Bell's Inequality and Detector Inefficiency

Bell’s inequality and detector inefficiency Jan-Åke Larsson

The self-archived postprint version of this journal article is available at Linköping University Institutional Repository (DiVA): http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-87169

N.B.: When citing this work, cite the original publication. Larsson, J., (1998), Bell’s inequality and detector inefficiency, Physical Review A. Atomic, Molecular, and Optical Physics, 57(5), 3304-3308. https://doi.org/10.1103/PhysRevA.57.3304

Original publication available at: https://doi.org/10.1103/PhysRevA.57.3304 Copyright: American Physical Society http://www.aps.org/

PHYSICAL REVIEW A VOLUME 57, NUMBER 5 MAY 1998

Bell’s inequality and detector inefﬁciency

Jan-A˚ ke Larsson* Department of Mathematics, Linko¨ping University, S-581 83 Linko¨ping, Sweden ͑Received 9 December 1997͒ In this paper, a method of generalizing the Bell inequality is presented that makes it possible to include detector inefficiency directly in the original Bell inequality. To enable this, the concept of ‘‘change of ensemble’’ will be presented, providing both qualitative and quantitative information on the nature of the ‘‘loophole’’ in the proof of the original Bell inequality. In a local hidden-variable model lacking change of ensemble, the generalized inequality reduces to an inequality that quantum mechanics violates as strongly as the original Bell inequality, irrespective of the level of efficiency of the detectors. A model that contains change of ensemble lowers the violation, and a bound for the level of change is obtained. The derivation of the bound in this paper is not dependent upon any symmetry assumptions such as constant efficiency, or even the assumption of independent errors. ͓S1050-2947͑98͒07405-8͔

PACS number͑s͒: 03.65.Bz

I. INTRODUCTION suppressing the parentheses in the last expression ͑this will be done in the following when no confusion can arise͒. Fur- The Bell inequality ͓1͔ and its descendants have been the thermore, a,b,... are the detector orientations used in the main argument on the Einstein-Podolsky-Rosen paradox various measurements, and to shorten the presentation a no- ͓2,3͔ for the past 30 years. A new research field of ‘‘experi- tation will be used where A,B,... are the RV’s correspond- mental metaphysics’’ has formed, where the goal is to show ing to the above orientations. The RV’s describe measure- that the concept of local realism is inconsistent with quantum ment results from one detector if it is unprimed (A), and mechanics, and ultimately with the real world. The experi- from the other when primed (AЈ), so that E(ABЈ) is the ments that have been performed to verify this have not been correlation between A and BЈ. completely conclusive, but they point quite decisively in a I have chosen the ‘‘deterministic’’ case here, and will not certain direction: Nature cannot be described by a realistic discuss the ‘‘stochastic’’ case as the generalization is local hidden-variable theory ͑see Refs. ͓4–6͔, for instance͒. straightforward. In this formalism, the Bell inequality can be The reason for saying ‘‘not been completely conclusive’’ stated as follows. is that there is an implicit assumption in the proof of the Bell Theorem 1 (the Bell inequality). The following four pre- inequality that the detectors are 100% effective. There has requisites are assumed to hold except at a null set. been considerable discussion in the literature on this ͑see (i) Realism. Measurement results can be described by Refs. ͓7–10͔, among others͒, and the main issue is to obtain probability theory, using ͑two families of͒ RV’s, a limit for the detector inefficiency, but previously inequalities other than the original Bell inequality had to be used, for A͑a,b͒: ⌳ V → A͑a,b,␭͒ۋexample, the Clauser-Horne inequality in ͓7͔, which in itself ␭ contains the case of inefficient detectors, or the Clauser- ᭙a,b. BЈ͑a,b͒: ⌳ V Horne-Shimony-Holt ͑CHSH͒ inequality first presented in → BЈ͑a,b,␭͒ۋwhich is generalized to the inefficient case in ͓9͔. ␭ ,11͔͓ Since a hidden-variable model is really a probabilistic model, formalism and terminology from probability theory (ii) Locality. A measurement result should be independent will be used in this paper ͑see, e.g., Ref. ͓12͔͒. The sample of the detector orientation at the other particle, space ⌳ is the mathematical analog to the state space used in physics, and a sample ␭ is a point in that space correspond- def ing to a certain value of the ‘‘hidden variable.’’ The mea- A͑a,␭͒ϭA͑a,b,␭͒ independently of b, surement results are described by random variables ͑RV’s͒ X(␭), which take their values in the value space V. def To be a probabilistic model, a probability measure P on BЈ͑b,␭͒ϭBЈ͑a,b,␭͒ independently of a. the space ⌳ is needed, by which we can define the expectation value E as (iii) Measurement result restriction. Only the results ϩ1 and Ϫ1 should be possible: def E͑X͒ϭ X͑␭͒dP͑␭͒ϭ XdP, ͑1͒ Vϭ͕Ϫ1,ϩ1͖. ͵⌳ ͵⌳ (iv) Perfect anticorrelation. A measurement with equally oriented detectors must yield opposite results at the two de- *Electronic address: [email protected] tectors,

AϭϪAЈ, ᭙a,␭. (i) Realism. Measurement results can be described by probability theory, using ͑two families of͒ RV’s, which may Then, be undefined on some part of ⌳, corresponding to measurement inefficiency, ͉E͑ABЈ͒ϪE͑ACЈ͉͒р1ϩE͑BCЈ͒. A͑a,b͒:⌳ ͑a,b͒ V To include detector inefficiency in the above inequality, A → A͑a,b,␭͒ۋ␭ previously two approaches have been used. The first is to use ᭙a,b. BЈ͑a,b͒:⌳ ͑a,b͒ V probabilities instead of correlations and derive an inequality BЈ → BЈ͑a,b,␭͒ۋon the probabilities ͑see Ref. ͓7͔͒. The second is to assign the ␭ measurement result 0 ͑zero͒ to an undetected particle, which makes the original Bell inequality inappropriate because of (ii) Locality. A measurement result should be independent prerequisite ͑iii͒ in Theorem 1. Thus the CHSH inequality of the detector orientation at the other particle, must be used and subsequently generalized to obtain an inequality valid in this case ͑see Ref. ͓9͔͒. def def A third approach, presented here, uses correlations but A͑a,␭͒ϭA͑a,b,␭͒ on ⌳A͑a͒ϭ⌳A͑a,b͒ makes no assignment of any measurement result to the undetected particles. Thus it is possible to obtain a direct gen- independently of b, eralization of the original Bell inequality.

II. GENERALIZATION OF THE ORIGINAL def def B b, ϭB a,b, on b ϭ a,b BELL INEQUALITY Ј͑ ␭͒ Ј͑ ␭͒ ⌳BЈ͑ ͒ ⌳BЈ͑ ͒

The measurement results are modeled as RV’s, which in independently of a. the ideal case would be defined as in prerequisite ͑i͒ in Theo- rem 1. In the case with inefficiency the situation is quite (iii) Measurement result restriction. Only the results ϩ1 different, as there are now points of ⌳ where the particle and Ϫ1 should be possible, would be undetected. To avoid the quite arbitrary assignment used in the second approach above, the RV’s will be said to be undefined at these points, i.e., they will only be defined at Vϭ͕Ϫ1,ϩ1͖. subsets of ⌳, which will be denoted ⌳A(a,b) and ⌳BЈ(a,b), respectively. (iv) Perfect anticorrelation. A measurement with equally In this setting, a new expression for the expectation value oriented detector must yield opposite results if both particles is needed. The averaging must be restricted to the set where are detected, the RV in question is defined, and the probability measure adjusted accordingly, AϭϪAЈ,on⌳AAЈϭ⌳Aപ⌳AЈ .

def Deﬁne the bound on the ensemble change when the measure- EX͑X͒ϭ ͵ XdPX , where PX͑E͒ϭP͑E͉⌳X͒. ͑2͒ ment setup is changed as ⌳X

The correlation is in this case E (AB ), the expectation of ␦ϭ inf PAB ͑⌳CD ͒ ͑ 0р␦р1͒. ABЈ Ј Ј Ј ⇒ ABЈ on the set at which both factors in the product are de- A,BЈ,C,DЈ fined, the set ⌳ABЈϭ⌳Aപ⌳BЈ . This is the correlation that would be obtained from an experimental setup where the Then, coincidence counters are told to ignore single particle events. In an experiment the pairs that are detected are the ones with ͉EABЈ͑ABЈ͒ϪEACЈ͑ACЈ͉͒р3Ϫ2␦ϩEBCЈ͑BCЈ͒. ␭’s in ⌳ABЈ , so the ensemble is restricted from ⌳ to ⌳ABЈ . It is now easy to see what makes the proof of the original Proof. The proof consists of two steps. The first part is Bell theorem break down. The start of the proof is similar to the proof of Theorem 1, using the ensemble ⌳ABBЈCЈ , on which all the RV’s A, B, BЈ, and CЈ are de- ͉EABЈ͑ABЈ͒ϪEACЈ͑ACЈ͉͒ fined. This is to avoid the problem mentioned above. This ensemble may be empty, but only when ␦ϭ0 and then the ϭ ABЈdPABЈϪ ACЈdPACЈ . ͑3͒ inequality is trivial, so ␦Ͼ0 can be assumed in the rest of the ͯ͵⌳ ͵⌳ ͯ ABЈ ACЈ proof. Now ͑i͒–͑iv͒ yields The integrals on the right-hand side cannot easily be added E AB ϪE AC 1ϩE BC . when ⌳ Þ⌳ , so a generalization of Theorem 1 is ͉ ABBЈCЈ͑ Ј͒ ABBЈCЈ͑ Ј͉͒р ABBЈCЈ͑ Ј͒ ABЈ ACЈ ͑4͒ needed. Theorem 2 (the Bell inequality with ensemble change). The second step is to translate this into an expression with The following four prerequisites are assumed to hold except EABЈ(ABЈ) and so on. Using ͑i͒–͑iii͒ and the triangle in- at a P-null set. equality C is the complement of , ͑⌳BDЈ ⌳BDЈ͒ 3306 JAN-A˚ KE LARSSON 57

volving measurement inefficiency would be useful. Ϫ ͯEACЈ͑ACЈ͒ ␦EABCЈDЈ͑ACЈ͒ͯ Corollary 3 (the Bell inequality with measurement inefficiency). Assume that Theorem 2͑i͒–͑iv͒ hold except on a р P͑⌳C ͒E ͑ACЈ ⌳C ͒ ͯ BDЈ ACЈ ͉ BDЈ ͯ P-null set. Two ways of defining measurement efficiency are used. (a) Detector efficiency. The least probability that a par- ϩͯP͑⌳BDЈ͒EACЈ͑ACЈ͉⌳BDЈ͒Ϫ␦EABCЈDЈ͑ACЈ͒ͯ ticle is detected is C C ϭPAC ͑⌳ ͒ EAC ͑ACЈ͉⌳ ͒ Ј BDЈ ͯ Ј BDЈ ͯ def ␩1ϭinf P͑⌳A͑i͒͒, ϩ͓PACЈ͑⌳BDЈ͒Ϫ␦͔ͯEABCЈDЈ͑ACЈ͒ͯ A,i рP ͑⌳C ͒E ͑ ACЈ ⌳C ͒ where the infimum is taken over all orientations of both de- ACЈ BDЈ ACЈ ͯ ͉ͯ BDЈ tectors. If ␩1у3/4, then ϩ͓PACЈ͑⌳BDЈ͒Ϫ␦͔EABCЈDЈ͑ ACЈ ͒ ͯ ͯ 3Ϫ2␩1 ͉EABЈ͑ABЈ͒ϪEACЈ͑ACЈ͉͒р ϩEBCЈ͑BCЈ͒. р1Ϫ␦. ͑5͒ 2␩1Ϫ1

The inequalities ͑4͒ and ͑5͒ together with the triangle in- (b) Coincidence efﬁciency. The least probability that a equality yield the desired result. ᭿ particle is detected at one detector given that it is detected at the other one is III. CHANGE OF ENSEMBLE def An important concept to understand the above inequality ␩2,1ϭ inf PA͑i͒͑⌳B͑ j͒͒, is ‘‘change of ensemble.’’ Assume that a sequence of experi- A,B,iÞ j ments is performed, where the ‘‘hidden variable’’ has the values ␭1 ,␭2 ,...,␭n . If the orientations of the detectors were where the inﬁmum is taken over all orientations of A,BЈ and a and b, the detected pairs would be the ones with ␭i’s in AЈ,B.If␩2,1у2/3, then ⌳ABЈ . Now if the orientations instead were c and d, the detected pairs would be the ones with ␭i’s in ⌳CDЈ . Then, if 4 , different pairs would be detected if it was ͉EABЈ͑ABЈ͒ϪEACЈ͑ACЈ͉͒р Ϫ3ϩEBCЈ͑BCЈ͒. ⌳ABЈÞ⌳CDЈ ␩2,1 possible to do the same run of ␭i’s in different setups, i.e., the ensemble would change. Proof. First, to prove ͑b͒, use the simple inequality The importance of this is most easily seen in the following example. Assume that the nondetections are distributed PCЈ͑⌳AB͒ independently of the detector orientations, i.e., PACЈ͑⌳B͒ϭ PCЈ͑⌳A͒

⌳A is independent of a, PC ͑⌳A͒ϩPC ͑⌳B͒ϪPC ͑⌳Aഫ⌳B͒ ͑6͒ ϭ Ј Ј Ј ⌳B is independent of b, Ј PCЈ͑⌳A͒ which yields ⌳ABЈϭ⌳CDЈ and ␩2,1Ϫ1 1 у1ϩ у2Ϫ , ͑9͒ PC ͑⌳A͒ ␩2,1 ␦ϭPABЈ͑⌳CDЈ͒ϭ1. ͑7͒ Ј Now, it is easy to see that when ␦ϭ1, the result resembles which gives that of the original Bell inequality: PACЈ͑⌳BDЈ͒ϭPACЈ͑⌳B͒ϩPACЈ͑⌳DЈ͒ϪPACЈ͑⌳Bഫ⌳DЈ͒ ͉E ͑ABЈ͒ϪE ͑ACЈ͉͒р1ϩE ͑BCЈ͒. ͑8͒ ABЈ ACЈ BCЈ 1 2 у2 2Ϫ Ϫ1ϭ3Ϫ . ͑10͒ Evidently, for this kind of model, inequality ͑8͒ is valid at all ͩ ␩ ͪ ␩ 2,1 2,1 levels of inefficiency. But we have discarded all events but coincidences in the correlations, so quantum mechanics with This means that detector inefficiency would violate inequality ͑8͒. Thus, to exploit the ‘‘loophole’’ in the Bell inequality, the nondetec- 2 ␦ϭ inf PABЈ͑⌳CDЈ͒у3Ϫ , ͑11͒ tions must be distributed in such a way that the ensemble ␩2,1 changes. They cannot be simple independent errors, but must A,BЈ,C,DЈ be included in the model at a deeper level. and when the right-hand side is non-negative (␩2,1у2/3), the ͑b͒ part follows from Theorem 2. Now, to prove ͑a͒ the IV. MEASUREMENT INEFFICIENCY same approach gives Ͼ In Theorem 2, ␦ 3/4 is sufficient for a violation to occur 1 from the quantum-mechanical correlation. But experiment ␩2,1ϭ inf PA͑i͒͑⌳B͑ j͒͒у2Ϫ . ͑12͒ does not yield an estimate of ␦ easily, so an inequality in- A,B,iÞ j ␩1 57 BELL’S INEQUALITY AND DETECTOR INEFFICIENCY 3307

If ␩1у3/4, the above inequality yields ␩2,1у2/3, and ͑a͒ (b) Coincidence efficiency. then follows from ͑b͒. ᭿ The two different definitions of measurement efficiency ͉EACЈ͑ACЈ͒ϪEADЈ͑ADЈ͉͒ϩ͉EBCЈ͑BCЈ͒ϩEBDЈ͑BDЈ͉͒ are motivated by models for which ␩1Þ␩2,1 ͑see, e.g., Ref. 4 ͓10͔, Chap. 6͒, and as these results are aimed to be as general р Ϫ2. as possible, the definitions do not need auxiliary assump- ␩2,1 tions. There is also the fact that it is possible to estimate ␩2,1 The proof is simply to apply the inequalities ͑11͒ and ͑12͒ to from coincidence data, whereas ␩1 is more difficult to ob- Theorem 4. The ‘‘coincidence efficiency’’ part in ͑b͒ is simi- tain. A quantum-mechanical violation of these inequalities is lar to the generalization presented Ref. ͓9͔ and the bound is obtained when ␩ Ͼ9/10 or ␩ Ͼ8/9Ϸ0.889. Both of these 1 2,1 of the same size, ␩2,1Ͼ2(&Ϫ1)Ϸ0.828. In Ref. ͓9͔, the bounds are higher than the previous bound in Ref. ͓9͔. assumptions of independent errors and constant detector efficiency are used, and then ␩2,1ϭ␩1ϭ␩, and Corollary 5 yields the same bound as in Ref. ͓9͔, ␩Ͼ2(&Ϫ1) V. PREVIOUS RESULTS Ϸ0.828. Note that these assumptions are not needed when The reason for the bound to be higher than previously is using the formal detector efficiency definitions above. that quantum mechanics violates the CHSH inequality more strongly than Theorem 1. Using this inequality, the following VI. CONCLUSIONS is obtained. The original Bell inequality is possible to generalize itself Theorem 4 (the CHSH inequality with ensemble change). to the inefficient case, although the bound on the inefficiency The following three prerequisites are assumed to hold except is not as low as obtained from the CHSH inequality. This is at a P-null set. because of the stronger violation of the CHSH inequality by (i) Realism. As in Theorem 2͑i͒. quantum mechanics. The generalization of the CHSH in- (ii) Locality. As in Theorem 2͑ii͒. equality in Ref. ͓9͔ is possible to obtain in this approach, and (iii) Measurement result restriction. The results may only the inefficiency bound is the same, ␩Ͼ82.8%. range from Ϫ1toϩ1, The formal definition of the term ‘‘measurement ineffi- Vϭ͕x෈R;Ϫ1рxрϩ1͖. ciency’’ in this paper uses no auxiliary assumptions such as constant efficiency or independent errors. It is nevertheless With ␦ as in Theorem 2, this yields possible to obtain the bounds. An estimate of the coincidence efficiency is possible to extract from the coincidence data, so ͉EACЈ͑ACЈ͒ϪEADЈ͑ADЈ͉͒ϩ͉EBCЈ͑BCЈ͒ϩEBDЈ͑BDЈ͉͒ that an easy check of the bound ␩2,1Ͼ82.8% is possible. ‘‘Change of ensemble’’ is an essential property needed in р4Ϫ2␦. the local hidden-variable model to enable it to approach the The proof is similar to that of Theorem 2. quantum-mechanical correlation. If the model does not have A quantum-mechanical violation of this inequality would this property, the generalized inequality reduces to the inequality ͑8͒, which resembles the original Bell inequality. demand ␦Ͼ2Ϫ&Ϸ0.586. This is, as expected, significantly lower than obtained from Theorem 2. The measurement in- The quantum-mechanical predictions including measurement efficiency result is as follows. inefficiency violate that inequality as strongly as quantum Corollary 5 (the CHSH inequality with measurement in- mechanics without measurement inefficiency violates the original Bell inequality. A quantitative bound on the change efficiency). Assume that Theorem 4͑i͒–͑iii͒ hold except on a of ensemble for quantum-mechanical violation in Theorem 2 P-null set, and define ␩ and ␩ as in Corollary 3. Then, 1 2,1 ͑generalized Bell inequality͒ is ␦Ͼ75% and in Theorem 4 we get the following. ͑generalized CHSH inequality͒, ␦Ͼ58.6%. (a) Detector efficiency. ACKNOWLEDGMENT ͉EACЈ͑ACЈ͒ϪEADЈ͑ADЈ͉͒ϩ͉EBCЈ͑BCЈ͒ϩEBDЈ͑BDЈ͉͒ The author would like to thank R. Riklund for inspiring 2 lectures on the foundations of quantum mechanics and for р . 2␩1Ϫ1 discussions on the subject.

͓1͔ J. S. Bell, Physics ͑Long Island City, NY͒ 1, 195 ͑1964͒. ͓5͔ A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49,91 ͓2͔ A. Einstein, N. Rosen, and B. Podolsky, Phys. Rev. 47, 777 ͑1982͒. ͑1935͒. ͓6͔ A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, ͓3͔ D. Bohm and Y. Aharonov, Phys. Rev. 108, 1070 ͑1957͒. 1804 ͑1982͒. ͓4͔ A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47, 460 ͓7͔ J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 ͑1974͒. ͑1981͒. ͓8͔ A. Shimony, Search for a Naturalistic World View ͑Cambridge 3308 JAN-A˚ KE LARSSON 57

University Press, Cambridge, 1993͒, Vol. II. ͓11͔ J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. ͓9͔ A. Garg and N. D. Mermin, Phys. Rev. D 35, 3831 ͑1987͒. Rev. Lett. 23, 880 ͑1969͒. ͓10͔ T. Maudlin, Quantum Non-Locality and Relativity ͑Blackwell ͓12͔ K. L. Chung, A Course in Probability Theory, 2nd ed. ͑Aca- Publishers, Oxford, 1994͒. demic Press, London, 1974͒.