Università degli Studi di Perugia

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Triennale in Fisica

Tesi di Laurea Triennale

EPR Paradoxand

Bell’s Theorem

Candidato:

Danny Laghi

Relatore:

Prof. Gianluca Grignani

Anno Accademico 2012-2013Sessione di Laurea 30 Settembre 2013

Alla mia famiglia, senza della

quale questo (piccolo) traguardo

non sarebbe stato raggiunto

�Entanglement is not one, but rather the characteristic

trait of Quantum Mechanics.�

—Erwin Schrödinger

Contents

1 Introduction 7

2 The EPR Paradox 11

2.1 Definition of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Reality, Locality and Completeness . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 EPR Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Bohm-Aharonov’s formulation of the paradox . . . . . . . . . . . . . . . . . 23

3 Bell’s Theorem 28

3.1 Bell’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 CHSH Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Optical version of Bell’s Gedankenexperiment . . . . . . . . . . . . . . . . . 38

3.4 A. Aspect’s experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.1 Experiment with single-channel polarizers . . . . . . . . . . . . . . . 47

3.4.2 Experiment with two-channel polarizers . . . . . . . . . . . . . . . . 48

3.4.3 Experiment with time-varying polarizers . . . . . . . . . . . . . . . . 49

3.4.4 Towards the impossibility of FTL communications . . . . . . . . . . 51

3.5 No-Communication Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.1 Some outlines about the “FLASH Project” . . . . . . . . . . . . . . 58

4 Conclusions on Quantum Mechanics 61

4.1 Interpretations of Quantum Mechanics: Copenhagen, Bohm . . . . . . . . . 63

4.1.1 Copenhagen interpretation . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2 Bohm’s interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Open questions on Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 68

A Some expectation values of Quantum Mechanics 71

Bibliography 77

6

Chapter 1

Introduction

Quantum Mechanics, as its own peculiarity, introduces some ideas that depart from ev-

eryday experience and common sense.

Until the end of the 19th century, the interpretation of macroscopic physical phe-

nomena was based on Newton’s Laws, which explained mechanical, acoustic and thermal

phenomena, and on Maxwell’s Equations, which governed the electric, magnetic and opti-

cal ones. These two large classes of physical phenomena yielded a distinction between the

wave-like behavior (or the electromagnetic field radiation) and the corpuscolar character

of matter, but from the beginning of the last century, physicists realized that there were

some phenomena that could not be organized within the system of classic laws, and some

of them voided the distinction between particle and field: for instance, the atom structure

and the mechanism of emission and absorption of matter radiation. Gradually, there was

the necessity of introducing a theory, without vanishing the excellent results obtained with

classical mechanics until then, which was capable to extend the latter in order to describe

even those phenomena that were remaining not understood.

There were too facts for whom there were not any explanations, according to classical

laws and statistics which were incredibly losing their own descriptive and predictive power

for certain physical phenomena, like the black body radiation spectrum and the spectral

lines of hydrogen atom. It is in this scenary that the first quantum theory was born,

laying its foundations with the introduction of the quantum of energy, introduced in 1900

by Max Planck in order to give a plausible explanation for the spectrum of black body ra-

diation. Planck postulated (and then showed) that the energy exchanged between matter

and radiation does not show itself in a continuous manner, but by discrete and indivisible

quantities, or quanta of energy, that must be proportional to the frequency of the radi-

ation; he obtained an expression for the spectrum, in accordance with the experimental

distribution, by properly adjusting the constant of proportionality. So the energy quanti-

7

zation was the first step toward a quantum theory able to give reason for the experimental

data, incompatible with the classical theory.

The second important step was taken in 1913 by Niels Bohr, who, basing his works

on the emission spectrum of hydrogenoid atoms, succeeded in generalizing the well-known

Balmer’s empirical formula for the spectrum of hydrogen atom, obtaining a first, funda-

mental, model for hydrogenoid atoms. This model was based on some postulates, the first

of them imposing a new discretization, this time for another physical quantity that was

always been considered like a continuous variable: the angular momentum. When the

quantization of the angular momentum is considered, some orbits were imposed to the

electron (second postulate): in these orbits (called stationary levels) the electron could

not emit energy quanta.

It will be only with the introduction of two formalisms, the matrix mechanics and the

wave mechanics, introduced by Werner Heisenberg (1925) and Erwin Schrödinger (1926)

respectively, that began to emerge a real quantum theory, which was able to clear up in

a more complete way the experiments, one over all, the double-slit experiment — that

Richard Feynman was used to say it holds the essential mystery, rather, the only mystery

of Quantum Mechanics.

This theory shows a double nature — wave-like and corpuscular — of elementary

particles, confirms the presence of discrete energy levels as assumed by Planck, and gives

an inferior limit about how much precisely it is possible to investigate on physical world,

because of the well-known Uncertainty Principle.

But undoubtedly the most peculiar feature of Quantum Mechanics is the introduction

of a wave function ψ, solution of the Schrödinger equation and representation of the

physical state of a quantum system, for which it is the complete description.

And it is the completeness of the wave function, that is, the quantum theory behind it,

that was called into question in 1935 by Albert Einstein, Boris Podolsky and Nathan Rosen

(EPR in the following), in their famous paper “Can Quantum-Mechanical Description of

Physical Reality Be Considered Complete?”, where they showed through a Gedankenex-

periment their so famous argument, later baptized as EPR paradox, pointing out how the

quantum theory couldn’t simultaneously satisfy three principles: reality, locality and com-

pleteness. Being assumed the first and the second as obvious and incontrovertible (it was

indeed the impossibility to accept “an istantaneous spooky action at a distance” one of

their key point), they inferred that it was the third principle to be abandoned, that is, the

wave function could not contain all of the possible information about quantum system.

Many controversies arose from this paper, mostly of them contesting the hypotheses

on the basis of their demonstration, so that the paradox became one of the principal

8

subject of discussion. Two main tendencies of opposite thought standed from that: on

one hand, those belonging to the Copenhagen school, which supported a probabilistic and

undetermined vision of the reality and thought that within this interpretation the paradox

was based on inconsistent hypotheses; on the other hand, there were those which, accord-

ing to the EPR paradox conclusions, were convinced of the incompleteness of Quantum

Mechanics, so that the latter had to be completed with the introduction of additional

parameters, that could explain the queer correlations that, like in EPR demonstration,

seemed to emerge between remote systems.

In this way came up the hypothesis of the introduction of additional degrees of freedom

with respect to those considered in Quantum Mechanics, which allowed to make it a

deterministic and complete theory.

Nevertheless in 1964 an Irish physicist, John Stewart Bell, demonstrated that consid-

ering local theories with additional hidden variables, there would arise some inequalities

which generally are violated by Quantum Mechanics.

The importance of this work is that the experiments have the possibility to determine

if such a theory is possible or not, according to the results stemming from laboratory;

thus the possibility of choosing what interpretation to assume for Quantum Mechanics,

either a local deterministic theory, or a nonlocal probabilistic one, is no longer arbitrary,

but it is told by the physics itself; as we shall see, by means of the experimental results

obtained and comparated with Bell’s predictions, Nature has showed an exclusion about

the possibility of being local.

Since 1972 several experiments, more refined as they were executed, have been con-

ceived and conducted to verify the effectiveness of Bell’s inequalities (in particular, as we

shall see, a series of decisive experiments has been performed in 1980-82 by Alain Aspect

et al.).

The experiments, within an acceptable range of experimental errors, proved that:

(a) nonlocal correlations predicted by Quantum Mechanics exist;

(b) Bell’s inequalities are violated.

Generally the arguments we are going to deal with are still not treated in a complete

uniform manner by Quantum Mechanics books, mainly because of the subtlety of the ques-

tions — and, of course, the difficulty to verify them experimentally — that arise out of

these subjects and because these arguments, though some sure points have been reached,

9

maybe have not been completely understood in their many facets yet 1.

The following work just wants to run through the fundamental stages in the develop-

ment of EPR problem, pointing out and describing as well the several possible solutions

for this problematic argument, and analyzing the main significative experiments that have

been made in order to verify some possible theoretic models, based on alternative principles

instead of orthodox Quantum Mechanics.

It is a sure thing that the concept that Einstein involved for questioning about the

completeness of quantum theory, i.e. the entanglement of two particles far apart from

each other, is a real physical phenomenon, whatever the meaning one could assume for the

latter terms.

The life and work of Einstein, hard critic of quantum theory, just because he was not

able to accept that Nature expresses itself in a probabilistic way (�God doesn’t play dice

with the world �), showed that curiously the German physicist ‘was right’ even when he

thought he was wrong (e.g. about the cosmological constant).

And as for the quantum world, Einstein’s paper of 1935, was actually the seed for one

of the most important discoveries in physics in the twentieth century: the actual discovery

of entanglement through physical experiments.

Bell’s contribution, whose works have been of fundamental importance in the develop-

ment of this subject, as we shall see, set out to prove that the Einstein-Podolsky-Rosen

thought experiment was far from being an absurd idea to be used just to invalidate the

completeness of the quantum theory, but rather the description of a real phenomenon. The

existence of this phenomenon had to provide a new proof in favour of Quantum Mechanics,

while against a limiting view of reality.

We shall see how the experiments have actually shown that.

1For instance, see [10].

10

Chapter 2

The EPR Paradox

Quantum Mechanics has been founded on radical revisions of many classical concepts.

For example, in order to take into account the wave-particle duality, it had to give up the

concept of classical trajectory, because of Heisenberg’s Uncertainty Principle: choosing as

canonically conjugate observables the position and momentum of a particle, it describes

quantitatively the impossibility of defining precisely and simultaneously its position and

its velocity.

According to that, the thing which Einstein thought was that quantum formalism was

incomplete. The paper of 1935 by A. Einstein, B. Podolsky and N. Rosen [2] wants to

demonstrate the non-completeness of the wave function, thought as description of real-

ity. What EPR show is substantially a thought experiment (Gedankenexperiment is the

Deutch term) that proofs how, assuming two principles — indicated as reality and locality

principle, Quantum Mechanics formalism leads to a contradiction, unless one introduces

the existence of hidden variables.

EPR show that quantum formalism can allow for the existence of states describing

two particles, for which one is able to expect strong correlations either for the velocity or

the position, even if the particles are widely separated and no longer interacting to each

other. The position measurements would always give symmetric values with respect to

the origin, so that a measure of one particle would allow one to know with certainty the

value of the position of the other.

Similarly, measurements of the momentum of the two particles would lead to opposite

values, so that the measurement on one of them would be sufficient for having knowledge

of the value of the other with certainty.

Undoubtedly, one has to choose between a precise measurement of position or momen-

tum for one particle, because of Heisenberg’s Uncertainty Principle. But the measurement

on the first particle cannot disturb the second (distant) particle, and it is here that the

11

principle of locality is introduced, by means of which EPR conclude that the second par-

ticle must have had, long before the measurement, predetermined values of position and

momentum. Since for Quantum Mechanics it is not possible to know precisely and simul-

taneously the values of these quantitities, Einstein and his co-authors inferred that the

theory is incomplete.

Following up the publication of this paper, E. Schrödinger coined the term ‘entan-

glement’, just to describe the impossibility of factorizing a quantum state like the one

assumed by EPR.

But before looking in detail the EPR paper and the paradox that arises from it, it is

best to define what we intend for entangled states and, in general, for completeness, reality

and locality.

2.1 Definition of entanglement

Entanglement is a purely quantum phenomenon, i.e. it is not supplied with a classical

counterpart; it is in this way that the quantum state of two or more physical systems

depends on the states of everyone of the systems that compose it. A first definition of

entanglement that we propose here is just the first absolute definition of the term, due to

E. Schrödinger, who conied it himself following up EPR paper. But in order to get the

Schrödinger’s definition, it is primarily necessary to define the important concepts of pure

states and mixed states.

Definition 1. Given a vector space X , a finite linear combination∑

i αixi is called con-

vex if αi ∈ [0, 1] and∑

i αi = 1 .

Moreover, C ⊂ X is called convex if for any pair x, y ∈ C, λx + (1 − λ)y ∈ C,∀λ ∈ [0, 1] (and thus any convex combination of elements in C belongs to C).

Definition 2. If C is convex, an element e ∈ C is called extreme if it cannot be writtenas e = λx+ (1− λ)y, with λ ∈ [0, 1], x, y ∈ C \{e}.

Definition 3. Let H be a separable Hilbert space ( H is separable ⇔ H has finite dimen-sion). Let S(H) be a convex subset of H.

Then the extreme elements in S(H) are called pure states. Non extreme-states aremixed states, or nonpure states.

12

Schrödinger defined as entangled those quantum pure states |Ψ〉, from an ensamble ofsystems, that cannot be represented in a form of simple tensorial products of eigenstates

of the systems, that is,

|Ψ〉 6= |ψ1〉 ⊗ |ψ2〉 ⊗ · · · |ψn〉 ,

where ⊗ denotes the tensorial product and |ψi〉 are some vectors that represent the statesof some Hilbert spaces. The states of composed systems, that can be instead represented

as tensorial products of the subset states, constitute the complement set of pure states,

known as product states.

It is easy to determine whether a pure state of a system, composed of only two sub-

systems, is an entangled state making use of Schmidt’s decomposition theorem [1], that is

always valid for these systems 1. Indeed any bifactorized pure state |Ψ〉 can be written asa sum of bi-orthogonal terms: it is always possible to write |Ψ〉 in a form

|Ψ〉 =∑i

ci |ui〉 ⊗ |vi〉 ,

with ci ∈ C, where the sets of vectors {|ui〉} and {|vi〉} consist of orthonormal unit vectorsspanning the space of possible-state vectors for the system, the index i running up to the

smaller of the dimensions of the two subsystem Hilbert spaces.

On the basis of our previous considerations, it is easy to get a second more formal

definition:

Definition 4. Let A and B be two non-interacting systems with their respective Hilbert

spaces HA and HB, and let |ψ〉A and |φ〉B be two states in which there are the first andthe second system respectively. Then the state of the composed system, belonging to the

Hilbert space given by the tensorial product HA ⊗HB, is |ψ〉A ⊗ |φ〉B.The states of the composed system that can be written in this way are called product

states or separable states.

Generally, not all of the states of a composed system are product states; let {|ui〉A}and {|vi〉B} be some bases in their respective Hilbert spaces HA and HB . According to

1In the simple case of finite dimensional spaces, Schmidt’s decomposition theorem asserts that anydouble sum Ψ =

∑klXklxmyn, can be converted into a single sum Ψ =

∑α aαξαηα, by means of unitary

transformations ξα =∑k Uαkxk and ηα =

∑l Vαlyl. If {xk} and {yl} are two orthonormal bases for

two distinct vector spaces, then {ξα} and {ηα} are also two, possibly complete, orthonormal vector basesfor these spaces. The absolute values |aα| are called the singular values of the matrix X, and are easilycalculated by noting that |aα|2 are the nonvanishing eigenvalues of the Hermitian matrices XX† and X†X.The corresponding sets of eigenvectors are {ξα} and {ηα}, respectively.

13

Schmidt’s decomposition, any state in the composed system can readily be written as

|ψ〉AB =∑i

ci |ui〉A ⊗ |vi〉B . (2.1)

Then the state (2.1) is a product state, or a non-entangled state, if there is one and

only one i so that ci 6= 0, and all of the other cj = 0, j 6= i; that is, a state is non-entangledif and only if

|ψ〉AB = ci |ui〉A ⊗ |vi〉B , i fixed. (2.2)

If instead there are more than one ci 6= 0 for the state 2.1, then such a state is calledentangled state.

If i = 1, . . . , N and if ci =1√N, ∀i, the state (2.1) is called to be maximally entangled.

In other words, a state is entangled if and only if it cannot be factorized.

As an explanatory example of this definition, let’s consider two bases {|0〉A , |1〉A} ∈ HAand {|0〉B , |1〉B} ∈ HB.

Then an entangled state of the composed system HA ⊗HB is represented by

1√2

(|0〉A ⊗ |1〉B − |1〉A ⊗ |0〉B) ,

that is, omitting tensorial product symbols,

1√2

(|0〉A |1〉B − |1〉A |0〉B) ,

where it is evident the impossibility of attributing separately a quantum state either to

the system A or to the system B without involving the other.

A similar state to the one above is a singlet state for a pair of electrons. Thus if we

knew, by experimental measurements, that for example the system A would be in the state

|1〉A, then, unless of coefficients that we can neglect, we will able to know automaticallythat the system B will be in the state |0〉B.

How to obtain entangled states

There are some different ways for achieving entangled states of two systems.

14

For instance, in the decay of a spin 0 particle, the final products are two spin 1/2

particles in a singlet state (like π0 → γ + e+ + e−).Another example is in the e+e− annihilation, that can produce a φ meson, that can

decay, for instance, in a couple of K0K̄0 in a spin singlet state.

Similarly it is possible to create photons entangled in polarization as result of positro-

nium annihilation (e+ + e− → γ + γ).One more example of entangled state is the above mentioned singlet spin of two elec-

trons near an atomic nucleus.

Entanglement has numerous applications beyond its use in the paradox topic as well;

for instance an application of such states is in Quantum Information Theory, in Quantum

Cryptography, where in the one time pad transmission, a couple of entangled particles is

used to exchange a random coding key between sender and receiver, in a completely secure

way from eavesdropping attacks.

2.2 Reality, Locality and Completeness

As we already said, EPR paper relies substantially upon the assumption of some principles,

i.e. the completeness, the reality and the locality. So it is good to find for these concepts

some definitions applicable to our contest [2].

Principle of reality

The elements of physical reality cannot be determined by a priori philosophical consider-

ations, but they are to be found using measurements and experimental results. Far from

being an exhaustive definition, EPR consider as satisfactory the following definition for

an element of physical reality.

If, without in any way disturbing a system, we can predict with certainty (i.e., with

probability equal to unity) the value of a physical quantity, then there exist an element of

physical reality corresponding to this physical quantity, that is, an objective property of

the system, independent of any eventual external observer.

Thus, all of the possible physical observables must have some preexistent values for

any possible measurement, before of carrying out that measurement.

15

Principle of locality

Accordingly to the existence of an upper limit for the velocity of propagation of a signal,

coincident with the speed light c in vacuum, an object is affected only directly by his

immediate neighbourhood. This lead us to a definition of locality.

Given two physical systems, let’s suppose that during a certain time interval they

remain isolated from each other. Then the temporal evolution of the physical properties

of one of them during that time interval cannot be influenced by operations executed on

the other (Einstein’s locality).

Condition of completeness of a theory

According to the definition adopted by EPR in their paper (called by them “condition of

completeness”):

In order for a theory to be complete, it is necessary that any elements of physical

reality must have a counterpart in a physical theory.

Whether an element of physical reality has no counterpart in the theory, then we shall

say that the theory is not complete.

2.3 EPR Paradox

Given two noncommuting physical quantities, because of the Uncertainty Principle, the

exact knowledge of one of them precludes the exact knowledge of the other. In fact, any

attempt of determining a quantity by means of measurement changes the system state, so

that the value of the other quantity is undetermined.

In other words, two noncommuting physical quantities cannot have both expected

values.

Starting from this assumption, EPR state that:

1. either the Quantum Mechanics description of reality given by the wave function is

not complete;

2. or the physical quantities associated with noncommuting operators cannot have si-

multaneous reality.

16

It can be easily shown that one of the above conclusions has to be necessarily true, for

there are no other possible situations.

Actually, if we supposed ab absurdum that both the assumptions were wrong, we

could consider two noncommuting physical quantities A, B, that have simultaneous reality

(negation of 2.). Then, for the principle of reality, both the solutions should have some

defined values that, by means of the completeness condition, would be part of the complete

description of reality. But then, if the wave function was able to provide such complete

description of the reality (negation of 1.), it would include those values, which would

be therefore predictable. But this is an absurd, for it would contradict the Uncertainty

Principle.

In this way, we have showed that 1. and 2. are complementary: whether we assume

the first as false, then the second must be necessarily true, and vice versa.

EPR create the paradox assuming the completeness of the wave function and, at the

same time, deducing the physical reality for two quantities, whose associated operators do

not commute, that is, inferring that both 1. and 2. are false, thus obtaining an absurd,

as we have seen above.

So they conclude that, in order to solve the paradox, one of the three assumptions has

to be dropped out, that is one amongst:

• the principle of reality

• the principle of locality

• the completeness of Quantum Mechanics.

Since it seemed that the first two hypotheses could not be debated, unless of having a

senseless theory (but actually it is just the debate on these assumptions to open to other

possible solutions of the paradox), EPR drop out the validity of the remaining assumption,

that is, Quantum Mechanics is an incomplete theory.

We now shall see what are the arguments of EPR paper and how the paradox arises

from it, i.e. how to get both the negation of 1. and 2..

Paradox formulation

It is helpful to run through the sheer introductory passages of the EPR paper [2].

Let’s consider a particle having a single degree of freedom, i.e. in a unidimensional

space. The fundamental concept of the theory is the concept of state, which is supposed

17

to be completely characterized by the wave function ψ, which is a function of the variables

chosen to describe the particle’s behaviour. Corresponding to each physically observable

quantity A there is an operator, which may be designated by the same letter (with a ‘hat’).

If ψ is an eigenfunction of the operator Â, that is, if:

Âψ = aψ, (2.3)

where a ∈ R, then the physical quantity A has with certainty the value a whenever theparticle is in the state described by ψ. In accordance with the principle of reality, for this

particle there is an element of physical reality corresponding to the physical quantity A.

For instance, let’s consider,

ψ = ei~p0x, (2.4)

where p0 is a numerical constant and x the independent variable. Taking the operator

corresponding to the linear momentum of a particle,

p̂ = −i~ ∂∂x

, (2.5)

we obtain

p̂ψ = −i~∂ψ∂x

= p0ψ, (2.6)

and therefore, in the state (2.4), the momentum assume with certainty the value p0. The

momentum of the particle therefore has physical reality.

Now let’s consider the case in which the eigenvalue equation (2.3) is not valid. In this

case we can speak no longer of the quantity A as if it had a definite value. Such a case

is that, for example, of the position of the particle. Its corresponding operator, x̂, is the

operator of multiplication by the independent variable x. Hence,

x̂ψ = xψ 6= aψ. (2.7)

In accordance to Quantum Mechanics, it is just possible to say that the relative prob-

ability of a coordinate measurement giving a result lying between a and b is

P (a, b) =

∫ baψ∗ψdx =

∫ badx = b− a. (2.8)

Since this probability is independent of a, but depends only upon the difference b− a,

18

we see that all of the values for the coordinate are equally probable. From another point

of view, we could have obtained such a conclusion without any computation, noting that

the state (2.4) is representative of a unidimensional free wave of definite momentum p0,

so it is not possible to localize exactly a free wave, which extends as far as infinity, for it

has an equivalent probability of being in any point of the (unidimensional) space.

So a defined value of the position of a particle in a state given by (2.4) is unpredictable,

but it can be reached only by direct measurement. In accordance with Quantum Mechan-

ics, this action alters the particle state, that becomes a position eigentstate (and no longer

of momentum). Therefore, after the determination of the position, the particle will be no

longer in the state given by Eq. (2.4).

We can conclude that whenever the momentum of a particle is known, its position has

no physical reality.

Now, supposing that the wave function of a certain physical system is complete (nega-

tion of 1.), yet Einstein, Podolsky and Rosen found that it is possible to have two simul-

taneous real physical quantities in spite of the fact that their respective operators do not

commute (negation of 2.), that is an absurd.

In order to deduce such an absurd, EPR built the following reasoning.

Let’s consider two systems I and II (which could be punctiform or hard rigid bodies, or

whatever other generic physical systems), described by the variables x1 and x2 respectively.

We suppose that they interact with each other from the time t = 0 to the time t = T .

We suppose further that the states of the two systems before t = 0 are known. Then, by

means of the wave function Ψ, it is possible to describe the state of the entangled system

I + II for any successive time t > T using the Schrödinger’s equation, even if, after the

interaction, it is no longer possible to work out the states of I or II, for they are no longer

factorable states.

Now let’s consider the physical quantity A, relative to the system I, which has some

eigenvalues a1, a2, a3, . . . and eigenfunctions u1(x1), u2(x1), u3(x1), . . . .

The wave function Ψ, considered as function of x1, at a certain fixed time t > T , can

then be expressed as:

Ψ(x1, x2) =

∞∑n=1

ψn(x2)un(x1), (2.9)

where x2 stands for the variables used to describe the second system. Here ψn(x2) are to

be regarded merely as the coefficients of the expansion of Ψ into a series of orthogonal

functions un(x1), i.e. as Fourier’s coefficients (un(x1),Ψ).

19

Now we want to measure A; suppose that it is found that it has the value ak.

According to the precipitation postulate, after a measurement, the state of the system

is given by Ψ′ = PΨ, where P is a projection operator on the eigenspace relative to

the found eigenvalue. Since the system I ‘has fallen’ in the state described by the wave

function uk(x1), we get:

Ψ(x1, x2) = ψk(x2)uk(x1). (2.10)

Since for systems formed by independent subsystems the wave function is the product

of the wave functions of the single subsystems, from Eq. (2.10) we deduce that, after the

measurement, the second system will ‘collapse’ into a definite state too, which is described

by the wave function ψk(x2).

So the wave packet given by the infinite series (2.9) is reduced to a single term given

by (2.10).

We now propose to consider, instead of the physical quantity A, another physical quan-

tity, say B, relative to the system I as well, with eigenvalues b1, b2, b3, . . . and eigenfunctions

v1(x1), v2(x1), v3(x1), . . . .

In this way, analogously to Eq. (2.9), we can write

Ψ(x1, x2) =∞∑m=1

φm(x2)vm(x1), (2.11)

where φm(x2) are the new coefficients.

In analogy with the previous passages, we measure B and suppose we find it in a

value bj ; then the system collapses into the state described by the wave function vj(x1);

consequently we obtain

Ψ(x1, x2) = φj(x2)vj(x1), (2.12)

from which we deduce that system II has collapsed into the state described by the wave

function φj(x2).

We see therefore that, as a consequence of the two different measurements performed

upon the first system, the second system may be left in states with two different wave

functions: ψk(x2) and φj(x2), and, since at the time of measurement the two systems I,

II no longer interact, no real change can take place in the second system in consequence

of anything that may be done in the first system.

20

Thus, it is possible to assign two different wave functions, ψk and φj , to the same

reality (the second system after the interaction with the first). Therefore, if we were able

to choose two functions for the eigenfunctions belonging to two noncommuting operators,

relative to two, non compatible, physical quantities P and Q, which assume the values pk

and qj respectively, then we would automatically obtain the proof that two noncommuting

quantities are simultaneously real (negation of 2.), that is exactly our aim.

To show this, let’s consider two generic systems, for example two point-like particles.

We suppose that the wave function of the total system is:

Ψ(x1, x2) =

∫ +∞−∞

ei~ (x1−x2+x0)pdp, (2.13)

where x0 is a constant.

Choosing as physical quantity A the observable linear momentum P , to which is nat-

urally associated the operator P̂ of the form

P̂ = −i~ ∂∂x

,

and measuring that quantity for the particle I, it will assume a value p; consequently the

particle will fall into a state described by the eigenfunction

up(x1) = ei~x1p. (2.14)

From now on, we consider the case of a continuous spectrum; hence Eq.(2.9), repre-

sentative of the wave function of the total system, is

Ψ(x1, x2) =

∫ +∞−∞

ψp(x2)up(x1)dp; (2.15)

the projection on up(x1) will naturally lead to

ψp(x2) = e− i~ (x2−x0)p, (2.16)

that is nothing but the eigenfunction of the operator P̂ = −i~ ∂∂x2 , relative to the eigen-value −p for the momentum of the second particle.

On the other hand, if we choose as physical quantity B the observable position Q,

whose operator is the operator Q̂ of multiplication by x, and performing a measurement

on the particle I, the latter will assume a value x, and consequently it will fall into the

21

state described by the eigenfunction

vx(x1) = δ(x1 − x), (2.17)

where δ(x1 − x) is the well-known Dirac delta-function.Eq. (2.11), representative of the wave function of the total system, now becomes

Ψ(x1, x2) =

∫ +∞−∞

vx(x1)dx1

∫ +∞−∞

ei~ (x−x2+x0)pdp; (2.18)

so we clearly have

φx(x2) =

∫ +∞−∞

ei~ (x−x2+x0)pdp = hδ(x− x2 + x0), (2.19)

that is the eigenfunction of the operator Q̂ = x2, relative to the eigenvalue x+ x0 for the

position of the second particle.

At this point the paradox is obtained. In fact it is immediate to verify that P̂ and Q̂ are

a pair of quantum operators associated with a couple of canonically conjugate variables,

which do not commute:

[P̂ , Q̂

]= i~. (2.20)

Finally, we have obtained two wave functions, ψk and φj , associated with two non

compatible quantities, that is, two noncommuting quantities P and Q, belonging to the

second system, that became simultaneously real: thus it follows the paradox.

Starting from the assumption that the wave function gave a real complete description

of physical reality, EPR came to the conclusion that two physical quantities, with noncom-

muting operators, could have simultaneous reality. Consequently the negation of 1. leads

to the negation of the only alternative 2.. Thus we conclude that the quantum mechanical

description of physical reality given by the wave function is not complete.

EPR conclude their paper saying that one could object them that the criterion of

reality they have chosen is not sufficiently restrictive. Indeed, we would not arrive to

a paradox if we just insisted on the point that two or more physical quantities could be

viewed as simultaneous elements of reality only whether they could be measured or predicted

simultaneously. From this point of view, since either one or the other, but not both

simultaneously of the quantitites P and Q could be expected, they are not simultaneously

real and the paradox therefore would not arise.

Nevertheless, this procedure is such as to make the reality of P and Q depending on

22

the measuring process that is executed on the first system, which does not interfere in

any way with the second one. And it is here that the principle of locality makes its part,

preventing from taking into account that possibility. With EPR words: �No reasonable

definition of reality could be expected to permit this � [2].

The paper ends opening to the search of new possible methods capable to describe

completely the physical reality, i.e. seeking for a new theory comprehensive of the lacking

information, so that, with the addition of additional parameters it would be able to com-

pletely describe the physical reality.

2.4 Bohm-Aharonov’s formulation of the paradox

Now we shall examine another formulation of the EPR paradox, following the example

ideated by David Bohm and Yakir Aharonov [4]. In this model, a quantum system is con-

sidered from the point of view of its own spin variables. This formulation of the paradox

is more handy and comfortable from a mathematical point of view, since, working with

spins, the operators and the eigenfunctions will become matrices and vectors respectively,

reducing all of the problem to a simple matricial calculus; moreover, the theoretical for-

mulation of the problem is nearer to the experimental situations where it has been really

examinated.

Let’s consider a diatomic molecule of spin 0, whose atoms (I and II) are our two

systems taken into account, having spin 1/2. It is straightforward that the wave function

of the composed system will be given by a singlet state of spin. We suppose that the

couple of atoms interact for a certain time and then become entangled; after that time

they are separated. From the time of separation on, there is no longer any interaction

between them.

We can certainly define the wave function of the composed system, even if we do

not know the functions that describe the state of each single atom separately; omitting

the orbital part, which is quite inessential for the description of the problem, the state

describing the composed system is a singlet state of spin:

Ψ =1√2

[Ψ+(I)Ψ−(II) − Ψ−(I)Ψ+(II)] , (2.21)

where Ψ±(I) e Ψ±(II) are the vectors that represent the state of the first and the second

particle respectively, with spin ±~2 on the direction of the axis z. In that state, for theconservation of the angular momentum, the total spin always must vanish; whether one

23

of the two atoms assumes a positive value, the other will be certainly negative.

Given the description of Bohm-Aharonov’s model, it is easy to see that the quantities

to be measured are two of the three components of the spin between one of the two atoms,

that we know from Quantum Mechanics to be incompatible; that is, the relative operators

generally obey to the relation:

[Si, Sj ] = i~εijkSk.

In order to see this, we maintain an analogy with EPR steps, choosing as physical

quantity the Sx component of the spin for I; measuring it, we shall obtain two possible

values: +~2 and −~2 .

In the first case, as we know from the knowledge of the eigenvectors of Ŝx, the atom I

collapses into the state given by

Ψ+(I) =1√2

(1

1

),

eigenvector of the operator Ŝx, relative to the assumed eigenvalue +~2 .

Therefore, for the conservation of angular momentum, it is necessarily

Ψ = Ψ+(I)Ψ−(II), (2.22)

from which we also deduce the state of the second atom, which will have a wave function

given by

Ψ−(II) =1√2

(1

−1

),

eigenvector of the operator Sx, relative to the eigenvalue −~2 of the spin for atom II.In the second case all is analogous, but with inverted eigenvalues and eigenvectors for

the two atoms.

Choosing as physical quantity B the component Sz of the spin for I, and measuring

it, we will get the same two possible values of the previous case.

So we see that whether the observable assumes a value +~2 , then the atom I collapses

into the state given by

Ψ+(II) =

(1

0

),

24

eigenvector of the operator Ŝz, relative to the assumed value +~2 .

Then the state of the total system will be

Ψ = Ψ+(I)Ψ−(II),

from which we can work out a state for the second atom, that will be equal to

Ψ−(II) =

(0

1

),

eigenvector of the operator Ŝz, relative to the eigenvalue −~2 of spin for atom II.In accordance with what we saw for Ŝx, when the measure of the observable B assumes

the second of the two possible values, it is all analogous, but with inverted eigenvalues and

eigenvectors for the atoms.

In this way Bohm and Aharonov demonstrate that the generic quantities A and B

are reduced to a pair of observables (in our example: Sx and Sz), whose operators do not

commute. Then the two wave functions, called by Einstein ψk and φj , represent two states

of simultaneous reality for these operators, related to the second atom (in the example:

Ψ+/−(II) for the Sx-spin and ψ+/−(II) for the Sz-spin).

The result obtained is the same as that obtained by Einstein, Podolsky and Rosen:

the steps followed are equivalent. Indeed both start considering a pair of incompatible

quantities, and they successively measure these observables on one of the two considered

systems. Both of them finally demonstrate that, measuring these quantities on one of the

two systems, they are able to determine these values on the other system too, but without

altering its state; that is, both demonstrate that the wave functions for the quantitites

of the second systems, could correspond with the eigenfunctions of two noncommuting

operators, related to two incompatible physical quantities.

In order to solve the contradiction of both the examined formulation, one can drop

one of the three hypotheses assumed, that we list here again:

i) Principle of reality,

ii) Principle of locality,

iii) Completeness of Quantum Mechanics.

25

According to EPR, the hypothesis to be abandoned is nothing but the completeness

of Quantum Mechanics, since, as they have demonstrated, the quantum description of

reality, provided by the wave function, comes out to be inconsistent with the assumed

principles. As well, since completeness of Quantum Mechanics means “indeterminism”,

they are sure of the need to complete the latter by means of a more fundamental theory,

in which the incompleteness is overwhelmed introducing additional dynamical parameters,

the so-called “hidden variables”: in this way, in addition to the solution of the paradox,

we would get back an essentially deterministic vision of the world, in which it is possible

to associate with certainty a definite value for every quantity, i.e. a definite element of

reality. As a matter of fact, the weak point of their proof is precisely to take for granted

the assumptions i) and ii), as we already said, but this is not to be considered as a gross

mistake of the authors, but rather a historical limit of their work.

It is better to stress again the importance of one of the critical point of the argumen-

tation of Einstein and collaborators. Expressing the principle of reality for the Gedanken-

experiment of Bohm-Aharonov, it states that, if one can execute an operation, that allows

him to predict with certainty the value of a measurement for a quantity without disturbing

the measured spin, then the measurement has a definite value, apart from the fact that this

operation is actually executed or not.

In accordance with the Copenhagen interpretation — that is the same of Bohr who,

by the way, replied with an article [3] to that of EPR as follows — the concept of reality

cannot be legitimately applied to a property unless there is a device able to measure it,

whereas Einstein intended this vision as anthropocentric, assering that instead physical

systems have intrinsic properties, apart from the fact that they are observed or not.

So it is here that arises the most large difference between the two currents of thought,

each of them defending their own position with their arguments.

But it is important to point out at once the concept of simultaneity, used more than

once by EPR, since this word was a ‘surprising’ expression for people who knew very

well that this term was undefined in the teory of relativity. Let us examine this issue

with Bohm’s singlet model. [5] One observer, conventionally called Alice, measures the

z-component of the spin for her particle and find +~/2. Then she immediately knowsthat if another distant observer, Bob, measures (or has measured, or will measure) the

z-component of the spin for his particle, the result is certainly −~/2. One could then askwhen Bob’s particle acquires the state with sz = −~/2. This question is meaningless, butit has a definite answer: Bob’s particle acquires this state instantaneously in the Lorentz

26

frame that we arbitrarily choose to perform our calculations. Of course, Lorentz frames

are not material objects: they exist only in our imagination. When Alice measures her

spin, the information she gets is localized at her position, and will remain so until she

decides to broadcast it. Absolutely nothing happens at Bob’s location. From Bob’s point

of view, any spin directions are equally probable, as can be verified experimentally by

repeating the experiment many times with a large number of singlets without taking in

consideration Alice’s results. Thus, after each one of her measurements, Alice assigns a

definite pure state to Bob’s particle, while from Bob’s point of view the state is completely

random. It is only if and when Alice informs Bob of the result she got (by mail, telephone,

radio, or by means of any other material carrier, which is naturally restricted to the speed

of light) that Bob realizes that his particle has a definite pure state. Until then, the two

observers can legitimately ascribe different quantum states to the same system. For Bob,

the state of his particle suddenly changes, not because anything happens to that particle,

but because Bob receives information about a distant event. So reality might differ for

different observers.

Anyway, the conclusions reached by EPR raised a discussion that nowadays is still —

in other ways — object of heated debates.

Is it really possible to complete Quantum Mechanics?

In other terms: is it possible to consider quantum states as averages over those states

for which the results of any possible measures are, in principle, completely determined?

27

Chapter 3

Bell’s Theorem

There were two principal currents of thought that wanted to give an explanation to the

question opened with the paper of Einstein, Podolsky and Rosen: on one hand there

were the determinists, i.e. those people who believed that Quantum Mechanics should

be extended in a theory with local hidden variables 1; on the other hand, there were

the indeterminists, that instead did not consider as right the formulation of the paradox

because of its wrong assumptions.

In 1964 the Irish physicist John Stewart Bell published the paper “On the Einstein-

Podolsky-Rosen Paradox” [6], where he provided an elegant solution that could be verified

experimentally, in order to consolidate the more or less validity of EPR argumentations.

He proved, through his inequalities, that any possible theory with hidden variables,

with the requirement of locality, is in contradiction with the statistical previsions of Quan-

tum Mechanics, e.g. the former cannot be considered as the completion of the latter, but

they are to be set on different planes. As we shall emphasize several times, it is best to

notice from now on that a possible interpretation with nonlocal hidden variables is not

forbidden; in such a case, the conflict does not arise.

We can illustrate and summarize the Bell’s Theorem as follows:

(Bell’s) Theorem. There is no local hidden variable theory that can reproduce (all) the

predictions of Quantum Mechanics.

Bell demonstrated that for certain phenomena local reality implies some constraints

— i.e. the inequalities — , which are not required but violated by Quantum Mechanics.

The experiments that have been carried so far have shown an evident violation of Bell’s

1Here we assume local in the sense of Einstein, that is, remote systems do not interact with each othersand they behave as if they were independent.

28

inequalities; so we get an empiric evidence against local reality, showing that some of the

“spooky actions at a distance” expected by EPR, actually occur.

Nevertheless, the principles of special relativity are not violated. Indeed, it has been

proved theoretically and experimentally that, because of the No-Communication Theorem,

it is impossible for one experimenter to use these special quantum effects to communicate

information to another with a velocity faster than light.

As a first point, Bell was interested in formulating Einstein’s theory within a math-

ematical frame. In order to do that, he based his work on Bohm’s formulation of the

paradox, considering a couple of spin-1/2 particles in a singlet spin state. The EPR ad-

ditional variables should have had the aim of making the theory causal and local. What

Bell was trying to (and finally) show is that such an operation is incompatible with the

statistical previsions of Quantum Mechanics.

3.1 Bell’s Inequalities

We consider a couple of 12 -spin particles, moving freely in opposite directions and forming a

system in a singlet spin state, of the form (2.21). It is possible to make measurements, e.g.

by Stern-Gerlach magnets, on selected components of the spins ~σ1 and ~σ2. From Quantum

Mechanics we know that, whether a measurement of the component ~σ1 · â, where â is aunit vector, yields value +1, then measurement of ~σ2 · â will give with certainty value –1and vice versa.

We now introduce the principle of locality seen in the last chapter, that establishes that

whether the measurements are made at places remote from one another, the orientation of

one magnet does not influence the result obtained with the other. Since we can predict in

advance the result of measuring any chosen component of ~σ2 after a measurement of the

same component of ~σ1, it follows that the result of any such measurements must actually

be predetermined.

Since the wave function describing the initial state does not determine the result of

an individual measurement, this predetermination implies the possibility of adding some

new variables that would represent some properties intrinsic to any considered pair of

particles, and that are not considered by the wave function just because of their difference

from couple to couple. These additional parameters are the so-called hidden variables,

that Bell labels with the letter λ. It is the same argument whether λ stands for a single

variable or a set of variables, or even a set of functions, or if the variables are discrete or

continuous. We consider for simplicity λ as a single continuous parameter.

29

With such addition, the result A of the measurement of ~σ1 · â is determined by thevector â and by the parameter λ, and the result B of the measurement of ~σ2 · b̂ is in asimilar manner determined by b̂ and λ, that is

A(â, λ) = ±1;

B(b̂, λ) = ±1. (3.1)

The principle of locality, the crucial point in the argumentation of Einstein, Podolsky

and Rosen, implies that the result B for particle 2 does not depend from the arrangement

â of the magnet for particle 1, neither A from b̂.

Defining by ρ(λ) the probability distribution of λ, then the expectation value of the

product of the two components ~σ1 · â and ~σ2 · b̂ is

P (â, b̂) =

∫dλρ(λ)A(â, λ)B(b̂, λ). (3.2)

This value should be equal to the quantum mechanical expectation value, which for a

singlet spin state is 2

〈~σ1 · â ~σ2 · b̂

〉= −â · b̂. (3.3)

Actually, as will be shown below, these two expressions are not equal and they are

mutually exclusive.

Since ρ(λ) is a probability distribution, it is normalized,∫dλρ(λ) = 1, (3.4)

and because the relations (3.1) are valid, (3.2) cannot be less than –1. It could reach the

value –1 only for â = b̂ if

A(â, λ) = −B(b̂, λ), (3.5)

except for a set of points λ of zero probability. Thus (3.2) becomes

P (â, b̂) = −∫dλρ(λ)A(â, λ)B(b̂, λ). (3.6)

Now define ĉ, another unit vector; therefore, with the help of (3.1),

2For an explicit evaluation, see Appendix A.

30

P (â, b̂)− P (â, ĉ) = −∫dλρ(λ)

[A(â, λ)A(b̂, λ)−A(â, λ)A(ĉ, λ)

]=

∫dλρ(λ)A(â, λ)A(b̂, λ)

[A(b̂, λ)A(ĉ, λ)− 1

], (3.7)

from which ∣∣∣P (â, b̂)− P (â, ĉ)∣∣∣ ≤ ∫ dλρ(λ) [1−A(b̂, λ)A(ĉ, λ)] . (3.8)For (3.6), the second term of the RHS of the last Eq. is equal to P (b̂, ĉ), then

1 + P (b̂, ĉ) ≥∣∣∣P (â, b̂)− P (â, ĉ)∣∣∣ . (3.9)

The inequality (3.9) is the first of a family of inequalities that are collectively called

‘Bell’s Inequalities’.

This inequality sets a constraint on the expectation values with hidden variables, val-

uated along the three considered directions.

Indeed, generally the RHS is of order |b̂− ĉ| for small |b̂− ĉ|. Consequently, the valueP (b̂, ĉ) cannot be stationary at the minimum value (-1 for b̂ = ĉ) and therefore cannot

be equal to the quantum mechanical result (3.3) in these conditions.

Moreover neither quantum mechanical correlation (3.3) can be approximated arbitrar-

ily by (3.2).

Just to prove it, we now suppose to make more repeated experiments. The directions

of unit vectors â, b̂, ĉ will not be strictly determined, but will lie into right circular cones

of small width — since physically the directions are always affected by certain errors.

Accordingly, instead of (3.2) and (3.3) we consider their averaged functions

P (â, b̂) −â · b̂, (3.10)

where the bar denotes independent average of P (â′, b̂′) and −â′ · b̂′ over vectors â′ andb̂′ within specified small angles of â and b̂. We suppose that for any unit vector â and

b̂, P (â, b̂) approximates quantum mechanical results well, that is, the difference between

the averages is bounded on the upper side by a small quantity ε:∣∣∣P (â, b̂) + â · b̂∣∣∣ ≤ ε. (3.11)31

Our aim is to show that ε cannot be made arbitrarily small, making the impossibility

of (3.2) to be equal to (3.3).

Let’s consider that for all â and b̂∣∣∣â · b̂− â · b̂∣∣∣ ≤ δ; (3.12)then ∣∣∣P (â, b̂) + â · b̂∣∣∣ ≤ ∣∣∣P (â, b̂) + â · b̂∣∣∣+ ∣∣∣â · b̂− â · b̂∣∣∣ ≤ ε+ δ. (3.13)

From (3.2) we can define

P (â, b̂) =

∫dλρ(λ)A(â, λ)B(b̂, λ) ≤ 1, (3.14)

where

∣∣A(â, λ)∣∣ ≤ 1 e ∣∣∣B(b̂, λ)∣∣∣ ≤ 1. (3.15)From (3.13) and (3.14), taking â = b̂, we get∫

dλρ(λ)[A(b̂, λ)B(b̂, λ) + 1

]≤ ε+ δ. (3.16)

Now, from definition (3.14) it turns out that

P (â, b̂)− P (â, ĉ) =∫dλρ(λ)

[A(â, λ)B(b̂, λ)−A(â, λ)B(ĉ, λ)

]=

∫dλρ(λ)A(â, λ)B(b̂, λ)

[1 +A(b̂, λ)B(ĉ, λ)

]−

∫dλρ(λ)A(â, λ)B(ĉ, λ)

[1 +A(b̂, λ)B(b̂, λ)

].

With the help of the two conditions (3.15),

∣∣∣P (â, b̂)− P (â, ĉ)∣∣∣ ≤ ∫ dλρ(λ) [1 +A(b̂, λ)B(ĉ, λ)]+

∫dλρ(λ)

[1 +A(b̂, λ)B(b̂, λ)

]. (3.17)

Making use of (3.14) and (3.16),

32

∣∣∣P (â, b̂)− P (â, ĉ)∣∣∣ ≤ 1 + P (b̂, ĉ) + ε+ δ. (3.18)Finally, using inequalities (3.13) we have for the RHS,

1 + P (b̂, ĉ) + ε+ δ = 1 + P (b̂, ĉ) + b̂ · ĉ− b̂ · ĉ + ε+ δ

≤ 1 +∣∣∣P (b̂, ĉ) + b̂ · ĉ∣∣∣− b̂ · ĉ + ε+ δ

≤ 1− b̂ · ĉ + 2(ε+ δ), (3.19)

while for the LHS,

∣∣∣P (â, b̂)− P (â, ĉ)∣∣∣ = ∣∣∣P (â, b̂) + â · b̂− â · b̂− P (â, ĉ) + â · ĉ− â · ĉ∣∣∣≥

∣∣∣â · ĉ− â · b̂∣∣∣− ∣∣∣P (â, b̂) + â · b̂∣∣∣− ∣∣P (â, ĉ) + â · ĉ∣∣≥

∣∣∣â · ĉ− â · b̂∣∣∣− 2(ε+ δ), (3.20)from which, for (3.19) and (3.20), (3.18) becomes∣∣∣â · ĉ− â · b̂∣∣∣− 2(ε+ δ) ≤ 1− b̂ · ĉ + 2(ε+ δ), (3.21)that is

4(ε+ δ) ≥∣∣∣â · ĉ− â · b̂∣∣∣+ b̂ · ĉ− 1. (3.22)

The interesting result here is that this inequality is not always verified for small values

of ε (as on the contrary it should be in order to make equal the expectation values (3.2)

and (3.3)). Actually, if we consider, for instance, â · ĉ = 0 and â · b̂ = b̂ · ĉ = 1√2, then

from (3.22) we obtain

4(ε+ δ) ≥√

2− 1, (3.23)

from which we conclude that, for little δ, ε cannot be made arbitrarily small. This is

the proof that the difference between the expectation values (3.2) and (3.3) is necessarily

finite, so that their corresponding theories are not compatible with each other.

Therefore, it could be reasonable to debate the principle of locality, or even to state

that if a theory with additional parameters wants to explain in a right manner quantum

mechanical rules, it necessarily has to be nonlocal, e.g. it should be in such a way that it

33

could allow for any measurement made on a system to influence any further measurement

made on other systems at any distance. Finally it is now justified the statement of Bell’s

Theorem given at the beginning of this chapter.

It is important to emphasize that Bell does not exclude local realistic theories, but

simply demonstrates that these latter would be in contradiction with quantum previsions,

so these cannot be considered as a complement of a quantum mechanical theory. Admitting

the existence of hidden variables automatically cancels the hypothesis of locality, but this

does not mean that there would be absolutely no possibility of making a different model

of additional parameters which could resolve the paradox and, at the same time, observe

quantum mechanical predictions.

For the moment, the problem raised by Bell is only the need to choose one or the other,

otherwise we have to renounce the concept — so intuitive and “taken for granted” — of

locality, the price to pay for allowing the introduction of a new theory with hidden variables

that would be consistent with Quantum Mechanics.

3.2 CHSH Inequalities

John Clauser, Michael Horne, Abner Shimony and Richard Holt (in the following CHSH),

in a paper of 1969 [7] generalized Bell’s inequalities in a way so that these could be applied

to realizable experiments. Successively (1971), a generalization of Bell’inequalities would

be given by Bell himself as well [6].

The CHSH inequalities are based upon a combination of four correlation coefficients

measured on four different orientations of the polarizers — a definition for coefficient of

polarization will be given in the next section as well.

Let’s consider a set of couple of entangled particles that move one in the opposite

direction of the other, so that one enters apparatus Ia and the other apparatus IIb, where

a and b are adjustable apparatus parameters. In each apparatus, a particle must select

one of the two channels labeled +1 and −1 respectively. Let the results of selections berepresented by A(a) and B(b), where a and b represent the adjustable parameters of the

two measurement apparatus. So A(a) and B(b) can assume values ±1 according as thefirst or second channel is selected.

Suppose now that there is a statistical correlation of A(a) and B(b), due to information

carried by and localized within each particle, and that at some time in the past the particles

constituting one pair were in contact and communication regarding this information. The

34

information, which emphatically is not quantum mechanical, is part of the content of a

set of hidden variables, denoted collectively by λ.

The results of the two selections must be deterministic functions, denoted by A(a, λ)

and B(b, λ).

A reasonable request of locality requires A(a, λ) to be independent of the parameter

b, and B(b, λ) to be likewise independent of a, since the two selections may occur at an

arbitrarily great distance from each other.

Finally, since the pair of particles is generally emitted by a source in a manner phys-

ically independent of the adjustable parameters a and b, we assume that the normalized

probability distribution, ρ(λ), characterizing the ensamble is independent of a and b.

We can now define a correlation function as

P (a, b) =

∫Γdλρ(λ)A(a, λ)B(b, λ), (3.24)

where Γ is the total space of the hidden variables λ.

Using the definition (3.24) it follows that

|P (a, b)− P (a, c)| ≤∫

Γdλρ(λ) |A(a, λ)B(b, λ)−A(a, λ)B(c, λ)|

=

∫Γdλρ(λ) |A(a, λ)B(b, λ)| [1−B(b, λ)B(c, λ)]

=

∫Γdλρ(λ) [1−B(b, λ)B(c, λ)]

= 1−∫

Γdλρ(λ)B(b, λ)B(c, λ). (3.25)

Let’s suppose that for some parameters b′ and b we have P (b′, b) = 1 − δ, where0 ≤ δ ≤ 1. Experimentally interesting cases will have δ close to but not equal to zero.Note as well that in this point of the argumentation we are avoiding the experimentally

unrealistic restriction of Bell that for some couple of b′ and b there is perfect correlation

(i.e., δ = 0).

Dividing the region Γ into two regions Γ+ and Γ− such that Γ± = {λ |A(b′, λ) = ±B(b, λ)},from (3.24) we have

35

P (b′, b) =

∫Γdλρ(λ)A(b′, λ)B(b, λ)

=

∫Γ+

dλρ(λ)A(b′, λ)B(b, λ) +

∫Γ−

dλρ(λ)A(b′, λ)B(b, λ)

=

∫Γ+

dλρ(λ) [B(b, λ)]2 −∫

Γ−

dλρ(λ) [B(b, λ)]2

=

∫Γ+

dλρ(λ)−∫

Γ−

dλρ(λ)

= 1− δ =∫

Γdλρ(λ)− δ =

∫Γ+

dλρ(λ) +

∫Γ−

dλρ(λ)− δ,

from which ∫Γ−

dλρ(λ) =δ

2.

We can therefore write the integral of RHS of (3.25) as

∫Γdλρ(λ)B(b, λ)B(c, λ) =

∫Γ+

dλρ(λ)B(b, λ)B(c, λ) +

∫Γ−

dλρ(λ)B(b, λ)B(c, λ)

=

∫Γ+

dλρ(λ)A(b′, λ)B(c, λ)−∫

Γ−

dλρ(λ)A(b′, λ)B(c, λ)

=

∫Γ+

dλρ(λ)A(b′, λ)B(c, λ) +

∫Γ−

dλρ(λ)A(b′, λ)B(c, λ) +

− 2∫

Γ−

dλρ(λ)A(b′, λ)B(c, λ)

=

∫Γdλρ(λ)A(b′, λ)B(c, λ)− 2

∫Γ−

dλρ(λ)A(b′, λ)B(c, λ)

≥ P (b′, c)− 2∫

Γ−

dλρ(λ)∣∣A(b′, λ)B(c, λ)∣∣

= P (b′, c)− δ = P (b′, c) + P (b′, b)− 1, (3.26)

therefore from (3.25) and (3.26),

|P (a, b)− P (a, c)| ≤ 2− P (b′, b)− P (b′, c); (3.27)

this is an expression for CHSH inequalities.

In principle entire measuring devices, each consisting of a filter followed by a detector,

36

could be used for Ia and IIb, and the values ±1 of A(a) and B(b) would denote detection ornondetection of the particles. Inequalities (3.27) would then apply directly to experimental

counting rates.

Nevertheless, if photons are used, this manner could not make to a very check of

(3.27), for the photomultipliers have a little efficiency; so from now on we shall interpret

A(a) = ±1 and B(b) = ±1 as the coming out or less of photons from their respectivefilters, that for example could be linear polarizers having orientations defined by a and b.

At this point it is worthwhile to introduce an exceptional value, ‘∞’, taken fromparameters a and b for representing polarizer remotion; clearly, we will necessarily have

A(∞) = B(∞) = +1.

Since P (a, b) is a correlation function of emission (of photons), it has to be made a

further assumption in order to derive an experimental prediction: that if a pair of photons

emerges from Ia and IIb, the probability of their joint detection is independent of a and b.

Then if the flux into Ia and IIb is a constant independent of a and b, the rate of coincidence

detection R(a, b) will be proportional to w [A(a)+, B(b)+], where w [A(a)±, B(b)±] is the

probability that A(a) = ±1 and B(b) = ±1.Letting

R0 = R(∞,∞), R1(a) = R(a,∞), R2(b) = R(∞, b),

and making use of the evident formulas

P (a, b) = w [A(a)+, B(b)+]− w [A(a)+, B(b)−]− w [A(a)−, B(b)+] + w [A(a)−, B(b)−] ,

and

w [A(a)+, B(∞)+] = w [A(a)+, B(b)+] + w [A(a)+, B(b)−] ,

w [A(∞)+, B(+)+] = w [A(a)+, B(b)+] + w [A(a)−, B(b)+] ,

w [A(∞)+, B(∞)+] = P (a, b) + 2w [A(a)+, B(b)−] + 2w [A(a)−, B(b)+] ,

we get

P (a, b) =4R(a, b)

R0− 2R1(a)

R0− 2R2(b)

R0+ 1. (3.28)

37

Finally, we can now express (3.27) in terms of experimental quantities, namely coinci-

dence rates with both polarizers in, and with one and then the other removed. Supposing

that R1(a) and R2(b) are constants equal to R1 and R2 experimentally determined, CHSH

inequalities are

|R(a, b)−R(a, c)|+R(b′, b) +R(b′, c)−R1 −R2 ≤ 0. (3.29)

This is the generalized formulation of Bell’s inequalities by CHSH, in a favourable

manner to be applied to experiments with photons. In the next section we shall see how

to rewrite these in a more explicit manner, thus applying them on the optical version of

Bell’s Gedankenexperiment.

3.3 Optical version of Bell’s Gedankenexperiment

As far as we have seen until now, Bell’s Theorem states that local realistic theories are in

disagreement with Quantum Mechanics, but there is nothing which tells us that these the-

ories are to be rejected. On the contrary, the disagreement between the theories illustrated

by Bell arises from rather unusual situations.

It is just for this reason that an empirical testing — by means of appropriately designed

experiments — is required to test those critical regions in which conflicts have origin.

The more significative experiments, as well as complete, are those by Alain Aspect,

who was interested in the individuation of critical regions and in the analysis of photon

behavior in situations built up ad hoc. Section 3.4 will deal with the experimental aspect

and the results obtained by Aspect.

What we wish to study now is a reformulation of Bell’s Theorem [9], adopted by As-

pect himself as starting point for his whole experimental examinations: a rewriting of

the inequalities in terms of correlation coefficients between the directions of the photons,

making more clear the importance of the hypothesis of locality within the theorem.

In the optical version of Gedankenexperiment of Einstein, Podolsky and Rosen, due to

Bohm (see Fig. 3.1), a source S emits pairs of photons with different frequencies, ν1 and

ν2, which propagate along opposite directions ±O~z, in a way similar to that of a pair of12 -spin particles in a singlet spin state.

Let’s suppose that the state entangled which describes polarization of the two photons

is given by the ket

38

Figure 3.1: The Einstein, Podolsky, Rosen and Bohm Gedankenexperiment.

|Ψ(ν1, ν2)〉 =1√2

(|x, x〉+ |y, y〉) , (3.30)

where |x〉 and |y〉 represent states describing two different orthogonal directions of linearpolarization. This state is not a product state, therefore it is not possible to fix any single

photons with arbitrary polarization, instead we have to see the system in its whole entirety.

Two linear polarizers I and II, placed at the sizes of the source and oriented in the

directions given by unit vectors â and b̂, have the aim to analyse photons and see their

tracks. So it is possible to make measurements on linear polarization of two photons

through the analysis of two detectors (represented by + and – in the Figure) placed next

to each of the polarizers. Both will give result +1 or –1 depending on whether the photon

polarization occurs in a direction parallel or perpendicular to the one of the polarizer itself.

Labeled P±(â) the probability of obtaining the result ±1 for ν1, and P±(b̂) that oneof obtaining ±1 for ν2, for these measurements quantum mechanical predictions for singledetection are 3

P+(â) = P−(â) =1

2,

P+(b̂) = P−(b̂) =1

2, (3.31)

according to the fact that no polarization cannot to be established for single photons;

therefore any measurement will give a random result.

Defining now P±±(â, b̂) to be the probability of combined detection of ν1 in channel

± of I (oriented along â), and ν2 in channel ± of II (b̂), quantum mechanical previsionsfor combined detection are 4

3For an explicit evaluation, see Appendix A.4See note 3.

39

P++(â, b̂) = P−−(â, b̂) =1

2cos2 ϑab,

P+−(â, b̂) = P−+(â, b̂) =1

2sin2 ϑab, (3.32)

where ϑab is the angle between â and b̂.

In particular, we can see that, in the case for ϑab = 0, we get

P++(â, b̂) = P−−(â, b̂) =1

2,

P+−(â, b̂) = P−+(â, b̂) = 0, (3.33)

that is, whether the photon ν1 gives result +1 (whose probability is 50%), then photon ν2

as well will give certainly result +1 (analogously for -1), that it means total correlation.

This prevision is perfectly in accordance with all we have discussed so far, since mea-

suring the polarization — along the same direction — of two photons emicted by the same

source at the same time, we obtain two results equal and opposite, as it happens with the

spin of two atoms belonging to the same molecule.

In this contest it is very useful to define the following quantity.

Definition (Correlation coefficient of polarization). It is called correlation coef-

ficient of polarization the quantity

E(â, b̂) = P++(â, b̂) + P−−(â, b̂)− P+−(â, b̂)− P−+(â, b̂). (3.34)

Hence, substituting (3.32), in the quantum mechanical case this coefficient becomes:

EMQ(â, b̂) = cos (2ϑab) . (3.35)

For ϑab = 0 we have instead

EMQ(0) = 1,

i.e. total correlation.

It is evident that the value obtained effectively explains strong correlations that relate

measurement results of ν1 and ν2. Generally, the correlation coefficient provides a quanti-

tative criterion in order to quantify the correlation between random results obtained from

any individual measurement.

40

So we have to admit that there are some properties typical for any pair of photons —

Einstein called them “elements of physical reality” — that explain correlations, according

to the type of correlated results one obtains. These properties, different from pair to pair,

are not taken in consideration by the vector state |Ψ(ν1, ν2)〉, that instead is the same forany pair of photons.

Here it is evident again the reason that guided Einstein to conclude that Quantum

Mechanics was not complete. And here there is the need of giving account of these

properties introducing some additional parameters, or hidden variables.

We can finally explain such interations through a classical description, and we can

hope of finding again quantum previsions by taking the average of the expectation values

over the hidden variables (procedure that brought Bell to his inequalities)

Now we shall try to work out Bell’s inequalities using the model described above.

Labelling λ the set of additional parameters and A(λ, â) and B(λ, b̂) the results re-

spectively obtained from analyser I oriented respectively along â, and analyser II oriented

along b̂, these quantities can only assume values ±1, hence the quantity 12 [1 +A(λ, â)]could assume only values +1 (in case of result +) and 0 (otherwise), and analogously, the

quantity 12 [1−A(λ, â)] could assume only values +1 (in case of result –) and 0 (other-wise). Hence, given the probability distribution of λ, that is ρ(λ), the expectation values

for single detection are found to be:

P±(â) =1

2

∫dλρ(λ) [1±A(λ, â)] ,

P±(b̂) =1

2

∫dλρ(λ)

[1±B(λ, b̂)

],

whereas for combined detection:

P++(â, b̂) =1

4

∫dλρ(λ) [1 +A(λ, â)]

[1 +B(λ, b̂)

],

P−−(â, b̂) =1

4

∫dλρ(λ) [1−A(λ, â)]

[1−B(λ, b̂)

],

P+−(â, b̂) =1

4

∫dλρ(λ) [1 +A(λ, â)]

[1−B(λ, b̂)

],

P−+(â, b̂) =1

4

∫dλρ(λ) [1−A(λ, â)]

[1 +B(λ, b̂)

].

Substituting these quantities in definition (3.34), after some straightforward passages,

one finds that the correlation coefficient, averaged over the distribution of λ, is given by:

41

E(â, b̂) =

∫dλρ(λ)A(λ, â)B(λ, b̂). (3.36)

Now we define a new quantity, that allows us to write the inequalities representing in

explicit form the correlation coefficient, quantity which can be denoted by s(λ, â, â′, b̂, b̂′

):

s(λ, â, â′, b̂, b̂′

)def= A(λ, â)B(λ, b̂)−A(λ, â)B(λ, b̂′) +A(λ, â′)B(λ, b̂) +A(λ, â′)B(λ, b̂′)

= A(λ, â)[B(λ, b̂)−B(λ, b̂′)

]+A(λ, â′)

[B(λ, b̂) +B(λ, b̂′)

]. (3.37)

Since A and B can assume only the values ±1, then

s(λ, â, â′, b̂, b̂′

)= ±2,

from which, averaging over the distribution of λ one gets

− 2 ≤∫dλρ(λ) s

(λ, â, â′, b̂, b̂′

)≤ +2, (3.38)

that is, defining

S := E(â, b̂)− E(â, b̂′) + E(â′, b̂) + E(â′, b̂′), (3.39)

we obtain the inequalities

− 2 ≤ S(â, â′, b̂, b̂′) ≤ +2. (3.40)

(3.40) are well-known as BCHSH inequalities, i.e. inequalities of Bell generalized by

Clauser, Horne, Shimony, Holt. These inequalities are based on a combination of four

correlation coefficients of polarization, measured along four orientations of the polarizers.

Then S is a measurable quantity.

However, BCHSH inequalities in some particular situations (which will be specified

shortly) are in conflict with Quantum Mechanics. Indeed if we put the system in the

configuration illustrated by Figure 3.2, with

ϑab = ϑba′ = ϑa′b′ =π

8,

ϑab′ = ϑab + ϑba′ + ϑa′b′ ,

42

substituting the quantities E in (3.39) with their quantum mechanical values (3.35), we

get

SMQ = 2√

2.

This quantum mechanical prevision deeply violates the upper limit of inequalities

(3.40).

Figure 3.2: Orientations with ϑab = ϑba′ = ϑa′b′ =π8 .

In this particular situation it turns out that quantum previsions cannot be obtained

from theories with hidden variables, therefore we can conclude that a local deterministic

theory does not exist, according to the — very general — model showed in this Section,

which reproduces all quantum mechanical previsions.

It is reasonable to ask what are precisely the critical regions and for which angles one

could have the maximum conflict. Deriving S with respect to the three independent angles

ϑab, ϑba′ , ϑa′b′ , we have that SMQ is extreme in correspondence with

ϑab = ϑba′ = ϑa′b′ = ϑ, (3.41)

and for (3.35),

SMQ(ϑ) = 3 cos(2ϑ)− cos(6ϑ). (3.42)

Finally, deriving this quantity with respect to ϑ and letting equal to zero,

dSMQdϑ

= 6 sin(6ϑ)− 6 sin(2ϑ) = 0,

we obtain the angles for which SMQ has its maximum and minimum values, that are

respectively

43

SmaxMQ = 2√

2 per ϑ =π

8,

SminMQ = −2√

2 per ϑ =3π

8,

(situations illustrated in Figure 3.2 and 3.3 respectively).

Figure 3.3: Orientations with ϑab = ϑba′ = ϑa′b′ =3π8 .

At last, with a brief study of function, it is possible to plot a graphic representing the

behaviour of S varying with the angle ϑ, as shown in Figure 3.4.

In conclusion, Bell’s Theorem brings out a conflict between theories with hidden vari-

ables and certain quantum mechanical previsions (according to the model expounded in

this section) and provides a quantitative criterion to clarify this conflict.

Discussing the hypotheses, the fundamental assumptions on the bases of the described

model are three: the existence of hidden variables, determinism and the locality condi-

tion. These hypotheses, if simultaneously assumed, give rise to an incoherence between

the created theory and Quantum Mechanics. The first two of them are not to be under

discussion since they are parts of the theory itself. The third instead is a “handy” and

“obvious” assumption that it seems absurd not to accept. And yet, whether in the case

that the first two hypotheses do not want to be dropped out, if one wish to complete

44

Figure 3.4: S(ϑ) = 3 cos(2ϑ) − cos(6ϑ) as expected by Quantum Mechanics for pairs inthe state (3.30). The conflict arises in the hatched zone.

Quantum Mechanics by a theory with hidden variables, then this theory will have to be

necessarily nonlocal. Indeed if the condition of locality is abandoned, it can be easily

shown that with quantities like A(λ, â, b̂) or ρ(λ, â, b̂), the demonstration that brings to

BCHSH inequalities drops, thus voiding their result.

3.4 A. Aspect’s experiments

Between 1980 and 1982 Alain Aspect and his équipe tried to create more complex experi-

mental apparatuses, compared with those used until that time, to verify Bell’s inequalities,

and they made three experiments to study the validity of these constraints using atomic

cascades [9].

The first experiment was been organized as a direct check for inequalities. Nevertheless,

it had some limits, e.g. single-channel polarizers were used , so that only photons polarized

parallel to â (or b̂) could pass, while those polarized orthogonally were blocked. Therefore

only the results + could be revealed, and measurements of coincidence could only provide

an evaluation for P++(â, b̂). So one could not know whether the missed measure of

correlation between a pair of photons was due to a real absence of correlation (i.e. to the

action of the polarizer), or to a scarce efficiency of the detector.

For this reason a second experiment was performed, in this case preserving a configura-

tion like that of the Bohm-Aharonov’s thought experiment. This experiment implemented

45

two-channel polarizers that made possible to detect the correlation of every photons that

reached the polarizer, thus even allowing for obtain a valuation for P−−(â, b̂).

Finally, the third experiment, which made use of polarizers with orientation variable

in time, can be considered as the more accurate and complete experiment, to which the

others can be reconducted. Moreover this experiment proves experimentally the impossi-

bility of faster-than-light communications.

As common source for each of the three equipments, it has been used an atomic cascade

(J = 0)→ (J = 1)→ (J = 0). The main feature of these experiments with respect to theprevious was the usage of a very powerful and stable source, that allowed to cut the data

recording times from several hours to a few minutes. The source exploited the cascade

4p2 1S0 → 4s4p 1P 1 → 4s2 1S0 of 40Ca.

Figure 3.5: Relevant energy levels for a 40Ca cascade. The atoms, excited by absorptionof two photons νK and νD, emit visible photons ν1 and ν2 correlated in polarization.

This cascade, represented in Fig. 3.5, produces two visible photons ν1 and ν2 corre-

lated in polarization. The atoms of 40Ca are thus excited from the ground state to the

upper energetic level by absorption of two photons, νK and νD, generated by two laser

beams. The first one (λK = 406.7 nm) is generated by ions of Kryptons, as the second

was a dye laser, brought to resonance for the two-photon process (λD = 581 nm); lasers

have parallel polarizations. First of all, there is a feedback cycle that checks on the dye

laser wavelenght, in order to have the maximum signal of fluorescence by the cascade.

Then a second feedback cycle checks on the power emitted by the Krypton laser in order

to guarantee a constant emission from the cascade. Strictly speaking, as a result of the

atom excitation, one electron of each atoms is led to jump up to two energy levels beyond

its ground state. When the electron falls from two energy levels, it sometimes emits a

pair of entangled photons. In this way, using a power of 40 mW for each laser, the typical

46

emission efficiency of the cascade is of 4× 107 photons per second.

3.4.1 Experiment with single-channel polarizers

In their first experiment, Aspect and collegues have mounted a pile of two polarizers made

up of ten glass plates to the Brewster’s angle; in front of each of these polarizer, a linear

polarizer has been placed, in order to transmit photons polarized in a way parallel with

the axes of the polarizer, and to stop those polarized perpendicularly.

In this type of experiment, one talks about single-channel polarizers because only the

value +1 is measured for each photon belonging to the couples of photons emitted by the

source. Therefore, the use of single-channel polarizers allows one to determine only the

result R(â+, b̂+) = R(â, b̂) 5, since it cannot be established if the result –1 for a photon

was due just to an orthogonal polarization with respect to the polarizer axes, or to a scarce

efficiency of the counting system. Accordingly, auxiliary recordings with one or both the

polarizers removed were needed, thus obtaining the following quantities:

R(∞,∞) = R0 R(â+,∞) = R1(â) R(∞, b̂+) = R2(b̂),

from which new BCHSH inequalities are obtained:

− 1 ≤ S′ ≤ 0, (3.43)

with

S′ =1

R0

[R(â, b̂) +R(â′, b̂) +R(â′, b̂′)−R(â, b̂′)−R1(â′)−R2(b̂)

]. (3.44)

Experimental test of Bell’s inequalities has provided

S′exp = 0.126± 0.014, (3.45)

that violates inequalities (3.43) by 9 standard deviations, and is on good agreement with

quantum mechanical previsions

SMQ = 0.118± 0.005, (3.46)5With analogy to the notation used in Section 3.2, by R(â+, b̂+) we intend the counting result of the

detection of coincidences parallel to the single-channel polarizers (+) oriented with respect to â and b̂.

47

(in this case the error accounts for the uncertainty in the measurements of the polarizer

efficiences).

3.4.2 Experiment with two-channel polarizers

When a photon is stopped by the polarizer in the single-channel experiment, it is lost and

there is no way to establish whether and how it was correlated to another photon. This is

the reason why, in the second experiment made by Aspect, two channels have been used.

Indeed, if a photon is stopped by the polarizer, then the photon is reflected by it and so

could be still detected. In this way the coincidence rate was much greater and leaded to

a more precise experiment.

With two-channel polarizers, the experiment implemented is more similar to that of

the scheme in Figure 3.1. The polarizers used were polarizing cubes that transmit a

polarization (par