Comment on " Proposed molecular test of local hidden-variables theories

Lo and Shimony' have proposed an experiment which should allow the refutation of local hidden-variables theories (LHVT) without a loophole arising in the analysis of experiments of polarization correlations of photons. 2' Their proposal is to measure spin correlations of two Na atoms coming from the dissociation of a Na2 molecule in a singlet state. The authors derive the following inequality, valid for any LHVT:

It is assumed that 8;"is due to accidental coincidences.The purpose of this Comment is to point out that the proposed background subtraction is incorrect.
In the first place, according to (5) the minimum of P++(a, b) is zero by definition, which excludes a priori all LHVT having strictly positive (nonzero) P++(a, b).In a similar way we might refute quantum mechanics by just ad- ding a constant to Eq. (5).In fact, according to quantum mechanics the minimum of P++(a, b) should be zero.This clearly shows that Eq. ( 5) cannot be used.
It may be suggested the use of one of the two standard methods to make corrections for accidental coincidences, namely, the study of the time-delay spectrum and the calcu- lation from the single counts rate." The time-delay spec- trurn method rests upon the assumption that the coincidence rate is independent of the delay between the first where q is the efficiency of the atom detectors, and P++ is the probability of detecting a pair of atoms in the + chan- nels of both analyzers conditional upon its being detected at all, and similarly for P, P+, and P +.The condition- al probabilities are related with total detection probabilities through and similar relations for P+, P +, and P The prediction of quantum mechanics for the proposed experiment is, for some choice of experimental setup, ' P", + -(a, b) = -dX &, dX'p(h., A. ')PP (a, X)P2 (b, A. ') (6) which parallels the expression for the probability of a true coincidence [Eq.(3c) of Ref. 1].Here, we have two dif- ferent parameters, A. , A. ', corresponding to two different Na2 molecules and p(A., A. ') is the joint probability distribution of these parameters.
(For the sake of clarity, a slight change in notation has been made with respect to Ref. 1.) It can be realized that the standard methods for calculat- ing the probability of an accidental coincidence are equivalent to Eq. ( 6) plus the assumption In fact, (7) means that the two Na2 molecules giving rise to an accidental coincidence dissociated in an uncorrelated manner, so that the probability (6) becomes the product of two probabilities for single counts.This is quite plausible for accidental coincidences in a delayed channel.Indeed, the agreement found between the accidenta1 coincidence rate calculated from the single counts rate and the coincidence rate in a delayed channel can be explained as a consequence of the validity of Eqs. ( 6) and ( 7) for delayed coincidences.
I and the second counts of an accidental coincidence.The calculation from the single counts rate assumes that the pro- bability of an accidental coincidence is the product of the probabilities of two single counts.(A good agreement has been reported between both procedures.) With any one of these methods, the correction again consists of subtracting a background independent of the relative orientation of the analyzers.In the following we show that this procedure is also not adequate.
According to a general methodological principle, if one at- tempts to test a theory, the full experiment must be inter- preted according to that theory.The subtraction of a con- stant background may be adequate for a test of quantum mechanics but, for a test of LHVT, it is necessary to calculate the probability of an accidental coincidence from an as- sumption valid for any LHVT.We propose the following one: In sharp contrast, molecules dissociated at the same (or near the same) time may have correlated hidden variables, Therefore, Eq. ( 7) may not be valid and the standard methods of background subtraction are incorrect.The max- imum possible correlation is obtained if we write, instead of ( 7), (8) where 8( ) is the Dirac delta.In this case, the accidental coincidence probability has the same dependence on a and b as the true (i.e. , nonaccidental) coincidence probability.
[However, the rate averaged over a and b may coincide with the one obtained from ( 7).]Then, the correction should no more consist of subtracting a constant (independent on a and b ) background, but rather in multiplying the experi- mental rate times a factor between G and l.Other choices somewhat intermediate between ( 7) and ( 8) are also possi- ble, and it is not easy to find the adequate method of mak- ing the correction for accidental coincidences.
It might be assumed that a test of the validity of Eq. ( 7)and, therefore, the correctness of performing a correc- tion by background subtractioncould be made by varying the dissociation rate (e.g. , by a variation of the laser intensi- ty).In fact, Eq. ( 7) leads to specific predictions about the statistics of coincidences when the dissociation rate changes.(For instance, the rate of accidental coincidences would P++(a, b) R,+", + (a, b), etc. (9) As a consequence, the function S of (1) cannot be identified with that of (3), but both can be written in terms of and R m;".Alternatively, we may write ( 1) and (3) in terms of R;"and the reduced rates R++,R+, R +, and R, defined by the second Eq. ( 5).
Choosing the last procedure, Eq. (1) becomes remain independent of a and b, and proportional to the square of the single rate.) However, the confirmation of these predictions is not an absolute proof of the validity of (7).Indeed, it is enough to assume that the functions Pt(h.,a) and P2(X, b) of Eq. ( 6) [and Eq. (3a) of Ref. 1] depend on the dissociation rate, in order for the reproduc- tion of any empirical rates to be possible, except if the Bell inequalities are violated even with the uncorrected rates.' It must be stressed that the commented experiment will not be very useful if it needs an untested hypothesis in its analysis because, in these conditions, numerous refutations of LHVT have been reported.
The conclusion of our analysis is that Eq. ( 5) may be valid only for the calculation of the probabilities used in the test of quantum mechanics.In contrast, for the test of the entire family of LHVT, the following relation should be For simplicity we assume that y is independent of a and b.Now, Eq. ( 10) can be compared with the quantum predictions (3) and ( 5), which show that a violation of (10) is possible only if q & (5.06+37)/(5.62+y)=0.9+0.37' .(13) as a necessary condition for a reliable test of LHVT through the experiment proposed by Lo and Shimony.' It is important to point out that no experiment performed till now have refuted the full family of LHVT.In particular, the atomic-cascade experimental tests have been strongly criticized.As a consequence, a test along the lines proposed by Lo and Shimony' should be very welcome, if it could be made reliable.
5A. Fine, Phys.Rev. Lett. 48, 291 (1982).In this paper it is shown that, for any quantum correlation experiment which does not violate the Bell inequalities, there exists a local hidden-variables model compatible with the data.