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Fig. 19.3 An optical implementation of the EPR experiment. Calcite prisms replace the
Stern“Gerlach magnets shown in Fig. 19.1. The¬source emits an entangled state of two oppo-
«
sitely-directed photons, such as the Bell state ¬Ψ’ . The birefringent prisms split the light
into ordinary ˜o™ and extraordinary ˜e™ rays. The vertical dotted lines inside the prisms in-
dicate the optic axes of the calcite crystals. Coincidence-counting circuitry connecting the
Geiger counters is not shown.

The counters on each side of the apparatus are mounted rigidly with respect to
the calcite prisms, so that they corotate with the prisms. Thus the four counters will
constantly monitor the o and e outputs of the calcite prisms for all values of ± and β.
The azimuthal angles ± and β are examples of what are called parameter set-
tings, or simply parameters, of the EPR experiment. The experimentalist on the right
side of the apparatus, Alice, is free to choose the parameter setting ± (the azimuthal
angle of rotation of calcite prism A) as she pleases. Likewise, the experimentalist on
the left side, Bob, is free to choose the parameter setting β (the azimuthal angle of
rotation of calcite prism B) as he pleases, independently of Alice™s choice.

19.3.2 Conditions de¬ning locality and realism
Bell™s seminal paper has inspired many proposals for realizations of the metaphysical
notions of realism and locality, including both deterministic and stochastic forms of
hidden variables theories. In this section we present a general class of realizations by
specifying the conditions that a theory must satisfy in order to be called local and
realistic.
We will say that a theory is realistic if it describes all required elements of physical
reality for a system by means of a space, Λ, of completely speci¬ed states »”i.e. the
states of maximum information”satisfying the following two conditions.

Objective reality
Λ is de¬ned without reference to any measurements. (19.7)

Spatial separability
The state spaces ΛA and ΛB for the spatially-separated systems
A and B are independently de¬ned. (19.8)

The only other condition imposed on Λ is that it must support probability distributions
ρ (») in order to describe situations in which maximum information is not available.
Local realism

The only conditions imposed on an admissible distribution ρ (») are that it be
positive de¬nite, i.e. ρ (») 0, normalized to unity,

d»ρ (») = 1 , (19.9)

and independent of the parameter values ± and β. The last condition incorporates the
intuitive idea that the states » are determined at the source S, before any encounters
with the measuring devices at A and B.
One possible example for Λ would be the classical phase space involved in the simple
model of radioactive decay presented above. In this case, the completely speci¬ed states
» are simply points in the phase space, and a probability distribution ρ (») would be
the usual phase space distribution.
A much more surprising example comes from a disentangled version of quantum
theory, which is de¬ned by excluding all entangled states of spatially-separated sys-
tems. This mutilated theory violates the superposition principle, but by doing so it
allows us to identify Λ with the Hilbert space H for the local system. An individual
state » is thereby identi¬ed with a pure state |ψ .
According to the standard interpretation of quantum theory, this choice of » gives
a complete description of the state of an isolated system. In this case ρ (») is just the
distribution de¬ning a mixed state. The fact that the disentangled version of quantum
theory is realistic illustrates the central role played by entanglement in di¬erentiating
the quantum view from the local realistic view.
We next turn to the task of developing a quantitative realization of locality. For this
purpose, we need a language for describing measurements at the spatially-separated
stations A and B, shown in Fig. 19.3. For the sake of simplicity, it is best to consider
experiments that have a discrete set of possible outcomes {Am , m = 1, . . . , M } and
{Bn , n = 1, . . . , N } at the stations A and B respectively, e.g. A1 could describe a
detector ¬ring at station A during a certain time interval. With each outcome Am , we
associate a numerical value, Am , called an outcome parameter. The de¬nition of
the output parameters is at our disposal, so they can be chosen to satisfy the following
convenient conditions:
’1 Am +1 and ’ 1 Bn +1 . (19.10)
For the two-calcite-prism experiment, sketched in Fig. 19.3, the indices m and n
can only assume the values o and e, corresponding respectively to the ordinary and the
extraordinary rays emerging from a given prism. The source S emits a pair of photons
prepared at birth in some state ». The experimental signals in this case are clicks in
one of the counters, so one useful de¬nition of the outcome parameters is
Ae = 1 for outcome Ae (Alice™s e-counter clicks) ,
Ao = ’1 for outcome Ao (Alice™s o-counter clicks) ,
(19.11)
Be = 1 for outcome Be (Bob™s e-counter clicks) ,
Bo = ’1 for outcome Bo (Bob™s o-counter clicks) .
The outcome Ae occurs when a rightwards-propagating photon from the source S
is de¬‚ected through the e port of the calcite prism A, and subsequently registered by
Bell™s theorem and its optical tests

Alice™s e-counter, etc. In this thought experiment we imagine that all counters have
100% sensitivity; consequently, if an e-counter does not click, we can be sure that the
corresponding o-counter will click.
The following conditional probabilities will be useful.
p(Am |», ±, β) ≡ probability of outcome Am , given
the system state » and parameter settings ±, β . (19.12)
p(Bn |», ±, β) ≡ probability of outcome Bn , given
the system state » and parameter settings ±, β . (19.13)
p(Am |», ±, β, Bn ) ≡ probability of outcome Am , given
the system state », parameter settings ±, β ,
and outcome Bn . (19.14)
p(Bn |», ±, β, Am ) ≡ probability of outcome Bn , given
the system state », parameter settings ±, β ,
and outcome Am . (19.15)
p(Am , Bn |», ±, β) ≡ joint probability of outcomes Am and Bn ,
given the system state » and
the parameter settings ±, β . (19.16)
Following the work of Jarrett (1984), as presented by Shimony (1990), we will say
that a theory is local if it satis¬es the following conditions.
Parameter independence
p(Am |», ±, β) = p(Am |», ±) , (19.17)
p(Bn |», ±, β) = p(Bn |», β) . (19.18)

Outcome independence
p(Am |», ±, β, Bn ) = p(Am |», ±, β) , (19.19)
p(Bn |», ±, β, Am ) = p(Bn |», ±, β) . (19.20)
Parameter independence states that the parameter settings chosen by one observer
have no e¬ect on the outcomes seen by the other. For example, eqn (19.17) tells us that
the probability distribution of the outcomes observed by Alice at A does not depend
on the parameter settings chosen by Bob at B.
This apparently innocuous statement is, in fact, extremely important. If parameter
independence were violated, then Bob”who might well be space-like separated from
Alice”could send her an instantaneous message by merely changing β, e.g. twisting
his calcite crystal. Such a possibility would violate the relativistic prohibition against
sending signals faster than light. Likewise, eqn (19.18) prohibits Alice from sending
instantaneous messages to Bob.
The principle of outcome independence states that the probability of outcomes seen
by one observer does not depend on which outcomes are actually seen by the other.
This is what one would expect for two independent coin tosses”since the outcome of
one coin toss is clearly independent of the outcome of the other”but eqns (19.19) and
(19.20) also seem to prohibit correlations due to a common cause, e.g. in the source S.
Local realism

This incorrect interpretation stems from overlooking the assumption that » is a
complete description of the state, including any secret mechanism that builds in corre-
lations at the source (Bub, 1997, Chap. 2). With this in mind, the conditions (19.19)
and (19.20) simply re¬‚ect the fact that the actual outcomes Bn or Am are super¬‚uous,
if » is given as part of the conditions. We will return to the issue of correlations after
deriving Bell™s strong-separability condition.
It is also important to realize that the individual events at A and B can be truly
random, even if they are correlated. This situation is exhibited in the experiment
sketched in Fig. 19.3. When the polarizations of photons γA and γB , in the Bell state
|Ψ’ , are measured separately”i.e. without coincidence counting”they are randomly
polarized; that is, the individual sequences of e- or o-counts at A and B are each as
random as two independent sequences of coin tosses.
Finally, we note that a violation of outcome independence does not imply any viola-
tions of relativity. The conditional probability p(Am |», ±, β, Bn ) describes a situation
in which Bob has already performed a measurement and transmitted the result to Al-
ice by a respectably subluminal channel. Thus protecting the world from superluminal
messages and the accompanying causal anomalies is the responsibility of parameter
independence alone.

19.3.3 Strong separability
Bell™s theorem is concerned with the strength of correlations between the random out-
comes at A and B, so the ¬rst step is to ¬nd the constraints imposed by the combined
e¬ects of realism and locality”in the form of parameter and outcome independence”
on the joint probability p(Am , Bn |», ±, β) de¬ned by eqn (19.16).
We begin by applying the compound probability rule (A.114) to ¬nd

p(Am , Bn |», ±, β) = p(Am |», ±, β, Bn )p(Bn |», ±, β) . (19.21)

In other words, the joint probability for outcome Am and outcome Bn is the product
of the probability for outcome Am (conditioned on the occurrence of the outcome Bn )
with the probability that outcome Bn actually occurred. All three probabilities are
conditioned by the assumption that the state of the system was » and the parameter
settings were ± and β. The situation is symmetrical in A and B, so we also ¬nd

p(Am , Bn |», ±, β) = p(Bn |», ±, β, Am )p(Am |», ±, β) . (19.22)

Applying outcome independence, eqn (19.19), to the right side of eqn (19.21) yields

p(Am , Bn |», ±, β) = p(Am |», ±, β)p(Bn |», ±, β) , (19.23)

and applying parameter independence to both terms on the right side of this equation
results in the strong-separability condition:

p(Am , Bn |», ±, β) = p(Am |», ±)p(Bn |», β) . (19.24)

This is the mathematical expression of the following, seemingly common-sense,
statement: for a given speci¬cation, », of the state, whatever Alice does or observes
Bell™s theorem and its optical tests

must be independent of whatever Bob does or observes, since they could reside in
space-like separated regions.
Before using the strong-separability condition to prove Bell™s theorem, we return
to the question of correlations that might be imposed by a common cause. In typical
experiments, the complete speci¬cation of the state represented by » is not available”
for example, the values of the hidden variables cannot be determined”so the strong-
separability condition must be averaged over a distribution ρ (») that represents the
experimental information that is available.
The result is

p(Am , Bn |±, β) = d»ρ (») p(Am |», ±)p(Bn |», β) , (19.25)

where
p(Am , Bn |±, β) = d»ρ (») p(Am , Bn |», ±, β) . (19.26)

The corresponding averaged probabilities for single outcomes are

p(Am |±) = d»ρ (») p(Am |», ±) ,
(19.27)
p(Bn |β) = d»ρ (») p(Bn |», β) ;

consequently, the condition for statistical independence,

p(Am , Bn |±, β) = p(Am |±)p(Bn |β) , (19.28)

can only be satis¬ed”for general choices of Am and Bn ”when ρ (») = δ (» ’ »0 ).
A closer connection with experiment is a¬orded by de¬ning Bell™s expectation
values.
(1) The expectation value of outcomes seen by Alice is

p(Am |», ±)Am .
E(», ±) = (19.29)
m

(2) The expectation value of outcomes seen by Bob is

p(Bn |», β)Bn .
E(», β) = (19.30)
n

(3) The expectation value of joint outcomes seen by both Alice and Bob is

p(Am , Bn |», ±, β)Am Bn .
E(», ±, β) = (19.31)
m,n

The quantity E(», ±, β) is the average value of joint outcomes as measured, for
example, in a coincidence-counting experiment. The bounds |Am | 1 and |Bn | 1,
together with the normalization of the probabilities, imply that the absolute values of
all these expectation values are bounded by unity.
Bell™s theorem

From Bell™s strong-separability condition, it follows that the joint expectation
value”for a given complete state »”also factorizes:
E(», ±, β) = E(», ±)E(», β) , (19.32)
but in the absence of complete state information, the relevant expectation values are

E (±) ≡ p(Am |±)Am ,
d»ρ (») E(», ±) = (19.33)
m

etc. Thus the correlation function
C (±, β) = E(±, β) ’ E (±) E (β) (19.34)
can only vanish in the extreme case, ρ (») = δ (» ’ »0 ), of perfect information.

19.4 Bell™s theorem
An evaluation of any one of Bell™s expectation values, e.g. E(», ±), would depend
on the details of the particular local realistic theory under consideration. One of the
consequences of Bell™s original work (Bell, 1964) has been the discovery of various
linear combinations of expectation values, which have the useful property that upper
and lower bounds can be derived for the entire class of local realistic theories de¬ned
above. We follow Shimony (1990), by considering the particular sum
S (») ≡ E(», ±1 , β1 ) + E(», ±1 , β2 ) + E(», ±2 , β1 ) ’ E(», ±2 , β2 ) , (19.35)
which was ¬rst suggested by Clauser et al. (1969). With a ¬xed value, », of the hid-
den variables, the four combinations (±1 , β1 ), (±1 , β2 ), (±2 , β1 ), and (±2 , β2 ) represent
independent choices ±1 or ±2 by Alice and β1 or β2 by Bob, as shown in Fig. 19.4.
For the typical situation in which the complete state » is not known, S (») should
be replaced by the experimentally relevant quantity:
S ≡ E(±1 , β1 ) + E(±1 , β2 ) + E(±2 , β1 ) ’ E(±2 , β2 ) . (19.36)
Bell™s theorem is then stated as follows.

Bob Alice
Correlation -(±1,β1)
±1
β1

Correlation -(±2,β1) Correlation -(±1,β2)
β2 ±2
Anticorrelation ’-(±2,β2)

Two choices of Two choices of
Bob's settings Alice's settings

Fig. 19.4 The four terms in the sum S de¬ned in eqn (19.35). The dependence of the
expectation values E(», ±, β) on the system state » has been suppressed in this ¬gure.
¼ Bell™s theorem and its optical tests

Theorem 19.1 For all local realistic theories,
’2 E(», ±1 , β1 ) + E(», ±1 , β2 ) + E(», ±2 , β1 ) ’ E(», ±2 , β2 ) +2 . (19.37)


Averaging over the distribution of states produces the Bell inequality:
’2 E(±1 , β1 ) + E(±1 , β2 ) + E(±2 , β1 ) ’ E(±2 , β2 ) +2 . (19.38)
This result limits the total amount of correlation, as measured by S, that is allowed
for a local realistic theory. Experiments using coincidence-detection measurements
performed on two-photon decays have shown that this bound can be violated.

19.4.1 Mermin™s lemma
In order to prove Bell™s theorem, we ¬rst prove the following lemma due to Mermin.
Lemma 19.2 If x1 , x2 , y1 , y2 are real numbers in the interval [’1, +1], then the sum
S ≡ x1 y1 + x1 y2 + x2 y1 ’ x2 y2 lies in the interval [’2, +2], i.e. |S| 2.

Proof Since S is a linear function of each of the four variables x1 , x2 , y1 , y2 , it must
take on its extreme values when the arguments of the function themselves are extrema,
i.e. when (x1 , x2 , y1 , y2 ) = (±1, ±1, ±1, ±1), where the four ±s are independent. There
are four terms in S, and each term is bounded between ’1 and +1; consequently,
|S| 4. However, we can also rewrite S as
S = (x1 + x2 ) (y1 + y2 ) ’ 2x2 y2 . (19.39)
The extrema of x1 + x2 are 0 or ±2, and similarly for y1 + y2 . Therefore the extrema
of the product (x1 + x2 ) (y1 + y2 ) are 0 or ±4. The extrema for 2x2 y2 are ±2. Hence
the extrema for S are ±2 or ±6. The latter possibility is ruled out by the previously
determined limit |S| 4; therefore, the extrema of S are ±2, i.e. |S| 2.

19.4.2 Proof of Bell™s theorem
Proof Bell™s theorem now follows as a corollary of Mermin™s lemma. With the iden-
ti¬cations
x1 = E(», ±1 ) , where |E(», ±1 )| 1 ,
x2 = E(», ±2 ) , where |E(», ±2 )| 1 ,
(19.40)
y1 = E(», β1 ) , where |E(», β1 )| 1 ,
where |E(», β2 )|
y2 = E(», β2 ) , 1,
Lemma 19.2 implies
|E(», ±1 )E(», β1 ) + E(», ±1 )E(», β2 ) + E(», ±2 )E(», β1 ) ’ E(», ±2 )E(», β2 )| 2 .
(19.41)
Using the strong-separability condition (19.32) for each term, i.e. E(», ±, β) =
E(», ±)E(», β), we now arrive at
’2 E(», ±1 , β1 ) + E(», ±1 , β2 ) + E(», ±2 , β1 ) ’ E(», ±2 , β2 ) +2 , (19.42)
and averaging over » yields eqn (19.38).
½
Quantum theory versus local realism

19.5 Quantum theory versus local realism
As a prelude to the experimental tests of local realism, we ¬rst support our previous
claim that quantum theory violates outcome independence and satis¬es parameter in-
dependence. In addition, we give an explicit example for which the quantum prediction
of the correlations violates Bell™s theorem.

19.5.1 Quantum theory is not local
The issues of parameter independence and outcome independence will be studied by
considering an experiment simpler than the one presented in Section 19.3.1. In this
arrangement, shown in Fig. 19.5, pairs of polarization-entangled photons are produced
by down-conversion, and Alice and Bob are supplied with linear polarization ¬lters and
a single counter apiece. This reduces the outcomes for Alice to: Ayes (Alice™s detector
clicks) and Ano (there is no click). The corresponding outcome parameters are Ayes = 1
and Ano = 0. Bob™s outcomes and outcome parameters are de¬ned in the same way.
We begin by assuming that the source produces the entangled state
|χ = F |hA , vB + G |vA , hB , (19.43)
where
|hA , vB ≡ a† A h a† B v |0 , |vA , hB ≡ a† A v a† B h |0 , (19.44)
k k k k
kA and kB are directed toward Alice and Bob respectively, and h and v label orthog-
onal polarizations: eh (horizontal ) and ev (vertical ). The parameters are the angles ±
and β de¬ning the linear polarizations e± and eβ transmitted by the polarizers.
Since akA h ∝ ekA h · E (+) , etc., the annihilation operators in the (h, v)-basis are
related to the annihilation operators in the (±, ± = π/2 ’ ±)-basis by
ak A h
ak A ± cos ± sin ±
= . (19.45)
’ sin ±
ak A ± cos ± ak A v
The corresponding relation for Bob follows by letting ± ’ β and kA ’ kB .

A Parameter independence
For this experiment, the role of p(Am |», ±, β) in eqn (19.17) is played by p(Ayes |χ, ±, β),
the probability that Alice™s detector clicks for the given state and parameter settings.
This is proportional to the detection rate for e± -polarized photons, i.e.


* )




Fig. 19.5 Schematic of an apparatus to measure the polarization correlations of the entan-
gled photon pair γA and γB emitted back-to-back from the source S. The coincidence-counting
circuitry connecting the two Geiger counters is not shown.
¾ Bell™s theorem and its optical tests


p(Ayes |χ, ±, β) ∝ G(1) (rA , tA ; rA , tA ) ∝ χ a† A ± akA ± χ . (19.46)
k
±


A calculation”see Exercise 19.2 ”using eqns (19.43)“(19.45) yields

p(Ayes |χ, ±, β) ∝ |F |2 cos2 ± + |G|2 sin2 ± . (19.47)

Thus the quantum result for the probability of a click of Alice™s detector is independent
of the setting β of Bob™s polarizer, although it can depend on her own polarizer setting
±. In other words, quantum theory”at least in this example”satis¬es parameter
independence. The symmetry of the experimental arrangement guarantees that the
probability, p(Byes |χ, ±, β), seen by Bob is independent of ±.
This single example does not constitute a general proof that quantum theory sat-
is¬es parameter independence, but the features of the calculation provide guidance
for crafting such a proof. In general, the calculation of outcome probabilities for Alice
take the same form as in the example, i.e. the expectation value of an operator”which
may well depend on Alice™s parameter settings”is evaluated by using the state vector
determined by the source. Neither Alice™s operator nor the state vector depend on
Bob™s parameter settings; therefore, parameter independence is guaranteed for quan-
tum theory. √
For the special values F = ’G = 1/ 2, the entangled state |χ becomes the
singlet-like Bell state

1
Ψ’ = √ {|hA , vB ’ |vA , hB } , (19.48)
2

¬rst de¬ned in Section 13.3.5. In this case, p(Ayes |χ, ±, β) is independent of ± as well
as β, so that Alice™s singles-counting measurements are the same as expected from an
unpolarized beam. This supports our previous claim that the individual measurements
can be as random as coin tosses.

B Outcome independence
Checking outcome independence requires the evaluation of the conditional probability
p(Ayes |», ±, β, Brslt ) that Alice hears a click, given that Bob has observed the outcome
Brslt , where rslt = yes, no. In this case, we will simplify the calculation by setting
|χ = |Ψ’ at the beginning.
With the usual assumption of 100% detector sensitivity, both possible outcomes

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