- Email: [email protected]
Constraints for the violation of the Bell inequality in Einstein-Podolsky-Rosen-Bohm experiments
10 April 1995 PHYSICS
ELSJZVIER
LETTERS
A
Physics Letters A 200 ( 1995 ) l-6
Constraints for the violation of the Bell inequality in Einstein-Podolsky-Rosen-Bohm experiments Emilio Santos Departarnento de Fisica Moderna. Universidad de Cantabriu, Santander, Spain Received 2 May 1994: revised manuscript
received 30 January 1995; accepted for publication Communicated by J.P. Vigier
31 January
1995
Abstract
A proof of Bell’s theorem is presented involving a pair of nonrelativistic spin-& particles in a singlet state. A detailed study of the spatial part of the wave function is made and several constraints are derived for the existence of reliable experiments.
It is well known that the proof of Bell’s theorem [ l] has two parts. The first one is the derivation of (Bell) inequalities valid for any local hidden-variables (LHV) theory. We shall not be concerned with this here. The second part consists of exhibiting a particular experiment where the quantum predictions violate a Bell inequality, thus showing the incompatibility between quantum mechanics and LHV theories. In the 30 years elapsed since Bell’s seminal paper [ 11, hundreds of gedanken experiments have been proposed showing the violation of a Bell inequality, but no incontrovertible (loophole-free) violation has been shown in a real experiment. (Compare, for instance, with the history of parity violation, where a crucial experiment [ 21 was performed within one year of the theoretical paper [ 3 1.) Proposals for loophole-free experiments are beginning to appear [4-6], but none of them is clearly feasible with present technology. Furthermore, the difficulties for performing empirical tests of the Bell inequalities cannot be just attributed to nonidealities of the available measuring devices. Indeed, LHV models have been found [7] which perfectly agree with the ideal quantum predictions for atomic cascade experiments, like Aspect’s one [ 81, which are currently con0375.9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10375-9601(95)00097-6
sidered the most reliable test of Bell’s inequalities. The sharp contrast between so many gedanken and so few (none in fact) real violations of Bell’s inequalities surely requires an explanation, and the obvious one is that gedanken experiments contain oversimplifications that make them unphysical. Bell’s theorem states that there is a contradiction between quantum theory and local hidden-‘variables theories. It is a mathematical statement which, as such, does not require any real experiment to be proved or disproved. Only if we want to investigate how nature behaves we require actual experiments. It is generally agreed that the contradiction may be proved by considering a gedanken experiment where the quantum predictions contradict a Bell inequality: I also agree, but claim that any experiment (gedanken or real) should involve three steps: the preparation of the system, the evolution, and the measurement. Almost eve#y gedanken experiment considered in the literature for the proof of Bell’s theorem considers just the last step. Indeed, usual proofs begin like: “Let us assume that we have two particles in an entangled spin state, placed at two macroscopically separated regions.. ,’ ‘. I strongly disagree with that. In my opinion the (gedanken) exper-
iment should include the process of preparation of the particle pair at the source and the free evolution towards the two macroscopically separated regions, both processes being treated according to quantum theory. This is the essential point that I want to stress in the present paper. With this purpose I shall analyze the requirements to make physical one of the most popular gedanken experiments: The version of Bohm [9] of the experiment considered by Einstein, Podolsky and Rosen [ lo]. The experiment of Einstein-Podolsky-RosenBohm (EPRB) , consisting of measuring the spin correlation of two spin-i particles prepared in a singlet state, was the example used by John Bell [ I] for the proof of his celebrated theorem. It has been considered in many articles and books, and it is still the standard example of “quantum nonlocality”. However, the experiment has never been discussed in enough detail. We will show that (1) the standard proofs of Bell’s theorem are incomplete because no care is taken of the space part of the wave function, (2) according to nonrelativistic quantum mechanics some spin singlet states are able to violate a Bell inequality, but not all such states do so, (3) EPRB experiments are extremely difficult due to several constraints that we shall derive. According to Bell [ 111, locality means the impossibility of causal (i.e. not superluminal) communication between spatially separated (in the sense of relativity theory) regions, where some measurements are performed. Bell’s theorem is striking because it seems to imply that, if quantum mechanics is correct, nonlocal correlations exist. Consequently, a rigorous proof of Bell’s theorem should necessarily involve considering experiments where measurements are made at several spatially separated regions (we shall consider only two regions throughout this article). Furthermore, the measurements should be made on subsystems (particles) resulting from the evolution of a system prepared in the common past (again, relativistic meaning) of both measurements. These conditions lead us to what we will call EPR [ lo] experiments, described in the following. We shall prepare some system of two particles in a region of space, S, that we shall call the source. At a later time, we shall measure some observable A of the first particle in region L (say on the left), and observable B on the second particle in region R (on the right). By repeating the preparations and the measurements
(in the same conditions) many times, we may obtain the joint probability distribution, P( A, B), of A and B. If our experiment consists of just this, local hiddenvariables models certainly exist able to reproduce that probability. As a matter of fact, the question of locality does not even arises because nonlocality is revealed only when a change of the observable measured in L influences the result of measuring observable B on region R. We must, therefore, perform new runs of the experiment with exactly the same preparation, but now measuring observable C (instead of A) on particle 1 in region L together with observable B on particle 2 in R. Then, in another run we measure C on 1 in L and D on 2 in R, and so on. In this way we may prove that no noncontextual (see Ref. [ 121 for a careful study of noncontextuality) hidden-variables model exists able to reproduce the prediction of quantum mechanics for P( A, B) and P( C, B) and P( A, D), etc. But, still, this would not prove Bell’s theorem if we do not state conditions about the causal connection between the measurement regions, that is without locality requirements. Locality puts stringent constraints on EPR experiments. In fact, the measurement on L (R) should be made during some finite time interval [tL, t,+ AfL] ( [ tR. f, + AfK] ), which defines a space-time “region of measurement” (L, tL, AtL) ((R, tR, AtR)). We require that the regions are spatially separated, and that the source, S, during the preparation [ ts, ts + Ats] lies completely within the common past light cones of both regions. As a last constraint, the experimenter is not allowed to modify the state of the state of the particles during the travel from S to L or from S to R, so that the evolution is “free”. (Any intervention of the experimenter should be considered as a part of either the preparation or the measurement.) Our first task will be to prove the following (Bell) theorem: Local hidden variables are incompatible with elementary quantum mechanics. By elementary we mean nonrelativistic quantum mechanics of systems with a finite number of degrees of freedom. Obviously, a proof not resting upon nonrelativistic approximations would be desirable because locality is an essentially relativistic concept. However, the relativistic treatment will not be considered in the present paper. A proof of Bell’s theorem should consist of finding a particular EPR experiment where the predictions of quantum mechanics violate a Bell inequality. The appropriate Bell inequality is [ 131
E. Santos /Physics
B=P,(a)
+P,(b) -P,*(a,
-P,?(C, b)
b) -P,,(a,
+p,,cc,d) 20.
4 (1)
Here (a, b, c, d} are four unit vectors corresponding to four possible spin projection observables. As explained above, we must prepare an ensemble of particle pairs and measure, on a subensemble, the probability P,,(a, b) that the first particle has positive spin projection along a and it is in region L, and the second along b in region R, then we should measure P,,(a, c) on another subensemble, etc. P, (P2) corresponds to a measurement on particle 1 (2) without measuring anything on 2 ( 1) . Inequality ( 1) is fulfilled by any local hiddenvariables model so that its violation in an EPRB experiment refutes the whole family of such theories. I shall point out that there are many different Bell inequalities, with quite diverse meaning. Some of the inequalities, like ( 1) , are derived only from Bell’s postulates, defining LHV theories and therefore they hold for all of them. An example is the first CH inequality (derived in Section III of Ref. [ 131). In other cases, however, the derivation involves the use of auxiliary assumptions and, consequently, they hold only for restricted families of LHV theories. An example is the second CH inequality (section V of Ref. [ 131) The latter inequalities have the advantage that they are more easily tested empirically than the former (and only they have been actually tested till now), but they do not have the fundamental relevance of the genuine Bell inequalities, like ( I), and will not be considered here. A usual proof of Bell’s theorem consists of showing that ( 1) is violated by the predictions of quantum mechanics if the ensemble of particle pairs is in a state with a wavefunction consisting of the product of a spin part times a spatial part of the form
T ( 1 ) being the spin state with positive (negative) spin projection along the Z axis, and & (&) a spatial wavefunction with support in L (R). In most of the published proofs of Bell’s theorem, the spatial part of the wavefunction is not written explicitly, usually adding instead a sentence like “the detectors are localized far away”. We may assume that what the authors have in the mind is either something like (2) or the idea that
3
Letters A 200 (1995) 14
finding an appropriate spatial wavefunction is trivial. In the first case the proof is incorrect, because (2) is not allowed for an EPRB experiment, as we shall show in the following. In the second case the proof is incomplete. A detailed study of the predictions of quantum mechanics for an EPRB experiment follows. ‘Without loss of generality we may work in an inertial reference frame where the measurements in regions L and R start simultaneously, that is r, = t, = T. For simpljicity we shall assume that the two particles are distinguishable but have the same mass, m, and we sHal1 take Ar, = At, = AT. If a particle pair is produced in the source at time t = 0, in the quantum state & and we measure, instantanously at time T, whether particle 1 is found in L with positive spin projection along a and particle 2 is found in L with positive spin projection along b, then the probability of the answer yes in both cases is given by P%(a,
b) = (rGh I exp(iHTlh)M,(a)M,Cb)
X exp( -iffTIh)
I h)
,
(3)
where H = H, + H2 is the Hamiltonian for the' free evolution of the particle pair and M, (a) is the projection operator corresponding to the observable “particle 1 is found in L with positive spin projection along a “, and similar for M2( b) . Analogous expressions should be used for the other probabilities involved in ( 1). Actually, as real measurements are not instantaneous, we have a difficulty because there is no simple: quantum rule to compute probabilities involving mea$urements that last some time. Although this problem deserves a more careful study, we shall solve the difficulty by assuming that quantum mechanics predicts the same probability for detecting between T and T+ AT in region L as for detecting the particle instantaneously at time Tin a greater region, appropriately chosen. In this sense we may use Eq. (3). Now the proof of Bell’s theorem consists’of finding a particular initial state $() of the particle pair at the source, such that the probabilities computed from (3) violate inequality (1) when the time T is sq large that the distance traveled by the particles from the source is macroscopic. If the initial state of the pair of particles is a spin singlet, then the initial wavefunction should be of the form
4
E. Santos/Physics
the function &,(ri, r2) having support in the source S. It is obvious that locality is not violated at T= 0 because the particles are not “far apart”. Then, an essential question for the proof of Bell’s theorem is the following: Is the correlation preserved by the quantum evolution so that the Bell inequality is violated at some time T? It is taken for granted that spin correlation is preserved over macroscopic distances. However, crucial for the violation of ( 1) is also a strong spatial correlation of the particles at that time. The projection operator M,(a) of Eq. (3) is the product of the spin projection operator i( 1 +a, *a}, where vi is the vector of the Pauli matrices of particle 1, times the projection operator characteristic of region L. This is defined as taking value 1 if the particle is found inside region L, and zero if it is not found there. In order to see whether the Bell inequality ( 1) is violated by the prediction of quantum mechanics, we shall compute the left hand side of (3) using the projection operators M,(a), M,(c), M,(b), and M,( d). In a simplified nonrelativistic treatment, we may assume that the Hamiltonian involved in (3) contains only the kinetic energy term. (This approximation certainly is not valid in relativistic regime, where the free Dirac Hamiltonian does not commute with the spin operators). With this assumption the spin part of (3) may be easily computed and the quantum prediction for the left hand side of ( 1) becomes go=
f(PP
+Py)
-aP?,(2+Q.b+a.d-tc.b-c*d),
(5)
where P$ is given by the integral of the probability density, i.e.
P$ = +=exp(
d3r, I
I
L
R
d3r,M12>
-iHTlfi)~#+,(ri,
r2)
Letters A 200 (1995) I-6
(2)). As is well known, if {Q, b, c, d] are four unit vectors, thequantit a*b+u*d+c*b--cadhasamaximum value of 2 $ 2. Therefore, the Bell inequality ( 1) can be violated by the quantum-mechanical predictions in an EPRB experiment if and only if PP-P9<(JZ+I)PR.
(7)
This inequality clearly shows that not all systems of two spin-i particles in a singlet spin state are capable of violating the Bell inequality in an EPR experiment. Indeed, condition (7) puts three kinds of constraints on possible violations in actual experiments: (a) the efficiencies of measuring devices (selectors and detectors) should be quite high, (b) a good angular correlation of the particles is needed, (c) the radial correlation should also be good. These constraints give rise to loopholes for the empirical refutation of local hidden variables theories, via Bell’s theorem. The low efficiency loophole has been extensively discussed, usually in connection with optical photon experiments (see, e.g., Ref. [ 141). It is seen as a purely practical problem of the experiments, irrelevant for the proof of Bell’s theorem, and it will not be discussed here. The problem of the angular correlation is also well known [ 13,7]. Finally, the need for a good radial correlation has never been discussed in relation with Bell’s theorem. In order to prove Bell’s theorem, all what remains is to find an initial wave function c#+,and two regions L and R fulfilling the following two conditions: ( 1) Inequality (7) holds when the integrals involved are performed using the wave function evolved from &, (2) the measurements are spatially separated (in the sense of relativity theory). I shall try an initial wave function, which is the product of a center of mass function and a relative position function, both with spherical symmetry (so that the orbital angular momentum is L = 0). For example, we may use the initial wave function & =N exp( -m*A~r~lfL*)
(6)
X and P?,Py ,by similar expressions (except that the integrals over either R or L regions should be extended to the whole space). As is explicitly shown, + is the spatial wavefunction of the particle pair at time T.(We emphasize the sharp contrast between this function 4 and the usual - but wrong - spatial wave function of
I
Xr-’
dk [(2kh/m-v)*+A:,/4]-’ exp( - ikr) ,
(8)
where N is a normalization factor, r,, = f (r, +r2) and r=r, -ri are the center of mass and relative position vectors respectively (r,, = 1I-, I, r = (r ( ), u is the aver-
E. Santos /Physics
age relative velocity of the particles and A,, (A,) is easily related to the dispersion of the relative (center of mass) velocity. Although the choice (8) of a Gaussian for the center of mass and a Lorentzian for the relative motion is quite natural, our conclusions will not depend on the detailed form of &, which has been written as in (8) just for the sake of clarity. The angular correlation is never perfect, even in a two-body decay, because the center of mass motion gives rise to a fluctuation a0 in the angle between the directions of propagation of the two particles. It is easy to see that it is a/3= [(mdoT)-2~2+A~]“2rl--.
(9)
The fluctuation introduces the requirement that the solid angle covered by each measurement region, as seen from the source, has a half-angle grater than de. This demand, however, puts neither fundamental nor strong practical difficulties on the experiments. Another restriction for inequality (7) to hold is the need of a radial correlation between the particle positions. Actually, the requirement of radial correlation may be diminished with a method proposed by Fry [ 41. It consists of using big measurement regions (L and R above) but insuring space-like separation by a very short measurement time, using a short pulse of laser light for a simultaneous position and spin measurement. We shall discuss elsewhere the possible problems of this procedure, but here we shall analyze only the conventional method of using thin detectors but relatively long measurement times (as proposed, e.g., in Ref. [ 151 ). In this case the detection time window aT should not be smaller than the fluctuation of the distance between the particles, at time T, divided by the particle velocity. If this were not so, whenever a particle arrives at the left (right) detector in the time interval [T, T+ aT], the other particle would arrive at the right (left) detector outside this time interval, and the ratios P,,/P,, i = 1 or 2, would be too small to fulfil (7). After some algebra it can be shown that for inequality (7) to hold, the following inequality should be fulfilled, aT>33rl -‘(tl’m
-2A,?
+ A;T2)
I/?.
( 10)
But the condition of spatial separation of the measurements requires dT to be smaller than the time needed for light to travel from one region to another, that is Tcilc. This gives the constraint
5
Letters A 200 (I 995) I4
x= fuT>3hc[
1- (3cA,lu’)
*] -“‘/(2nwA,,)
, (11)
where x is the distance from the source to one of the measurement regions. We see that ( 11) involves two inequalities, which may be written in term4 of the energy due to the motion of the center of mass, E,, = $rnAi, and the energy of the relative motion, E E=~rnc~‘. We have n>3&ic/8\lEE, T>9hmc21(4E)‘.
,
E,, <4E’/3mc’. (12)
The third inequality, which is a trivial consequence of the other two, shows that locality cannot be violated until the particles have traveled from the source during some relatively long time interval. We may illustrate the above constraints with the example of a nonrelativistic EPRB experiment, proposed some time ago, consisting of measuringithe spin correlation of two atoms produced in the dissociation of a Na, molecule [ 151. The values of the relevant quantities involved in the experiment are m=2.2X101”c~’ eV and E = 0.06 eV. Using ( 12)) this gives the requirements T> 2 X lop3 s, x> 1 m, E, < 2 X lo- ” eV. In particular. the center of mass energy E, expected in the actual experiment (due to thermal fluctuations) was 11 orders of magnitude greater than this value. Clearly, this experiment would not be a reliable test of LHV theories. In contrast, an experiment has been proposed recently [ 161 where all the above constraints have been carefully taken into account. It involves the scattering of slow neutrons and requires that the detectors are 100 m away from the scattering region. The experiment seems feasible although the authors recognize that it is extremely difficult. In spite of the practical difficulties for actual experiments, which we have shortly reviewed, our specific example (8) is able to violate the Bell inequality ( 1 ) This completes the proof of Bell’s theorem as stated above. We see that the proof is not as trivial as current (incomplete) treatments suggest. In particular, the problem of the spatial correlation of the particles is quite delicate even for two particles. It is to be expected that the problem becomes much worse for systems with three or more particles, as in the fashionable Oreenberger-Horne-Zeilinger [ 171 experiments.
E. Santos /Physics
6
We may summarize our study of the EPRB experiments with the following conclusions. In the nonrelativistic domain, quantum mechanics puts strong constraints for a practical experimental test of LHV theories via EPRB experiments, but no fundamental difficulty exists. The obvious next step will be to try to generalize the proof to the relativistic domain by using the appropriate Hamiltonian in Eq. (3). Before this is done no rigorous proof will exist of the incompatibility between LHV theories and relativistic quantum theory. I acknowledge financial support from DGCYT, project number PB92-0507 (Spain).
References [l] J.S. Bell, Physica 1 (1964) 195. [Z] C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes and R.P. Hudson, Phys. Rev. 105 (1957) 1413. [3] T.D. Lee and C.N. Yang, Phys. Rev. 104 (1956) 254.
Letters A 200 (1995) Id
[4] E.S. Fry, in: Proc. Int. Conf. on Lasers, 1993, ed. C.P. Wang (STS Press, McLean, VA, 1993). 151 P.G. Kwiat, P.H. Eberhard, A.M. Seinberg and R.Y. Chiao, Phys. Rev. A 49 ( 1994) 3209. [61 SF. Huelga, M. Ferrer0 and E. Santos, Europhys. Lett. 27 (1994) 181. 171 E. Santos, Phys. Rev. A 46 ( 1992). 3646. [81 A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49 (1982) 1804. 191 D. Bohm, Quantum theory (Prentice-Hall, Englewood Cliffs, NJ, 1951). [ lo] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 ( 1935) 777. [ 111 J.S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge Univ. Press, Cambridge, 1987). [ 121 N.D. Mermin, Rev. Mod. Phys. 65 ( 1993) 803. [ 13 1F.J. Clauser and M.A. Home, Phys. Rev. D 10 (1974) 526. L141 F. Selleri, ed. Quantum mechanics versus local realism. The Einstein-Podolsky-Rosen paradox (Plenum. New York, 1988). [ 151 T.K. Lo and A. Shimony, Phys. Rev. A 23 (1981) 3003. [ 161 R.T. Jones and E.G. Adelberger, Phys. Rev. Lett. 72 ( 1994) 2675. [ 17] D.M. Greenberger, M. Home, A. Shimony and A. Zeilinger, Am.J.Phys.58(1990) 1131.
ELSJZVIER
LETTERS
A
Physics Letters A 200 ( 1995 ) l-6
Constraints for the violation of the Bell inequality in Einstein-Podolsky-Rosen-Bohm experiments Emilio Santos Departarnento de Fisica Moderna. Universidad de Cantabriu, Santander, Spain Received 2 May 1994: revised manuscript
received 30 January 1995; accepted for publication Communicated by J.P. Vigier
31 January
1995
Abstract
A proof of Bell’s theorem is presented involving a pair of nonrelativistic spin-& particles in a singlet state. A detailed study of the spatial part of the wave function is made and several constraints are derived for the existence of reliable experiments.
It is well known that the proof of Bell’s theorem [ l] has two parts. The first one is the derivation of (Bell) inequalities valid for any local hidden-variables (LHV) theory. We shall not be concerned with this here. The second part consists of exhibiting a particular experiment where the quantum predictions violate a Bell inequality, thus showing the incompatibility between quantum mechanics and LHV theories. In the 30 years elapsed since Bell’s seminal paper [ 11, hundreds of gedanken experiments have been proposed showing the violation of a Bell inequality, but no incontrovertible (loophole-free) violation has been shown in a real experiment. (Compare, for instance, with the history of parity violation, where a crucial experiment [ 21 was performed within one year of the theoretical paper [ 3 1.) Proposals for loophole-free experiments are beginning to appear [4-6], but none of them is clearly feasible with present technology. Furthermore, the difficulties for performing empirical tests of the Bell inequalities cannot be just attributed to nonidealities of the available measuring devices. Indeed, LHV models have been found [7] which perfectly agree with the ideal quantum predictions for atomic cascade experiments, like Aspect’s one [ 81, which are currently con0375.9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10375-9601(95)00097-6
sidered the most reliable test of Bell’s inequalities. The sharp contrast between so many gedanken and so few (none in fact) real violations of Bell’s inequalities surely requires an explanation, and the obvious one is that gedanken experiments contain oversimplifications that make them unphysical. Bell’s theorem states that there is a contradiction between quantum theory and local hidden-‘variables theories. It is a mathematical statement which, as such, does not require any real experiment to be proved or disproved. Only if we want to investigate how nature behaves we require actual experiments. It is generally agreed that the contradiction may be proved by considering a gedanken experiment where the quantum predictions contradict a Bell inequality: I also agree, but claim that any experiment (gedanken or real) should involve three steps: the preparation of the system, the evolution, and the measurement. Almost eve#y gedanken experiment considered in the literature for the proof of Bell’s theorem considers just the last step. Indeed, usual proofs begin like: “Let us assume that we have two particles in an entangled spin state, placed at two macroscopically separated regions.. ,’ ‘. I strongly disagree with that. In my opinion the (gedanken) exper-
iment should include the process of preparation of the particle pair at the source and the free evolution towards the two macroscopically separated regions, both processes being treated according to quantum theory. This is the essential point that I want to stress in the present paper. With this purpose I shall analyze the requirements to make physical one of the most popular gedanken experiments: The version of Bohm [9] of the experiment considered by Einstein, Podolsky and Rosen [ lo]. The experiment of Einstein-Podolsky-RosenBohm (EPRB) , consisting of measuring the spin correlation of two spin-i particles prepared in a singlet state, was the example used by John Bell [ I] for the proof of his celebrated theorem. It has been considered in many articles and books, and it is still the standard example of “quantum nonlocality”. However, the experiment has never been discussed in enough detail. We will show that (1) the standard proofs of Bell’s theorem are incomplete because no care is taken of the space part of the wave function, (2) according to nonrelativistic quantum mechanics some spin singlet states are able to violate a Bell inequality, but not all such states do so, (3) EPRB experiments are extremely difficult due to several constraints that we shall derive. According to Bell [ 111, locality means the impossibility of causal (i.e. not superluminal) communication between spatially separated (in the sense of relativity theory) regions, where some measurements are performed. Bell’s theorem is striking because it seems to imply that, if quantum mechanics is correct, nonlocal correlations exist. Consequently, a rigorous proof of Bell’s theorem should necessarily involve considering experiments where measurements are made at several spatially separated regions (we shall consider only two regions throughout this article). Furthermore, the measurements should be made on subsystems (particles) resulting from the evolution of a system prepared in the common past (again, relativistic meaning) of both measurements. These conditions lead us to what we will call EPR [ lo] experiments, described in the following. We shall prepare some system of two particles in a region of space, S, that we shall call the source. At a later time, we shall measure some observable A of the first particle in region L (say on the left), and observable B on the second particle in region R (on the right). By repeating the preparations and the measurements
(in the same conditions) many times, we may obtain the joint probability distribution, P( A, B), of A and B. If our experiment consists of just this, local hiddenvariables models certainly exist able to reproduce that probability. As a matter of fact, the question of locality does not even arises because nonlocality is revealed only when a change of the observable measured in L influences the result of measuring observable B on region R. We must, therefore, perform new runs of the experiment with exactly the same preparation, but now measuring observable C (instead of A) on particle 1 in region L together with observable B on particle 2 in R. Then, in another run we measure C on 1 in L and D on 2 in R, and so on. In this way we may prove that no noncontextual (see Ref. [ 121 for a careful study of noncontextuality) hidden-variables model exists able to reproduce the prediction of quantum mechanics for P( A, B) and P( C, B) and P( A, D), etc. But, still, this would not prove Bell’s theorem if we do not state conditions about the causal connection between the measurement regions, that is without locality requirements. Locality puts stringent constraints on EPR experiments. In fact, the measurement on L (R) should be made during some finite time interval [tL, t,+ AfL] ( [ tR. f, + AfK] ), which defines a space-time “region of measurement” (L, tL, AtL) ((R, tR, AtR)). We require that the regions are spatially separated, and that the source, S, during the preparation [ ts, ts + Ats] lies completely within the common past light cones of both regions. As a last constraint, the experimenter is not allowed to modify the state of the state of the particles during the travel from S to L or from S to R, so that the evolution is “free”. (Any intervention of the experimenter should be considered as a part of either the preparation or the measurement.) Our first task will be to prove the following (Bell) theorem: Local hidden variables are incompatible with elementary quantum mechanics. By elementary we mean nonrelativistic quantum mechanics of systems with a finite number of degrees of freedom. Obviously, a proof not resting upon nonrelativistic approximations would be desirable because locality is an essentially relativistic concept. However, the relativistic treatment will not be considered in the present paper. A proof of Bell’s theorem should consist of finding a particular EPR experiment where the predictions of quantum mechanics violate a Bell inequality. The appropriate Bell inequality is [ 131
E. Santos /Physics
B=P,(a)
+P,(b) -P,*(a,
-P,?(C, b)
b) -P,,(a,
+p,,cc,d) 20.
4 (1)
Here (a, b, c, d} are four unit vectors corresponding to four possible spin projection observables. As explained above, we must prepare an ensemble of particle pairs and measure, on a subensemble, the probability P,,(a, b) that the first particle has positive spin projection along a and it is in region L, and the second along b in region R, then we should measure P,,(a, c) on another subensemble, etc. P, (P2) corresponds to a measurement on particle 1 (2) without measuring anything on 2 ( 1) . Inequality ( 1) is fulfilled by any local hiddenvariables model so that its violation in an EPRB experiment refutes the whole family of such theories. I shall point out that there are many different Bell inequalities, with quite diverse meaning. Some of the inequalities, like ( 1) , are derived only from Bell’s postulates, defining LHV theories and therefore they hold for all of them. An example is the first CH inequality (derived in Section III of Ref. [ 131). In other cases, however, the derivation involves the use of auxiliary assumptions and, consequently, they hold only for restricted families of LHV theories. An example is the second CH inequality (section V of Ref. [ 131) The latter inequalities have the advantage that they are more easily tested empirically than the former (and only they have been actually tested till now), but they do not have the fundamental relevance of the genuine Bell inequalities, like ( I), and will not be considered here. A usual proof of Bell’s theorem consists of showing that ( 1) is violated by the predictions of quantum mechanics if the ensemble of particle pairs is in a state with a wavefunction consisting of the product of a spin part times a spatial part of the form
T ( 1 ) being the spin state with positive (negative) spin projection along the Z axis, and & (&) a spatial wavefunction with support in L (R). In most of the published proofs of Bell’s theorem, the spatial part of the wavefunction is not written explicitly, usually adding instead a sentence like “the detectors are localized far away”. We may assume that what the authors have in the mind is either something like (2) or the idea that
3
Letters A 200 (1995) 14
finding an appropriate spatial wavefunction is trivial. In the first case the proof is incorrect, because (2) is not allowed for an EPRB experiment, as we shall show in the following. In the second case the proof is incomplete. A detailed study of the predictions of quantum mechanics for an EPRB experiment follows. ‘Without loss of generality we may work in an inertial reference frame where the measurements in regions L and R start simultaneously, that is r, = t, = T. For simpljicity we shall assume that the two particles are distinguishable but have the same mass, m, and we sHal1 take Ar, = At, = AT. If a particle pair is produced in the source at time t = 0, in the quantum state & and we measure, instantanously at time T, whether particle 1 is found in L with positive spin projection along a and particle 2 is found in L with positive spin projection along b, then the probability of the answer yes in both cases is given by P%(a,
b) = (rGh I exp(iHTlh)M,(a)M,Cb)
X exp( -iffTIh)
I h)
,
(3)
where H = H, + H2 is the Hamiltonian for the' free evolution of the particle pair and M, (a) is the projection operator corresponding to the observable “particle 1 is found in L with positive spin projection along a “, and similar for M2( b) . Analogous expressions should be used for the other probabilities involved in ( 1). Actually, as real measurements are not instantaneous, we have a difficulty because there is no simple: quantum rule to compute probabilities involving mea$urements that last some time. Although this problem deserves a more careful study, we shall solve the difficulty by assuming that quantum mechanics predicts the same probability for detecting between T and T+ AT in region L as for detecting the particle instantaneously at time Tin a greater region, appropriately chosen. In this sense we may use Eq. (3). Now the proof of Bell’s theorem consists’of finding a particular initial state $() of the particle pair at the source, such that the probabilities computed from (3) violate inequality (1) when the time T is sq large that the distance traveled by the particles from the source is macroscopic. If the initial state of the pair of particles is a spin singlet, then the initial wavefunction should be of the form
4
E. Santos/Physics
the function &,(ri, r2) having support in the source S. It is obvious that locality is not violated at T= 0 because the particles are not “far apart”. Then, an essential question for the proof of Bell’s theorem is the following: Is the correlation preserved by the quantum evolution so that the Bell inequality is violated at some time T? It is taken for granted that spin correlation is preserved over macroscopic distances. However, crucial for the violation of ( 1) is also a strong spatial correlation of the particles at that time. The projection operator M,(a) of Eq. (3) is the product of the spin projection operator i( 1 +a, *a}, where vi is the vector of the Pauli matrices of particle 1, times the projection operator characteristic of region L. This is defined as taking value 1 if the particle is found inside region L, and zero if it is not found there. In order to see whether the Bell inequality ( 1) is violated by the prediction of quantum mechanics, we shall compute the left hand side of (3) using the projection operators M,(a), M,(c), M,(b), and M,( d). In a simplified nonrelativistic treatment, we may assume that the Hamiltonian involved in (3) contains only the kinetic energy term. (This approximation certainly is not valid in relativistic regime, where the free Dirac Hamiltonian does not commute with the spin operators). With this assumption the spin part of (3) may be easily computed and the quantum prediction for the left hand side of ( 1) becomes go=
f(PP
+Py)
-aP?,(2+Q.b+a.d-tc.b-c*d),
(5)
where P$ is given by the integral of the probability density, i.e.
P$ = +=exp(
d3r, I
I
L
R
d3r,M12>
-iHTlfi)~#+,(ri,
r2)
Letters A 200 (1995) I-6
(2)). As is well known, if {Q, b, c, d] are four unit vectors, thequantit a*b+u*d+c*b--cadhasamaximum value of 2 $ 2. Therefore, the Bell inequality ( 1) can be violated by the quantum-mechanical predictions in an EPRB experiment if and only if PP-P9<(JZ+I)PR.
(7)
This inequality clearly shows that not all systems of two spin-i particles in a singlet spin state are capable of violating the Bell inequality in an EPR experiment. Indeed, condition (7) puts three kinds of constraints on possible violations in actual experiments: (a) the efficiencies of measuring devices (selectors and detectors) should be quite high, (b) a good angular correlation of the particles is needed, (c) the radial correlation should also be good. These constraints give rise to loopholes for the empirical refutation of local hidden variables theories, via Bell’s theorem. The low efficiency loophole has been extensively discussed, usually in connection with optical photon experiments (see, e.g., Ref. [ 141). It is seen as a purely practical problem of the experiments, irrelevant for the proof of Bell’s theorem, and it will not be discussed here. The problem of the angular correlation is also well known [ 13,7]. Finally, the need for a good radial correlation has never been discussed in relation with Bell’s theorem. In order to prove Bell’s theorem, all what remains is to find an initial wave function c#+,and two regions L and R fulfilling the following two conditions: ( 1) Inequality (7) holds when the integrals involved are performed using the wave function evolved from &, (2) the measurements are spatially separated (in the sense of relativity theory). I shall try an initial wave function, which is the product of a center of mass function and a relative position function, both with spherical symmetry (so that the orbital angular momentum is L = 0). For example, we may use the initial wave function & =N exp( -m*A~r~lfL*)
(6)
X and P?,Py ,by similar expressions (except that the integrals over either R or L regions should be extended to the whole space). As is explicitly shown, + is the spatial wavefunction of the particle pair at time T.(We emphasize the sharp contrast between this function 4 and the usual - but wrong - spatial wave function of
I
Xr-’
dk [(2kh/m-v)*+A:,/4]-’ exp( - ikr) ,
(8)
where N is a normalization factor, r,, = f (r, +r2) and r=r, -ri are the center of mass and relative position vectors respectively (r,, = 1I-, I, r = (r ( ), u is the aver-
E. Santos /Physics
age relative velocity of the particles and A,, (A,) is easily related to the dispersion of the relative (center of mass) velocity. Although the choice (8) of a Gaussian for the center of mass and a Lorentzian for the relative motion is quite natural, our conclusions will not depend on the detailed form of &, which has been written as in (8) just for the sake of clarity. The angular correlation is never perfect, even in a two-body decay, because the center of mass motion gives rise to a fluctuation a0 in the angle between the directions of propagation of the two particles. It is easy to see that it is a/3= [(mdoT)-2~2+A~]“2rl--.
(9)
The fluctuation introduces the requirement that the solid angle covered by each measurement region, as seen from the source, has a half-angle grater than de. This demand, however, puts neither fundamental nor strong practical difficulties on the experiments. Another restriction for inequality (7) to hold is the need of a radial correlation between the particle positions. Actually, the requirement of radial correlation may be diminished with a method proposed by Fry [ 41. It consists of using big measurement regions (L and R above) but insuring space-like separation by a very short measurement time, using a short pulse of laser light for a simultaneous position and spin measurement. We shall discuss elsewhere the possible problems of this procedure, but here we shall analyze only the conventional method of using thin detectors but relatively long measurement times (as proposed, e.g., in Ref. [ 151 ). In this case the detection time window aT should not be smaller than the fluctuation of the distance between the particles, at time T, divided by the particle velocity. If this were not so, whenever a particle arrives at the left (right) detector in the time interval [T, T+ aT], the other particle would arrive at the right (left) detector outside this time interval, and the ratios P,,/P,, i = 1 or 2, would be too small to fulfil (7). After some algebra it can be shown that for inequality (7) to hold, the following inequality should be fulfilled, aT>33rl -‘(tl’m
-2A,?
+ A;T2)
I/?.
( 10)
But the condition of spatial separation of the measurements requires dT to be smaller than the time needed for light to travel from one region to another, that is Tcilc. This gives the constraint
5
Letters A 200 (I 995) I4
x= fuT>3hc[
1- (3cA,lu’)
*] -“‘/(2nwA,,)
, (11)
where x is the distance from the source to one of the measurement regions. We see that ( 11) involves two inequalities, which may be written in term4 of the energy due to the motion of the center of mass, E,, = $rnAi, and the energy of the relative motion, E E=~rnc~‘. We have n>3&ic/8\lEE, T>9hmc21(4E)‘.
,
E,, <4E’/3mc’. (12)
The third inequality, which is a trivial consequence of the other two, shows that locality cannot be violated until the particles have traveled from the source during some relatively long time interval. We may illustrate the above constraints with the example of a nonrelativistic EPRB experiment, proposed some time ago, consisting of measuringithe spin correlation of two atoms produced in the dissociation of a Na, molecule [ 151. The values of the relevant quantities involved in the experiment are m=2.2X101”c~’ eV and E = 0.06 eV. Using ( 12)) this gives the requirements T> 2 X lop3 s, x> 1 m, E, < 2 X lo- ” eV. In particular. the center of mass energy E, expected in the actual experiment (due to thermal fluctuations) was 11 orders of magnitude greater than this value. Clearly, this experiment would not be a reliable test of LHV theories. In contrast, an experiment has been proposed recently [ 161 where all the above constraints have been carefully taken into account. It involves the scattering of slow neutrons and requires that the detectors are 100 m away from the scattering region. The experiment seems feasible although the authors recognize that it is extremely difficult. In spite of the practical difficulties for actual experiments, which we have shortly reviewed, our specific example (8) is able to violate the Bell inequality ( 1 ) This completes the proof of Bell’s theorem as stated above. We see that the proof is not as trivial as current (incomplete) treatments suggest. In particular, the problem of the spatial correlation of the particles is quite delicate even for two particles. It is to be expected that the problem becomes much worse for systems with three or more particles, as in the fashionable Oreenberger-Horne-Zeilinger [ 171 experiments.
E. Santos /Physics
6
We may summarize our study of the EPRB experiments with the following conclusions. In the nonrelativistic domain, quantum mechanics puts strong constraints for a practical experimental test of LHV theories via EPRB experiments, but no fundamental difficulty exists. The obvious next step will be to try to generalize the proof to the relativistic domain by using the appropriate Hamiltonian in Eq. (3). Before this is done no rigorous proof will exist of the incompatibility between LHV theories and relativistic quantum theory. I acknowledge financial support from DGCYT, project number PB92-0507 (Spain).
References [l] J.S. Bell, Physica 1 (1964) 195. [Z] C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes and R.P. Hudson, Phys. Rev. 105 (1957) 1413. [3] T.D. Lee and C.N. Yang, Phys. Rev. 104 (1956) 254.
Letters A 200 (1995) Id
[4] E.S. Fry, in: Proc. Int. Conf. on Lasers, 1993, ed. C.P. Wang (STS Press, McLean, VA, 1993). 151 P.G. Kwiat, P.H. Eberhard, A.M. Seinberg and R.Y. Chiao, Phys. Rev. A 49 ( 1994) 3209. [61 SF. Huelga, M. Ferrer0 and E. Santos, Europhys. Lett. 27 (1994) 181. 171 E. Santos, Phys. Rev. A 46 ( 1992). 3646. [81 A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49 (1982) 1804. 191 D. Bohm, Quantum theory (Prentice-Hall, Englewood Cliffs, NJ, 1951). [ lo] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 ( 1935) 777. [ 111 J.S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge Univ. Press, Cambridge, 1987). [ 121 N.D. Mermin, Rev. Mod. Phys. 65 ( 1993) 803. [ 13 1F.J. Clauser and M.A. Home, Phys. Rev. D 10 (1974) 526. L141 F. Selleri, ed. Quantum mechanics versus local realism. The Einstein-Podolsky-Rosen paradox (Plenum. New York, 1988). [ 151 T.K. Lo and A. Shimony, Phys. Rev. A 23 (1981) 3003. [ 161 R.T. Jones and E.G. Adelberger, Phys. Rev. Lett. 72 ( 1994) 2675. [ 17] D.M. Greenberger, M. Home, A. Shimony and A. Zeilinger, Am.J.Phys.58(1990) 1131.