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 1997! 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 en- semble’’ will be presented, providing both qualitative and quantitative information on the nature of the ‘‘loop- hole’’ 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 assump- tion of independent errors. @S1050-2947~98!07405-8# PACS number~s!: 03.65.Bz I. INTRODUCTION suppressing the parentheses in the last expression ~this will be done in the following when no confusion can arise!. Fur- The Bell inequality @1# 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- @2,3# 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 (A8), so that E(AB8) is the ments that have been performed to verify this have not been correlation between A and B8. 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. @4–6#, for instance!. 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. @7–10#, among others!, and the main issue is to obtain probability theory, using ~two families of! RV’s, a limit for the detector inefficiency, but previously inequali- ties other than the original Bell inequality had to be used, for A~a,b!: L V → !A~a,b,lۋexample, the Clauser-Horne inequality in @7#, which in itself l contains the case of inefficient detectors, or the Clauser- ;a,b. B8~a,b!: L V Horne-Shimony-Holt ~CHSH! inequality first presented in → !B8~a,b,lۋwhich is generalized to the inefficient case in @9#. l ,11#@ 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. @12#!. The sample of the detector orientation at the other particle, space L is the mathematical analog to the state space used in physics, and a sample l is a point in that space correspond- def ing to a certain value of the ‘‘hidden variable.’’ The mea- A~a,l!5A~a,b,l! independently of b, surement results are described by random variables ~RV’s! X(l), which take their values in the value space V. def To be a probabilistic model, a probability measure P on B8~b,l!5B8~a,b,l! independently of a. the space L is needed, by which we can define the expecta- tion value E as (iii) Measurement result restriction. Only the results 11 and 21 should be possible: def E~X!5 X~l!dP~l!5 XdP, ~1! V5$21,11%. EL EL (iv) Perfect anticorrelation. A measurement with equally oriented detectors must yield opposite results at the two de- *Electronic address: [email protected] tectors, 1050-2947/98/57~5!/3304~5!/$15.0057 3304 © 1998 The American Physical Society 57 BELL’S INEQUALITY AND DETECTOR INEFFICIENCY 3305 A52A8, ;a,l. (i) Realism. Measurement results can be described by probability theory, using ~two families of! RV’s, which may Then, be undefined on some part of L, corresponding to measure- ment inefficiency, uE~AB8!2E~AC8!u<11E~BC8!. A~a,b!:L ~a,b! V To include detector inefficiency in the above inequality, A → !A~a,b,lۋl previously two approaches have been used. The first is to use ;a,b. B8~a,b!:L ~a,b! V probabilities instead of correlations and derive an inequality B8 → !B8~a,b,lۋon the probabilities ~see Ref. @7#!. The second is to assign the l measurement result 0 ~zero! to an undetected particle, which makes the original Bell inequality inappropriate because of (ii) Locality. A measurement result should be independent prerequisite ~iii! in Theorem 1. Thus the CHSH inequality of the detector orientation at the other particle, must be used and subsequently generalized to obtain an in- equality valid in this case ~see Ref. @9#!. def def A third approach, presented here, uses correlations but A~a,l!5A~a,b,l! on LA~a!5LA~a,b! makes no assignment of any measurement result to the un- detected 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, 5B a,b, on b 5 a,b BELL INEQUALITY 8~ l! 8~ l! LB8~ ! LB8~ ! The measurement results are modeled as RV’s, which in independently of a. the ideal case would be defined as in prerequisite ~i! in Theo- rem 1. In the case with inefficiency the situation is quite (iii) Measurement result restriction. Only the results 11 different, as there are now points of L where the particle and 21 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 V5$21,11%. subsets of L, which will be denoted LA(a,b) and LB8(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, A52A8,onLAA85LAùLA8 . def Define the bound on the ensemble change when the measure- EX~X!5 E XdPX , where PX~E!5P~EuLX!. ~2! ment setup is changed as LX The correlation is in this case E (AB ), the expectation of d5 inf PAB ~LCD ! ~ 0<d<1!. AB8 8 8 8 ) AB8 on the set at which both factors in the product are de- A,B8,C,D8 fined, the set LAB85LAùLB8 . 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 uEAB8~AB8!2EAC8~AC8!u<322d1EBC8~BC8!. l’s in LAB8 , so the ensemble is restricted from L to LAB8 . 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 LABB8C8 , on which all the RV’s A, B, B8, and C8 are de- uEAB8~AB8!2EAC8~AC8!u fined. This is to avoid the problem mentioned above. This ensemble may be empty, but only when d50 and then the 5 AB8dPAB82 AC8dPAC8 . ~3! inequality is trivial, so d.0 can be assumed in the rest of the UEL EL U AB8 AC8 proof. Now ~i!–~iv! yields The integrals on the right-hand side cannot easily be added E AB 2E AC 11E BC . when L Þ⌳ , so a generalization of Theorem 1 is u ABB8C8~ 8! ABB8C8~ 8!u< ABB8C8~ 8! AB8 AC8 ~4! 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 EAB8(AB8) and so on.