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