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Jan-Åke Larsson

University Institutional Repository (DiVA):

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/

Bell's inequality and detector inef®ciency

Jan-AÊke Larsson*

Department of Mathematics, LinkoÈping University, S-581 83 LinkoÈping, Sweden ~Received 9 December 1997!

In this paper, a method of generalizing the Bell inequality is presented that makes it possible to include

detector inef®ciency 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 ef®ciency 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 ef®ciency, or even the assump-

tion of independent errors.@S1050-2947~98!07405-8#

PACS number~s!: 03.65.Bz

I. INTRODUCTIONThe Bell inequality@1#and its descendants have been the main argument on the Einstein-Podolsky-Rosen paradox @2,3#for the past 30 years. A new research ®eld of ``experi- mental metaphysics'' has formed, where the goal is to show that the concept of local realism is inconsistent with quantum mechanics, and ultimately with the real world. The experi- ments that have been performed to verify this have not been completely conclusive, but they point quite decisively in a certain direction: Nature cannot be described by a realistic local hidden-variable theory~see Refs.@4±6#, for instance!. The reason for saying ``not been completely conclusive'' is that there is an implicit assumption in the proof of the Bell inequality that the detectors are 100% effective. There has been considerable discussion in the literature on this~see Refs.@7±10#, among others!, and the main issue is to obtain a limit for the detector inef®ciency, but previously inequali- ties other than the original Bell inequality had to be used, for example, the Clauser-Horne inequality in@7#, which in itself contains the case of inef®cient detectors, or the Clauser- Horne-Shimony-Holt~CHSH!inequality ®rst presented in @11#, which is generalized to the inef®cient case in@9#. Since a hidden-variable model is really a probabilistic model, formalism and terminology from probability theory will be used in this paper~see, e.g., Ref.@12#!. The sample space Lis the mathematical analog to the state space used in physics, and a samplelis a point in that space correspond- ing to a certain value of the ``hidden variable.'' The mea- surement results are described by random variables~RV's! X(l), which take their values in the value spaceV. To be a probabilistic model, a probability measurePon the spaceLis needed, by which we can de®ne the expecta- tion valueEas E~X!5 def E L

X~l!dP~l!5

E LXdP,~1!suppressing the parentheses in the last expression~this will be done in the following when no confusion can arise!. Fur- thermore,a,b,... are thedetector orientations used in the various measurements, and to shorten the presentation a no- tation will be used whereA,B,... are theRV's correspond- ing to the above orientations. The RV's describe measure- ment results from one detector if it is unprimed (A), and from the other when primed (A

8), so thatE(AB8) is the

correlation betweenAandB 8. I have chosen the ``deterministic'' case here, and will not discuss the ``stochastic'' case as the generalization is straightforward. In this formalism, the Bell inequality can be stated as follows. Theorem 1 (the Bell inequality).The following four pre- requisites are assumed to hold except at a null set. (i) Realism.Measurement results can be described by probability theory, using~two families of!RV's,

A~a,b!:L!V

l°A ~a,b,l!

B8~a,b!:L!V

l

°B8~a,b,l!

;a,b. (ii) Locality.A measurement result should be independent of the detector orientation at the other particle, A ~a,l!5 def

A~a,b,l!independently ofb,

B

8~b,l!5

def

B8~a,b,l

!independently ofa. (iii) Measurement result restriction.Only the results11 and21 should be possible: V5 $21,11%. (iv) Perfect anticorrelation.A measurement with equally oriented detectors must yield opposite results at the two de- tectors, *Electronic address: jalar@mai.liu.sePHYSICAL REVIEW AMAY 1998VOLUME 57, NUMBER 5 57

1050-2947/98/57~5!/3304~5!/$15.00 3304 © 1998 The American Physical Society

A52A8,;a,l.

Then, uE~AB8!2E~AC8!u<11E~BC8!. To include detector inef®ciency in the above inequality, previously two approaches have been used. The ®rst is to use probabilities instead of correlations and derive an inequality on the probabilities~see Ref.@7#!. The second is toassignthe measurement result 0~zero!to an undetected particle, which makes the original Bell inequality inappropriate because of prerequisite~iii!in Theorem 1. Thus the CHSH inequality must be used and subsequently generalized to obtain an in- equality valid in this case~see Ref.@9#!. A third approach, presented here, uses correlations but makes no assignment of any measurement result to the un- detected particles. Thus it is possible to obtain a direct gen- eralization of the original Bell inequality.

II. GENERALIZATION OF THE ORIGINAL

BELL INEQUALITY

The measurement results are modeled as RV's, which in the ideal case would be de®ned as in prerequisite~i!in Theo- rem 1. In the case with inef®ciency the situation is quite different, as there are now points ofLwhere the particle would be undetected. To avoid the quite arbitrary assignment used in the second approach above, the RV's will be said to beunde®nedat these points, i.e., they will only be de®ned at subsets ofL, which will be denotedL A (a,b) andL B 8 (a,b), respectively. In this setting, a new expression for the expectation value is needed. The averaging must be restricted to the set where the RV in question is de®ned, and the probability measure adjusted accordingly, E X ~X!5 def E L X XdP X , whereP X ~E!5P~EuL X !.~2!

The correlation is in this caseE

AB 8 (AB8), the expectation of AB

8on the set at which both factors in the product are de-

®ned, the setL

AB 8 5L A ùL B 8 . This is the correlation that would be obtained from an experimental setup where the coincidence counters are told to ignore single particle events. In an experiment the pairs that are detected are the ones with l's inL AB 8 , so the ensemble is restricted fromLtoL AB 8 It is now easy to see what makes the proof of the original Bell theorem break down. The start of the proof is uE AB 8 ~AB8!2E AC 8 ~AC8!u 5 U E L AB8 AB8dP AB 8 2 E L AC8 AC8dP AC 8 U .~3! The integrals on the right-hand side cannot easily be added whenL AB 8 ÞL AC 8 , so a generalization of Theorem 1 is needed. Theorem 2 (the Bell inequality with ensemble change). The following four prerequisites are assumed to hold except at aP-null set.(i) Realism.Measurement results can be described by probability theory, using~two families of!RV's, which may be unde®ned on some part ofL, corresponding to measure- ment inef®ciency, A ~a,b!:L A ~a,b! l B

8~a,b!:L

B 8 ~a,b!quotesdbs_dbs27.pdfusesText_33

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