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).