From: danforth@riacs.edu (Douglas G. Danforth)
Subject: RESULTS: Bell dethrowned
Keywords: EPR, Quantum mechanics, Bell, Aspect, Locality, efficiency
Organization: RIACS, NASA Ames Research Center
Date: Mon, 27 Jul 92 04:13:10 GMT


In the spirit of John Baez's recent post asking for more real news I offer
the following preprint.

I wish to publicly thank Bill Taylor (wft@math.canterbury.ac.nz) for his Jun
4th posting which crystalized my thinking on this subject.

Comments, suggestions, and criticisms are welcomed.

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    <<DRAFT>> <<DRAFT>> <<DRAFT>> <<DRAFT>> <<DRAFT>> <<DRAFT>> <<DRAFT>>
     
     A LOCAL HIDDEN VARIABLES MODEL THAT EXPLOITS DETECTOR INEFFICIENCY
	 WHICH EXACTLY REPRODUCES THE QUANTUM MECHANICAL PREDICTION
			FOR TWO PARTICLE CORRELATIONS
			     
			     
			     Douglas G. Danforth
				July 24, 1992
 
 
 
I. INTRODUCTION

J. S. Bell [1] derived an inequality which any local theory of quantum
mechanics must satisfy. The inequality is incompatible with standard quantum
mechanical predictions and with experiment [2].  His conclusion was that
any theory that reproduces exactly the quantum mechanical predictions must
have a grossly nonlocal structure.

It is claimed here that Bell's inequality does not represent all local
quantum theories due to the loophole of detector efficiency. An explicit
model is herein constructed that does reproduce exactly the quantum
mechanical predictions. The detector efficiency of the model is 4/(pi+2)
(77.8%).


II. PRELIMINARIES

Bell introduces the quantities A(a,v), B(b,v) to represent the results of
measurements by detector A with orientation 'a' and hidden variables v and
detector B with orientation 'b' and hidden variables v. His equation (1)
states 

        A(a,v) = +-1, B(b,v) = +-1.                                     (1)

The hidden variables are assumed to be governed by a probability density
function f(v) which is normalized,

        /dv f(v) = 1.                                                   (2)

The expected correlation between measurements A and B is defined as:

	P(a,b) = /dv f(v) A(a,v)B(b,v).					(3)

Aspect [2] defines a quantity E(a,b) to estimate P(a,b) using detector counts
R(a,b) as:
		 
		 R++(a,b) + R--(a,b) - R+-(a,b) - R-+(a,b)
	E(a,b) = -----------------------------------------
		 R++(a,b) + R--(a,b) + R+-(a,b) + R-+(a,b)



He finds the following constraints satisfied by his measurements:

	     R++(a,b) + R+-(a,b) = R-+(a,b) + R--(a,b) = constant	(4)
	
	     R++(a,b) + R+-(a,b) + R-+(a,b) + R--(a,b) = constant	(5)

That is, the marginal counts (marginal given joint detection) for detector
A are independent of angle 'a' and equal to the marginal counts of 
detector B which is independent of angle 'b'. Secondly, he finds the total
joint counts to be constant and independent of detector angles.

The experimental results and the standard quantum mechanical prediction 
agree to high accuracy and give a cos2(a-b) dependency:

	E(a,b) = cos2(a-b).						(6)

To be in accord with experiment, any local quantum mechanical model must
satisfy (4,5, and 6).


III. CROWN MODEL

Let the hidden variables form a Euclidean 3-space. Picture in this space a
King's crown; flat base and scalloped top. The crown is partitioned into 4
equal quadrants with alternating colors of red and green; red, green,
red, green, around the crown. The partition lines run vertically from base
to top. There are four identical scallops, one in each quadrant. The lowest
point of each scallop is halfway between the base and the highest point of
the scallops. The equation for the form of a scallop is given by the function
g where,
	       
	       1
	g(x) = - sin2x,      0 <= x <= pi/2 .				(7)
	       8

The crown represents the results of measurments for detector A at
a fixed orientation.  As the detector rotates the crown rotates in
accord, around its vertical axis of symmetry (the base stays level).

Points in this hidden variable space outside the crown represent 
events of nondetection.  Red points represent the detection of a
particle with + polarization.  Green points represent detection of
a particle with - polarization.

Detector B is constructed identically to A except that it is flipped
upside down. Take these two crowns and superimpose them so they have
the same axis of symmetry with base touching top and top touching base.

The resultant composite is a band (the hidden variables density function, f,
is uniform over this band). The scallops of the crowns fall in the
unscalloped region of the other crown. When the detector angles are equal,
a=b, the crowns align with red overlapping only red and green overlapping
only green.


The choice of the function g is dictated by the way a partition line
cuts across the function as one crown rotates. A base region forms a
rectangle on top of g.  When aligned all of g is enclosed.  When partially
aligned the area of g enclosed is the integral under the curve of g within
the enclosure.  Let t=a-b then  
	 
	 pi/2	         2
	/dx sin(2x) = cos(t)					     
       t
									(8)
	 t   	         2
	/dx sin(2x) = sin(t),					   
       0

(note the symmetry in the constuction so that one obtains the same result
whether a>b or a<b). Since red and green regions are 90 degrees apart a
decrease of overlap in one causes and increase in overlap of the other but
shifted by 90 degrees. If red-red and green-green overlap is weighted
positively and red-green and green-red overlap is weighted negatively
then the weighted integral of the 8 (top and bottom) overlap regions is
precisely the average correlation function defined by Bell, namely

	A(a,v) = region of crown A (with signs)
	
	B(b,v) = region of crown B (with signs)

	f(v)   = uniform over band (A UNION B).

	P(a,b) = /dv f(v) A(a,v) B(b,v) 

                   pi/2        a-b    
               = 8/dx g(x) - 8/dx g(x)
                a-b          0
                    
                    2          2
               = cos(a-b) - sin(a-b) 

               = cos2(a-b).

The crown model exactly reproduces the quantum mechanical predictions. It
also satisfies (4) and (5).

IV. EFFICIENCY

If p is the probability that a particle incident on a detector will be
observed and N events occur then K = Np is the expected number of
observations for a single detector.  If whether a particle is observed at
detector A is a process independent of whether a particle is observed at
detector B then the probability that both detectors will observe a particle
given that one is present at each detector is p^2.  For N events the
expected number of joint observations is J = Np^2.  From J and K one can
estimate the efficiency of a detector to be

	p = J / K	(estimated detector efficiency).
 

For the crown model, the probability of an observation at a single detector 
is equal to the area of a crown (band area equals 1). This area is (1/2 +
1/pi) (81.8%). The probability of a joint detection is equal to the area
under the scallops (from the scallop curve to the mid line of the crown).
This area is 2/pi (63.6%).  The ratio of these two numbers provides a
measured estimate of the detector efficiency
		     
		     		          2/pi         4
	measureable crown  efficiency = ---------- = ------ = 77.5%.
		  		        (1/2+1/pi)   (pi+2)


V. DETECTION CORRELATIONS

Several comments need to be made.  The actual probability of detecting a
particle (81.8%) is not the same as the measureable efficiency (77.5%).
This is due to the fact that the assumption of independence of observations
between the detectors is false for the crown model.  Nothing mysterious
causes this detection correlation.  It is a local phenomena.  The information
can be carried in the hidden variables in exactly the same way that the
polarization of the particles is carried from the source to the detectors,
thereby inducing correlations in polarization measurements.


VI. DISCUSSION

A local hidden variable model has been introduced that exactly reproduces
the quantum mechanical predictions for two particle polarization
correlations.  

The model does not violate Bell's inequality simply because Bell's assuptions
do not apply to this model. Bell assumed 100% efficiency. The crown model is
77.8% efficient.

With perfect detectors one is forced to assume that A(a,v) = B(a,v). This is
not true for imperfect detectors since the functions A and B differ on points
v where partial detection takes place.

Since imperfect detection occurs one can no longer assume B^2=1 as Bell
does in the derivation of his inequality (futher discussion is necessary
on this point but will not be carried out here).

The crown model is a member of a class of models in n-dimensional hidden
variable space where thick loops of variable cross section slide within tubes
of fixed cross section. All that is needed to satisfy the quantum mechanical
correlation function is that the integrated volume within a quadrant along a
loop obeys a cos^2 law. All members of this class have the efficiency of the
crown model.

It is an open question whether other classes exist with higher efficiencies.


CONCLUSION

Neither theory [1] nor experiment [2] has ruled out the possibility of local
realisitic models of quantum mechanics due to detector inefficiency. Aspect's
[2] experiment fails to meet the perfect detector assumption used by Bell [1]
and so any conclusions based on Aspect's results are inappropriate.

The discovery of a class of models with 77.8% efficiency which exactly
reproduces the quantum mechanical results shows that further theoretical work
is needed to place an upper bound on the efficiency of local realistic models
capable of reproducing the quantum mechanical predictions. Once this has been
accomplished experimentation can begin again to determine whether nature
violates locality.



REFERENCES

[1] Bell, J.S. (1964). On the Einstein Podolsky Rosen Paradox. Physics
    Vol. I, No. 3, pp. 195-200.

[2] Aspect, A., P. Grangier, and G. Roger (1982). Experimental Realization 
    of Einstein-Podolsky-Rosen-Bohm, Gedankenexperiment: A New Violation of 
    Bell's Inequalities.  Physical Review Letters, Vol. 49, No. 2, pp. 91-94.