Bounds on Quantum Probabilities - TU Wientph.tuwien.ac.at/~svozil/publ/boundprob.pdf · EPR-paradox [9] as a ﬁrst incentive to the search for hidden variables, mentioning von Neumann’s

DIPLOMARBEIT

Bounds on Quantum Probabilities

ausgefuhrt am

Institut fur Theoretische Physik,

Technische Universitat Wien

Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria

unter der Anleitung von

Ao. Univ.-Prof. Dr. Karl Svozil

durch

Stefan Filipp

Hauptstrasse 28, 2125 Bogenneusiedl

15. Janner 2003

2

Abstract

Since the emergence of quantum mechanics there have been doubtsabout its completeness and thoughts about its possible extensions yieldinga deterministic description of Nature. Bell’s Theorem [12] is the most fa-mous argument against a (local) hidden-variable theory that would solveproblems like “uncertainty” or “spooky action at a distance” [41, p. 158].We shall give a short overview about “no-go” theorems stating that quan-tum mechanics has to be considered complete as it is, starting from theEPR-paradox [9] as a first incentive to the search for hidden variables,mentioning von Neumann’s proof [10] and its refutation and discussing theBell-Kochen-Specker Theorem [12, 15] as well as Bell-type inequalities [2].

Furthermore, we shall show a method of generalizing Bell-type inequal-ities in terms of classical and quantum correlation polytopes introducedby I. Pitowsky [6, 8, 32] and apply these concepts to depict the violationof Bell-type inequalities using arbitrary four-dimensional quantal states.We will depict upper bounds of violations and investigate into the relationbetween the degree of mixedness and the possible violation. Different setsof possible probability values will be introduced that can be visualized ascuts through quantum correlation polytopes.

CONTENTS 3

Contents

1 Introduction 5

2 Hidden Variables Theories 7

2.1 EPR - Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Von Neumann’s “no-go” theorem . . . . . . . . . . . . . . . . . . . 10

2.3 Gleason’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Bell’s “continuum” no-go theorem . . . . . . . . . . . . . . . . . . 14

2.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Contextuality . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Kochen-Specker Theorem . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Peres’ variant of the KS-Theorem . . . . . . . . . . . . . . . . . . 17

2.7 Mermin’s variants of the KS-Theorem[11] . . . . . . . . . . . . . . 19

2.7.1 Four dimensions . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7.2 Eight dimensions . . . . . . . . . . . . . . . . . . . . . . . 20

2.8 Implications of the KS-Theorem . . . . . . . . . . . . . . . . . . . 20

2.9 Bell’s Theorem - from non-contextuality to locality . . . . . . . . 22

2.9.1 Link between the KS-Theorem and Bell’s Theorem . . . . 23

2.9.2 Proof of Bell’s theorem - with inequalities . . . . . . . . . 24

3 Correlation Polytopes 29

3.1 Simple Urn Model . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Geometrical interpretation . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Minkowski-Weyl representation theorem . . . . . . . . . . . . . . 30

3.4 From vertices to inequalities . . . . . . . . . . . . . . . . . . . . . 31

3.5 From inequalities to vertices . . . . . . . . . . . . . . . . . . . . . 32

3.6 Quantum mechanical context . . . . . . . . . . . . . . . . . . . . 32

4 Quantum Correlation Polytopes 35

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Classical probabilities - the set c(n) . . . . . . . . . . . . . 35

4.1.2 Quantum probabilities - the set bell(n) . . . . . . . . . . . 36

4.1.3 More general quantum probabilities - the set q(n) . . . . . 37

4.2 Violation of inequalities . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Generation of states . . . . . . . . . . . . . . . . . . . . . 39

4.2.2 Measurement operators . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Clauser-Horne-(CH)-inequality - bell(2) . . . . . . . . . . . 41

4.2.4 CHSH-inequality . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.5 Boole-Bell-type inequality out of bell(3) . . . . . . . . . . 47

4.3 Bounds of bell(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

CONTENTS 4

4.3.1 Representation of bell(1) . . . . . . . . . . . . . . . . . . . 49

4.3.2 Representation of bell(2) . . . . . . . . . . . . . . . . . . . 50

4.4 Generalizing bell(n) - the set q(n) . . . . . . . . . . . . . . . . . . 52

4.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Summary 56

6 Acknowledgments 58

A Characterization of states 59

A.1 Pure vs. mixed states . . . . . . . . . . . . . . . . . . . . . . . . . 59

A.2 Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.3 Composed systems . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.3.1 Pure states . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.3.2 Schmidt-Decomposition . . . . . . . . . . . . . . . . . . . 61

A.3.3 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.3.4 Composed mixed states . . . . . . . . . . . . . . . . . . . 63

A.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.4.1 Pure state entanglement . . . . . . . . . . . . . . . . . . . 63

A.4.2 Mixed state entanglement . . . . . . . . . . . . . . . . . . 64

B Dispersion-free states 65

1 INTRODUCTION 5

1 Introduction

“Die immer wieder ausgesprochenen Hoffnungen, die wesentlich statis-

tische Beschreibungsweise der Quantenmechanik durch die Annahme

eines den atomaren Phanomenen unterliegenden, aber unseren bish-

erigen Beobachtungen unzuganglichen kausalen Mechanismus zu ver-

meiden, durften in der Tat ebenso vergeblich sein wie jede Hoffnung,

mit den gewohnlichenen Vorstellungen von absolutem Raum und ab-

soluter Zeit der durch die allgemeine Relativitatstheorie gewonnenen

Vertiefung unseres Weltbildes gerecht zu werden.”[1]

Although the starting point of quantum mechanics lies already one century

in the past, the question whether or not quantum theory is complete has not

been decided yet. However, this does not pose a threat to the prolongation of its

success-story, since the question concerns mainly the interpretation of quantum

mechanics, not its predictions. From the operational point of view there do not

arise any difficulties, the theoretical calculations are in perfect agreement with

experimental results.

However, there have been numerous attempts to add components to the theo-

retical framework in order to achieve a complete description of Nature and to get

rid of the – to most researchers, most notably A. Einstein –“unpleasant”property

of indeterminism. In return, there have been numerous arguments that quantum

mechanics is already the best description of Nature without any add-ons.

The best known argument is definitely Bell’s Theorem [2], rejecting any en-

hancement of quantum theory in terms of a local hidden variables theory, i. e. any

theory consistent with quantum mechanical predictions but based on a determin-

istic foundation might be non-local, that is to say, not Lorentz-covariant1.

We will also review the EPR paradox as the most famous ’gedankenexperi-

ment’ pointing out the inconsistencies between quantum mechanics and special

relativity. Furthermore, von Neumann’s “no-go” theorem declining dispersion free

states, which are essential for a deterministic theory, and the Kochen-Specker The-

orem declining non-contextual hidden variables theories by an algebraic approach

will be reviewed.

However, Bell’s theorem predicting a quantum mechanical violation of the

so called Bell’s inequality is the most important experimental tool to test the

correctness and completeness of quantum theory, therefore many variants of this

particular sort of inequality have been found and many generalizations have been

devised. In this work a particular approach to find such Bell-type inequalities

(also referred as Boole-Bell-type inequalities2) will be adopted, namely to exploit

1There are notable exceptions - cf. I. Pitosky [3] or D. A. Meyer [4].2In the middle of the 19th century the English mathematician George Boole formulated

1 INTRODUCTION 6

the correspondence between convex polytopes and probability calculus introduced

by I. Pitowsky [6].

The main aim is to calculate the violations of specific Boole-Bell-type inequali-

ties for arbitrary quantal states, since usually only special kinds of quantum states

are used, for example Bell states. We will investigate, if the degree of violation

is dependent on the choice of measurements and if upper bounds of violations

appear – as proven by Tsirelson [7] for the CHSH-inequality. The arbitrary nu-

merically generated states will be classified into pure states and mixed states, and

we will analyze the dependence between the degree of mixedness and the degree

of violation.

A second topic will be an attempt to visualize the sets c(n), bell(n), and q(n),defined by Pitowksy in [8], consisting of different kinds of possible probability

values for events and their conjunction, by considering purely classical probability

values satisfying any corresponding Boole-Bell-type inequality, and also general

quantum mechanical probabilities, that can be achieved by using entangled states

as well as “entangled measurements”. Unfortunately, the number of dimensions

of the associated polytopes increases rapidly with the number of different events,

and therefore problems arise when trying to numerically generate usable states

for a two-dimensional visualization.

his theory on ”conditions of possible experience” [5]. These conditions are related to relativefrequencies of (logically connected) events and are expressed by certain equations or inequalities.Some of these inequalities are in turn violated by quantum mechanics and therefore called Boole-Bell-type inequalities.

2 HIDDEN VARIABLES THEORIES 7

2 Hidden Variables Theories

As already pointed out, quantum mechanics (QM) has always been a theory

with many different types of interpretation, although the predictions made by

quantum theory perfectly agree with the experimental results. The crucial point

is the interpretation of the mathematical formalism that is in most cases far

away from common sense. A main point of discussion is certainly the inherent

indeterminism, i. e. that predictions derived from quantum theory can only be

given in terms of probability values. When we have a quantized system (think for

instance of a free particle) described by a state vector |Ψ〉 and we want to know

specific properties of the particle like position or momentum, we can only assign

probability values for measuring a specific position or momentum. If we prepare

an ensemble of systems in the same state |Ψ〉 we can find out the probability

distribution of position or momentum by performing the same measurement on

each system of the ensemble.

Since momentum and position are not commuting, we cannot measure them

both on the same system, since for non-commuting observables Heisenberg’s un-

certainty relation holds, which imposes a lower bound on the product of two

non-commuting, self-adjoint operators representing observables of the system.

In what follows we shall concentrate now on the question if it is possible to find

an extension to QM that enables us to predict with certainty any desired property

of a quantum system. We can think of such an extension as additional variables,

which would impose definite values for position and momentum, if we only knew

the values of the variables. Such quantities are called “hidden variables”, because

they are not experimentally accessible within the QM context, therefore we can

only theoretically construct a theory that comprises hidden variables and predicts

the same measurement outcomes as QM.

According to the conventional interpretation of quantum mechanics, the state

vector is the most complete possible description of the system, which implies that

nature is fundamentally probabilistic (i.e., non-deterministic). What is the differ-

ence between conventional quantum theory and a quantum theory supported by

additional hidden variables? In a hidden variables (HV) theory a state is in prin-

ciple fully determined by a set of variables, so that the knowledge of the values of

all variables implies the possibility to predict the future evolution of the system

and also the result of any thinkable measurement with certainty. In QM this is

fundamentally impossible: Even if all – quantum mechanically – relevant prop-

erties are known, one cannot achieve more than statistical results. By preparing

a system in a particular state the evolution can be calculated accurately if the

governing Hamiltonian is known, but it is impossible to calculate the result of

any thinkable measurement operation.


A successful hidden variables theory would bear analogy to the relation be-

tween classical mechanics and classical statistical mechanics: Although it is not

feasible to measure position and momentum of each particle of a large system

and, therefore, we only get statistical predictions of the evolution and properties

of the system, there is no fundamental restriction to such an operation. We could

in principle measure position and momentum of each single particle and we would

acquire then all possible information about the whole system. But QM prohibits

us to do this; we have to check if there is an extension to QM to safe the idea of

fully deterministic Nature. In the next section we will provide an overview about

some theorems, stating that HV theories compatible with predictions of QM and

compatible with other reasonable assumptions are not possible.

2.1 EPR - Paradox

Already in 1935 Einstein and his coworkers Podolsky and Rosen concluded in

their often cited paper entitled“Can Quantum-Mechanical Description of Physical

Reality Be Considered Complete?” [9] that either QM cannot be complete (thus

additional (hidden) variables are needed for a full specification of a quantum state)

or that the assertion“the real states of spatially separated objects are independent

of each other”3 has to be relinquished. But that would imply a violation of the

Einstein’s causality principle4, which excludes signal transmission with a velocity

exceeding the speed of light. The latter was indisputable for Einstein and is still

indisputable to us at present.

Formulation The basis of their consideration is that “every element of the

physical reality must have a counterpart in the physical theory,” [9] also called

the condition of completeness, i. e. every theory demanding to be complete must

contain (mathematical) objects for any property of Nature accessible to experi-

ments. Assuming this, it does not make sense to talk about some metaphysical

properties common to a philosophical framework, since these are not measur-

able and therefore beyond every physical theory, but one has to concentrate on

properties accessible to physical experiments.

A criterion for an “element of physical reality” is the following: “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 exists an element of

physical reality corresponding to this physical quantity.” [9].

They then considered a quantum system consisting of two subsystems A and B

far apart, so that there cannot be any (classical) physical interaction in accordance

3A. Einstein, in Albert Einstein, Philosopher-Scientist, ed. by P. A. Schilpp, Library ofLiving Philosophers, Evanston (1949), p. 682

4Events localized in relatively space-like space-time regions must be causally independent.


with special relativity, i. e. A and B have moved outside each others light cones and

are therefore space-like separated. As a typical example, let us consider two spin-

1/2 particles in a singlet state |Ψ〉 = 1/√

2(| ↑↓〉− | ↓↑〉), where | ↑〉 denotes the

eigenvector of the spin-operator σz to the eigenvalue +1 and | ↓〉 to the eigenvalue

−15, sent in opposite directions to two observers waiting at the points A and B

in space-time.

If now the observer in A measured the property σz of the first particle finding

the particle in the spin-up state, he could predict with certainty the value of the

same property σz of the second particle, namely that an observer in B would find

his particle in the spin-down state. Furthermore, if the observer in A measured

another - not necessarily compatible - property, for instance σx, he could again

predict the value of the property σx of the particle in B. This can easily be seen

due to the rotational invariance of |Ψ〉.It follows that the observer A could predict with certainty the value of either

σz or σx of the far-away system. Assuming that there occurs no “spooky action

at a distance”, as Einstein put it, the values of σz or σx of the particle in B are

both ’elements of reality’, since the observer A“can predict with certainty without

in any way disturbing the value of [the] physical quantity” σz and σx in B, thus it

lies in the hands of the observer A to decide which property becomes an ’element

of reality’.

Counterfactual reasoning One can try to escape the reasoning above by ar-

guing that either σz or σx is an element of reality, but not both simultaneously,

because due to the incompatibility of these properties one cannot perform a si-

multaneous measurement of both. EPR refute this by noting that “this makes

the reality of [σz] and [σx] depend upon the process of measurement carried out

on the first system, which does not disturb the second system in any way. No

reasonable definition of physical reality could be expected to permit this.” In other

words, if σz or σx is an element of reality, and the choice of which one is real is

determined by an action that cannot affect faraway realities, then both properties

must be real. But if both properties had simultaneously definite values (i. e. are

simultaneously real), a complete physical theory, in this case quantum mechanics,

must take account of this fact.

Here occurs a concept known as counterfactual reasoning for the first time.

Contrary to expectation this is nothing spectacular, but it merely means that

since measurements of non-commuting observables6 do not make sense on a single

5EPR used the incompatible properties position and momentum for their gedankenexperi-ment, but the situation given by two spin-1/2 particles in a singlet state - first considered byBohm - is easier to handle.

6For example, position and momentum are not commuting, but also spin measurements indifferent non-orthogonal directions like σx and σz


system, one has to test such properties on different quantum systems prepared in

the same state. In the case discussed above σz can be tested by the observer A,

but in the following he cannot test σx on the same particle, because the operators

are not commuting. He has to take another particle in the same state to measure

σx. The crucial point is here that the particles are “in the same state”, thus it

is totally justified to speak about incompatible tests on one particle, although in

reality one has to take more particles into account.

No superluminal information It has to be mentioned that if we believe in

the completeness of quantum theory and therefore do not deny the possibility of

“spooky” action at a distance, it is still impossible to transport any information

from A to B faster than the speed of light, i. e. the observer in B does not have

any clue what the observer in A has measured and only from his results he cannot

derive what measurement has been performed in A. Therefore the principle of

Einstein causality is not violated, it is not possible to detect an effect earlier than

its cause.

To sum up, EPR claimed to have shown that the “wave function [description

of QM] does not provide a complete description of the physical reality” assuming

that there is no “action at a distance”, but “left open the question of whether

or not such a description exists. [They] believe, however, that such a theory is

possible.”

2.2 Von Neumann’s “no-go” theorem

The first “no-go” theorem, i. e. a proof refuting the possibly to construct a hid-

den variables theory has been given by John v. Neumann in his famous book

“Mathematische Grundlagen der Quantenmechanik” [10] in 1932. His proof has

been cited a lot until Bell (re)discovered7 that von Neumann’s proof is based on

a “silly” [11] assumption.

“Silly” proof The proof is based on properties of self-adjoint operators in

Hilbert space as the mathematical objects representing observables on quantum

systems. It is well known that if A and B are self-adjoint (=Hermitian) operators

in Hilbert space, then any linear combination of them

C = αA+βB, α,β ∈ C (2.1)

is also self-adjoint. Now if A and B (represented by A, B∈H ) are observables on

a system, then C (represented by the self-adjoint operator C) is also an observable

7Grete Hermann already pointed out a deficiency in the proof in 1935, but has been entirelyignored.


of the same system. The only allowed results ν(A) of a measurement represented

by the Hermitian operator A are its eigenvalues ai , i. e. ν(A) = ai . If the operators

A,B and C are mutually commuting, they have a common set of eigenfunctions

and can therefore be measured simultaneously, thus the only allowed results of

a simultaneous measurement of A,B and C are a set of simultaneous eigenval-

ues ν(A), ν(B) and ν(C). If we consider now the equation C = A+B, the values

assigned to the operators in an individual system must fulfill

ν(C) = ν(A)+ν(B), if [A,B] = 08. (2.2)

Von Neumann’s assumption was now, that Eq. (2.2) holds also for non-

commuting operators A and B. Applying this restriction to HV theories he proved

[10] that the dispersion (or variance) of at least one observable is not equal to

zero, i. e. dispersion free states are not possible (cf. Appendix B). That means

that the measurement of at least one observable yields a non-deterministic result,

consequently no HV theory can uniquely determine the result of all possible mea-

surements, and we have to accept that Nature is fundamentally non-deterministic.

But if [A,B] 6= 0, A and B do not have simultaneous eigenvalues and there is

no reason for this assumption. Von Neumann was led to it, because QM dictates

that

〈Ψ|C|Ψ〉= 〈Ψ|A|Ψ〉+ 〈Ψ|B|Ψ〉 (2.3)

is valid for any state |Ψ〉 and for arbitrary - also non-commuting - observables A

and B. In other words Eq. (2.2) holds in the mean: For non-commuting A, B we

cannot measure A+B simultaneously, but we can prepare an ensemble of systems

in the same state |Ψ〉 and perform measurements A+ B on some of them, and

measurements of A or B on some others, then the averages of A, B, and C = A+B

will be related according to Eq. (2.3).

The crucial point is now that an ensemble of systems in the same quantum

state |Ψ〉 is in general not in the same state denoted, for simplicity, by |Ψ,HV〉in a hidden variables theory. The HV can take different values still yielding

the same quantum state, thus many |Ψ,HV〉 map to the same quantum state

|Ψ〉: |Ψ,λ1〉, |Ψ,λ2〉, . . . → |Ψ〉. A sufficient condition for Eq. (2.3) is imposing

Eq. (2.2) to each single system in a state |Ψ,λm〉 out of the ensemble of systems

all in the quantum state |Ψ〉, i. e. if the relation ν(C) = ν(A)+ ν(B) is true for

every individual system than clearly the same relation has to be fulfilled by the

averages. But this is not a necessary condition, the relation for the averages in

Eq. (2.3) can also be fulfilled if the relation for the values in Eq. (2.2) is not true

for every individual system.

8[A,B] 6= 0 ↔ A and B non-commuting


Example 1 A trivial set of data shows that we do not have to assume Eq. (2.2)

to get to Eq.(2.3) (cf. Table 1).

System QM-State HV-State ν(A) ν(B) ν(C) = ν(A+B)

1 |Ψ〉 |Ψ,1〉 2 5 42 |Ψ〉 |Ψ,2〉 4 7 143 |Ψ〉 |Ψ,1〉 2 5 44 |Ψ〉 |Ψ,2〉 4 7 14

Table 1: Simple set of data to refute von Neumann’s “no-go” theorem

We have an ensemble of quantum systems (namely 4 systems) in the same

quantum state |Ψ〉. Each system has attributed a variable λ, which determines

the values of the observables A, B, and C, i. e. the knowledge of λ would make it

possible to distinguish the identical systems as seen from the quantum mechanical

point of view. We can see that due to

〈Ψ|C|Ψ〉=(2∗4+2∗14)/4= 9=(2∗2+2∗4)/4+(2∗5+2∗7)/4= 〈Ψ|A|Ψ〉+〈Ψ|B|Ψ〉

Eq. (2.3) holds, but Eq. (2.2) is not fulfilled for each individual system (2+5 6= 4

and 4+7 6= 14).

This shows that von Neumann’s assumption that ν(C) = ν(A)+ ν(B) should

be valid for each individual system is too strong.

Example 2 Another example has been given by Bell [12]: Take A = σx B = σy

as example and C = 1√2(σx + σy) as the composite operator bisecting the x and

y-axis. Since the result of these spin-measurements on a spin-1/2 particle are

both ν(A), ν(B) = ±1 and, furthermore, ν(C) can also only have values ±1, the

condition ν(C) = ν(A) + ν(B) above results in ±1 = 1/√

2(±1+±1), which is

definitely not true.

From these examples one can readily deduce that non-commuting operators

A and B make troubles. In Table 1 each individual row had to fulfill Eq. (2.2), if

[A,B] = 0, the current value assignment is only consistent if [A,B] 6= 0.

2.3 Gleason’s theorem

In a (successful) attempt to base quantum mechanics on a smaller set of axioms

Gleason [13] derived that on a Hilbert space of dimensionality greater or equal to

3 the only possible probability measures are the measures

〈Pα〉= Tr[PαW], (2.4)


where Pα is a projection operator onto a (pure) state |α〉, W is a density (sta-

tistical) operator characterizing the actual state of the system, Tr stands for the

trace of a matrix and 〈P〉 denotes the expectation value of the operator P. To

establish a link to general observables one has to use the spectral decomposition

theorem, namely that every observable A of a quantum system can be built up

from mutually orthogonal projection operators A = ∑i aiPi . Then from Eq. (2.4)

the conventional probability measure 〈A〉= Tr[AW] can be derived.

The fundamental axioms are now [14, p. 190f.]:

(i) Elementary tests (yes-no questions) are represented by projectors in a com-

plex vector space.

(ii) Compatible tests (yes-no questions that can be answered simultaneously)

correspond to commuting projectors.

(iii) If Pu and Pv are orthogonal projectors, their sum Puv = Pu + Pv, which is

itself a projector, has expectation value

〈Puv〉= 〈Pu〉+ 〈Pv〉,

where 〈P〉 ≡ Tr[PW] according to Eq. (2.4).

Example The projection operator Pα can be interpreted as a yes-no question

to the system, thus a test if the system has the property α or “not α” according

to the eigenvalues 1 and 0 of Pα.

Take for instance a spin-1/2 particle “polarized in the positive z-direction”9

given by the state |ψ〉 = | ↑〉. The projection operator P+z corresponding to the

question “Does the particle’s spin point in the positive z-direction?” reads as

P+z = | ↑〉〈↑ | and the “answer” is given by

Tr[P+z | ↑〉〈↑ |] = 〈ψ|P+

z |ψ〉= 〈↑ | ↑〉〈↑ | ↑〉= 1,

i. e. with probability 1 the particle is in the state |ψ〉. Asking the question whether

the particle’s spin points in the positive x-direction yields a probability of 1/2,

since from

P+x = 1/2(| ↑〉〈↑ |+ | ↑〉〈↓ |+ | ↓〉〈↑ |+ | ↓〉〈↓ |)

it follows

Tr[P+x |ψ〉〈ψ|] = 〈ψ|P+

x |ψ〉=12.

9This means, when performing a Stern-Gerlach experiment, which deflects the particles inthe positive or negative z-direction, we would see particles only in the beam deflected in thepositive z-direction. In quantum theory the Hilbert space associated with the spin-space doescertainly not have a z-direction, but in this case we can make the correspondence between thetwo-dimensional Hilbert space for the spin and R3.


We can see that in this case the “answer to the yes-no question” can be given

only with a probability of 1/2. In other words, in this formalism an experiment

performed to measure P+x would give the answer “yes, the particle has spin-up in

x-direction” for half of the tested particles and “no” for the other half. A hidden-

variables theory should correct this “deficiency” of quantum theory and give a

definite answer to each question, i. e. the result 0 or 1 (and nothing in between)

for the measurement Tr[PαW] for arbitrary α.

Sum of orthogonal projectors The last axiom (iii) from above reappears in

the subsequent discussion about Bell’s “no-go” theorem and the Kochen-Specker

theorem and is used to derive contradictory assertions. It means that a projector

Puv can be split in infinitely many ways into a sum of mutually orthogonal pro-

jectors, in other words, we can assign infinitely many orthonormal bases to the

subspace Puv(H ). We could for example take the Pu = |u〉〈u| and Pv = |v〉〈v| as

projection operators on the vectors |u〉, |v〉 ∈ H with 〈u|v〉 = 0. Another pair of

orthogonal vectors can then be chosen as

|x〉= 1/√

2(|u〉+ |v〉), |y〉= 1/√

2(|u〉− |v〉) (2.5)

with the belonging projectors Px = |x〉〈x| and Py = |y〉〈y|. Trivially

Puv = Pu +Pv = Px +Py (2.6)

and with the axim (iii) this relation holds also for the expectation values

〈Puv〉= 〈Pu〉+ 〈Pv〉= 〈Px〉+ 〈Py〉. (2.7)

2.4 Bell’s “continuum” no-go theorem

Bell showed in [12] that choosing two vectors |φ〉 and |ψ〉 in H such that for a

quantum system in the state W

〈Pφ〉 ≡ Tr[|φ〉〈φ|W] = 1, (2.8)

〈Pψ〉 ≡ Tr[|ψ〉〈ψ|W] = 0 (2.9)

is only possible under the restriction that |φ〉 and |ψ〉 cannot be arbitrarily close,

in fact

||φ〉− |ψ〉| ≥ 1/2〈ψ|ψ〉1/2. (2.10)

Assuming a dispersion free state W as required by a HV theory every pro-

jection operator must have the definite expectation value 0 or 1, i. e. for every

physical property we can definitely determine whether the system has it or not.


Consider now a complete set of orthogonal basis vectors |ϕi〉 ∈ H , then

∑i

Pϕi = 1l (2.11)

and from (iii) follows

∑i〈Pϕi〉= 1. (2.12)

In a hidden-variables theory the only possible values for 〈Pϕi〉 are 0 or 1, thus

exactly for one Pϕ j , 〈Pϕ j 〉 = 1 must be valid, whereas for other projectors in

Eq. (2.12) 〈Pϕi〉 is equal to zero for i 6= j.

As one can think of infinitely many alternative basis sets |ϕ′i〉 fulfilling the same

conditions (2.11) and (2.12) as |ϕi〉, Bell concluded that there must be arbitrarily

close pairs |φ〉, |ψ〉 with different expectation values 0 or 1, respectively, which

yields a contradiction. Consequently, there are no dispersion free states.

2.4.1 Example

For spin-1 particles, because then we can describe the Pi as squares of the com-

ponents of the spin along various directions, i. e. we have a set of observables

{S2x,S

2y,S

2z} with eigenvalues 0 and 1, where the indices x,y,z denote mutually

orthogonal vectors |x〉, |y〉, |z〉 ∈ H . For a spin 1 particle

S2 = S2x +S2

y +S2z = s(s+1) = 2

must be valid for every triad of orthogonal vectors, furthermore S2x,S

2y, and S2

y are

mutually commuting, which means that theoretically a simultaneous measure-

ment of all three observables can be performed and consequently

〈S2u〉+ 〈S2

v〉+ 〈S2w〉= 2

has to be fulfilled as well.

The resulting problem here is to assign numbers {1,1,0} to the triple {S2x,S

2y,S

2z},

which we can convert to a geometrical problem by associating |x〉, |y〉, |z〉 with or-

thogonal rays (= directions) x, y, z in R3. These rays build an orthogonal triad

and the task is now to “color” these rays so that we have a “red” one and two

“blue” ones in each triad in lieu of ascribing the values 0 or 1 to the expectation

values of S2x, S2

y, and S2z [cf. Figure 1(a), where the thick arrowa are colored “blue”

and the thin arrows “red”]. There are infinitely many possible orthogonal triads,

each of them has to be colored consistently, and the angle between two rays in dif-

ferent colors cannot be arbitrarily small (due to Bell’s condition - cf. Eq. (2.10))

one can intuitively see that this is no feasible. A recipe how to explicitly reach a


contradiction has been given by Mermin in [11].

z

x

y

z'

x'

y'

(a)

z = z'

x

y

x'y'

(b)

Figure 1: Coloring of a sphere

2.4.2 Contextuality

Bell was sceptical about this result and objected “that so much follows from such

apparently innocent assumptions leads us to question their innocence” [12]. He

points out that “it was tacitly assumed that measurement of an observable must

yield the same value independently of what other measurements may be made

simultaneously”, which is a formulation of the principle of “non-contextuality”.

In other words, the projection operators corresponding to the x-y-z – triad can

be measured simultaneously, since the {S2x,S

2y,S

2z} are mutually commuting, but

these are not necessary commuting with the projectors {S2x′,S

2y′,S

2z′} corresponding

to the x′-y′-z′ – triad in Figure 1(a).

In the particular case when z= z′ (S2z = S2

z′) depicted in Figure 1(b), we“assume

tacitly” that the measurement of S2z does not depend on the choice of the rays x′

and y′, as long as they are orthogonal to each other and orthogonal to the ray

pointing in the z-direction. In terms of projection operators that means S2z = S2

z′

can be measured simultaneously with S2x and S2

y, or simultaneously with S2x′ and

S2y′ and the result of S2

z should be the same. Clearly one cannot measure both

sets simultaneously, which is again an evidence of counterfactuality.

2.5 Kochen-Specker Theorem

While Bell’s proof depends on value assignments for a continuum of vectors in

Hilbert space, Kochen and Specker presented a discrete and finite set of observ-

ables in Hilbert space for which a value assignment by a HV-theory leads to an

inconsistency. However, the assumption of non-contextuality is still remaining in

their proof.


The Kochen-Specker theorem states that in a Hilbert space H with dimen-

sionality dimH ≥ 3, it is impossible to associate definite numerical values, 0 or

1, with every projection operator Pi ∈H , such that for every set of commuting Pi

satisfying ∑i Pi = 1

∑i

ν(Pi) = 1 with ν(Pi) = 0,1 (2.13)

is valid. The Kochen-Specker theorem [15] consists of an algebraic proof cir-

cumventing the difficulties arising when taking a continuum of vectors in Hilbert

space. They present a discrete and finite set (117 operators in their original work)

of observables in Hilbert space for which a theory using hidden variables would

lead to inconsistencies.

2.6 Peres’ variant of the KS-Theorem

The original proof of Kochen and Specker [15] has been simplified by numerous

people. We will present here a proof by A. Peres [14] using 33 vectors belonging

to 16 distinct bases in R3. The vectors used are depicted in Figure 2 and labeled

by xyz, where x,y,z∈ {0,1, 1(≡−1),2(≡√

2), 2(≡−√

2}. For example the vector

112 connects the origin (0,0,0) with the point (1,1,−√

2). Opposite vectors,

such as 112 and 112 are counted only once, because they correspond to the same

projector and will therefore called rays. From these 33 rays 16 orthogonal triad

can be formed, where each ray can belong to several triads.

Assuming a HV-theory it must be possible to assign to one vector the value 1

and to the other vectors 0 for each possible triad. As already mentioned above an

equivalent approach is to try to “color” the rays: blue is associated to the value

“1” and red to “0”, thus in every triad there must be one blue and two red rays.

In the beginning we can color now arbitrarily some of the rays blue due to the

invariance under interchanges of the x, y and z axis, and under reversal of the

direction of each axis. In Table (2) you can see the coloring of the rays: In each

line the first entry (in boldface) is colored blue, the following two rays are red

and they form an orthogonal triad. Additional rays in the category “Other rays”

are also orthogonal to the first one and therefore red. The list of orthogonal rays

to the first (blue) ray is not extensive, only rays needed for the proof are listed.

A ray printed in italic letters has already been used before.

By looking at the first, fourth and last line we notice that 100, 021 and 012

are all red, although they are forming an orthogonal triad, which definitely is

a contradiction to the claim of an HV-theory that every triad can be colored

blue-red-red.


�� 21

0

�� 2

� 1

z

x

y

Figure 2: Construction of Peres’ KS-theorem variant

Orthogonal triad Other rays Reason of first ray being blue

001 100 010 110 110 arbitrary (choice of z axis)101 101 010 arbitrary (choice of x vs. −x axis)011 011 100 arbitrary (choice of y vs. −y axis)112 112 110 201 021 arbitrary (choice of x vs y axis)102 201 010 211 orthogonality to 2nd and 3rd ray211 011 211 102 orthogonality to 2nd and 3rd ray201 010 102 112 orthogonality to 2nd and 3rd ray112 110 112 021 orthogonality to 2nd and 3rd ray012 100 021 121 orthogonality to 2nd and 3rd ray121 101 121 012 orthogonality to 2nd and 3rd ray

Table 2: Proof of Kochen-Specker theorem in three dimensions


Physical interpretation We have already given a physical interpretation of

the coloring problem in Section 2.4.1. Here 16 distinct orthogonal bases are given

instead of a continuum of bases vectors, but the main idea is the same being more

accessible and constructible in the discrete case.

2.7 Mermin’s variants of the KS-Theorem[11]

Although the latter proof was considerably easier than the original proof of

Kochen and Specker, there are some more arrangements of observables enabling

simpler proofs, but with the drawback that they are only valid for higher dimen-

sional Hilbert spaces.

2.7.1 Four dimensions

We get a four dimensional Hilbert space H = H 1⊗H 2 - as a demonstrative

example - by considering two spin-1/2 particles living in the two-dimensional

Hilbert spaces H 1 and H 2, respectively. The observables on this system are then

represented as the familiar Pauli spin-operators σ1α ∈H 1 and σ2

β ∈H 2 to measure

the spin in an arbitrary direction α or β, respectively. The upper index denotes

the Hilbert space in which the operator lives, i. e. σ1∈H 1 acts on the first particle

and σ2 ∈ H 2 acts on the second. The eigenvalues of σ1α and σ2

β are ±1 and any

component of σ1α commutes with any component of σ2

β since they are belonging

to another subspace of Hilbert space. Other relations are that σiα anti-commutes

with σiβ and σi

ασiβ = iσi

γ for i = 1,2 and α,β,γ specifying orthogonal directions.

By taking now the nine observables from Fig. (3) we can easily convince us

that it is impossible to consistently assign values to all of them:

(i) In each row and each column the operators are mutually commuting.

(ii) The product of the operators in each row and in each column except in the

column on the right is +1l4. In the column on the right the product of the

three operators is −1l4, where 1l4 stands for the four dimensional identity

operator.

(iii) According to QM the values assigned to a set of mutually commuting op-

erators have to satisfy the same relations as the operators themselves. Ac-

cording to (ii) the product of the values assigned to the three observables

in each row and in each column has to be +1, except for the third column,

where we should get −1.

But (iii) cannot be fulfilled, because the row identities require the product of

all nine operator values to be +1, while the column identities require it to be −1,

which yields a contradiction.


σ1x⊗1l 1l⊗σ2

x σ1x⊗σ2

x

σ1y⊗1l 1l⊗σ2

y σ1y⊗σ2

y

σ1x⊗σ2

y σ1y⊗σ2

x σ1z⊗σ2

z

Figure 3: Observables for Mermin’s KS-variant

2.7.2 Eight dimensions

With nearly the same arguments as before we can construct an eight dimensional

Hilbert space H = H 1⊗H 2⊗H 3 with three spin-1/2 particles. Ten observables

are arranged in groups of four on five intersecting lines that form a five-pointed

star.

(i) The four operators on each line are again mutually commuting.

(ii) The product of the operators on each line of the star but the horizontal line

is +1l8. The product of the operators on the horizontal line is −1l8.

(iii) The values assigned to the operators must again satisfy the same relations

as the operators themselves, therefore we should get −1 for the product of

the operator values on the horizontal line and +1 for the other lines.

(iv) Consequently the product of the values over all five lines must be equal to

−1.

But like in the four dimensional case this cannot hold, if we consider that each

operator appears exactly twice when we multiply all products of the operator

values of all five lines, since each lies on the intersection of two lines. From (iv)

we know that for the operator values the result is always −1, but we also know

(or if not, it can easily be calculated) that the square of each operator is the

identity +18, and therefore in this case the product is +18. Again the operator

values must fulfill the same relations as the operators, consequently we have a

contradiction to the value −1 in (iv).

2.8 Implications of the KS-Theorem

From all the arguments above it follows that QM cannot be embedded into a

non-contextual hidden-variables theory, because the “tacit” assumption of non-

contextuality runs through all proofs introduced above. The question is then, if


σ1y

σ1xσ2

xσ3x σ1

yσ2yσ3

x σ1yσ2

xσ3y σ1

xσ2yσ3

y

σ3x σ3

y

σ1x

σ2y σ2

x

Figure 4: Observables for Mermin’s KS-variant

it makes sense to impose the restriction of non-contextuality to a theory, or if this

is merely an assumption stemming from our usual perception of Nature.

Due to the KS-Theorem we are left with two possibilities10:

(i) Non-contextuality is a reasonable assumption. But an underlying hidden

variables theory can only be a contextual one, thus we conclude that quan-

tum theory is already a complete description of physical reality without

any additional hidden variables and the inherent indeterminism cannot be

circumvented.

(ii) The assumption of non-contextuality is only based on intuition and common

sense, but does not withstand a proper scientific investigation. In this

case nothing hinders us from proposing contextual hidden-variables theories

replacing QM and indeterminism.

Applying two distinct ways to measure a property A of a quantum system

“there is no a priori reason to believe that the results for [A ] should be the same

[in both cases]. The result of an observation may reasonably depend not only

on the state of the system (including hidden variables) but also on the complete

disposition of the apparatus. [12]”Suppose we measure in one experimental setup

A together with the observables B,C corresponding to the set of mutually com-

muting operators {A,B,C} and in another setup we have the operators {A,B′,C′},which are again mutually commuting. These two sets belong to two physically

different arrangements of an experimental setup distinguished for example by dif-

ferent positions of some detectors responsible for detecting B or B′, and C or C′,

respectively. Since one can presume that different physical arrangements produce

different interactions between the measurement apparatus itself and between the

10Actually there are of course more subtle escapes from the KS argument not presented inthe limited scope of this work. See for example [16] for further discussion and references.


apparatus and the particles, it is not clear anymore, why we should insist on a

non-contextual hidden-variables theory.

Such a contextual hidden-variables approach has been given by Bohm [17],

which is not only contextual but also non-local, but where “Quantum-mechanical

probabilities are regarded (like their counterparts in classical statistical mechan-

ics) as only a practical necessity and not as a manifestation of an inherent lack

of complete determination in the properties of matter at the quantum level.” We

will soon show that due to Bell this non-local behavior must be inherent to a

hidden-variables theory consistent with quantum theory.

Mermin [11] argues that non-contextuality is nevertheless a reasonable as-

sumption:

If A is in both setups the first property measured and after measuring A we

do some other tests for B,C or B′,C′ our intuition tells us that the results for A

must be the same for both experiments, since {A,B,C} and {A,B′,C′} are mu-

tually compatible and, furthermore, the measurement of A is not influenced by

anything that comes afterwards. Beyond that, even if we allow some kind of re-

flections from the operations B,C or B′,C′ to A it is still a basic feature of quantum

mechanics that the statistical predictions for A alone neglecting the “context”B,C

or B′,C′, respectively, are exactly the same for both sets of observables and that a

“contextual hidden-variables account for this fact would be as mysteriously silent

as the quantum theory on the question of why Nature should conspire to arrange

for the marginal distributions to be the same for the two different experimental

arrangements.” [11, p. 812]

2.9 Bell’s Theorem - from non-contextuality to locality

The previously introduced variants proofing the KS-theorem are based on the

assumption of non-contextuality, but we have seen that this assumption cannot

be justified without hesitation.

Using the example above, as long as we cannot exclude any interactions be-

tween the observables A , B , and C - or better between the parts of the measure-

ment apparatus responsible for detecting these properties - A can in general be

written as A = A(B,C ) or A = A(B ′,C ′), respectively, and A(B,C ) = A(B ′,C ′)is not necessarily a physical imperative.

Consequently it would be better to replace the concept of non-contextuality

by locality. The different measurements must be space-like separated to prevent

any interaction between the measurement of the observables, i. e. the measure-

ment processes must not be in the forward light cone of each other so that one

observable cannot be influenced by another.


2.9.1 Link between the KS-Theorem and Bell’s Theorem

Mermin [11] showed that by using only spatially separated operators in the

eight-dimensional KS-proof in Section 2.7.2 one can establish a link from non-

contextuality to locality, and therefore from the KS-Theorem to Bell’s Theorem.

If we consider the operators in Figure 4 now as spin measurements on three

far apart spin-1/2 particles eliminating all operators that act on more than one

single particle, we are left with six operators situated on the non-horizontal lines.

These operators can be interpreted as measurements on exactly one out of three

particle located in different space-time regions depiced in Figure 5 and “for any

of these six local observables, the assumption that the value assigned it should

not depend on which pair of faraway components are measured with it is justified

not by possibly dubious assumption of non-contextuality, but by the condition of

locality.” [11, p. 812].

σ3y

σ1x

σ1y

σ2x

σ2y

σ3x

Figure 5: Bell’s Theorem without inequalities

The other four non-local operators (from the horizontal line in Figure 4) act-

ing on all three particles are commuting, thus we can prepare a particular sys-

tem in a common eigenstate of all four observables on the horizontal line. In a

HV-theory each of these four observables has a definite value, namely its eigen-

value. We can now proceed in measuring independently the local observables of

one non-horizontal line on the three particles and due to quantum theory the

product of these values must be equal to the eigenvalue of the corresponding

operator-product appearing in the horizontal line11. If we continue in this way

by measuring the observables of the other non-horizontal lines we find definite

values for all of the ten operators.

Here of course counterfactual reasoning comes into play again: In reality we

11Take for example the line from the lower right corner to the top consisting of the one-particleoperators σ2

y,σ3x,σ1

y. The corresponding element in the horizontal row is in this case σ1yσ2

yσ3x.


can only perform measurements corresponding to one non-horizontal line on one

triple of particles, for the next line we have to take a “fresh” triple, but since this

new triple is in the same state as the old one and because we assume that the

results of a measurement of a local operator is independent of the particular choice

of the operators measured elsewhere (i. e. the measurement is non-contextual) this

assumption is justified.

In this case we can mend the broken chain of arguments, which yielded a con-

tradiction in the previous proof (cf. Section 2.7.2): The product of the eigenvalues

on the non-horizontal lines will always be +1 and the product of the eigenvalues

on the horizontal line will always be −1, hence the product of the eigenvalues of

all lines is −1. When calculating the product of all operators we have seen that

the result is 1l8, it follows that the product of all operator values has to be 1.

Contradiction!

Thus, we only have to restrict us to a specific state in the proof of the KS-

Theorem, and we can derive a contradiction replacing the assumption of non-

contextuality by the stronger assumption of locality.

Bell’s Theorem reads now:

“In a theory in which parameters are added to quantum mechanics

to determine the results of individual measurements, without chang-

ing the statistical predictions, there must be a mechanism whereby

the setting of one measuring device can influence the reading of an-

other instrument, however remote. Moreover, the signal involved must

propagate instantaneously, so that such a theory could not be Lorentz

invariant.” [2]

Actually Bell proved that there is an upper limit to the correlation of distant

events, if one just assumes the validity of the principle of locality [2], by deriving

an inequality valid for local hidden-variables theories, but violated by quantum

mechanical predictions.

2.9.2 Proof of Bell’s theorem - with inequalities

Consider a pair of spin-1/2 particles in a singlet state |Ψ〉 = 1/√

2(| ↑↓〉− | ↓↑〉)moving freely in opposite directions. For each particle we can perform a mea-

surement on selected components of the spin nξ ·σ and nχ ·σ for particle 1 and

2, respectively, by Stern-Gerlach magnets, where nξ is some unit vector (cf. Fig-

ure 6). In the particular state |Ψ〉, if measurement on the first particle of nξ ·σyields the value +1, the same measurement nξ ·σ on the second particle must

yield the value -1 and vice versa. If the locations of measurement are far apart,

the measurement cannot - due to locality - influence each other, thus the result


must be predetermined by some hidden variables, or the concept of locality must

be abandoned.

y

x

y

x

δ

γα

β

Figure 6: Experimental setup for testing the CHSH-inequality

Correlation function Consider as a classical analogue (cf. [14, p. 160f.]) an

object at rest that is disrupted into two parts with opposite angular momentum

J1 = −J2. An observer can now measure the variable sgn(nξ · J1) on the first

fragment. The result of this variable (denoted by x) can only take the values ±1.

A second observer can now measure sgn(nχ · J2) on the second fragment of the

initial object and will get values y =±1. If several repetitions of the experiment

are done the observers get the average values

〈x〉= ∑j

x j/N and 〈y〉= ∑j

y j/N, (2.14)

where N is the total number of repetitions. If the directions of the Ji are randomly

distributed both values are close to zero, but if we compare the results of both

observers we can construct the correlation function

E(ξ,χ)class= 〈xy〉= ∑j

x jy j/N, (2.15)

which does not vanish in general. Take for example nξ = nχ, then each x jy j =−1

and therefore 〈xy〉 = −1. For arbitrary nξ and nχ (i. e. arbitrary measurement

directions) the correlation function can be calculated to

E(ξ,χ)class= 〈xy〉=−1+2θ/π, (2.16)

where θ is the angle between nξ and nχ.

Quantum mechanical correlation Using a singlet state |Ψ〉 the QM corre-

lation function for the measurement of spin components on two spin-1/2 particles

can be written as

E(ξ,χ)qm = 〈Ψ|nξ ·σ⊗nχ ·σ|Ψ〉=−cosθ, (2.17)


where θ is again the angle between nξ and nχ and the σ = (σx,σy,σz)T are the

Pauli spin matrices. Here we can see, that the QM correlation is stronger than

the classical one, except at the points θ = 0 and θ = π, where E(ξ,χ) =−1,1, and

θ = π/2, where E(ξ,χ) = 0 (cf. Figure 7).

Eclass

Eqm

θππ

20

1

0

-1

Figure 7: Correlation functions

Bell’s theorem Suppose now that two observers independently perform mea-

surements on one particle out of the two-particle singlet state each, and each

observer can choose between two directions of measurements. Observer A can se-

lect one out of two directions denoted by nα and nβ and observer B has the choices

nγ and nδ (cf. Figure 6). The outcome of each measurement is a,b,c,d =±1 when

a stands for the result of nα and similar for b,c and d. Classically these results

are not dependent on the actual measurement on the opposite side, since there is

no interaction between separated particles located outside each others light-cone.

If we take now the quantity

ac+bc+bd−ad = (a+b)c+(b−a)d

we can see that due to a,b,c,d =±1 either (a+b) = 0 or (a−b) = 0 and it follows

in either case that

ac+bc+bd−ad =±2. (2.18)

It is important to note that we consider here the situation that a,b,c and d are

predetermined, i. e. after the generation of the state each measurement result

is fixed by a hidden variable. Furthermore, we cannot measure all properties

simultaneously, a measurement along nα excludes a measurement along nβ, thus


assigning values to a and b for one particle-pair is not possible, and similarly this

is valid for c and d.

Nevertheless we can make statistical predictions: For each pair of particles

Eq. (2.18) must be valid, thus we can write the average value of ac+bc+bd−ad

〈ac+bc+bd−ad〉 = ∑j(a jc j +b jc j +b jd j −a jd j)/N (2.19)

=1N ∑

j±2

≤ 2

Splitting up the left hand side of Eq. (2.19) into correlation functions of two

properties we get

〈ac〉+ 〈bc〉+ 〈bd〉−〈ad〉 ≤ 2, (2.20)

which is a variant of the original Bell inequality known as the CHSH -inequality.

These correlation functions in Eq. (2.20) can now be calculated in the QM for-

malism, although quantum theory is not able to predict single values a j ,b j ,c j and

d j for each measurement, but the average values can be calculated. In the partic-

ular case of pairs of polarized photons with the correlation E(ξ,χ) = cos2(ξ−χ),where ξ and χ denote the angles of polarization measurements, Equation (2.20)

becomes

cos2(α− γ)+cos2(β− γ)+cos2(β−δ)−cos2(α−δ)≤ 2. (2.21)

For the special choice of measurement directions α = 0, β = π/4, γ = π/8, δ = 3π/8

the inequality reads

1√2

+1√2

+1√2− 1√

2= 2

√2 6≤ 2. (2.22)

Obviously quantum theory violates the CHSH-inequality.

This behavior has already been tested experimentally [18, 19] and it seems

that Nature does not obey the Bell-inequality and many variants thereof.

We are left with only a few choices to “solve” this problem. The most common

are:

(i) Non-locality is inherent to quantum mechanics (and to Nature), although

if QM is accurate, the non-locality is clearly of a sort that should not allow

faster than light communication.

(ii) All ”possible” outcomes really occur. (A many worlds interpretation.)

(iii) Strong determinism - particles in region A can behave according to what


all the particles and detector settings in region B are doing, because that is

predetermined and A shares a past history with them.

(iv) The appropriate Bell inequality was not violated, but ”loopholes” allow low

detector efficiencies to give that illusion. This possibility could be tested

with better detectors.

3 CORRELATION POLYTOPES 29

3 Correlation Polytopes

Bell’s inequality and the CHSH-variant are not the only inequalities that can be

used to test quantum mechanical predictions, similar equations for a particular

setup have been discussed by Clauser and Horne [20], Mermin [21] and others.

Pitowsky has given a geometrical interpretation in terms of correlation polytopes

[6, 22, 23, 24] to derive an infinite hierarchy of correlation inequalities for different

setups. Many of the following definitions and examples are taken from these

publications of Pitowsky.

3.1 Simple Urn Model

To get a basic idea how to get to these correlation polytopes we first consider a

simple urn model.

Consider an urn containing some balls of different colors and styles: Each ball

can be described by two properties, let us say ”yellow” and ”wooden”, so each ball

can have the property ”yellow” or the property ”wooden”, but it can also have

both ”yellow and wooden”. The state of the urn can be given by the probabilities

to draw a ball with one of these properties: p1 is the proportion of yellow balls

in the urn, p2 the proportion of wooden ones and p12 denotes the proportion of

yellow and wooden balls. If there are enough balls in the urn, these proportions

are in fact the relative frequencies of drawing a ball with the special property.

Clearly the inequalities

0≤ p12≤ p2 ≤ 1 and 0≤ p12≤ p1 ≤ 1 (3.1)

are fulfilled by the proportions and so p1, p2 and p12 can be seen as probabilities

of two events and their joint event only if these inequalities are satisfied. Simply

by taking some appropriate numbers (p1 = 0.6, p2=0.72 and p12=0.32) we can

see, that equations (3.1) are not sufficient. If we take the probability to draw a

ball which is either yellow or wooden (p1 + p2 - p12) into consideration, a new

inequality can be found that is not satisfied by the numbers chosen:

0≤ p1 + p2− p12≤ 1 (3.2)

It can be shown that the inequalities (3.1) and (3.2) are necessary and sufficient

for the numbers p1, p2 and p12 to represent probabilities of two events and their

joint [6].


3.2 Geometrical interpretation

Itamar Pitowsky [22, 6, 23, 24] has suggested a geometric interpretation. Consider

the truth table 3 of the above urn model, in which a1 and a2 represent the

statements that “the ball drawn from the urn is yellow,”“the ball drawn from the

urn is wooden,” and in which a12 represents the statement that “the ball drawn

from the urn is yellow and wooden.” The third “component bit” of the vector

a1 a2 a12

0 0 01 0 00 1 01 1 1

Table 3: Truth table for two propositions a1,a2 and their joint proposition a12 =a1∧a2

is a function of the first components. Actually, the function is a multiplication,

since we are dealing with the classical logical “and” operation here. Let us take

the set of all numbers (p1, p2, p12) satisfying the inequalities stated above as

a set of vectors in a three-dimensional real space. This amounts to interpreting

the rows of the truth table as vectors; the entries of the rows being the vector

components. This procedure yields a closed convex polytope with vertices (0,0,0),

(1,0,0), (0,1,0) and (1,1,1) (cf. Figure 8). The extreme points (vertices) can be

interpreted as follows:

(0,0,0) is a case where no yellow and no wooden balls are in the urn at all,

(1,0,0) is representing the configuration that all balls are yellow and no one is

wooden.

(0,1,0) is representing the configuration that all balls are wooden and no one is

yellow.

(1,1,1) is a case with only yellow and at the same time wooden balls.

3.3 Minkowski-Weyl representation theorem

The Minkowski-Weyl representation theorem (e.g., [25, p. 29]) states that com-

pact convex sets are “spanned” by their extreme points; and furthermore that the

representation of this polytope by the inequalities corresponding to the planes of

their faces is an equivalent one.

Stated differently, every convex polytope in an Euclidean space has a dual

description: either as the convex hull of its vertices (V-representation), or as the

intersection of a finite number of half-spaces, each one given by a linear inequality

(H-representation) This equivalence is known as the Weyl-Minkowski theorem.


(0,0,0)

(1,0,0)

(0,1,0)

(1,1,1)

Figure 8: Polytope associtated with the urn model

The problem to obtain all inequalities from the vertices of a convex polytope

is known as the hull problem. One solution strategy is the Double Description

Method [26] used for example in [27].

3.4 From vertices to inequalities

For the above simple urn model, the inequalities are rather intuitive and can

be easily obtained by guessing. This is impossible in the general case involving

more events and more joint probabilities thereof. In order to obtain the rele-

vant inequalities—Boole’s “conditions of possible experience”—we have to find a

hopefully constructive way to derive them.

Recall that a vector is an element of the polytope if and only if it can be

represented as a certain bounded convex combination, i.e., a bounded linear span,

of the vertices. More precisely, let us denote the convex hull conv(K) of a finite

set of points K = {x1, . . . ,xn} ∈ Rd by

conv(K) =

{λ1xi + · · ·+λnxn

∣∣∣ n≥ 1,λi ≥ 0,n

∑i=1

λi = 1

}. (3.3)

In the probabilistic context, the coefficients λi are interpreted as the probability

that the event represented by the extreme point xi occurs, whereby K represents

the complete set of all atoms of a Boolean algebra. The geometric interpretation

of K is the set of all extreme points of the correlation polytope.

In summary, the connection between the convex hull of the extreme points

of a correlation polytope and the inequalities representing its faces is guaranteed

by the Minkowski-Weyl representation theorem. A constructive solution of the

corresponding hull problem exists (but is NP-hard [23]).

For the special urn model introduced above this means that any three numbers


(p1, p2 and p12) must fulfill an equation dictated by Kolmogorov’s probability

axioms [28]:

(p1, p2, p12)= λ1(0,0,0)+λ2(0,1,0)+λ3(1,0,0)+λ4(1,1,1)= (λ2+λ4,λ3+λ4,λ4).(3.4)

It is important to realize that these logical possibilities are exhaustive. By def-

inition, there cannot be any other classical case which is not already included

in the above possibilities (0,0,0),(1,0,0),(0,1,0),(1,1,1). Indeed, if one or more

cases would be omitted, the corresponding polytope would not be optimal; i.e.,

it would be embedded in the optimal one. Therefore, any “state” of a physical

system represented by a probability distribution has to satisfy the constraint

λ1 +λ2 +λ3 +λ4 = 1. (3.5)

The four extreme cases λi = 1,λ j = 0 for i ∈ {1,2,3,4} and j 6= i just correspond to

the vertices spanning the classical correlation polytope as the convex sum (3.3).

A generalization to arbitrary configurations is straightforward. To solve the

hull problem for more general cases, an efficient algorithm has to be used. There

are some algorithms to solve this problem, but they run in exponential time in

the number of events, thus it can be solved only for small enough cases to get a

solution in conceivable time.

3.5 From inequalities to vertices

Conversely, a vector is an element of the convex polytope if and only if its coor-

dinates satisfy a set of linear inequalities which represent the supporting hyper-

planes of that polytope. The problem to find the extreme points (vertices) of the

polytope from the set of inequalities is dual to the hull problem considered above.

3.6 Quantum mechanical context

In the quantum mechanical case the elementary irreducible events are clicks in

particle detectors and the probabilities have to be calculated using the formal-

ism of quantum mechanics. It is by no means trivial that these probabilities

satisfy Eq. (3.5), in particular if one realizes that quantum Hilbert lattices are

nonboolean and have an infinite number of atoms. As it turns out, Boole’s“condi-

tions of possible experience” are violated if one considers probabilities associated

with complementary events, thereby assuming counterfactuality. (This is a de-

velopment and a generalization Boole could have hardly forseen!)

As an example we take a source that produces pairs of spin-12 particles in a

singlet-state (|ψ〉= 1√2(| ↑↓〉− | ↓↑〉)). The particles fly apart along the z axis and


y

x

y

x

ab

α2

β1

β2 α1

Figure 9: Experimental setting to test the violation of Boole - Bell type inequal-ities

after the particles have separated, measurements on spin components along one

out of two directions are made. If, for simplicity, the measurements are made in

the x-y plane perpendicular to the trajectory of the particles, the direction of the

measurement can be given by angles measured from the vertical x axis (α1 and α2

on the one side, β1 and β2 on the other side). On each side the measurement angle

is chosen randomly for each pair of incoming particles and each measurement can

yield two results - in h2 units: “+1” for spin up and “-1” for spin down (cf. Figure

9).

Deploying this configuration we get probabilities to find a particle measured

along the axis specified by the angles α1, α2, β1 and β2 either in spin up or in

spin down state denoted as pa1, pa2, pb1, pb2 - and we also take the joint event of

finding a particle on one side at the angle α1 (α2) in a specific spin state and the

other particle on the other side along the vector β1 (β2) in a specific spin state,

denoted as pa1b1, pa2b1, pa1b2 and pa2b2. To construct the convex polytope to this

experiment we build up a truth table of all possible events using a “1” as “spin

up is detected along the specific axis” and a “0” as “spin down is detected along

the specific axis” (table 4). The rows of this table are then identified with the

vertices of the convex polytope. By using the Minkowski-Weyl theorem and by

solving the hull problem, the vertices determine the hyper-planes confining the

polytope, i.e. the inequalities which the probabilities have to satisfy. As a result

the following inequalities are obtained:

0≤ paibi ≤ pai ≤ 1,0≤ paibi ≤ pbi ≤ 1 i = 1,2

pai + pbi− paibi ≤ 1 i = 1,2(3.6)

−1≤ pa1b1 + pa1b2 + pa2b2− pa2b1− pa1− pb2 ≤ 0




(3.7)

The last four inequalities are known as Clauser-Horne inequalities. As noticed

above the probabilities have to be seen in a quantum mechanical context. If we


α1 α2 β1 β1 α1β1 α1β2 α2β1 α2β2

0 0 0 0 0 0 0 01 0 0 0 0 0 0 00 1 0 0 0 0 0 01 1 0 0 0 0 0 00 0 1 0 0 0 0 01 0 1 0 1 0 0 00 1 1 0 0 0 1 01 1 1 0 1 0 1 00 0 0 1 0 0 0 01 0 0 1 0 1 0 00 1 0 1 0 0 0 11 1 0 1 0 1 0 10 0 1 1 0 0 0 01 0 1 1 1 1 0 00 1 1 1 0 0 1 11 1 1 1 1 1 1 1

Table 4: Truth table for four propositions

consider the singlet state of spin-12 particles |ψ〉= 1√

2(| ↑↓〉−| ↓↑〉) it is well known

that the probability to find the particles both either in spin up or in spin down

states is given by P↑↑(θ) = P↓↓(θ) = 12sin2(θ/2) - where θ is the angle between

the measurement directions. The single event probability is clearly pi = 12. By

choosing

a1 =−π3

a2 = b1 =π3

b2 =π3

(3.8)

as measurement directions, we get for p=(pa1, pa2, pb1, pb2, pa1b1, pa2b1, pa1b2, pa2b2):

p = (12,12,12,12,38,38,0,

38) (3.9)

and one of the inequalities found in (3.7) is violated:

pa1b1 + pa1b2 + pa2b2− pa2b1− pa1− pb2 =38

+38

+38−0− 1

2− 1

2=

18

> 0 (3.10)

The generalization is straightforward. Violations of certain inequalities involv-

ing classical probabilities - Boole’s “conditions of possible experience” [5] - also

appear in higher dimensions in configurations containing more particles and/or

more measurement directions.

4 QUANTUM CORRELATION POLYTOPES 35

4 Quantum Correlation Polytopes

We have seen in the previous section that some Boole-Bell-type inequalities are

violated when using quantum probabilities instead of classical relations. In terms

of the polytope formalism we can say, that vectors given by p= (p1, . . . , pi , . . . , pi j ),where the probabilities are calculated - conforming to quantum mechanics - as

pi = Tr[WEi ] pi j = Tr[W(Ei ⊗E j)] (4.1)

point outside of the corresponding classical correlation polytope. Here W denotes

the state of the quantum system and Ei are projection operators representing

quantum mechanical measurements. In the following we will give some basic

definitions to classify sets of probabilities corresponding to classical or quantum

mechanical probability calculus.

4.1 Definitions

In contrast to Section 3 we shall adopt a more modern approach introduced by

I. Pitowsky [8] for further discussion: Instead of vectors p = (p1, . . . , pi , . . . , pi j )we arrange the probabilities pi j in (n+ 1)× (n+ 1) matrices. The space of all

(n+1)× (n+1) real matrices is denoted by Rn+1, where the indices of a matrix

ai j ∈ Rn+1 have ranges 0≤ i, j ≤ n. Until now there are no further conditions

imposed on the matrices ai j , but we can define special subsets of the space Rn+1

according to different types of probability values:

4.1.1 Classical probabilities - the set c(n)

Definition 1 c(n) is the set of all matrices pi j ∈ Rn+1 with the following prop-

erties: p00 = 1 and there exists a probability space (X,Σ,µ), events A1,A2, . . . ,An,

B1,B2, . . . ,Bn ∈ Σ, such that pi0 = µ(Ai), p0 j = µ(B j), pi j = µ(Ai ∩B j) for i, j =1,2, . . . ,n.

This sounds a little bit complicated, but it is only a rearrangement from the vec-

tors p, which we have already used when discussing classical correlation polytopes

(cf. Section 3), to matrices. Stated in terms of the simple urn model from (3.1)

containing ’yellow’, ’wooden’ and ’yellow and wooden’ balls we have the matrix

c(n) 3 pi j =

(p00 p01

p10 p11

)=

(1 P1

P2 P12

),

where P1 = p01 is the proportion of yellow balls in the urn, P2 = p10 the proportion

of wooden ones and P12 = p11 denotes the proportion of yellow and wooden balls,


that is to say, the probability to draw a yellow ball out of the urn is given by p01

and similar for p10 and p11.

The set c(n) can be interpreted as a convex polytope and we have already

discussed the properties of such correlation polytopes in Section 3.

4.1.2 Quantum probabilities - the set bell(n)

In Section 3.6 we have already found out that using quantum mechanical pre-

scriptions to calculate the probability values pi j we get some matrices (or vectors

in ”old speech”) that lie outside the corresponding polytope (violating therefore

one or more Boole-Bell-type inequalities). We can define now a set of matrices

comprising quantum probabilities as well:

Definition 2 bell(n) is the set of all matrices pi j ∈ Rn+ with the following prop-

erties: p00 = 1 and there exist a finite dimensional Hilbert space H , projections

E1,E2, . . . ,En,, F1,F2, . . . ,Fn⊂H , and a density operator W on the tensor product

H ⊗H such that pi0 = Tr[W(Ei⊗1l)], p0 j = Tr[W(1l⊗Fj)], and pi j = Tr[W(Ei⊗Fj ])for i, j = 1, . . . ,n, where 1l denotes the unit matrix in H .

The meaning of this definition is that we consider a quantum system in the

state W ∈H ⊗H consisting of two subsystems Wi ∈H i , i = 1,2, where measure-

ments on the first subsystem are given by the projection operators Ei and on

the second subsystems by the Fj . This definition is more or less derived from

the experimental setup, where the composition of a total Hilbert space out of

two subspaces by means of a tensor product is implemented as a two-particle

system and the projection operators Ei and Fj are measurements on one particle

each (cf. Figure 12). The generalization of this definition is straightforward to a

many-particle system by considering (n+1)× (n+1)× . . .× (n+1) matrices.

Clearly we get the probability that the system has the property Ei or Fj by

calculating pi0 = Tr[W(Ei⊗1l)] or p0 j = Tr[W(1l⊗Fj)], respectively. The expression

pi j = Tr[W(Ei⊗Fj)] denotes the joint probability that we measure property Ei and

Fj simultaneously, thus measuring Ei ⊗Fj describes the process of measuring Ei

’on the left’ and Fj ’on the right’. This is in analogy with the classical case where

Ai ∩Bi is the event Ai and B j .

It is important to note that the projections Ei ,Fj are not fixed, i. e. we are

not considering any specific (experimental) setting with given measurement pa-

rameters and/or given input state, but we form the set of all real matrices having

in common that the single entries can be (re-)produced by at least one choice of

projection operators Ei ,Fj and input state W following the construction recipe in

Definition 2.

It has been proven by I. Pitowsky [8] that the set bell(n) is convex and that

bell(n)⊃ c(n).


4.1.3 More general quantum probabilities - the set q(n)

Since the tensor product is not the most general form for quantum mechanical

representation of the conjunction (logical ’and ’ operator), we can define another

set of matrices being a superset of c(n) and bell(n) by applying the suggestion of

Birkhoff and von Neumann [29] that subspace intersection should be the quantum

analogue of the conjunction.

Definition 3 q(n) is the set of all matrices pi j ∈ Rn+1 with the following prop-

erties: p00 = 1 and there exist a Hilbert space H , not necessarily commuting

projections E1,E2, . . . ,En, F1,F2, . . . ,Fn ∈ H and a density operator W on H such

that pi0 = Tr[WEi ], p0 j = Tr[WFj ] and pi j = Tr[W(Ei ∧Fj)] for i, j = 1, . . . ,n, where

Ei ∧Fj is the projection on Ei(H )∩Fj(H ).

The difference to Definition 2 is that here we consider projection operators Ei

and Fj not necessarily located in the subspaces Ei ∈ H e⊂ H and Fj ∈ H f ⊂ H ,

but each being defined on the total Hilbert space Ei ,Fj ∈ H = H e⊗H f . Thus the

projection operators are generalizations of the ones considered in Definition 2 in

the sense that we do not have to deal here with projection operators constructed

as tensor products like Ei⊗Fj , but the Ei and Fj can now be arbitrary projection

operators defined in H . The conjunction (as the analogue to the “and” operator

to form joint probabilities) is now the subspace intersection onto the subspace

spanned by Ei(H )∩Fj(H ), where Ei(H ),Fj(H )⊂ H .

The set q(n) is convex but not closed, its closure in Rn+1 is again a polytope

like c(n)12 (but unlike bell(n), which is not a polytope in general) and c(n) ⊂bell(n)⊂ q(n) is valid [8].

Polytope representation of q(1) If we take as an example q(1) thinking

back to the vector representation in Section 3 for c(1), we can construct the

corresponding polytope (Figure 10) by augmenting the c(1)-polytope (Figure 8)

by the additional vertex (1,1,0). Remember that the vertices of the polytope

have been derived from the probability values of the simple urn model (Section

3.1) (p1, p2, p12) with p1 and p2 denoting the single probabilities of drawing a

’yellow’ (property A1) or ’wooden’ (property A2) ball, respectively, and the joint

probability p12 = A1∧A2 for drawing a ’yellow and wooden’ ball. From this point

of view the vertex (1,1,0) corresponds to the statement that although we get hold

of a yellow ball with certainty and we also draw a wooden ball with certainty, we

never get a ’yellow and wooden’ ball. In Section 4.4 we will attempt to interpret

this result.

12Since the difference between q(n) and its closure q(n) is not relevant for our discussion wewill treat q(n) and q(n) as equivalent.


(0,0,0)

(1,0,0)

(0,1,0)

(1,1,1)

(a) c(1)

→(0,0,0)

(1,0,0)

(0,1,0)

(1,1,0)

(1,1,1)

(b) q(1)

Figure 10: Corresponding polytope to q(1)

Illustration of c(n) ⊂ bell(n) ⊂ q(n) Even though the polytopes belonging to

c(1) and q(1) as the simplest cases are already three dimensional and the di-

mensionality of c(n) and q(n) is quadratically increasing for larger n13, one can

get a rough idea by considering a two-dimensional projection, i. e. a cut through

the polytope. The structure of the three sets c(n), bell(n) and q(n) can be pic-

tured as in Figure 11, where the convexity of c(n), bell(n) and q(n), the relation

c(n) ⊂ bell(n) ⊂ q(n) and the polytope-like character of c(n) and q(n) has been

taken into account.

bell� n

�

c � n �

q � n �

Figure 11: Illustration of c(n)⊂ bell(n)⊂ q(n) as a two-dimensional projection

4.2 Violation of inequalities

In Section 3 we analyzed the violation of some inequalities for a special input

state, namely the singlet state |ψ〉 = 1/√

2(| ↑↓〉− | ↓↑〉), where | ↑〉 is an eigen-

13dimc(2) = 8, dimc(3) = 15, dimc(4) = 24,. . . , dimc(n) = n2 +2n, where dimc(n) denotes thedimensionality of the polytope associated to c(n) and dimc(n) = dimq(n).


state to the Pauli matrix σz to the eigenvalue +1 and | ↓〉 to the eigenvalue −1.

Now we examine the behavior of such violations for any state W ∈ H that can

be mixed or pure, but we restrict us to the case where dim(H ) = 4 14. Since

this is the only restriction, we can generate randomly 4×4 matrices W (equiv-

alent to quantum states) that are Hermitian (W = W†), positive (semi-) definite

(〈ϕ|W|ϕ〉 ≥ 0, ∀|ϕ〉 ∈ H ) and normalized (Tr[W] = 1).

4.2.1 Generation of states

States are generated according to two different mechanisms guaranteeing that we

obtain valid quantum states :

Mixed states: A definition of positive definiteness is the following:

A Hermitian matrix W is positive definite if and only if it can be written as

the square of another Hermitian matrix B; i. e. W = B2.

A parameterization of the Hermitian B is given by

B =

b1 b5 + ib6 b11+ ib12 b15+ ib16

b5− ib6 b2 b7 + ib8 b13+ ib14

b11− ib12 b7 + ib8 b3 b9 + ib10

b15− ib16 b13− ib14 b9− ib10 b4

, (4.2)

thus we have 16 real parameters bi to specify B. By squaring B we get a matrix

W′ = B2 that is positive definite and to ensure that the desired density matrix W

is normalized we only have to divide W′ by its trace, yielding W = W′/Tr[W′].By this method we should be able to reconstruct every possible quantum state

W∈H , dim(H ) = 4, but we cannot be sure that we get a uniform distribution over

all possible states. As we shall see this has the unpleasant side effect that some

types of states occur only with a very small probability in numerical evaluations.

Pure states: If we are only interested in pure states, we can consider the general

bipartite pure state

|Ψ〉= α|00〉+β|10〉+ γ|01〉+δ|11〉, α,β,γ,δ ∈ C, (4.3)

which transforms to the associated density matrix

W′|Ψ〉 = |Ψ〉〈Ψ|=

aa? ab? ac? ad?

ba? bb? bc? bd?

ca? cb? cc? cd?

da? db? dc? dd?

. (4.4)

14Thus we are considering bipartite systems consisting of two particles


Dividing by its trace we get the normalized pure state W|Ψ〉 =W′|Ψ〉/Tr[W′

|Ψ〉] in its

density matrix representation.

4.2.2 Measurement operators

The projection operators Ei and Fj are chosen to lie in the x− y plane if we

consider the states represented as particles flying apart along the z-axis in positive

or negative direction, respectively. Such projection operators can in general be

written as

M(θ,φ)± =12(1l±n·σ) =

12

(1±cosθ sinθe−iφ

sinθeiφ 1∓cosθ

), (4.5)

with the unit vector n = (cosθ,sinθcosφ,sinθsinφ)T pointing in the direction of

measurement and the sign determines whether we perform a spin-up (+) or spin-

down (−) measurement. The eigenvalues of M(θ,φ)± are {0,1}, i. e. if a spin-up

measurement is performed, the eigenvalue 0 corresponds to the result“the particle

has spin-up in the direction given by n”. In our special case, n lies in the x− y

plane φ = 0 and only spin-up measurements are made; consequently M(θ,φ)±

reads as

M(θ,φ)± → M(θ)+ =12(1l+n·σ) =

12

(1+cosθ sinθ

sinθ 1−cosθ

)(4.6)

and is dependent merely on θ. Ei = M(θ)+ (Fj = M(θ)+) denotes then the spin

measurement on the left (right) hand side, where the direction of the measurement

is given by angle θ∈ [0,2π] in the x−y plane (cf. Figure (12)) and the lower index

denotes a particular angle θ.

The probability for finding the spin-1/2 particle in the spin-up state along

the angle θ is given as usual by P(Ei) = Tr[W(Ei ⊗1l)], if the particle is moving

in the negative z-direction. In the same manner, P(Fj) = Tr[W(1l⊗Fj)] is the

probability for finding the particle on the right hand side having spin-up, and

P(Ei⊗Fj) = Tr[W(Ei⊗Fj)] denotes the joint probability for finding the left particle

in the spin-up state when measured by Ei and the right particle in the spin-up

state when measured by Fj . The Ei and Fj can be written as operators of the

form given in Eq. (4.6). In Tr[W(Ei ⊗ 1l)] and Tr[W(1l⊗Fj)] with 1l as the 2-

dimensional identity we take account for the fact that we perform operations

only on one subspace and the other one is unaffected.


y

x

y

xθ

�

1θ1

θ�

2

θ�

3

E3

E1θ2

θ3E2 F2F3

F1

Figure 12: Projection measurements Ei and Fj

4.2.3 Clauser-Horne-(CH)-inequality - bell(2)

First we consider the CH inequality given by

−1≤ p11+ p12+ p22− p21− p10− p02 = CH(pi j )≤ 0, (4.7)

thus we consider the situation that there are two different spin measurements on

each particle, which are in the state W ∈ H 1⊗H 2. The sets of measurement

operators for each side are then {E1,E2} and {F1,F2}, For simultaneous measure-

ments on both sides we have {E1⊗F1,E1⊗F2,E2⊗F1,E2⊗F2}. Clearly with the

notation introduced above the set of probabilities is given by the 3×3 matrix

pi j =

p00 p01 p02

p10 p11 p12

p20 p21 p22

(4.8)

=

1 Tr[W(1l⊗F1)] Tr[W(1l⊗F2)]Tr[W(E1⊗1l)] Tr[W(E1⊗F1)] Tr[W(E1⊗F2)]Tr[W(E2⊗1l)] Tr[W(E2⊗F1)] Tr[W(E2⊗F2)]

.

and the pi j are element of bell(2).

One-parameter plot Now we make the special choice that we perform the

projection measurement in the x-direction on the left hand side, i. e. θ = 0 and

E1 = M(0)+. Furthermore, E2 = F1 = M(θ)+ and F2 = M(2θ)+, i. e. the 2nd angle

on the left hand side θ is equal to the 1st angle on the right hand side and the 2nd

angle on the right is twice the angle θ. For clarification see Figure 13, where solid

lines symbolize measurement directions on the left and dotted lines directions on

the right hand side.

The parameter θ runs from 0 to π, since for the region θ ∈ [π,2π] the setup

differs only in the sense of direction of θ, i. e. if θ is counted clockwise or counter-

clockwise. Now numerous (pure15) states W are generated randomly for each

parameter value of θ. For each state we can calculate the matrix pi j for fixed θ,

15The probability to achieve a maximal violation is higher for pure states.


θ

2θ

E1

E2� F1

F2

Figure 13: E1 = M(0)+, E2 = F1 = M(θ)+, F2 = M(2θ)+

Singlet stateMaximum ValuesClassical Bounds

θ

CH

�

p ij

�

32.521.510.50

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

Figure 14: CH-inequality

i. e. for fixed Ei , Fj . Plugging the probability values pi j into the CH-inequality in

Eq. (4.7) and computing the value of CH(pi j ) we can select the state yielding a

minimum and the state yielding a maximum of CH(pi j ). By varying θ from 0 to

π we get all the maxima and minima in dependence of θ (cf. Figure 14). We can

see that the extremal violations actually depend on θ, although we use arbitrary

states W, and not only a singlet state.

That means that there exist ’global’ extrema independent of the state used,

but dependent on the configuration of the projection operators. Furthermore, we

can see in Figure 14 that the violation achieved by using a singlet state |Ψ〉 =1/√

2(| ↑↓〉− | ↓↑〉) is not optimal in the sense that by using another state Wmax

we could achieve a better violation.

Two-parameters plot If we set the measurement directions according to

E1 = M(0)+ E2 = M(θ1)+

F1 = M(θ2)+ F2 = M(θ1 +θ2)+(4.9)


orE1 = M(0)+ E2 = M(θ1)+

F1 = M(θ1)+ F2 = M(θ1 +θ2)+,(4.10)

(cf. Figure 15) we can produce a contour plot of the violation in dependence of

the parameters θ1 and θ2.

θ2

E1

E2

F1

F2

θ1

θ1 �

θ2

(a) E1-E2-F1-F2: 0-θ1-θ2-(θ1 +θ2)

E2� F1

E1

F2

θ1� θ2

θ1

(b) E1-E2-F1-F2: 0-θ1-θ1-(θ1 +θ2)

Figure 15: Two-parameter measurement directions

The contours in these plots are quite blurry, because when randomly gen-

erating (pure) states W as described in Section 4.2.1 we do not always (for any

θ1,θ2) find exactly the state yielding a maximal violation. Nevertheless the rough

contours of violation of CH(pi j ) can be seen.

When reading off the values along the diagonal line in the graphs given by

θ1 = θ2 we get back to the one-parameter plot (cf. Figure 13).

θ1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

θ 2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

(a) E1-E2-F1-F2: 0-θ1-θ2-(θ1 +θ2)

0.00 0.05 0.10 0.15 0.20

Violation:

θ1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

θ 2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

(b) E1-E2-F1-F2: 0-θ1-θ1-(θ1 +θ2)

Figure 16: Two-parameter plots of violation of the CH-inequality


Dependence on ”mixedness” We will now investigate if the violation is de-

pendent on the ”mixedness” of the generated states. Until now we have only used

pure states to get a maximal violation, but we have to verify that these maxi-

mum values are really maxima for all states, thus if mixed states do not violate

an inequality to a higher degree than pure states.

As a measure of the mixedness m(W) of a state W we take the trace of the

squared density matrix, i. e.

m(W) = Tr[W2], (4.11)

since for a pure state W|Ψ〉, W|Ψ〉 =W2|Ψ〉 is valid and therefore m(W|Ψ〉) = Tr[W2

|Ψ〉] =Tr[W|Ψ〉] = 1. For a maximally mixed state Wmm= 1

41l ∈ H with dim(H ) = 4 the

mixedness is given by m(Wmm) = Tr[W2mm] = 1/4.

We generate now four different sets of states with different ranges of m(W):

{X0.25≤m<0.3,X0.3≤m<0.6,X0.6≤m<0.9,Xm=1},

where X0.25≤m<0.3 is the set of all states with 0.25≤m< 0.3, i. e. the set contain-

ing maximally mixed states, X0.3≤m<0.6 the set of states fulfilling 0.3≤ m(W) =Tr[W2] ≤ 0.6, etc. It is shown in Figure 17 that for maximally mixed states

(W ∈ X0.25≤m<0.3) we do not get any violation of the CHSH-inequality at all. For

states in X0.6≤m<0.9 the value for CH(pi j ) exceeds the classical boundaries for some

angles θ; and for pure states (W ∈ Xm=1) θ indicates only the degree of violation,

but for all θ except at θ = 0,π the CHSH-inequality is violated in some states.

Class. BoundsX0 � 25

�m � 0 � 3

X0 � 3�

m�

0 � 6X0 � 6

�m � 0 � 9Xm � 1

θ

CH

� p ij

�

32.521.510.50

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

Figure 17: Dependence on ”mixedness”


We can draw the conclusion that when the randomly generated states are all

pure the probability of producing a state violating the inequality maximally for

given Ei and Fj is much higher as when considering all possible states ranging from

maximally mixed to pure states. The higher the degree of mixedness the more

unlikely it is to get a violation at all, i. e. probability values belonging to mixed

states yield points in the center of the plot (cf. Figure 17). Although we cannot

say anything about the amount of states violating Boole-Bell-type inequalities

relative to all possible states, because we do not generate uniformly distributed

states, one can conjecture that pure states constitute the upper bound of violation.

In other words: no mixed state can violate a Boole-Bell-type inequality to a

higher degree than a pure state. Note also that Figure 17 is misleading: One

might speculate that a maximal violation cannot be obtained using mixed states,

but this has been refuted by S. Braunstein et al. in [30].

4.2.4 CHSH-inequality

The Clauser-Horne-Shimony-Holt-inequality (CHSH-inequality)[31] is given by

|CHSH(α,β,γ,δ)|= |E(α,γ)+E(β,γ)+E(β,δ)−E(α,δ)| ≤ 2, (4.12)

where E(ξ,χ) denotes the correlation function given by the expectation value

E(ξ,χ) = Tr[W(nξ ·σ⊗ nχ ·σ)]. nξ is a unit vector pointing in the direction of

measurement, σ = (σx,σy,σz)T are the usual Pauli matrices and W is the QM-

state. Although the CHSH-inequality cannot be described with the formalism

used above, because it consists of correlation functions instead of probability

values, we will nevertheless analyze this inequality in detail. The reason for our

particular interest in this inequality is that Tsirelson gave a proof [7] that there

exists an upper bound for the violation of the CHSH-inequality. This bound

cannot be surpassed by any QM-state.

Tsirelson’s inequality Suppose we have the same setup as for the CH-inequality,

thus a pair of spin-1/2 particles produced by a source and sent one to the left

(negative z-direction) and one to the right (positive z-direction). On each side the

spin expectation values in two different directions can be measured, the observ-

ables for these measurements are given by σα,σβ,σγ and σδ, whereas σα,σβ ∈H 1

and σγ,σδ ∈H 2 with H tot = H 1⊗H 2, i. e. either the measurement is done in the

left subsystem (H 1) or in the right subsystem (H 2), and α,β,γ,δ are the possible

angles for the direction of measurement in the corresponding subsytem. With

σξ = nξ ·σ these operators form the quantum mechanical correlation functions


E(ξ,χ) = Tr[W(nξ ·σ⊗nχ ·σ)]. They satisfy

σ2α = σ2

β = σ2γ = σ2

δ = 1l

and, furthermore, they fulfill the commutator relations

[σα,σγ] = [σα,σδ] = [σβ,σγ] = [σβ,σδ] = 0. (4.13)

When we define now the CHSH-operator

C = σασγ +σβσγ +σβσδ−σασδ, (4.14)

which has the same structure as the correlation functions on the left hand side of

Equation (4.12). Squaring C yields

C2 = 4+[σα,σβ][σγ,σδ]. (4.15)

Using the following identities which are valid for any two bounded QM operators

A and B

‖[A,B]‖ ≤ ‖AB‖+‖BA‖ ≤ 2‖A‖‖B‖ (4.16)

we get ‖C2‖ ≤ 8 and therefore

‖C‖ ≤ 2√

2. (4.17)

No violation beyond Tsirelson’s bound In Figure (18) we can see that

Tsirelson’s bound is not violated for any measurement direction and any QM

state W. The parameterization is given by

α = 0, β = 2θ,

γ = θ, δ = 3θ,

i. e. the observer on the left hand side can choose between the directions given

by α and β, and on the right γ and δ are the possible choices of measurement

directions. Again only pure states are generated since they seem to violate Boole-

Bell-type inequalities maximally.

As a particular well-known example the CHSH-inequality is calculated using

the singlet state |Ψ〉= 1/√

2(| ↑↓〉−| ↓↑〉) that yields a maximal violation of 2√

2

for α = 0, β = π/2, γ = π/4, δ = 3π/4, which is exactly Tsirelson’s bound.


Maximum ValuesSinglet state

Classical Bounds

θ

CH

SH

� α

�

β

�

γ�

δ�

32.521.510.50

4

3

2

1

0

-1

-2

-3

-4

Figure 18: Tsirelson’s bound

4.2.5 Boole-Bell-type inequality out of bell(3)

As a higher dimensional example from the set bell(3), an inequality by Pitowsky

and Svozil [32] is given by

PIT(θ) =−p10− p20− p01− p02− p11+ p12+ p13+ p21+ p23+ p31+ p32− p33≤ 0,

(4.18)

where pi j = Tr[W(Ei⊗Fj)], i, j = 1,2,3. Thus we are considering again a two spin-

1/2 particle system, but now with three possible measurement directions on each

side. When considering the symmetric setup Ei = Fi , the set of spin-measurement

operators is then given with regards to the general projection operator form in

Eq. (4.6) by

E1 = M(0)+, F1 = M(0)+,

E2 = M(θ)+, F2 = M(θ)+,

E3 = M(2θ)+, F3 = M(2θ)+.

. (4.19)

Generating again randomly pure states for varying θ, we can depict PIT(θ) in

Figure 19. Like in the CHSH-case the singlet state as a special pure state yields

a maximal violation of PIT(θ) = 1/4 for θ = 2π/3.

We notice that there is no violation for θ = π/2, although the associated

projection operators are non-commuting for this choice of θ. This is an example

of a particular choice of measurements that does not allow a violation of the

corresponding inequality16.

Formally we can prove that there must not be any violation for θ = π/2,

which describes the setup where consecutive measurement directions are orthog-

16apart from the trivial setup with Ei = Fj , ∀i, j, i. e. θ = 0.


Singlet stateMaximum Values

Classical Bound

θ

PIT

(θ)

32.521.510.50

0.5

0

-0.5

-1

-1.5

-2

-2.5

-3

Figure 19: Pitwosky-Svozil inequality

onal (E1 ⊥ E2 ⊥ E3 and F1 ⊥ F2 ⊥ F3): Consider the operator PSassociated with

the left hand side of Eq. (4.18) given by

PS = −E1⊗1l−E2⊗1l−1l⊗F1−1l⊗F2− (4.20)

E1⊗F1 +E1⊗F2 +E1⊗F3 +E2⊗F1 +E2⊗F3 +

E3⊗F1 +E3⊗F2−E3⊗F3.

Inserting the matrix expressions for the Ei , Fj from Eq. (4.6) we get PS in matrix

form (in the basis {| ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉}):

PS=

−3 0 0 0

0 0 0 0

0 0 0 0

0 0 0 −1

=−3| ↑↑〉〈↑↑ |−1| ↓↓〉〈↓↓ | (4.21)

Obviously the eigenvalues are λ1 = −3,λ2,3 = 0,λ4 = −1 and from the Her-

miticity of PS and the negativity of all eigenvalues, λi ≤ 0, it follows that PS is

negative semi-definite, thus 〈ϕ|PS|ϕ〉≤ 0, ∀|ϕ〉. Consequently the Pitowsky-Svozil

inequality in Eq. (4.18) cannot be violated by any state |ϕ〉 ∈ H for θ = π/2.

4.3 Bounds of bell(n)

We have to keep in mind that we cannot draw any conclusions about the shape of

bell(n) from the considerations above. For the set c(n) we know that it is a convex

polytope and we can in principle plot two- or three-dimensional intersections, but

for bell(n) we only know that is convex and embedded in q(n) (cf. Figure 11). So


here we try to find some kind of representation of bell(n) as far as this is feasible.

This approach is different from the previous calculations: In Section 4.2 only

the state was arbitrary, the set of measurements {Ei ,Fj} was fixed by one or

two parameters θk, but here both the projections {Ei ,Fj} and the state W are

generated randomly, because we try to find a representation of the set bell(n)itself, not only a “path” in bell(n). This “path” is traced out by the bi j ∈ bell(n)having maximal distance from the bounds of c(n) under the restriction that the

projectors Ei and Fj correspond to a special type of measurement, namely a spin-

up measurement along directions in the plane perpendicular to the direction of

propagation of a particle. This path is parameterized by the angle θ (in the one-

parameter plots), and by varying θ one can move along this path. However, it

does not represent the bounds of bell(n).Therefore, the task is now to release this constraint by creating all possible

combination of Ei , Fj and W to get all matrices pi j ∈ bell(n) fulfilling the condi-

tions in Definition 2.

4.3.1 Representation of bell(1)

It is obvious that bell(1) is equivalent to c(1), in other words, it is not possible

to get any violation of the classical bounds (cf. Figure 20), since there are only

commuting operators involved, which can be measured simultaneously; there are

no ”unperformed experiments [that] have no results” [14, p. 168]. Nevertheless,

this example is illustrative, because we can easily imagine a two-dimensional

cut through the polytope associated with the set q(1) (Figure 21). The dark-

grey shaded area in the plane E intersecting the q(1)-polytope depicts the two-

dimensional subset of c(1).The set bell(1) is given according to Definition 2 in Section 4.1 as

pi j =

(p00 p01

p10 p11

)=

(1 Tr[W(1l⊗F1)]

Tr[W(E1⊗1l)] Tr[W(E1⊗F1)]

), (4.22)

where W is a bipartite quantum state (two-particle system), E1 and F1 denote the

projection operators corresponding to spin-measurements ”on the left” and ”on

the right”, respectively.

The bounds on classical probabilities constituting the polytope c(1) are given

by

0≤ p11≤ p01≤ 1, 0≤ p11≤ p10≤ 1,

0≤ p10+ p01− p11≤ 1. (4.23)

We can now proceed by generating arbitrary (pure) bipartite states and cal-

culate the matrices pi j ∈ bell(1), i, j = 0,1 using random projection operators E1


and F1.

By selecting only such pi j for that p11 = c± ε with 0 ≤ c ≤ 1, we restrict

our considerations to elements of bell(1) on the intersecting plane E . The term

±ε is due to the fact that we have to select appropriate matrices pi j out of all

numerically generated pi j . Without a tolerance value ε sufficiently large we do

not get sufficiently many matrices for further processing. Thus one has to find a

trade-off between the number of pi j ’s (the more, the better) and the significance

of the calculations.

In terms of vertices (p10, p01, p11) the polytope associated to q(1) is given

by the set of extremal vertices {(0,0,0),(1,0,0),(0,1,0),(1,1,1),(1,1,0)}, fixing

p11 = c means a restriction to the plane E parallel to the plane B in distance

c, where the latter is spanned by p10 and p01 (i. e. by the vectors (1,0,0) and

(0,1,0)) as depicted in Figure 21.

In Figure 20, p10 over p01 of the matrices pi j ∈ bell(1) satisfying the condition

p11 = 0.1± 0.015 is depicted. The dashed lines represent the classical bounds

from Eq. (4.23). These lines are enclosed by dotted lines representing the tol-

erance value ε. Since there are no points outside the region limited by classical

inequalities apart from deviations due to ε,

c(1) = bell(1).

p02

p 12

10.80.60.40.20

1

0.8

0.6

0.4

0.2

0

p02

p 12

10.80.60.40.20

1

0.8

0.6

0.4

0.2

0

p02

p 12

10.80.60.40.20

1

0.8

0.6

0.4

0.2

0

p02

p 12

10.80.60.40.20

1

0.8

0.6

0.4

0.2

0

p02

p 12

10.80.60.40.20

1

0.8

0.6

0.4

0.2

0

� bell(1)

p02

p 12

10.80.60.40.20

1

0.8

0.6

0.4

0.2

0

Figure 20: bell(1) (c = 3/8, ε =±0.015)

H0,0,0L

H1,0,0L

H1,1,1L

E

B

Figure 21: Cut through polytope q(1)

4.3.2 Representation of bell(2)

The next aim is to draw the bounds of bell(2), i. e. for the eight dimensional case,

where we have the projections {E1,E2,F1,F2} and the state W ∈ H ⊗H yielding


the 3×3-matrix pi j stated in Equation (4.8).

The operational interpretation is that we have a two-particle system in the

state W ∈H tot = H 1⊗H 2 with measurements E1,E2 on the first subsystem and

F1,F2 on the second subsystem.

This special case represents the classical Clauser-Horne polytope already dis-

cussed in Section (3.6), the associated inequalities, i. e. bounds of classical prob-

abilities, read as follows:

0≤ pi j ≤ pi0 ≤ 1,0≤ pi j ≤ p0 j ≤ 1, i = 1,2,

pi0 + p0 j − pi j ,≤ 1 i = 1,2,(4.24)

−1≤ p11+ p12+ p22− p21− p10− p02≤ 0,

−1≤ p21+ p22+ p12− p11− p20− p02≤ 0,

−1≤ p12+ p11+ p21− p22− p10− p01≤ 0,

−1≤ p22+ p21+ p11− p12− p20− p01≤ 0.

(4.25)

Since the complete eight dimensional polytope is a little bit tricky to vi-

sualize when taking all pi j as free parameters, we again consider only a two-

dimensional cut through the polytope c(2) by setting p01 = p10 = p20 = a and

p22 = c, a,c const., i. e. we generate arbitrary density matrices and projection

operators, but we take only those with p01 = p10 = p20 = a±ε and p22 = c±ε for

some a,c. The probabilities p11 and p21 are also fixed to p11 = p21 = b± ε and

additionally b has to fulfill the inequalities 2a−1≤ p11, p21≤ a due to Eq. (4.24).

With this choice we are left with the free parameters p02 and p12, for which

we can now extract the inequalities from the Equations (4.24,4.25) above. In first

place, clearly for each pi j

0≤ pi j ≤ 1

must be valid. From Eq. (4.24) follows also

p12 ≤ a, (4.26)

p12 ≤ p02,

c ≤ p02,

p02 ≤ −a+c+1,

p02 ≤ p12−a+1,

and from Eq. (4.25)

p02+a−c−1 ≤ p12 ≤ p02+a−c,

2a−2b+c−1 ≤ p12 ≤ 2a−2b+c,

−2a+2b+c ≤ p12 ≤ 2a+2b+c+1.

(4.27)


We now choose a = 1/2,b = c = 3/8, and the tolerance ε = ±0.015. When

using p02 and p12 as coordinates we can visualize the two-dimensional cut through

bell(2) (cf. Figure 22).Again we have only used pure states, since we conjecture

that these build up the bounds of the set bell(n) [8].

violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1violation

p02

p 12

10.90.80.70.60.50.40.3

0.5

0.4

0.3

0.2

0.1

0

-0.1

Figure 22: bell(2) (a = 1/2, b = c = 3/8, ε =±0.015

c(2)⊂ bell(2): In Figure 22, dashed lines indicate the classical inequalities adapted

to this two-dimensional cut-through in Eq. (4.26,4.27), and dotted lines show

the same inequalities when taking the tolerance value ε = ±0.015 into account.

Although due to a quite large tolerance value of ε = ±0.015 we do not obtain a

clear shape of bell(2), it is obvious that there exist some pi j violating the classical

bounds on probabilities. Unfortunately, ε cannot be decreased, because then there

are not enough matrices pi j left fulfilling the conditions p01 = p10 = p20 = a= 0.5

and p11 = p21 = p22 = b= c= 3/8, i. e. there are too little points to plot. Already

for the given ε =±0.015 the shape of bell(2) is not very good rendered and one

can only estimate that the set is convex.

4.4 Generalizing bell(n) - the set q(n)

Finally, the question, what we can say about the set q(n), is left open. We have

seen the representation as a three-dimensional polytope q(1) in Figure 10 and

in principle we can construct polytopes q(n) also for n > 1 by adding additional

vertices to the classical polytopes [6]. For example, q(2) consists out of 47 vertices

versus 16 vertices for c(2).Nevertheless, already the meaning of the additional vertex (1,1,0) to c(1) is

quite obscure (cf. Section 4.1.3). Since we can interpret this vertex point as


extremal probability values this corresponds to an ”absurd” experiment where

properties A and B happen with certainty, but the joint property A∧B never

occurs.

Absurd experiment Assume now our usual implementation of a two-particle

system (cf. Figure 12) with only one measurement on each side E1 ∈ H 1 and

F1 ∈ H 2 with H = H 1⊗H 2, respectively. In this context the vertex (1,1,0)would stand for the result, that given measurement directions α on the left and βon the right there must exist a quantum state W fulfilling p10 = Tr[W(E1⊗1l)] = 1,

p01 = Tr[W(1l⊗F1)] = 1, and p11 = Tr[W(E1⊗F1)].The single probabilities would not impose a great difficulty, we just have

to take the state W = |Ψ〉〈Ψ|, |Ψ〉 = 1/√

2(| ↑↑〉+ | ↓↓〉) and for the particular

case α = β we get p10 = p01 = 1. For α 6= β we can certainly find a similar

state |Ψ〉 bearing in mind that any state can be written in its Schmidt-basis as

|Ψ〉= ∑i λi |i1〉|i2〉 (Section A.3.2) with orthonormal basis vectors |i1〉 and |i2〉.But the problems arise when we try to achieve p11 = 0, thus finding projection

operators E1 ∈ H 1, F1 ∈ H 2 and a state W ∈ H such that, for instance, we

measure spin-up on the left and spin-up on the right for each pair of particles,

but never both properties simultaneously. Because there are no non-commuting

measurements involved the situation is restricted by classical probability laws and

the claim p11 = 0 is in fact absurd.

Hence, we run into problems when attempting to produce matrices pi j ∈ q(n)demanding that the Ei and Fj are located in a subspace of the total Hilbert

space, although this is exactly the restriction imposed by the physical setup that

Ei corresponds to a measurement ’on the left’ and Fj to a measurement ’on the

right’.

From Definition 3 we know that the elements of the set q(n) can be calculated

using more general projection operators Ei , Fj ∈ H without any restriction to a

subspace of H . It is clear that these operators are not necessarily separable,

thus they can in general not be written as tensor products of two lower dimen-

sional projection operators, but they can be seen as an analogue of non-separable

entangled states.

4.4.1 Example

Theoretically, consider the following example to obtain a matrix pi j 6∈ c(1):A system in the singlet state |φ〉 can be described in terms of a general pure

state

|Ψ〉= α| ↑↑〉+β| ↑↓〉+ γ| ↓↑〉+δ| ↓↓〉, α,β,γ,δ ∈ C (4.28)


as

|φ〉= |Ψ〉α=δ=0, β=1/√

2, γ=−1/√

2 = 1/√

2(| ↑↓〉− | ↓↑〉). (4.29)

Since a projection operator does not differ formally from a pure state, we shall

take projection operators into consideration that have the same form, namely

Pi = |Ψ〉〈Ψ|, with α = δ = 0, β = xi/√|xi |2 + |yi |2, γ = yi/

√|xi |2 + |yi |2, (4.30)

where the factor 1/√|xi |2 + |yi |2 is due to normalization.

In the q(1) case, we need two Pi ’s and we choose them to be

P1 = |ϕ1〉〈ϕ1|, |ϕ1〉= 1/√|s|2 + |t|2(s| ↑↓〉+ t | ↓↑〉) (4.31)

P2 = |ϕ2〉〈ϕ2|, |ϕ2〉= 1/√|s|2 + |t|2(t | ↑↓〉+s| ↓↑〉). (4.32)

Now, p10 = Tr[P1|φ〉〈φ|] and p01 = Tr[P2|φ〉〈φ|]. Furthermore, the joint proba-

bility p11 of P1∧P2 is the projection on the subspace intersection P1(H )∩P2(H ),which can be calculated by

p11 = Tr[ limn→∞

(P1P2)n|φ〉〈φ|]. (4.33)

Eq. (4.33) is valid also for non-commuting projection operators, for commuting

Pi ’s it reduces to p11 = Tr[(P1P2)|φ〉〈φ|].Using this parameterization, we can calculate numerically projection operators

with s, t ∈ [−1,1] and approximating the limes limn→∞(P1P2)n by a large n, namely

n = 100000. Inserting the probabilities into the inequality p10 + p01− p11 ≤ 1

characteristic for the c(1) polytope, we can see that for some values p10+ p01−p11≥ 1 and that 2 is apparently the upper bound (cf. Figure 23).

-1

-0.5

0

0.5

1

s

-1

-0.5

0

0.5

1

t

0

0.5

1

1.5

2

-1

-0.5

0

0.5

1

s

Figure 23: p10+ p01− p11≥ 1 for q(1)

Taking, for example, s= 1, t =−0.9 results in a quite good approximation of


the vertex (1,1,0) of the q(1)-polytope:

(p10, p01, p11) = (0.997,0.997,0.6×10−481).

Entangled measurements We have seen that analogous to the step from c(n)to bell(n), where we had to take entangled states into account, we get from bell(n)to q(n). In order to find elements in q(n), but not in bell(n), we have to move on

from measurement operators that can be represented as tensor products of lower

dimensional projectors to “entangled” measurement operators. But this step is

even more dubious than the entanglement of states, and its physical meaning is

not clear at all. Maybe one can exploit the structural equivalence between pure

states and measurement operators, so that the measurement apparatus is coupled

to a pure entangled state so that all the properties of the state pass over to the

measurement operator.But this is beyond the scope of this treatise.

5 SUMMARY 56

5 Summary

Implications of the “no-go” theorems introduced In the discussion about

the completeness of quantum theory we considered a eventually possible enhance-

ment by so called hidden variables. The problem arising from the formalism of

QM is the inherent indeterminism that has a tremendous impact on philosoph-

ical questions. From the experience of classical physics, it is clearly not easy to

accept a physical theory predicting only probability values, while claiming that

every possible piece of knowledge has already been taken into account, meaning,

that there is nothing more to add to get definite predictions.

We have seen that it is possible to enlarge quantum theory by additional

hidden variables, but with a major drawback: Due to the Kochen-Specker theorem

such theories seem to be contextual , if the predictions of quantum mechanics are

reproduced. Since quantum theory has been confirmed by numerous experiments,

there is certainly no need to abandon QM.

Things get worse when considering Bell’s theorem: not only we have to restrict

possible alternative theories to contextual ones, but the concept of locality is

endangered. If we do not include the possibilities that our world is predetermined

from the very beginning or that everybody is living in only one universe out a

multitude of parallel-universes it seems that we have to accept “spooky actions

at a distance.” Quantum theory cannot be reconciled with special relativity in a

strong sense; i. e., superluminal transmission of information is still not possible,

but there occurs some kind of action between two far-away quantum systems.

Boole-Bell-type inequalities Bell’s theorem can be stated with and without

inequalities, but our further investigations considered mainly generalizations of

the original inequality presented by Bell [2]. We have reviewed a general frame-

work for the derivation of Boole-Bell-type inequalities in terms of correlation

polytopes. Due to the Weyl-Minkowsky theorem there is a one-to-one correspon-

dence between vertices of a convex polytope (in Euclidean space) and a finite

number of half-space intersections given by a linear inequality. These inequalities

are identical with the set of Boole-Bell-type inequalities for a specific configura-

tion and some of them are violated by quantum mechanical proposition calculus.

The vertices represent maximal probability values of propositions and their con-

junction.

We have then considered the three sets c(n), bell(n), and q(n) to distinguish

between vertices that are found in classical logic and vertices that are found in

quantum logic.

5 SUMMARY 57

Maximal violations for arbitrary states Usually, the violation of Boole-

Bell-type inequalities is calculated using special choices of quantum states, namely

Bell-states, that are pure and maximally entangled. The maximal violation can

be obtained by properly choosing the measurement operators characterized by

spin-measurements along an angle in a plane perpendicular to the direction of

propagation of the particles.

We have computed numerically the violations for three inequalities (CHSH-

inequality, CH-inequality, Pitowsky-Svozil-inequality) utilizing general pure and

mixed states. It has been shown that there exist configurations for which it is more

likely to obtain maximal violations for general quantum states. Conversely, for

some configurations - mainly when using perpendicular measurement directions

- it is not possible to achieve any violation at all, no matter which state is used.

Furthermore, the probability to obtain a violation of the CH-inequality is

minimal for maximally mixed states, a maximal violation can be achieved more

likely using pure states. But since the probability of generating a certain state is

dependent of the particular algorithm in use, and since, therefore, the distribution

is not uniform, we can only conjecture that the degree of violation is dependent

on the degree of mixedness. At least there is strong evidence that the maximal

degree of violation reached by pure states cannot be exceeded by mixed states.

Illustration of bell(n) In addition, an attempt has been made to visualize the

convexity of the set bell(n) for the case n = 2 by plotting a two-dimensional cut

through the higher dimensional polytope. Although for n = 1 we do not run into

any problems, for n= 2 the generation rate of suitable states is very low, since we

have to apply severe restrictions to the possible probability values. One can easily

imagine this by taking into consideration that we can use only configurations

(i. e. particular states and measurements) yielding probability values lying in the

intersection between a two-dimensional plane and an eight-dimensional object,

whereas without restrictions we could use any configuration inside the total eight-

dimensional object.

Therefore, we do not achieve a well-formed convex hull of the set bell(n),however, we can see that c(n) forms a subset of bell(n).

Entangled measurements Beyond that, we have analyzed the definition of

q(n) and found out that when using “entangled” measurement operators, one

can calculate probability values fitting only in the set q(n). Such “entangled”

measurements have to be regarded as the equivalent to entangled states, that is

to say, states that cannot be represented as a tensor product of lower dimensional

states. The operational meaning of this kind of measurements would be a matter

of future research, as well as the implementation of subspace intersection for the

6 ACKNOWLEDGMENTS 58

conjunction of joint events.

6 Acknowledgments

I would like to say thanks to my supervisor Karl Svozil for providing the oppor-

tunity to work on a topic concerning the highly interesting basics of quantum

mechanics and for rewarding discussions in a relaxed and friendly atmosphere.

Special thanks go to all my friends accompanying me through my studies as

well as through “trivial” life, namely (in alphabetical order) Andreas, Christoly,

Herbert, Maria, Michi, Philipp, Reinhard, Sandra, especially to Flo who also

bothered to look through this thesis, but over and above Claudia for all kinds of

support in all kinds of situations.

Last but not least I want to thank my parents for financial support with-

out hesitation and doubts making my studies possible, but beyond that for a

harmonious and cordial family life.

Finally, thanks to urbi et orbi for being and constituting a fascinating world.

Welch Schauspiel! Aber ach! ein Schauspiel nur!

Wo faß ich dich, unendliche Natur?

Goethe, Faust

A CHARACTERIZATION OF STATES 59

A Characterization of states

A.1 Pure vs. mixed states

A pure state in a finite dimensional Hilbert space H is a superposition of basis

states denoted by

|ψ(t)〉= ∑i

ai(t)|ϕi〉, ai(t) ∈ C , (A.1)

where the |ϕi〉 are the (orthonormal) basis vectors of H . The time-evolution is

given by the Schrodinger Equation ih ddt |ψ(t)〉= H|ψ(t)〉 with H as the governing

Hamiltonian of the system. These states are the fundamental objects for quantum

mechanics since the evolution of a closed quantum system can always be described

in terms of a unitary evolution of a pure state.

Unfortunately, mostly we have to deal with quantum systems, where we can-

not neglect the influences of the environment, or we cannot prepare a state prop-

erly, so we have to use the concept of mixed states. When considering the prepa-

ration of a state (for example to use later for some experiment), we can easily

imagine what can happen to a pure state: If a source produces the pure quantum

state |ψ〉 only with a certain probability p due to environmental effects, which

cannot be totally controlled, it is obvious that we can write our new state as a

weighted sum over the different output states. Thus the output (mixed) state is

given by

ρ = ∑i

pi |ψi〉〈ψi |, ∑i

pi = 1, 0≤ pi ≤ 1. (A.2)

This state represented by the so called density matrix (or density operator) ρ is

diagonal in the basis |ψi〉〈ψi |.In general a mixed state is represented by a matrix ρ that is Hermitian (ρ =

ρ†), positive definite (〈ψ|ρ|ψ〉 ≥ 0 ∀|ψ〉 ∈ H ) and Tr[ρ] = 1, thus a general form

of a mixed state is

ρ = ∑i, j

pi j |ψi〉〈ψ j |, (A.3)

that can always be diagonalised (since ρ is Hermitian) by a unitary matrix, i. e.

transformed to Eq. (A.2).

Clearly if ρ = |ψk〉〈ψk| (pi 6=k = 0, pk = 1), ρ represents a pure state and ρ is a

projection operator. By definition

ρ = ρ2 iff ρ = |ψk〉〈ψk|, (A.4)

which is equivalent to the statement that ρ has one eigenvalue equal to 1, the

others are 0 and to

Tr[ρ] = Tr[ρ2]. (A.5)


If Tr[ρ2] 6= 1 than we can be sure that we have a mixed state, thus we can use this

criterion also as a classification of “mixedness” dependent on the value of Tr[ρ2],where Tr[ρ2] = 1/dim(H ) represents a maximally mixed state.

A.2 Purification

As already mentioned above, the evolution of a quantum system can always be

given by a unitary evolution of a pure state. To see this, we have to introduce

the concept of purification:

We can consider the Hilbert space of the observed quantal system H s as part

of an extended Hilbert space

H ext = H s⊗H a, (A.6)

where H a is called the ancilla part of H ext. This extension is useful, because we

can reduce a pure state |Ψsa〉 ∈ H ext to a (density) operator in H s (or H a) by

performing a partial trace over the ancilla (or the system).

ρs = Tra[|Ψsa〉〈Ψsa|] (or ρa = Trs[|Ψsa〉〈Ψsa|]) (A.7)

Conversely, we can lift a density operator ρ ∈ H s to a pure state |Ψsa〉 ∈ H ext,

also called purification by attaching an ancilla.

Note that the dimension of the ancilla can be greater than the dimension

of the system. The concept of purification is used for example for quantum

error-correcting codes, where an ancilla (an auxiliary state of auxiliary qubits) is

attached to find out what errors occurred in the system part [33].

A.3 Composed systems

Until now we only considered quantum systems consisting of one particle17, but

since reality consists of more than one particle we need a description of systems

composed of more than one particle. This can easily be done by using the concept

indicated in the previous section about purification: each particle is represented

by a subspace H i of the total Hilbert space H tot, consequently H tot can be written

as a direct product of the n single subspaces (n being the number of particles under

consideration)

H tot = H 1⊗H 2⊗ . . .⊗H n. (A.8)

17In fact by definition a mixed state cannot be only one particle, because it can be describedas a weighted sum over pure states, which occur with a certain probability.


The dimensionality of H tot is given by

dimH tot = dimH 1dimH 2 . . .dimH n. (A.9)

Take for instance two spin-1/2 particles each living in its two-dimensional Hilbert

space H A and H B, respectively. According to Eq. (A.9) the dimension of H tot =H A⊗H B is therefore dimH tot = 4.

A.3.1 Pure states

A single particle in H A (system A) in a pure state can now be described by the

(not normalized) wave function (neglecting the spatial degrees of freedom)

|ϕi〉= α| ↑〉+β| ↓〉, α,β ∈ C (A.10)

where | ↑〉, | ↓〉 are the eigenstates of the usual σz spin operator to the eigenvalues

±1. The notation | ↑〉, | ↓〉 has been chosen to symbolize the correspondence to

spin-up and spin-down states of spin-1/2 particles and is commonly used in this

thesis, but sometimes it is better to use the notation | ↑〉 ≡ |0〉, | ↓〉 ≡ |1〉, so that

both notations are considered equivalent in the following.

Adding another particle (system B) living in H B to the setup the general form

of the wave function |Ψ〉 ∈ H tot describing this two-particle system is given by

|Ψ〉 = |ϕ1〉⊗ |ϕ2〉 (A.11)

= (α1|0〉+β1|1〉)⊗ (α2|0〉+β2|1〉 (A.12)

= α1α2|0〉⊗ |0〉+α1β2|0〉⊗ |1〉+β1α2|1〉⊗ |0〉+β1β2|1〉⊗ |1〉= ∑

i, jci j |i〉⊗ | j〉, i, j = 0,1, ci j ∈ C.

When we consider only normalized states (〈Ψ|Ψ〉= 1), the coefficients ci j have to

fulfill ∑i, j c2i, j = 1.

A.3.2 Schmidt-Decomposition

The |i〉⊗| j〉= |i〉| j〉= |i j 〉, i, j = 0,1 in Eq. (A.11) form a basis of the 4-dimensional

Hilbert space H tot, but it can be proven that the double-sum in the last line of

Eq. (A.11) can be reduced to a single sum, thus the two-particle system can be

represented by

|Ψ〉= ∑i

λi |i1〉|i2〉,

where λi (Schmidt coefficients) are non-negative real numbers satisfying ∑i λ2i = 1

and |i1〉, |i2〉 being orthonormal basis states in system 1 and system 2, respectively.


This is called Schmidt-decomposition.

The proof for the special case where the systems A and B have state spaces of

the same dimension is not difficult:

Given | j〉 and |k〉 as orthonormal bases for the systems A and B, respectively,

|ψ〉 can be written as

|Ψ〉= ∑jk

a jk| j〉|k〉, a jk ∈ C.

By polar (or singular value) decomposition the matrix a jk can be written as

a jk = u ji dii vik, where d is a diagonal matrix and u,v are unitary matrices. Using

this we can write

|ψ〉= ∑i jk

u ji dii vik| j〉|k〉

and when we define new basis states as |iA〉 ≡ ∑ j u ji | j〉 and |iB〉 ≡ ∑k vik|k〉 and

λi ≡ dii , we get

|Ψ〉= ∑i

λi |iA〉|iB〉. (A.13)

Since u (v) is unitary and due to the orthonormality of | j〉 (|k〉) the |iA〉 (|iB〉) form

an orthonormal set and are called the Schmidt bases for A and B, respectively,

and the number of non-zero values λi are called Schmidt numbers of the state

|Ψ〉.

A.3.3 Mixed states

We have already noted that the description of the system in terms of pure states

is not sufficient, but we have to introduce mixed states for a proper description

of physical systems. For the composition of a system out of several subsystems i,

each described by a density matrix ρi , we apply the same scheme as above. The

total system ρ can then be written as a direct product of the ρi , thus

ρ = ρi ⊗ρ j . (A.14)

Let us give an example for a two-dimensional quantal system: Here ρi can be

parameterized by ρi = 1/2(1l+ r i ·σ) with r i denoting a unit vector in R3 and the

usual Pauli matrices σ = (σx,σy,σz)T , therefore we can write

ρ =12(1l+ r1 ·σ)⊗ 1

2(1l+ r2 ·σ) (A.15)

=14(1l⊗1l+ r1,xσx⊗1l+ r1,yσy⊗1l+ . . . r1,zr2,zσz⊗σz).

The states B = {1l⊗1l,1l⊗σx,1l⊗σy,1l⊗σz,σx⊗1l,σx⊗σx,σx⊗σy,σx⊗σz,σy⊗1l,σy⊗σx,σy⊗σy,σy⊗σz,σz⊗1l,σz⊗σx,σz⊗σy,σz⊗σz} form a basis of H tot, but

it is evident that this cannot be the most general form of a mixed state ρ ∈H tot,


since the coefficients in Eq. (A.15) are too restrictive. Here we only have 6 real

parameters18, in contrast to a general state in four-dimensional Hilbert space

having 15 real parameters, since it corresponds to a Hermitian, non-negative

matrix with trace equal to 1. The states acquired by putting ρ = ρi⊗ρ j are only

a special class of mixed states called factorable, which are a subset of the separable

states.

A.3.4 Composed mixed states

General bipartite mixed state In general we can write a composite (bipar-

tite) mixed state as

ρ = ∑k

akρ1⊗ρ2, (A.16)

where ρi denotes the density operator belonging to the subsystem i and ak are

complex coefficients.

Separable states If the coefficients in Eq. (A.16) are real, positive and satisfy

∑k ak = 1, thus if it can be written as

ρ = ∑k

akρ1⊗ρ2, R 3 ak ≥ 0, ∑k

ak = 1, (A.17)

the state is called separable or classically correlated [34]. States belonging to this

class satisfy all possible Boole-Bell-type inequalities. Clearly, if ak = 1 for some k

we get a factorable state as described in (A.3.3).

A.4 Entanglement

Entanglement forms the basic ingredient to some of those puzzling effects ex-

hibited by quantum mechanics, for example Quantum Teleportation [35], Fault-

tolerant Quantum Computation [36], Quantum Cryptography [37], etc.

A.4.1 Pure state entanglement

Generally speaking a pure state is said to be entangled, if and only if it cannot be

written as a tensor product of states of the parts. For example the singlet state

|Ψ〉= 1/√

2(|01〉− |10〉) = 1/√

2(|0〉⊗ |1〉− |1〉⊗ |0〉) cannot be represented by a

state of the form |ϕ〉= (α1|0〉+β1|1〉)⊗(α2|0〉+β2|1〉). Such states are responsible

for producing nonlocal quantum effects like violation of the Bell inequalities or

quantum Teleportation.

18r1 = r(r1,θ1,φ1), r2 = r2(r2,θ2,φ2) with r(θ,φ) = (r cosφsinθ, r sinφsinθ, r cosθ)T


The amount of entanglement E(φ) of a bipartite quantum system A⊗B in the

pure state |φ〉 is defined as

E(φ) =−Tr(ρA log2ρ) =−Tr(ρB log2ρ), (A.18)

where ρ = |φ〉〈φ| and ρA (ρB) denotes the partial trace over system B (A), that is

ρA = TrB[|φ〉〈φ| and similar for ρB. E(φ) ranges from zero for a product state to

log2N for a maximally entangled state of two N-level systems.

By this definition the singlet state |Ψ〉 is maximally entangled, since with

ρA = TrB[|Ψ〉〈Ψ|] (A.19)

=12

TrB[|0〉〈0|⊗ |1〉〈1|− |1〉〈0|⊗ |0〉〈1|− |0〉〈1|⊗ |1〉〈0|+ |1〉〈1|⊗ |0〉〈0|]

=12(|0〉〈0|+ |1〉〈1|)

it follows

E(Ψ) = −Tr[ρA log2ρA] (A.20)

= −12

Tr[log22−1|0〉〈0|+ log22−1|1〉〈1|]= 1

= log22

A.4.2 Mixed state entanglement

A mixed state is said to be entangled if it is not separable, whereas a criterion

of separability can be given by Eq. (A.17), but there are also numerous other

criteria. For example:

• Werner [34] pointed out that separable states must must satisfy all possible

Bell inequalities.

• Peres [38] derived the necessary condition for a state being separable that

the state remains a positive operator when performing partial transposition

The partial transposition of a state ρ is defined as

ρTBmµ,nν ≡ ρmν,nµ (A.21)

with ρmµ,nν = 〈m|⊗ 〈µ|ρ|n〉⊗ |ν〉.

• Bennett et al. introduced in [39] a measure of entanglement E(W) of a

mixed state W called entanglement of formation, which is defined as the

least expected entanglement of any ensemble of pure states realizing W.

B DISPERSION-FREE STATES 65

B Dispersion free states19

Since there are many sources of errors in a “real” measurement process of an

observable X, not even in classical physics the result will always be ν(X) = x, but

there will of course be some deviations from the value x standing for the theoretical

result that can only be achieved under idealized conditions20. A measure of the

uncertainty is conveniently given by the standard deviation

∆X = (〈X2〉−〈X〉2)12 , (B.1)

where 〈X〉 is short for 〈Ψ|X|Ψ〉 denoting the expectation or mean value of the

observable X of a quantal system in state |Ψ〉. The square of this quantity, (∆X)2

is called dispersion or variance.

In classical physics this uncertainty is only due to improper preparation, in-

strumental errors, a. s. o. and it is in principle possible to achieve (∆X)2 → 0 in

the limit of optimal conditions.

In quantum mechanics the dispersion (∆X)2 plays a crucial role: Even if we

would use perfectly reliable instruments, after preparing an ensemble of particles

in exactly the same state the results of identical quantum tests would, in general,

be different. For instance, if a beam of spin-1/2 particles polarized in the z

direction is sent through a Stern-Gerlach magnet oriented along the x direction,

the observed magnetic moment µx of the particles would either be ±µ under

perfect conditions. The dispersion is then (∆µx)2 = 〈µ2x〉−〈µx〉2 = µ2.

If we could further specify the initial state of the spin-1/2 particles by adding

additional variables, so that we can predict the result +µ or −µ with certainty,

the dispersion would be zero. In conventional quantum mechanics the initial state

can be prepared according to a property like“polarized in the x direction”and this

is the best we can do, there is no possibility to prepare the initial state according

to the property “polarized in the positive x direction”. This would consequently

(with a proper definition of positive) determine the result +µ or −µ, and under

idealized experimental conditions for this state (∆µx)2 = 0 would be valid.

Projection measurements If we consider now only observables represented

by projection operators P in Hilbert space21 Eq. (B.1) reduces to

σ(P)≡ (∆P)2 = 〈P〉−〈P〉2, (B.2)

19According to J. Jauch in “Foundations of Quantum Mechanics”[40, p. 114ff.]20“In theory there is no difference between theory and practice. But in practice there is.”

(Jan L.A. van de Snepscheut)21Any other observable can be constructed out of projectors - cf. Section 2.3.

B DISPERSION-FREE STATES 66

due to the relation P2 = P. The expectation value 〈P〉 ranges from 0 to 1, it

follows that the dispersion function has the value of zero for 〈P〉 = 0 or 〈P〉 = 1

and its maximum value σ(P)2 = 1/4 for 〈P〉= 1/2 as shown in Figure 24.

Bearing in mind that 〈P〉 is conveniently interpreted as the probability that

a system in a specific state has the property P, σ(P)2 is an adequate measure of

the degree of uncertainty. If we define also an overall dispersion σ

σ ≡ supP∈H

σ(P), (B.3)

we can conclude that a state is dispersion free, if σ = 0.

Dispersion free states are the basic ingredients of a hidden-variables theory,

because for such states the result of every projection measurement is by definition

0 or 1 and this is in turn required from HV-theories.

1�4

11�20 �

P �

σ

� P

� 2

1�4

11�20 �

P �

σ

� P

� 2

1�4

11�20 �

P �

σ

� P

� 2

Figure 24: Dispersion function

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Bounds on Quantum Probabilities - TU Wientph.tuwien.ac.at/~svozil/publ/boundprob.pdf · EPR-paradox [9] as a ﬁrst incentive to the search for hidden variables, mentioning von Neumann’s