<< l there would be for the first time the nearly "single trial" demonstration of a Bell inequality. As in the "old" two-particle cases the premises would be "the experiment contradicts the local

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PHYSICS LETTERS A

classical theories a n d agrees with the local q u a n t u m theory". I suppose that the conclusion should be "local q u a n t u m theory is right in this case a n d local classical theories are wrong", i.e. in P o p p e r ' s words, the latter are falsified. The choice between two options, " n o n l o c a l classical" a n d "local q u a n t u m " , b o t h quite alien to comm o n physical sense, is a m a t t e r o f taste, b u t it seems that the t e r m " n o n l o c a l " is m i s u s e d nowadays. The frequently used formulations, such as " e x p e r i m e n t proves nonlocality", "nonlocal effects", "algebraic p r o o f o f n o n l o c a l i t y ' , " q u a n t u m nonlocality", ..., could be rather confusing for the uninitiated. I believe that the Q T o f optical E P R experiments is no m o r e nonlocal than the classical stochastic theory is. This is evident i f one uses the Heisenberg representation (see section 4), in which the initial entangled state-vector ( 1 3 ) describes the correlation properties o f the localized source. These correlations are t r a n s p o r t e d by the (essentially classical) propagators to the three measuring a p p a r a t u s A, B, C, see fig. I. Classical stochastic fields can possess analogous correlations, i.e. they are also " i n s e p a r a b l e " . Belinsky recently considered [ 21 ] simple classical analogs o f two or three-photon E P R experiments, using two p a r a m e t r i c generators a n d two o r three observables A ~ , = s i g n ( l j , + - I ~ , _ ) where I is the intensity. The m a x i m a l correlation V is 1 or 0.5 correspondingly ( b u t the interference p a t t e r n differs from c o s ( g ) , cf. the classical two-spin model, considered b y Bell [ 2,15 ] ).

18 January 1993

Acknowledgement I a m grateful to Dr. A.V. Belinsky for n u m e r o u s stimulating discussions.

References [ I ] D.M. Greenberger, M.A. Home and A. Zeilinger, in: Bell's theorem, quantum theory and conceptions of the universe, ed. M. Katafos (Kluwer, Dordrecht, 1989) p. 69. [2]D.M. Greenberger, M.A. Home, A. Shimony and A. Zeilinger, Am. J. Phys. 58 (1990) 113 I. [3] N.D. Mermin, Am. J. Phys. 58 (1990) 731. [4] N.D. Mermin, Phys. Rev. Lett. 65 (1990) 3373. [5] N.D. Mermin, Phys. Rev. Left. 65 (1990) 1838. [6 ] H.S. Choi, Phys. Lett. A 153 (1991 ) 285. [7] B. Yurke and D. Stoler, Phys. Rev. A 46 (1992) 2229. [8] M.D. Reid and W.J. Munro, Phys. Rev. Lett. 69 (1992) 997. [9] L. Hardy, Phys. Lett. A 160 ( 1991 ) 1. [ 10] M. Zukowski, Phys. Lett. A 157 ( 1991 ) 198, 203. [ 11 ] S. Kochen and E. Soecker, J. Math. Mech. 17 (1967) 59. [ 12] C. Pagonis, M.L.G. Redhead and R.K. Clifton, Phys. Lett. A 155 (1991) 441. [ 13 ] A. Peres, Phys. Len. A 151 (1990) 107. [ 14] L. Hardy, Phys. Lett. A 167 (1992) 17. [15] J.S. Bell, Physics 1 (1964) 195. [16] D.N. Klyshko, Phys. Lett. A 132 (1988) 299. [ 17 ] D.N. Klyshko, Phys. Lett. A 163 (1992) 349. [ 18 ] A.V. Belinsky and D.N. Klyshko, Laser Phys. 2 (1992) 112. [ 19] D.N. Klyshko, Laser Phys. 2 (1992) 997. [20] A.V. Belinsky and D.N. Klyshko, Phys. Letl. A 166 (1992) 303. [ 21 ] A. V. Belinsky, private communication; A.V. Belinsky and D.N. Klyshko, Usp. Fiz. Nauk, to be published.

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