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for Bob”Byes (click) or Bno (no click)”constitute a measurement. According to von
Neumann™s projection postulate, we must then replace the original state |Ψ’ by the
reduced state |Ψ’ rslt , to ¬nd

p(Ayes |», ±, β, Brslt ) ∝ rslt Ψ’ a† A ± akA ± Ψ’ . (19.49)
k rslt


The reduced state for either of Bob™s outcomes can be constructed by inverting
Bob™s version of eqn (19.45) to express the creation operators in the (h, v)-basis in
terms of the creation operators in the β, β -basis:
¿
Quantum theory versus local realism


a† B β
a† B h ’ sin β
cos β k
k = . (19.50)
a† β
a† B v ’ sin β cos β
k kB


Using this in the de¬nition (19.44) exhibits the original states as superpositions of
states containing β-polarized photons and states containing β-polarized photons.
For the outcome Byes ”Bob heard a click”the projection postulate instructs us
to drop the states containing the β-polarized photons, since they are blocked by the
polarizer. This produces the reduced state
1
Ψ’ = √ {sin β |hA , βB ’ cos β |vA , βB } , (19.51)
yes
2

where |βB = a† B β |0 . Substituting this into eqn (19.49) leads”by way of the calcu-
k
lation in Exercise 19.3”to the simple result

p(Ayes |», ±, β, Byes ) ∝ sin2 (± ’ β) . (19.52)

For the opposite outcome, Bno , the projection postulate tells us to drop the states
containing β-polarized photon states instead, and the result is

p(Ayes |», ±, β, Bno ) ∝ cos2 (± ’ β) . (19.53)

The conclusion is that quantum theory violates outcome independence, since the
probability that Alice hears a click depends on the outcome of Bob™s previous mea-
surement. The fact that Alice™s probabilities only depend on the di¬erence in polarizer
settings follows from the assumption that the source produces the special state |Ψ’ ,
which is invariant under rotations around the common propagation axis.
The violation of outcome independence implies that the two sets of experimental
outcomes must be correlated. The probability that both detectors click is proportional
to the coincidence-count rate, which”as we learnt in Section 9.2.4”is determined by
the second-order Glauber correlation function; consequently,

p Ayes , Byes Ψ’ , ±, β ∝ G±β (r1 t1 ; r2 t2 )
(2)


∝ Ψ ’ a† A ± a† B β ak B β ak A ± Ψ ’ . (19.54)
k k

The techniques used above give

p Ayes , Byes Ψ’ , ±, β = sin2 (± ’ β)
11
= ’ cos(2± ’ 2β) , (19.55)
22
which describes an interference pattern, e.g. if β is held ¬xed while ± is varied. Fur-
thermore, this pattern has 100% visibility, since perfect nulls occur for the values
± = β, β + π, β + 2π, . . ., at which the planes of polarization of the two photons are
parallel. The surprise is that an interference pattern with 100% visibility occurs in the
(2) (1) (1)
second-order correlation function G±β while the ¬rst-order functions G± and Gβ
display zero visibility, i.e. no interference at all.
Bell™s theorem and its optical tests

19.5.2 Quantum theory violates Bell™s theorem
The results (19.52), (19.53), and (19.55) show that quantum theory violates outcome
independence and the strong-separability principle; consequently, quantum theory does
not satisfy the hypothesis of Bell™s theorem. Nevertheless, it is still logically possible
that quantum theory could satisfy the conclusion of Bell™s theorem, i.e. the inequality
(19.37). We will now dash this last, faint hope by exhibiting a speci¬c example in
which the quantum prediction violates the Bell inequality (19.38).
For the experiment depicted in Fig. 19.3, let us now calculate what quantum theory
predicts for S (») when » is represented by the Bell state |Ψ’ . For general parameter
settings ± and β, the de¬nition (19.31) for Bell™s joint expectation value can be written
as

E(±, β) = pee (±, β)Ae Be + peo (±, β)Ae Bo + poe (±, β)Ao Be + poo (±, β)Ao Bo , (19.56)

where we have omitted the »-dependence of the expectation value, and adopted the
simpli¬ed notation
pmn (±, β) ≡ p(Am , Bn |», ±, β) (19.57)
for the joint probabilities.
In Exercise 19.4, the calculation of the probabilities is done by using the techniques
leading to eqn (19.55), with the result
1
sin2 (± ’ β) ,
pee (±, β) = poo (±, β) = (19.58)
2
1
cos2 (± ’ β) .
peo (±, β) = poe (±, β) = (19.59)
2
After combining these expressions for the probabilities with the de¬nition (19.11) for
the outcome parameters, Bell™s joint expectation value (19.56) becomes

E(±, β) = sin2 (± ’ β) ’ cos2 (± ’ β) = ’ cos (2± ’ 2β) . (19.60)

Our objective is to choose values (±1 , β1 , ±2 , β2 ) such that S violates the inequality
|S| 2. A set of values that accomplishes this,

±1 = 0—¦ , ±2 = 45—¦ , β1 = 22.5—¦ , β2 = ’22.5—¦ , (19.61)

is illustrated in Fig. 19.6.

±

β
Fig. 19.6 A choice of angular settings
± = 
±1 , ±2 , β1 , β2 in the calcite-prism-pair experi-
ment (see Fig. 19.3) that maximizes the viola-
tion of Bell™s bounds (19.42) by the quantum
β
theory.
Quantum theory versus local realism

For these settings, the expectation values are given by
1
E(±1 = 0, β1 = 22.5—¦ ) = ’ cos (45—¦ ) = ’ √ ,
2
1
E(±1 = 0, β2 = ’22.5—¦) = ’ cos (’45—¦ ) = ’ √ ,
2
(19.62)
1
—¦ —¦ —¦
E(±2 = 45 , β1 = 22.5 ) = ’ cos (’45 ) = ’ √ ,
2
1
E(±2 = 45—¦ , β2 = ’22.5—¦) = ’ cos (’135—¦) = + √ ,
2
√ √
so that S = ’2 2. This violation of the bound |S| 2 by a factor of 2 shows that
quantum theory violates the Bell inequality (19.38) by a comfortable margin.

Motivation for the de¬nition of the sum S
19.5.3
What motivates the choice of four terms and the signs (+, +, +, ’) in eqn (19.35)?
The answers to this question now becomes clear in light of the above calculation. The
independent observers, Alice and Bob, need to make two independent choices in their
respective parameter settings ± and β, in order to observe changes in the correlations
between the polarizations of the photons γA and γB . This explains the four pairs of
parameter settings appearing in the de¬nition of S, and pictured in Fig. 19.4.
The motivation for the choice of signs (+, +, +, ’) in S can be explained by refer-
ence to Fig. 19.6. Alice and Bob are free to choose the ¬rst three pairs of parameters
settings, (±1 , β1 ), (±1 , β2 ), and (±2 , β1 ), so that all three pairs have the same setting
di¬erence, 22.5—¦ , and negative correlations. In the quantum theory calculation of S for

the Bell state |Ψ’ , these choices yield the same negative correlation, ’1/ 2, since
the expectation values only depend on the di¬erence in the polarizer settings.
By contrast, the fourth pair of settings, (±2 , β2 ), describes the two angles that are
the farthest away from√ each other in Fig. 19.6, and it yields a positive expectation
value E(±2 , β2 ) = +1/ 2. This arises from the fact that, for this particular pair of
angles (±2 = 45—¦ , β2 = ’22.5—¦), the relative orientations of the planes of polarization
of the back-to-back photons γA and γB are almost orthogonal. The opposite sign of
this expectation value compared to the ¬rst three can be exploited by deliberately
choosing the opposite sign for this term in eqn (19.35). This stratagem ensures that
all four terms contribute with the same sign, and this gives the best chance of violating
the inequality.
It should be emphasized that the violation of this Bell inequality by quantum
theory is not restricted to this particular example. However, it turns out that this
special choice of angular settings de¬nes an extremum for S in the important case
of maximally entangled states. Consequently, these parameter settings maximize the
quantum theory violation of the Bell inequality (Su and W´dkiewicz, 1991).
o
Bell™s theorem and its optical tests

19.6 Comparisons with experiments
19.6.1 Visibility of second-order interference fringes
For comparison with experiments with two counters, such as the one sketched in Fig.
19.5, the visibility of the second-order interference fringes observed in coincidence
detection can be de¬ned”by analogy to eqn (10.26)”as
(2) (2)
’ G±β
G±β
V≡ max min
, (19.63)
(2) (2)
G±β + G±β
max min

(2) (2)
where G±β max and G±β min are respectively the maximum and minimum, with respect
to the angles ± and β, of the second-order Glauber correlation function. Let us assume
that data analysis shows that an empirical ¬t to the second-order interference fringes
has the form
(2)
G±β ∝ 1 ’ · cos (2± ’ 2β) , (19.64)
for some value of the ¬tting parameter ·. Given appropriate assumptions about the
curve-¬tting technique, one can show that

·=V. (19.65)

The physical meaning of a high, but imperfect (V < 1), visibility is that decoherence
of some sort has occurred between the two photons γA and γB during their propagation
from the source to Alice and Bob. Thus the entangled pure state emitted by the source
changes, for either fundamental or technical reasons, into a slightly mixed state before
arriving at the detectors.
Next, let us consider experiments with four counters, such as the one sketched in
Fig. 19.3. Again, using data analysis that assumes a ¬nite-visibility ¬tting parameter
·, the joint probabilities (19.58) and (19.59) have the following modi¬ed forms:
11
’ · cos(2± ’ 2β) ,
pee (±, β) = poo (±, β) = (19.66)
22
11
+ · cos(2± ’ 2β) ,
peo (±, β) = poe (±, β) = (19.67)
22
so that Bell™s joint expectation value becomes

E(±, β) = ’· cos(2± ’ 2β) . (19.68)

For the special settings in eqn (19.61), one ¬nds

4
S = ’ √ · = ’2 2· . (19.69)
2
This implies that the maximum amount of visibility Vmax permitted by Bell™s inequal-
ity |S| 2 is
1
Vmax = ·max = √ = 70.7% . (19.70)
2
Comparisons with experiments

19.6.2 Data from the tandem-crystal experiment violates the Bell
inequality |S| 2
For comparison with experiment, we show once again the data from the tandem two-
crystal experiment discussed in Section 13.3.5, but this time we superpose a ¬nite-
visibility, sinusoidal interference-fringe pattern, of the form (19.64), with the maximum
visibility Vmax = 70.7% permitted by Bell™s theorem. This is shown as a light, dotted
curve in Fig. 19.7.
One can see by inspection that the data violate the Bell inequality (19.38) by many
standard deviations. Indeed, detailed statistical analysis shows that these data violate
the constraint |S| 2 by 242 standard deviations. However, this data exhibits a high
signal-to-noise ratio, so that systematic errors will dominate random errors in the data
analysis.

19.6.3 Possible experimental loopholes
A The detection loophole
Since the quantum e¬ciencies of photon counters are never unity, there is a possible
experimental loophole, called the detection loophole, in most quantum optical tests
of Bell™s theorem. If the quantum e¬ciency is less than 100%, then some of the photons
will not be counted. This could be important, if the ensemble of photons generated by
the source is not homogeneous. For example, it is conceivable”although far-fetched”
that the photons that were not counted just happen to have di¬erent correlations than
the ones that were counted. For example, the second-order interference fringes for the
undetected photons might have a visibility that is less than the maximum allowable
amount Vmax = 70.7%. Averaging the visibility of the undetected photons with the
visibility of the detected photons, which do have a measured visibility greater than
70.7%, might produce a total distribution which just barely manages to satisfy the
inequality (19.38).

2000
Singles (10 s)
Coincidences (10 s)




1500

1000

500

0 0
’45 45 135 225
0 90 180 270 315
θ2 (θ1 = ’45 )


Fig. 19.7 Data from the tandem-crystals experiment (discussed in Section 13.3.5) compared
to maximum-visibility sinusoidal interference fringes with Vmax = 70.7% (light, dotted curve),
which is the maximum visibility permitted by Bell™s theorem. (Adapted from Kwiat et al.
(1999b).)
Bell™s theorem and its optical tests

This scenario is ruled out if one adopts the entirely reasonable, fair-sampling as-
sumption that the detected photons represent a fair sample of the undetected photons.
In this case, the undetected photons would not have substantially distorted the ob-
served interference fringes if they had been included in the data analysis. Nevertheless,
the fair-sampling assumption is di¬cult to prove or disprove by experiment.
One way out of this di¬culty is to repeat the quantum optical tests of Bell™s
theorem with extremely high quantum e¬ciency photon counters, such as solid-state
photomultipliers (Kwiat et al., 1994). This would minimize the chance of missing any
appreciable fraction of the photons in the total ensemble of photon pairs from the
source. To close the detection loophole, a quantum e¬ciency of greater than 83% is
required for maximally entangled photons, but this requirement can be reduced to
67% by the use of nonmaximally entangled photons (Eberhard, 1993).
Replacing photons by ions allows much higher quantum e¬ciencies of detection,
since ions can be detected much more e¬ciently than photons. In practice, nearly
all ions can be counted, so that almost none will be missed. An experiment using
entangled ions has been performed (Rowe et al., 2001). With the detection loophole
closed, the experimenters observed an 8 standard deviation violation of the Clauser“
Horne“Shimony“Holt inequality (Clauser et al., 1969)

|E (±1 , β1 ) + E (±2 , β1 )| + |E (±1 , β2 ) ’ E (±2 , β2 )| 2. (19.71)

This is one of several experimentally useful Bell inequalities that are equivalent in
physical content to the condition |S| 2 discussed above.

B The locality loophole
Another possible loophole”which is conceptually much more important than the ques-
tion of detector e¬ciency”is the locality loophole. Closing this loophole is especially
vital in light of the incorporation of the extremely important Einsteinian principle of
locality into Bell™s theorem.
Since photons travel at the speed of light, they are much better suited than atoms or
ions for closing the locality loophole. Using photons, it is easy to ensure that Alice™s and
Bob™s decisions for the settings of their parameters ± and β are space-like separated,
and therefore truly independent.
For example, Alice and Bob could randomly and quickly reset ± and β during the
time interval after emission from the source and before arrival of the photons at their
respective calcite prisms. There would then be no way for any secret machinery at
the source to know beforehand what values of ± and β Alice or Bob would eventu-
ally decide upon for their measurements. Therefore, properties of the photons that
were predetermined at the source could not possibly in¬‚uence the outcomes of the
measurements that Alice and Bob were about to perform.
The ¬rst attempt to close the locality loophole was an experiment with a separation
of 12 m between Alice and Bob. Rapidly varying the settings of ± and β, by means of
two acousto-optical switches (Aspect et al., 1982), produced a violation of the Clauser“
Horne“Shimony“Holt inequality (19.71) by 6 standard deviations.
However, the time variation of the two polarizing elements in this experiment was
periodic and deterministic, so that the settings of ± and β at the time of arrival of the
Comparisons with experiments

photons could, in principle, be predicted. This would still allow the properties of the
photons that led to the observed outcomes of measurements to be predetermined at
the source.
A more satisfactory experiment vis-`-vis closing the locality loophole was per-
a
formed with a separation of 400 m between the two polarizers. Two separate, ultrafast
electro-optic modulators, driven by two local, independent random number genera-
tors, rapidly varied the settings of ± and β in a completely random fashion. The result
was a violation of the Clauser“Horne“Shimony“Holt inequality (19.71) by 30 standard
deviations.
The two random number generators operated at the very high toggle frequency
of 500 MHz. After accounting for various extraneous time delays, the experimenters
concluded that no given setting of ± or β could have been in¬‚uenced by any event that
occurred more than 0.1 µs earlier, which is much shorter than the 1.3 µs light transit
time across 400 meters.
Hence the locality loophole was ¬rmly closed. However, the detection loophole was
far from being closed in this experiment, since only 5% of all the photon pairs were
detected. Thus a heavy reliance on the fair-sampling assumption was required in the
data analysis.

19.6.4 Relativistic issues
An experiment with a very large separation, of 10.9 km, between Alice and Bob has
been performed using optical ¬ber technology, in conjunction with a spontaneous
down-conversion light source (Tittel et al., 1998). A violation of Bell™s inequalities
by 16 standard deviations was observed in this experiment.
Relativistic issues, such as putting limits on the so-called speed of collapse of the
two-photon wave function, could then be examined experimentally using this type of
apparatus. Depending on assumptions about the detection process and about which
inertial frame is used, the speed of collapse was shown to be at least 104 c to 107 c
(Zbinden et al., 2001). Further experiments with rapidly rotating absorbers ruled out
an alternative theory of nonlocal collapse (Suarez and Scarani, 1997).

19.6.5 Greenberger“Horne“Zeilinger states
The previous discussion of experiments testing Bell™s theorem was based on constraints
on the total amount of correlation between random events observable in two-particle
coincidence experiments. These constraints are fundamentally statistical in nature.
Greenberger, Horne, and Zeilinger (GHZ) (Kafatos, 1989, pp. 69“72) showed that
using three particles, as opposed to two, in a maximally entangled state such as

|ψGHZ ∝ |a, b, c ’ |a , b , c , (19.72)

allows a test of the combined principles of locality and realism by observing, or failing
to observe, a single triple-coincidence click. Thus, in principle, the use of statistical
correlations is unnecessary for testing local realistic theories. However, in practice,
the detectors with quantum e¬ciencies less than 100% used in real experiments again
required the use of inequalities. Violations of these inequalities have been observed
in experiments involving nonmaximally entangled states generated by spontaneous
¼¼ Bell™s theorem and its optical tests

down-conversion (Torgerson et al., 1995; White et al., 1999). Once again, the results
contradict all local realistic theories.
For a review of these and other quantum optical tests of the foundations of physics,
see Steinberg et al. (2005).

19.7 Exercises
19.1 The original EPR argument
(1) Show that the EPR wave function, given by eqn (19.1), is an eigenfunction of
the total momentum pA + pB , with eigenvalue 0, and also an eigenfunction of the
operator xA ’ xB , with eigenvalue L.
(2) Calculate the commutator [pA + pB , xA ’ xB ] and use the result to explain why
(1) does not violate the uncertainty principle.
(3) If pA is measured, show that pB has a de¬nite value. Alternatively, if xA is mea-
sured, show that xB has a de¬nite value.
(4) Argue from the previous results that both xB and pB are elements of physical
reality, and explain why this leads to the EPR paradox.

19.2 Parameter independence for quantum theory
(1) Use eqns (19.43)“(19.45) to derive eqn (19.47).
(2) Verify parameter independence when |χ is replaced by any of the four Bell states
{|Ψ± , |¦± } de¬ned by eqns (13.59)“(13.62).

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