Bounds on Quantum Probabilities - D

Transcription

DIPLOMARBEIT
Bounds on Quantum Probabilities
ausgeführt am
Institut für Theoretische Physik,
Technische Universität 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. Jänner 2003
2
Abstract
Since the emergence of quantum mechanics there have been doubts about its
completeness and thoughts about its possible extensions yielding a deterministic
description of Nature. Bell’s Theorem [12] is the most famous argument against
a (local) hidden-variable theory that would solve problems like “uncertainty” or
“spooky action at a distance” [41, p. 158]. We shall give a short overview about
“no-go” theorems stating that quantum mechanics has to be considered complete
as it is, starting from the EPR-paradox [9] as a first incentive to the search for
hidden variables, mentioning von Neumann’s proof [10] and its refutation and
discussing the Bell-Kochen-Specker Theorem [12, 15] as well as Bell-type inequalities [2].
Furthermore, we shall show a method of generalizing Bell-type inequalities
in terms of classical and quantum correlation polytopes introduced by I. Pitowsky
[6, 8, 32] and apply these concepts to depict the violation of Bell-type inequalities
using arbitrary four-dimensional quantal states. We will depict upper bounds
of violations and investigate into the relation between the degree of mixedness
and the possible violation. Different sets of possible probability values will be
introduced that can be visualized as cuts through quantum correlation polytopes.
CONTENTS
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Contents
1 Introduction
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2 Hidden Variables Theories
2.1 EPR - Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Von Neumann’s “no-go” theorem . . . . . . . . . . . . . . . . . . . .
2.3 Gleason’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Bell’s “continuum” no-go theorem . . . . . . . . . . . . . . . . . . .
2.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Contextuality . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Kochen-Specker Theorem . . . . . . . . . . . . . . . . . . . . . . .
2.6 Peres’ variant of the KS-Theorem . . . . . . . . . . . . . . . . . . .
2.7 Mermin’s variants of the KS-Theorem[11] . . . . . . . . . . . . . . .
2.7.1 Four dimensions . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Eight dimensions . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Implications of the KS-Theorem . . . . . . . . . . . . . . . . . . . .
2.9 Bell’s Theorem - from non-contextuality to locality . . . . . . . . . .
2.9.1 Link between the KS-Theorem and Bell’s Theorem . . . . . .
2.9.2 Proof of Bell’s theorem - with inequalities . . . . . . . . . . .
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3 Correlation Polytopes
3.1 Simple Urn Model . . . . . . . . . . .
3.2 Geometrical interpretation . . . . . . .
3.3 Minkowski-Weyl representation theorem
3.4 From vertices to inequalities . . . . . .
3.5 From inequalities to vertices . . . . . .
3.6 Quantum mechanical context . . . . . .
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4 Quantum Correlation Polytopes
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Classical probabilities - the set c(n) . . . . . . .
4.1.2 Quantum probabilities - the set bell(n) . . . . .
4.1.3 More general quantum probabilities - the set q(n)
4.2 Violation of inequalities . . . . . . . . . . . . . . . . . .
4.2.1 Generation of states . . . . . . . . . . . . . . . .
4.2.2 Measurement operators . . . . . . . . . . . . . .
4.2.3 Clauser-Horne-(CH)-inequality - bell(2) . . . . .
4.2.4 CHSH-inequality . . . . . . . . . . . . . . . . .
4.2.5 Boole-Bell-type inequality out of bell(3) . . . . .
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CONTENTS
4.3
4.4
Bounds of bell(n) . . . . . . . . .
4.3.1 Representation of bell(1) .
4.3.2 Representation of bell(2) .
Generalizing bell(n) - the set q(n)
4.4.1 Example . . . . . . . . .
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5 Summary
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6 Acknowledgments
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A Characterization of states
A.1 Pure vs. mixed states . . . . . . .
A.2 Purification . . . . . . . . . . . .
A.3 Composed systems . . . . . . . .
A.3.1 Pure states . . . . . . . .
A.3.2 Schmidt-Decomposition .
A.3.3 Mixed states . . . . . . .
A.3.4 Composed mixed states . .
A.4 Entanglement . . . . . . . . . . .
A.4.1 Pure state entanglement .
A.4.2 Mixed state entanglement
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B Dispersion-free states
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64
1 INTRODUCTION
5
1 Introduction
“Die immer wieder ausgesprochenen Hoffnungen, die wesentlich statistische Beschreibungsweise der Quantenmechanik durch die Annahme eines
den atomaren Phänomenen unterliegenden, aber unseren bisherigen Beobachtungen unzugänglichen kausalen Mechanismus zu vermeiden, dürften in
der Tat ebenso vergeblich sein wie jede Hoffnung, mit den gewöhnlichenen Vorstellungen von absolutem Raum und absoluter Zeit der durch die
allgemeine Relativitätstheorie 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 theoretical
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 enhancement of quantum theory in terms of a local hidden variables theory, i. e. any theory
consistent with quantum mechanical predictions but based on a deterministic foundation might be non-local, that is to say, not Lorentz-covariant1 .
We will also review the EPR paradox as the most famous ’gedankenexperiment’
pointing out the inconsistencies between quantum mechanics and special relativity.
Furthermore, von Neumann’s “no-go” theorem declining dispersion free states (cf.
Appendix B), which are essential for a deterministic theory, and the Kochen-Specker
Theorem 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
1 There
are notable exceptions - cf. I. Pitosky [3] or D. A. Meyer [4].
1 INTRODUCTION
6
inequalities2) will be adopted, namely to exploit 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 inequalities for arbitrary quantum 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 numerically generated states will be classified into pure states and mixed states, and we will analyze the
dependence of the degree of violation on the degree of mixedness.
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.
2 In the middle of the 19th century the English mathematician George Boole formulated his theory on
"conditions of possible experience" [5]. These conditions are related to relative frequencies 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 |Ψi 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 |Ψi 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 uncertainty
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 difference between
conventional quantum theory and a quantum theory supported by additional hidden
variables? In a hidden variables (HV) theory a state is in principle 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 properties 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 between
8
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 a 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 experiments. Assuming this, it
does not make sense to talk about some metaphysical properties common to a philosophical framework, since these are not measurable 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
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
3 A.
Einstein, in Albert Einstein, Philosopher-Scientist, ed. by P. A. Schilpp, Library of Living
Philosophers, Evanston (1949), p. 682
4 Events localized in relatively space-like space-time regions must be causally independent.
9
√
particles in a singlet state |Ψi = 1/ 2(| ↑↓i−| ↓↑i), where | ↑i denotes the eigenvector
of the spin-operator σz to the eigenvalue +1 and | ↓i 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 |Ψi.
It follows that the observer A could predict with certainty the value of either σ z
or σx of the far-away system. Assuming that there occurs no “spooky action at a distance”, as Einstein put it, the values of σ z 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 arguing 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 simultaneous 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 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
5 EPR
used the incompatible properties position and momentum for their gedankenexperiment, but
the situation given by two spin-1/2 particles in a singlet state - first considered by Bohm - is easier to
handle.
6 For example, position and momentum are not commuting, but also spin measurements in different
non-orthogonal directions like σx and σz
10
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 hidden 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 ∈
(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
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 eigenvalues ν(A), ν(B) and ν(C).
If we consider now the equation C = A + B, the values assigned to the operators in an
7 Grete
Hermann already pointed out a deficiency in the proof in 1935, but has been entirely ignored.
11
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 measurements, 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
hΨ|C|Ψi = hΨ|A|Ψi + hΨ|B|Ψi
(2.3)
is valid for any state |Ψi 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 |Ψi 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
|Ψi is in general not in the same state denoted, for simplicity, by |Ψ, HV i in a hidden
variables theory. The HV can take different values still yielding the same quantum
state, thus many |Ψ, HV i map to the same quantum state |Ψi: |Ψ, λ1 i, |Ψ, λ2 i, . . . →
|Ψi. A sufficient condition for Eq. (2.3) is imposing Eq. (2.2) to each single system in
a state |Ψ, λm i out of the ensemble of systems all in the quantum state |Ψi, 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.
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).
We have an ensemble of quantum systems (namely 4 systems) in the same quantum
state |Ψi. 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
8 [A, B] 6= 0
↔ A and B non-commuting
System QM-State HV-State
1
2
3
4
|Ψi
|Ψi
|Ψi
|Ψi
|Ψ, 1i
|Ψ, 2i
|Ψ, 1i
|Ψ, 2i
12
ν(A) ν(B) ν(C) = ν(A + B)
2
4
2
4
5
7
5
7
4
14
4
14
Table 1: Simple set of data to refute von Neumann’s “no-go” theorem
the identical systems as seen from the quantum mechanical point of view. We can see
that due to
hΨ|C|Ψi = (2 × 4 + 2 × 14)/4 = 9
= (2 × 2 + 2 × 4)/4 + (2 × 5 + 2 × 7)/4
= hΨ|A|Ψi + hΨ|B|Ψi
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 = σ x , and B =
σy as example and C = √12 (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) from 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 has to fulfill Eq. (2.2), if [A, B] = 0, in
contrast 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
hPα i = Tr[PαW ],
(2.4)
where Pα is a projection operator onto a (pure) state |αi, W is a density (statistical)
operator characterizing the actual state of the system, Tr stands for the trace of a matrix
13
and hPi 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 ai Pi . Then from Eq. (2.4) the conventional probability measure hAi =
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 complex
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
hPuv i = hPu i + hPv i,
where hPi ≡ 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 |ψi = | ↑i. The projection operator Pz+ corresponding to the question
“Does the particle’s spin point in the positive z-direction?” reads as P z+ = | ↑ih↑ | and
the “answer” is given by
Tr[Pz+ | ↑ih↑ |] = hψ|Pz+ |ψi = h↑ | ↑ih↑ | ↑i = 1,
i. e. with probability 1 the particle is in the state |ψi. Asking the question whether the
particle’s spin points in the positive x-direction yields a probability of 1/2, since from
Px+ = 1/2(| ↑ih↑ | + | ↑ih↓ | + | ↓ih↑ | + | ↓ih↓ |)
follows
9 This
1
Tr[Px+ |ψihψ|] = hψ|Px+ |ψi = .
2
means, when performing a Stern-Gerlach experiment, which deflects the particles in the positive or negative z-direction, we would see particles only in the beam deflected in the positive z-direction.
In quantum theory the Hilbert space associated with the spin-space does certainly not have a z-direction,
but in this case we can make the correspondence between the two-dimensional Hilbert space for the spin
and 3 .
14
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 Px+ 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 projectors, in other words, we
can assign infinitely many orthonormal bases to the subspace Puv (H ). We could for
example take the Pu = |uihu| and Pv = |vihv| as projection operators on the vectors
|ui, |vi ∈ H with hu|vi = 0. Another pair of orthogonal vectors can then be chosen as
√
|xi = 1/ 2(|ui + |vi),
√
|yi = 1/ 2(|ui − |vi)
(2.5)
with the belonging projectors Px = |xihx| and Py = |yihy|. Trivially
Puv = Pu + Pv = Px + Py
(2.6)
and with the axim (iii) this relation holds also for the expectation values
hPuv i = hPu i + hPv i = hPx i + hPy i.
(2.7)
2.4 Bell’s “continuum” no-go theorem
Bell showed in [12] that choosing two vectors |φi and |ψi in the Hilbert space H such
that for a quantum system in the state W
hPφ i ≡ Tr[|φihφ|W ] = 1,
hPψ i ≡ Tr[|ψihψ|W ] = 0
(2.8)
(2.9)
is only possible under the restriction that |φi and |ψi cannot be arbitrarily close, in fact
||φi − |ψi| ≥ 1/2hψ|ψi1/2 .
(2.10)
Assuming a dispersion free state W as required by a HV theory every projection 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.
15
Consider now a complete set of orthogonal basis vectors |ϕi i ∈ H , then
∑ Pϕi = 1l
(2.11)
∑hPϕi i = 1.
(2.12)
i
and from (iii) follows
i
In a hidden-variables theory the only possible values for hPϕi i are 0 or 1, thus exactly
for one Pϕ j , hPϕ j i = 1 must be valid, whereas for other projectors in Eq. (2.12) hPϕi i
(i 6= j) is equal to zero.
As one can think of infinitely many alternative basis sets |ϕ0i i fulfilling the same
conditions (2.11) and (2.12) as |ϕi i, Bell concluded that there must be arbitrarily close
pairs |φi, |ψi 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 we can describe the Pi as squares of the components of the spin
along various directions, i. e. we have a set of observables {S x2 , Sy2 , Sz2 } with eigenvalues
0 and 1, where the indices x, y, z denote mutually orthogonal vectors |xi, |yi, |zi ∈ H .
For a spin 1 particle
S2 = Sx2 + Sy2 + Sz2 = s(s + 1) = 2
must be valid for every triad of orthogonal vectors, furthermore S x2 , Sy2 , and Sy2 are
mutually commuting, which means that theoretically a simultaneous measurement of
all three observables can be performed and consequently
hSu2 i + hSv2 i + hSw2 i = 2
has to be fulfilled as well.
The resulting problem here is to assign numbers {1, 1, 0} to the triple {S x2 , Sy2 , Sz2 },
which we can convert to a geometrical problem by associating |xi, |yi, |zi with orthogonal rays (= directions) x, y, z in 3 . 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 S x2 , Sy2 , and Sz2
[cf. Figure 1(a), where the thick arrows 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 different colors cannot be arbitrarily
small (due to Bell’s condition - cf. Eq. (2.10)), thus one can intuitively see that this
is not feasible. A recipe how to explicitly reach a contradiction has been given by
Mermin in [11].
16
z
z = z'
y'
z'
x
x
y'
y
x'
x'
y
(a)
(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
{Sx2 , Sy2 , Sz2 } are mutually commuting, but these are not necessary commuting with the
projectors {Sx20 , Sy20 , Sz20 } corresponding to the x0 -y0 -z0 – triad in Figure 1(a).
In the particular case when z = z0 (Sz2 = Sz20 ) depicted in Figure 1(b), we “assume
tacitly” that the measurement of S z2 does not depend on the choice of the rays x0 and
y0 , 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 Sz2 = Sz20 can be measured
simultaneously with Sx2 and Sy2 , or simultaneously with Sx20 and Sy20 and the result of Sz2
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 observables 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 [15] states that in a Hilbert space H with dimensionality dim H ≥ 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 satisfy-
ing ∑i Pi = 1
∑ ν(Pi) = 1
17
with ν(Pi ) = 0, 1
(2.13)
i
is valid. The Kochen-Specker theorem consists of an algebraic proof circumventing 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 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 3 . The vectors used are depicted in Figure 2 and labeled by xyz, where
√
√
¯ −1), 2 (≡ 2), 2¯(≡ − 2}. For example the vector 112¯connects
x, y, z ∈ {0, 1, 1(≡
√
¯
the origin (0, 0, 0) with the point (1, 1, − 2). Opposite vectors, such as 112¯and 1¯12
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
z
2
1
0
1
2
x
y
Figure 2: Construction of Peres’ KS-theorem variant
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
18
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.
Orthogonal triad Other rays
¯
001 100 010 110 110
¯
101 101
010
¯
011 011 100
¯
¯ 112
¯
021
112
110 201
¯
102 2̄01 010 211
¯
211 01̄1 2̄11 102
¯
201 010 1̄02 1¯12
¯
112 11̄0 1̄1̄2 021
¯
012 100 02̄1 121
¯
121 1̄01 12̄1 012
Reason of first ray being blue
arbitrary (choice of z axis)
arbitrary (choice of x vs. −x axis)
arbitrary (choice of y vs. −y axis)
arbitrary (choice of x vs y axis)
orthogonality to 2nd and 3rd ray
Table 2: Proof of Kochen-Specker theorem in three dimensions
¯ are all
By looking at the first, fourth and last line we notice that 100, 021 and 0 12
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.
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 dimensional 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 σ2β ∈ H 2 to measure the spin in an arbitrary
direction α or β, respectively. The upper index denotes the Hilbert space in which the
19
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 σ2β are ±1 and any component of σ1α commutes with any
component of σ2β since they are belonging to another subspace of Hilbert space. Other
relations are that σiα anti-commutes with σiβ and σiα σiβ = iσiγ 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 operators
have to satisfy the same relations as the operators themselves. According 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 ⊗ σ2x
σ1x ⊗ σ2x
1l ⊗ σ2y
σ1y ⊗ 1l
σ1y ⊗ σ2y
σ1x ⊗ σ2y σ1y ⊗ σ2x σ1z ⊗ σ2z
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.
The argumentation runs as follows:
(i) The four operators on each line are again mutually commuting.
20
(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 +1l8 , and therefore in this
case the product is +1l8 . Again the operator values must fulfill the same relations as
the operators, consequently we have a contradiction to the value −1 in (iv).
σ1y
σ1x σ2x σ3x
σ1y σ2y σ3x
σ1y σ2x σ3y
σ3x
σ1x σ2y σ3y
σ3y
σ1x
σ2y
σ2x
Figure 4: Observables for Mermin’s KS-variant
2.8 Implications of the KS-Theorem
From all the arguments above it follows that QM cannot be embedded into a noncontextual hidden-variables theory, because the “tacit” assumption of non-contextuality
runs through all proofs introduced above. The question is then, if 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:
10 Actually there
are of course more subtle escapes from the KS argument not presented in the limited
scope of this work. See for example [16] for further discussion and references.
21
(i) Non-contextuality is a reasonable assumption. But an underlying hidden variables theory can only be a contextual one, thus we conclude that quantum 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 commuting operators {A, B,C} and in another setup we have the operators {A, B0 ,C 0 }, which are again mutually commuting.
These two sets belong to two physically different arrangements of an experimental
setup distinguished for example by different positions of some detectors responsible
for detecting B or B0 , and C or C 0 , respectively. Since one can presume that different
physical arrangements produce different interactions between the measurement apparatus itself and between the 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 mechanics) 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 assumption:
If A is in both setups the first property measured and after measuring A we do some
other tests for B,C or B0 ,C 0 our intuition tells us that the results for A must be the same
for both experiments, since {A, B,C} and {A, B0,C 0 } are mutually compatible and, furthermore, the measurement of A is not influenced by anything that comes afterwards.
Beyond that, even if we allow some kind of reflections from the operations B,C or
B0 ,C 0 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 0 ,C 0 , 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
22
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 between
the observables A , B , and C - or better between the parts of the measurement apparatus
responsible for detecting these properties - A can in general be written as A = A (B , C )
or A = A (B 0 , C 0 ), respectively, and the equality A (B , C ) = A (B 0 , C 0 ) 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 measurement 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 eightdimensional 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].
The other four non-local operators (from the horizontal line in Figure 4) acting on
all three particles are commuting, thus we can prepare a particular system 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 eigenvalue. 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 line 11 . If
11 Take
for example the line from the lower right corner to the top consisting of the one-particle
23
σ̂2y
σ̂2x
σ̂1y
σ̂1x
σ̂3y
σ̂3x
Figure 5: Bell’s Theorem without inequalities
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 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 local) this assumption is justified.
In this case we can mend the broken chain of arguments, which yielded a contradiction 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 1l 8 , 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 changing the
statistical predictions, there must be a mechanism whereby the setting of
one measuring device can influence the reading of another instrument,
however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.” [2]
operators σ2y , σ3x , σ1y . The corresponding element in the horizontal row is in this case σ 1y σ2y σ3x .
24
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 |Ψi = 1/ 2(| ↑↓i−| ↓↑i) moving
freely in opposite directions. For each particle we can perform a measurement 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. Figure 6). In the particular
state |Ψi, if measurement on the first particle of nξ · σ yields 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
y
β
δ
α
γ
x
x
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 J 1 = −J 2 . An
observer can now measure the variable sgn(nξ · J 1 ) 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 · J 2 ) 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
hxi = ∑ x j /N and hyi = ∑ y j /N,
(2.14)
j
j
where N is the total number of repetitions. If the directions of the J i 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 = hxyi = ∑ x j y j /N,
j
(2.15)
25
which does not vanish in general. Take for example nξ = nχ , then each x j y j = −1 and
therefore hxyi = −1. For arbitrary nξ and nχ (i. e. arbitrary measurement directions)
the correlation function can be calculated to
E(ξ, χ)class = hxyi = −1 + 2θ/π,
(2.16)
where θ is the angle between nξ and nχ .
Quantum mechanical correlation Using a singlet state |Ψi the QM correlation
function for the measurement of spin components on two spin-1/2 particles can be
written as
E(ξ, χ)qm = hΨ|nξ · σ ⊗ nχ · σ|Ψi = − cos θ,
(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, and θ = π/2, where
E(ξ, χ) = 0 (cf. Figure 7).
1
0
Eqm
Eclass
-1
0
π
2
π
θ
Figure 7: Correlation functions
Bell’s theorem Suppose now that two observers independently perform measurements on one particle out of the two-particle singlet state each, and each observer can
choose between two directions of measurements. Observer A can select 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
26
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
hac + bc + bd − adi =
=
∑(a j c j + b j c j + b j d j − a j d j )/N
(2.19)
j
1
±2
N∑
j
≤ 2
Splitting up the left hand side of Eq. (2.19) into correlation functions of two properties
we get
haci + hbci + hbdi − hadi ≤ 2,
(2.20)
which is a variant of the original Bell inequality known as the CHSH-inequality.
The correlation functions in Eq. (2.20) can now be calculated in the QM formalism.
Although quantum theory is not able to predict single values a j , b j , c j and d j for each
measurement, the average values can be determined. In the particular case of pairs of
polarized photons the correlation function reads E(ξ, χ) = cos 2(ξ − χ), where ξ and χ
denote the angles of polarization measurements, Equation (2.20) becomes
cos 2(α − γ) + cos 2(β − γ) + cos 2(β − δ) − cos 2(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
1
1
1
√ + √ + √ − √ = 2 2 6≤ 2.
(2.22)
2
2
2
2
Obviously quantum theory violates the CHSH-inequality.
This behavior has already been tested experimentally [18, 19] and it seems that
27
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
28
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 obtain such 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. 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 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
29
“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 is a function of the first components.
a1
0
1
0
1
a2
0
0
1
1
a12
0
0
0
1
Table 3: Truth table for two propositions a1 , a2 and their joint proposition a12 = a1 ∧ a2
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 (p 1 , 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.
(1,1,1)
(0,1,0)
(0,0,0)
(1,0,0)
Figure 8: Polytope associtated with the urn model
30
3.3 Minkowski-Weyl representation theorem
The Minkowski-Weyl representation theorem (e.g., [25, p. 29]) states that compact
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.
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 relevant 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) =
(
)
n
λ1 xi + · · · + λn xn n ≥ 1, λi ≥ 0, ∑ λi = 1 .
(3.3)
i=1
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 (p 1 ,
31
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) = (λ3 +λ4 , λ2 +λ4 , λ4 ).
(3.4)
It is important to realize that these logical possibilities are exhaustive. By definition,
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 coordinates
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 formalism 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 “conditions of possible experience” are violated if one considers probabilities associated with complementary events, thereby
assuming counterfactuality. (This is a development and a generalization Boole could
have hardly forseen!)
As an example we take a source that produces pairs of spin- 12 particles in a singletstate (|ψi = √1 (| ↑↓i − | ↓↑i)). The particles fly apart along the z axis and after the
2
32
y
y
α2
β2
α1
β1
x
b
a
x
Figure 9: Experimental setting to test the violation of Boole - Bell type inequalities
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 2¯h
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” (cf Table 4). The rows of this table
are then identified with the vertices of the convex polytope. By using the MinkowskiWeyl 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
−1 ≤
−1 ≤
−1 ≤
−1 ≤
pa1b1 + pa1b2 + pa2b2 − pa2b1 − pa1 − pb2 ≤ 0,
pa2b2 + pa2b1 + pa1b1 − pa1b2 − pa2 − pb1 ≤ 0.
(3.6)
(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 consider
the singlet state of spin- 12 particles |ψi = √1 (| ↑↓i − | ↓↑i) it is well known that the
2
probability to find the particles both either in spin up or in spin down states is given
α1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
α2
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
β1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
β1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
33
α 1 β1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
α 1 β2
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
α 2 β1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
1
α 2 β2
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
Table 4: Truth table for four propositions
by P↑↑ (θ) = P↓↓ (θ) = 21 sin2 (θ/2) - where θ is the angle between the measurement
directions. The single event probability is clearly pi = 21 . By choosing
a1 = −
π
3
a2 = b 1 =
π
3
b2 = π
(3.8)
as measurement directions, we get for p = (pa1 , pa2 , pb1 , pb2 , pa1b1 , pa2b1 , pa1b2 , pa2b2 ):
1 1 1 1 3 3 3
p = ( , , , , , , 0, )
2 2 2 2 8 8 8
(3.9)
and one of the inequalities found in (3.7) is violated:
pa1b1 + pa1b2 + pa2b2 − pa2b1 − pa1 − pb2 =
1 1 1
3 3 3
+ + − 0 − − = > 0 (3.10)
8 8 8
2 2 8
The generalization is straightforward. Violations of certain inequalities involving
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
34
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[W Ei ]
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 properties:
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 vectors
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
35
to say, the probability to draw a yellow ball out of the urn is given by p 01 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 prescriptions 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 properties:
p00 = 1 and there exist a finite dimensional Hilbert space H , projections E 1 , 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 measurements
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 parameters
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).
36
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 properties: 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[W Ei ], p0 j = Tr[W Fj ] 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 E i 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 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 E i (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 A 1 ) 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.
12 Since
the difference between q(n) and its closure q̄(n) is not relevant for our discussion we will
treat q(n) and q̄(n) as equivalent.
37
(1,1,1)
(1,1,1)
(0,1,0)
(0,1,0)
→
(0,0,0)
(1,1,0)
(0,0,0)
(1,0,0)
(1,0,0)
(a) c(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 dimensionality
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 pictured 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.
qn
n
el l
b
cn
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 |ψi = 1/ 2(| ↑↓i − | ↓↑i), where | ↑i is an eigenstate of the
13 dim c(2)
= 8, dim c(3) = 15, dimc(4) = 24,. . . , dimc(n) = n 2 + 2n, where dimc(n) denotes the
dimensionality of the polytope associated to c(n) and dim c(n) = dimq(n).
38
Pauli matrix σz to the eigenvalue +1 and | ↓i 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 (equivalent to quantum states) that are Hermitian (W = W † ), positive (semi-) definite (hϕ|W |ϕi ≥ 0, ∀ |ϕi ∈ H ) and normalized
(Tr[W ] = 1).
4.2.1 Generation of states
States are generated according to two different mechanisms both 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 0 = B2
that is positive definite and to ensure that the desired density matrix W is normalized
we only have to divide W 0 by its trace, yielding W = W 0 / Tr[W 0 ].
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
|Ψi = α|00i + β|10i + γ|01i + δ|11i,
14 Thus
α, β, γ, δ ∈ ,
we are considering bipartite systems consisting of two particles
(4.3)
39
which transforms to the associated density matrix
0
W|Ψi



= |ΨihΨ| = 

αα?
βα?
γα?
δα?
αβ?
ββ?
γβ?
δβ?
αγ?
βγ?
γγ?
δγ?
αδ?
βδ?
γδ?
δδ?



.

(4.4)
0 / Tr[W 0 ] in its
Dividing by its trace we get the normalized pure state W|Ψi = W|Ψi
|Ψi
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
1
1
M(θ, φ) = (1l ± n · σ) =
2
2
±
1 ± cos θ sin θe−iφ
sin θeiφ 1 ∓ cos θ
!
,
(4.5)
where the unit vector n = (cos θ, sin θ cos φ, sin θ sin φ)T points 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, thus φ = 0, and only
spin-up measurements are made; consequently M(θ, φ)± reads as
1
1
M(θ, φ)± → M(θ)+ = (1l + n · σ) =
2
2
1 + cos θ
sin θ
sin θ
1 − cos θ
!
(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 zdirection. 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 F j . 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
40
operations only on one subspace and the other one is unaffected.
y
y
E3
E2
F3
θ3
θ3
θ2
θ2
E1
θ1
F2
F1
θ1
x
x
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 measurements 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

1

=  Tr[W (E1 ⊗ 1l)]
Tr[W (E2 ⊗ 1l)]



(4.8)

Tr[W (1l ⊗ F1 )] Tr[W (1l ⊗ F2 )]

Tr[W (E1 ⊗ F1 )] Tr[W (E1 ⊗ F2 )]  .
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 E 1 = 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.
41
E1
θ
E2
F1
2θ
F2
Figure 13: E1 = M(0)+ , E2 = F1 = M(θ)+ , F2 = M(2θ)+
0.2
0
-0.2
CH pi j
-0.4
Classical Bounds
Maximum Values
Singlet state
-0.6
-0.8
-1
-1.2
0
0.5
1
1.5
2
2.5
3
θ
Figure 14: CH-inequality
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, 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 |Ψi = 1/ 2(| ↑↓i − | ↓↑i)
is not optimal in the sense that by using another state Wmax we could achieve a better
violation.
15 The
probability to achieve a maximal violation is higher for pure states.
42
Two-parameters plot If we set the measurement directions according to
or
E1 = M(0)+ , E2 = M(θ1 )+ ,
F1 = M(θ2 )+ , F2 = M(θ1 + θ2 )+
(4.9)
E1 = 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 .
E1
E1
E2
θ1
θ1
F2
θ2
θ2
θ1
E2
θ1
F1
θ2
F1
F2
(a) E1 -E2 -F1 -F2 : 0-θ1 -θ2 -(θ1 + θ2 ) (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 generating
(pure) states W as described in Section 4.2.1 we do not always (for any θ 1 , θ2 ) find
exactly the state yielding a maximal violation. Nevertheless the rough contours of
violations of CH(pi j ) can be seen.
When reading off the values along the diagonal line in the graphs given by θ 1 = θ2
we get back to the one-parameter plot (cf. Figure 13).
Dependence on “mixedness” We will now investigate if the violation is dependent
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 maximum 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[W 2 ],
(4.11)
2 is valid and therefore m(W ) = Tr[W 2 ] =
since for a pure state W|Ψi , W|Ψi = W|Ψi
|Ψi
|Ψi
43
3.0
3.0
2.5
2.5
2.0
2.0
Violation:
1.5
0.00
0.05
0.10
0.15
0.20
1.0
0.5
θ2
θ2
1.5
1.0
0.5
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
θ1
θ1
(a) E1 -E2 -F1 -F2: 0-θ1 -θ2 -(θ1 + θ2 )
(b) E1 -E2 -F1 -F2 : 0-θ1 -θ1 -(θ1 + θ2 )
Figure 16: Two-parameter plots of violation of the CH-inequality
Tr[W|Ψi ] = 1. For a maximally mixed state Wmm = 41 1l ∈ H with dim(H ) = 4 the
2 ] = 1/4.
mixedness is given by m(Wmm ) = Tr[Wmm
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 containing
maximally mixed states, X0.3≤m<0.6 the set of states fulfilling 0.3 ≤ m(W ) = Tr[W 2 ] ≤
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.
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 E i
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].
44
0.2
0
-0.2
CH pi j
-0.4
-0.6
-0.8
-1
-1.2
0
0.5
1
1.5
2
2.5
3
θ
X0 6
Xm
1
m 09
X0 3
X0 25
m 06
m 03
Class. Bounds
Figure 17: Dependence on “mixedness”
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 CHSHinequality 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 observables 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
45
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 = σ2β = σ2γ = σ2δ = 1l
and, furthermore, they fulfill the commutator relations
[σα , σγ ] = [σα , σδ ] = [σβ , σγ ] = [σβ , σδ ] = 0.
(4.13)
We can define the CHSH-operator
C = σ α σγ + σ β σγ + σ β σδ − σ α σδ ,
(4.14)
which has the same structure as the correlation functions on the left hand side of Equation (4.12). Squaring C yields
C 2 = 4 + [σα , σβ ][σγ , σδ ].
(4.15)
Using the following identities which are valid for any two bounded QM operators A
and B
k[A, B]k ≤ kABk + kBAk ≤ 2kAkkBk
(4.16)
we get kC 2 k ≤ 8 and therefore
√
kCk ≤ 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 |Ψi = 1/ 2(| ↑↓i − | ↓↑i) 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.
46
4
3
2
CHSH α β γ δ
1
0
Classical Bounds
Singlet state
Maximum Values
-1
-2
-3
-4
0
0.5
1
1.5
2
2.5
3
θ
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 spin1/2 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
16 Apart
from the trivial setup with Ei = Fj , ∀i, j, i. e. θ = 0.
47
0.5
0
PIT(θ)
-0.5
Classical Bound
Maximum Values
Singlet state
-1
-1.5
-2
-2.5
-3
0
0.5
1.5
1
2
2.5
3
θ
Figure 19: Pitwosky-Svozil inequality
describes the setup where consecutive measurement directions are orthogonal (E 1 ⊥
E2 ⊥ E3 and F1 ⊥ F2 ⊥ F3 ): Consider the operator PS associated 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 {| ↑↑i, | ↑↓i, | ↓↑i, | ↓↓i}):



PS = 

−3
0
0
0
0
0
0
0
0 0
0 0
0 0
0 −1



 = −3| ↑↑ih↑↑ | − 1| ↓↓ih↓↓ |

(4.21)
Obviously the eigenvalues are λ1 = −3, λ2,3 = 0, λ4 = −1 and from the Hermiticity
of PS and the negativity of all eigenvalues, λi ≤ 0, it follows that PS is negative semidefinite, thus hϕ|PS|ϕi ≤ 0, ∀|ϕi. Consequently the Pitowsky-Svozil inequality in
Eq. (4.18) cannot be violated by any state |ϕi ∈ 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
48
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 b i 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 conditions 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 calculate
49
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 p 11 = 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 tolerance value ε.
Since there are no points outside the region limited by classical inequalities apart from
deviations due to ε,
c(1) = bell(1).
1
H1,1,1L
bell(1)
0.8
p12
0.6
E
0.4
0.2
H0,0,0L
0
0
0.2
0.4
0.6
0.8
1
p02
Figure 20: bell(1) (c = 3/8, ε = ±0.015)
B
H1,0,0L
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).
50
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 discussed
in Section (3.6), the associated inequalities, i. e. bounds of classical probabilities, read
as follows:
0 ≤ pi j ≤ pi0 ≤ 1, 0 ≤ pi j ≤ p0 j ≤ 1, i = 1, 2,
(4.24)
pi0 + p0 j − pi j ≤ 1,
i = 1, 2,
−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 visualize
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
51
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].
0.5
0.4
p12
0.3
0.2
0.1
0
violation
-0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p02
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 p i 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.
52
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 β on 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 = |ΨihΨ|, |Ψi = 1/ 2(| ↑↑i + | ↓↓i) and for the particular case α = β we
get p10 = p01 = 1. For α 6= β we can certainly find a similar state |Ψi bearing in mind
that any state can be written in its Schmidt-basis as |Ψi = ∑i λi |i1 i|i2 i (Section A.3.2)
with orthonormal basis vectors |i1 i and |i2 i.
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 p 11 = 0 is
in fact absurd.
Hence, we run into problems when attempting to produce matrices p i 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 dimensional 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 p i j 6∈ c(1):
A system in the singlet state |φi can be described in terms of a general pure state
|Ψi = α| ↑↑i + β| ↑↓i + γ| ↓↑i + δ| ↓↓i,
as
α, β, γ, δ ∈
√
|φi = |Ψiα=δ=0, β=1/√2, γ=−1/√2 = 1/ 2(| ↑↓i − | ↓↑i).
(4.28)
(4.29)
Since a projection operator does not differ formally from a pure state, we shall take
53
projection operators into consideration that have the same form, namely
Pi = |ΨihΨ|,
q
q
with α = δ = 0, β = xi / |xi |2 + |yi |2 , γ = yi / |xi |2 + |yi |2 , (4.30)
p
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 ihϕ1 |,
P2 = |ϕ2 ihϕ2 |,
q
|ϕ1 i = 1/ |s|2 + |t|2 (s | ↑↓i + t | ↓↑i)
q
|ϕ2 i = 1/ |s|2 + |t|2 (t | ↑↓i + s | ↓↑i).
(4.31)
(4.32)
Now, p10 = Tr[P1 |φihφ|] and p01 = Tr[P2 |φihφ|]. Furthermore, the joint probability
p11 of P1 ∧ P2 is the projection on the subspace intersection P1 (H ) ∩ P2 (H ), which can
be calculated by
p11 = Tr[ lim (P1 P2 )n |φihφ|].
(4.33)
n→∞
Eq. (4.33) is valid also for non-commuting projection operators, for commuting Pi ’s it
reduces to p11 = Tr[(P1 P2 )|φihφ|].
Using this parameterization, we can calculate numerically projection operators
with s,t ∈ [−1, 1] and approximating the limes limn→∞ (P1 P2 )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 the value 2 is apparently the upper bound (cf. Figure 23).
2
1.5
1
1
0.5
0
-1
0.5
0
t
-0.5
-0.5
0
s
0.5
1 -1
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).
54
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
55
5 Summary
Implications of the “no-go” theorems introduced In the discussion about the completeness of quantum theory we considered a eventually possible enhancement by so
called hidden variables. The problem arising from the formalism of QM is the inherent indeterminism that has a tremendous impact on philosophical 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 framework for the derivation of Boole-Bell-type inequalities in terms of correlation polytopes. Due to the WeylMinkowsky theorem there is a one-to-one correspondence 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 configuration and some of them are violated by quantum
mechanical proposition calculus. The vertices represent maximal probability values of
propositions and their conjunction.
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
56
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 conjunction of joint events.
6 ACKNOWLEDGMENTS
57
6 Acknowledgments
I would like to say thanks to my supervisor Karl Svozil for providing the opportunity
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 without 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
58
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
(A.1)
|ψ(t)i = ∑ ai (t)|ϕii, ai (t) ∈ C ,
i
where the |ϕi i are the (orthonormal) basis vectors of H . The time-evolution is given by
the Schrödinger Equation i¯hdtd |ψ(t)i = H|ψ(t)i 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 cannot
neglect the influences of the environment, or we cannot prepare a state properly, so
we have to use the concept of mixed states. When considering the preparation 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 |ψi 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 = ∑ pi |ψi ihψi |,
∑ pi = 1,
i
i
0 ≤ pi ≤ 1.
(A.2)
This state represented by the so called density matrix (or density operator) ρ is diagonal
in the basis |ψi ihψi |.
In general a mixed state is represented by a matrix ρ that is Hermitian (ρ = ρ † ),
positive definite (hψ|ρ|ψi ≥ 0 ∀|ψi ∈ H ) and Tr[ρ] = 1, thus a general form of a mixed
state is
ρ = ∑ pi j |ψi ihψ j |,
(A.3)
i, j
that can always be diagonalised (since ρ is Hermitian) by a unitary matrix, i. e. transformed to Eq. (A.2).
Clearly if ρ = |ψk ihψk | (pi6=k = 0, pk = 1), ρ represents a pure state and ρ is a
projection operator. By definition
ρ = ρ2
iff ρ = |ψk ihψk |,
(A.4)
which is equivalent to the statement that ρ has one eigenvalue equal to 1, the others are
59
0 and equivalent to
Tr[ρ] = Tr[ρ2 ].
(A.5)
If Tr[ρ2 ] 6= 1 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[ρ 2 ], 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 i ∈ 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 ihΨsa |] (or ρa = Trs [|Ψsa ihΨsa |])
(A.7)
Conversely, we can lift a density operator ρ ∈ H s to a pure state |Ψsa i ∈ 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 particle 17 , 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)
17
In fact by definition a mixed state cannot be only one particle, because it can be described as a
weighted sum over pure states, which occur with a certain probability.
60
The dimensionality of H tot is given by
dim H tot = dim H 1 dim H 2 . . . dim H n .
(A.9)
Take for instance two spin-1/2 particles each “living” in its two-dimensional Hilbert
space H A or H B , respectively. According to Eq. (A.9) the dimension of H tot = H A ⊗
H B is therefore dim H 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 i = α| ↑i + β| ↓i,
α, β ∈ C
(A.10)
where | ↑i, | ↓i are the eigenstates of the usual σz spin operator to the eigenvalues ±1.
The notation | ↑i, | ↓i 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 | ↑i ≡ |0i, | ↓i ≡ |1i, 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 |Ψi ∈ H tot describing this two-particle system is given by
|Ψi = |ϕ1 i ⊗ |ϕ2 i
(A.11)
= (α1 |0i + β1 |1i) ⊗ (α2 |0i + β2 |1i
(A.12)
= α1 α2 |0i ⊗ |0i + α1 β2 |0i ⊗ |1i + β1 α2 |1i ⊗ |0i + β1 β2 |1i ⊗ |1i
=
∑ ci j |ii ⊗ | ji,
i, j = 0, 1, ci j ∈ C.
i, j
When we consider only normalized states (hΨ|Ψi = 1), the coefficients c i j have to
fulfill ∑i, j c2i, j = 1.
A.3.2 Schmidt-Decomposition
The |ii ⊗ | ji = |ii| ji = |i ji, 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
|Ψi = ∑ λi |i1 i|i2 i,
i
61
where λi (Schmidt coefficients) are non-negative real numbers satisfying ∑i λ2i = 1 and
|i1 i, |i2 i 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 | ji and |ki as orthonormal bases for the systems A and B, respectively, |ψi
can be written as
|Ψi = ∑ a jk | ji|ki, a jk ∈ C.
jk
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
|ψi = ∑ u ji dii vik | ji|ki
i jk
and when we define new basis states as |iA i ≡ ∑ j u ji | ji and |iB i ≡ ∑k vik |ki and λi ≡ dii ,
we get
(A.13)
|Ψi = ∑ λi |iA i|iB i.
i
Since u (v) is unitary and due to the orthonormality of | ji (|ki) the |iA i (|iB i) 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 |Ψi.
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 ρ i can be parameterized by ρi = 1/2(1l + ri · σ) with ri denoting a unit vector in R3 and the usual Pauli
matrices σ = (σx , σy , σz )T , therefore we can write
1
1
(1l + r1 · σ) ⊗ (1l + r2 · σ)
2
2
1
(1l ⊗ 1l + r1,xσx ⊗ 1l + r1,y σy ⊗ 1l + . . .r1,z r2,z σz ⊗ σz ).
=
4
ρ =
(A.15)
62
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 the expression in (A.14) 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
state as
In general we can write a composite (bipartite) mixed
ρ = ∑ a k ρ1 ⊗ ρ 2 ,
(A.16)
k
where ρi denotes the density operator belonging to the subsystem i and a k 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 = ∑ a k ρ1 ⊗ ρ 2 ,
k
3 ak ≥ 0,
∑ ak = 1,
(A.17)
k
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 exhibited 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 |Ψi =
√
√
1/ 2(|01i − |10i) = 1/ 2(|0i ⊗ |1i − |1i ⊗ |0i) cannot be represented by a state of
18 r
1
= r(r1 , θ1 , φ1 ), r2 = r2 (r2 , θ2 , φ2 ) with r(θ, φ) = (r cos φ sin θ, r sin φ sin θ, r cos θ)T
63
the form |ϕi = (α1 |0i + β1 |1i) ⊗ (α2 |0i + β2 |1i). Such states are responsible for producing nonlocal quantum effects like violation of the Bell inequalities or quantum
Teleportation.
The amount of entanglement E(φ) of a bipartite quantum system A ⊗ B in the pure
state |φi is defined as
E(φ) = − Tr(ρA log2 ρ) = − Tr(ρB log2 ρ),
(A.18)
where ρ = |φihφ| and ρA (ρB ) denotes the partial trace over system B (A), that is ρA =
TrB [|φihφ| and similar for ρB . E(φ) ranges from zero for a product state to log2 N for a
maximally entangled state of two N-level systems.
By this definition the singlet state |Ψi is maximally entangled, since with
ρA = TrB [|ΨihΨ|]
(A.19)
1
=
TrB [|0ih0| ⊗ |1ih1| − |1ih0| ⊗ |0ih1| − |0ih1| ⊗ |1ih0| + |1ih1| ⊗ |0ih0|]
2
1
(|0ih0| + |1ih1|)
=
2
it follows
E(Ψ) = − Tr[ρA log2 ρA ]
1
= − Tr[log2 2−1 |0ih0| + log2 2−1 |1ih1|]
2
= 1
(A.20)
= log2 2
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
B
≡ ρmν,nµ
ρTmµ,nν
with ρmµ,nν = hm| ⊗ hµ|ρ|ni ⊗ |νi.
(A.21)
B DISPERSION-FREE STATES
64
• 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 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
1
∆X = (hX 2 i − hX i2 ) 2 ,
(B.1)
where hX i is short for hΨ|X |Ψi denoting the expectation or mean value of the observable X of a quantal system in state |Ψi. The square of this quantity, (∆X ) 2 is called
dispersion or variance.
In classical physics this uncertainty is only due to improper preparation, instrumental 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 or −µ under perfect conditions. The dispersion
is then (∆µx )2 = hµ2x i − hµx i2 = 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.
19 According
to J. Jauch in “Foundations of Quantum Mechanics”[40, p. 114ff.]
theory there is no difference between theory and practice. But in practice there is.” (Jan L.A.
van de Snepscheut)
20 “In
B DISPERSION-FREE STATES
65
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 = hPi − hPi2 ,
(B.2)
due to the relation P2 = P. The expectation value hPi ranges from 0 to 1, it follows that
the dispersion function has the value of zero for hPi = 0 or hPi = 1 and its maximum
value σ(P)2 = 1/4 for hPi = 1/2 as shown in Figure 24.
Bearing in mind that hPi 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 ≡ sup σ(P),
(B.3)
P∈H
2
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.
σP
1 4
0
1 2
1
P
Figure 24: Dispersion function
21 Any
other observable can be constructed out of projectors - cf. Section 2.3.
REFERENCES
66
References
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Kopenhagen 1936, chapter Kausalität und Komplementarität. F. Meiner Verlag
Leipzig, Levin & Munksgaard Verlag Kopenhagen, 1937.
[2] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1964.
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