Bell's theorem

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In theoretical physics,
Bell's theorem (AKA Bell's inequality) is a no-go
theorem, loosely stating that:

No physical theory of local hidden
variables can ever reproduce all of the predictions of quantum
mechanics.

It is the most famous legacy of the late physicist John S. Bell.

Bell's theorem has important implications for physics and the philosophy of
science as it indicates that every quantum theory must violate either locality or counterfactual
definiteness. In conjunction with the experiments verifying
the quantum mechanical predictions of Bell-type systems, Bell's theorem
demonstrates that certain quantum effects travel faster than
light and therefore restricts the class of tenable hidden variable theories
to the nonlocal variety.

Contents

1 Overview

2 Importance of the
theorem

3 Bell
inequalities

3.1 Original Bell's
inequality

3.2 CHSH
inequality

4 Bell inequalities are
violated by quantum mechanical predictions

5 Practical experiments
testing Bell's theorem

5.1 Two classes of Bell
inequalities

5.2 Practical
challenges

6 Theoretical
challenges

7 Final
remarks

8 See also

9 Notes

10 References

11 Further
reading

12 External
links

[edit]
Overview

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for verification.
Please
and removed.
(July 2010)

Bell’s theorem shows that the local hidden
variable interpretation of quantum mechanics,
also known as local realism, necessitates certain conditions which are violated
by measurements performed on entangled quantum
systems. A variety of physical experiments have verified the quantum mechanical
predictions, thus ruling out the local hidden variable theories while also
proving the existence of superluminal effects. The
theorem has several variations, but the most common involve systems of 2 Spin-½
particles (or equivalently, any pair of 2-state quantum systems or qubits), which were used in the
original proof and will be the primary focus of this article.

Illustration of Bell test for spin 1/2 particles.
Source produces spin singlet pair, one particle sent to Alice another to Bob.
Each performs one of the two spin measurements.

As in the Einstein–Podolsky–Rosen (EPR)
paradox, Bell considered an experiment in which a source produces pairs of
correlated particles. For
example, a pair of particles may be produced in a Bell state so that if the
spins are measured along the same axes they are certain to produce identical
results. The particles are then sent to two distant observers: Alice
and Bob. In each trial, both of the observers independently chooses to measure
the spin of their respective
particle along a particular axis, and each measurement yields a result of either
spin-up (+1) or spin-down (-1). Whether or not Alice and Bob obtain the same
result depends on the relationship between the orientations of the two spin
measurements, and in general is subject to some uncertainty. The classical
incarnation of Bell's theorem is derived from the statistical properties
observed over many runs of this experiment.

Mathematically the correlation between results is represented by their
product (thus taking on values of ±1 for a single run). While measuring the spin of
these entangled particles along the same axis always results in identical
(perfectly correlated) results, measurements along perpendicular directions have
a 50% chance of matching (uncorrelated). These basic cases are illustrated in
the table to the right.

Same axis:

pair 1

pair 2

pair 3

pair 4

... pair n

Alice, 0°:

+

−

−

+

...

Bob, 0°:

+

−

−

+

...

Correlation: (

+1

+1

+1

+1

... ) / n = +1

(100% identical)

Orthogonal axes:

pair 1

pair 2

pair 3

pair 4

... pair n

Alice, 0°:

+

−

+

−

...

Bob, 90°:

−

−

+

+

...

Correlation: (

−1

+1

+1

−1

... ) / n = 0

(50% identical)

In general, the measurements can be oriented at arbitrary angles, and the
quantum mechanical expectation of the correlation is equal to the cosine of the
angle.

Bell achieved his breakthrough by first assuming that a theory of local hidden
variables could reproduce these results. Without making any assumptions
about the specific form of the theory beyond basic consistency requirements, he
was able to derive an inequality that was clearly at odds with the result
described above, which is both predicted by quantum mechanics and observed
experimentally. Thus, Bell's theorem ruled out the idea of local realism as a
viable interpretation of quantum mechanics, though it still leaves the door open
for non-local realism.

Over the years, Bell's theorem has undergone a wide variety of experimental
tests. Two possible loopholes in the original argument have been proposed, the
detection loophole[1]
and the communication loophole[1],
each prompting a new round of experiments that re-verified the integrity of the
result[1].
To date, Bell's theorem is supported by an overwhelming body of evidence and is
treated as a fundamental principle of physics in mainstream quantum mechanics
textbooks[2]
[3].
Still, no principle of physics can ever be absolutely beyond question, and there
are some people who still do not accept the theorem's validity[4].

[edit]
Importance of the
theorem

Bell's theorem, derived in his seminal 1964 paper titled On the
Einstein Podolsky Rosen paradox,[5]
has been called "the most profound in science".[6]
The title of the article refers to the famous paper by Einstein, Podolsky and
Rosen[7]
purporting to prove the incompleteness of quantum mechanics. In his paper, Bell
started from essentially the same assumptions as did EPR, viz. i) reality
(microscopic objects have real properties determining the outcomes of quantum
mechanical measurements) and ii) locality (reality is not influenced by
measurements simultaneously performed at a large distance). Bell was able to
derive from these assumptions an important result, viz. Bell's
inequality, violation of which by quantum mechanics implying that at least
one of the assumptions must be abandoned if experiment would turn out to satisfy
quantum mechanics.

In two respects Bell's 1964 paper was a big step forward compared to the EPR
paper: i) it considered more general hidden variables
than the elements of physical reality
of the EPR paper, ii) more importantly, Bell's inequality was liable to be
experimentally tested, thus yielding the opportunity to lift the discussion on
the completeness of quantum mechanics from metaphysics to physics. Whereas
Bell's 1964 paper deals only with deterministic hidden variables theories,
Bell's theorem was later generalized to stochastic theories[8]
as well, and it was realized[9]
that the theorem can even be proven without introducing hidden variables.

After EPR (Einstein–Podolsky–Rosen), quantum mechanics was left in an
unsatisfactory position: either it was incomplete, in the sense that it failed
to account for some elements of physical reality, or it violated the principle
of finite propagation speed of physical effects. In a modified version of the
EPR thought experiment, two observers, now commonly
referred to as Alice and Bob, perform independent measurements of
spin on a pair of electrons, prepared at a source in a special state called a
spin singlet state. It
was equivalent to the conclusion of EPR that once Alice measured spin in one
direction (e.g., on the x axis), Bob's measurement in that direction was
determined with certainty, with opposite outcome to that of Alice, whereas
immediately before Alice's measurement, Bob's outcome was only statistically
determined. Thus, either the spin in each direction is an element of physical
reality, or the effects travel from Alice to Bob instantly.

In QM, predictions were formulated in terms of probabilities — for example,
the probability that an electron might be detected in a
particular region of space, or the probability that it would have spin up or
down. The idea persisted, however, that the electron in fact has a
definite position and spin, and that QM's weakness was its inability to
predict those values precisely. The possibility remained that some yet unknown,
but more powerful theory, such as a hidden variables theory, might be
able to predict those quantities exactly, while at the same time also being in
complete agreement with the probabilistic answers given by QM. If a hidden
variables theory were correct, the hidden variables were not described by
QM, and thus QM would be an incomplete theory.

The desire for a local realist theory was based on two
assumptions:

Objects have a definite state that determines the values of all other
measurable properties, such as position and momentum.

Effects of local actions, such as measurements,
cannot travel faster than the speed of light (as a result of special
relativity). If the observers are sufficiently far apart, a measurement
taken by one has no effect on the measurement taken by the other.

In the formalization of local realism used by Bell, the predictions of theory
result from the application of classical probability theory to an underlying
parameter space. By a simple argument based on classical probability, he then
showed that correlations between measurements
are bounded in a way that is violated by QM.

Bell's theorem seemed to put an end to local realist hopes for QM. Per Bell's
theorem, either quantum mechanics or local realism is wrong. Experiments were
needed to determine which is correct, but it took many years and improvements in
technology to perform them.

Bell test
experiments have been interpreted as showing that Bell inequalities are
violated, in favor of QM. The no-communication
theorem proves that the observers cannot use the inequality violations to
communicate information to each other faster than the speed of
light. But the ‘fair sampling’ and ‘no enhancement’ assumptions require more
careful consideration (below).

John Bell's paper examines both John von Neumann's 1932
proof of the incompatibility of hidden variables with QM and the seminal paper
on the subject by Albert Einstein and his
colleagues.

[edit]
Bell inequalities

Bell inequalities concern measurements made by observers on pairs of
particles that have interacted and then separated. According to quantum mechanics they
are entangled while
local realism limits the correlation of subsequent measurements of the
particles. Different authors subsequently derived inequalities similar to Bell´s
original inequality, collectively termed Bell inequalities. All Bell
inequalities describe experiments in which the predicted result assuming entanglement
differs from that following from local realism. The
inequalities assume that each quantum-level object has a well defined state that
accounts for all its measurable properties and that distant objects do not
exchange information faster than the speed of light. These well defined states
are often called hidden
variables, the properties that Einstein posited when he stated his
famous objection to quantum mechanics: "God does not play dice."

Bell showed that under quantum mechanics, which lacks local hidden variables,
the inequalities may be violated. Instead, properties of a particle are not
clear to verify in quantum mechanics but may be correlated with those of another
particle due to quantum
entanglement, allowing their state to be well defined only after a
measurement is made on either particle. That restriction agrees with the Heisenberg
uncertainty principle, a fundamental and inescapable concept in quantum
mechanics.

In Bell's work:

Theoretical physicists live in a classical world, looking out into a
quantum-mechanical world. The latter we describe only subjectively, in terms
of procedures and results in our classical domain. (...) Now nobody knows just
where the boundary between the classical and the quantum domain is situated.
(...) More plausible to me is that we will find that there is no boundary. The
wave functions would prove to be a provisional or incomplete description of
the quantum-mechanical part. It is this possibility, of a homogeneous account
of the world, which is for me the chief motivation of the study of the
so-called "hidden variable" possibility.

(...) A second motivation is connected with the statistical character of
quantum-mechanical predictions. Once the incompleteness of the wave function
description is suspected, it can be conjectured that random statistical
fluctuations are determined by the extra "hidden" variables — "hidden" because
at this stage we can only conjecture their existence and certainly cannot
control them.

(...) A third motivation is in the peculiar character of some
quantum-mechanical predictions, which seem almost to cry out for a hidden
variable interpretation. This is the famous argument of Einstein, Podolsky and
Rosen. (...) We will find, in fact, that no local deterministic
hidden-variable theory can reproduce all the experimental predictions of
quantum mechanics. This opens the possibility of bringing the question into
the experimental domain, by trying to approximate as well as possible the
idealized situations in which local hidden variables and quantum mechanics
cannot agree

In probability theory,
repeated measurements
of system properties can be regarded as repeated sampling of random variables. In
Bell's experiment, Alice can choose a detector setting to measure either A(a) or A(a') and Bob can choose a detector setting
to measure either B(b) or B(b'). Measurements of Alice and Bob may be
somehow correlated with each other, but the Bell inequalities say that if the
correlation stems from local random variables, there is a limit to the amount of
correlation one might expect to see.

[edit]
Original Bell's
inequality

The original inequality that Bell derived was:[5]

where C is the "correlation" of the particle pairs and a,
b and c settings of the apparatus. This inequality is not used in
practice. For one thing, it is true only for genuinely "two-outcome" systems,
not for the "three-outcome" ones (with possible outcomes of zero as well as +1
and −1) encountered in real experiments. For another, it applies only to a very
restricted set of hidden variable theories, namely those for which the outcomes
on both sides of the experiment are always exactly anticorrelated when the
analysers are parallel, in agreement with the quantum mechanical prediction.

There is a simple limit of Bell's inequality which has the virtue of being
completely intuitive. If the result of three different statistical coin-flips A,
B, and C have the property that:

A and B are the same (both heads or both tails) 99% of the time

B and C are the same 99% of the time

then A and C are the same at least 98% of the time. The number of mismatches
between A and B (1/100) plus the number of mismatches between B and C (1/100)
are together the maximum possible number of mismatches between A and C.

In quantum mechanics, by letting A, B, and C be the values of the spin of two
entangled particles measured relative to some axis at 0 degrees, θ
degrees, and 2θ degrees respectively, the overlap of the wavefunction
between the different angles is proportional to .
The probability that A and B give the same answer is ,
where
is proportional to θ. This is also the probability that B and C give the
same answer. But A and C are the same 1 − (2ε)2 of
the time. Choosing the angle so that ε = 0.1, A and B
are 99% correlated, B and C are 99% correlated and A and C are only 96%
correlated.

Imagine that two entangled particles in a spin singlet are shot out to two
distant locations, and the spins of both are measured in the direction A. The
spins are 100% correlated (actually, anti-correlated but for this argument that
is equivalent). The same is true if both spins are measured in directions B or
C. It is safe to conclude that any hidden variables which determine the A,B, and
C measurements in the two particles are 100% correlated and can be used
interchangeably.

If A is measured on one particle and B on the other, the correlation between
them is 99%. If B is measured on one and C on the other, the correlation is 99%.
This allows us to conclude that the hidden variables determining A and B are 99%
correlated and B and C are 99% correlated. But if A is measured in one particle
and C in the other, the results are only 96% correlated, which is a
contradiction. The intuitive formulation is due to David Mermin, while the
small-angle limit is emphasized in Bell's original article.

[edit]
CHSH inequality

Main article: CHSH
inequality

In addition to Bell's original inequality,[5]
the form given by John Clauser, Michael
Horne, Abner Shimony and R. A.
Holt,[10]
(the CHSH form) is especially
important[10],
as it gives classical limits to the expected correlation for the above
experiment conducted by Alice and Bob:

where C denotes correlation.

Correlation of observables X, Y is defined as

This is a non-normalized form of the correlation
coefficient considered in statistics (see Quantum
correlation).

In order to formulate Bell's theorem, we formalize local realism as
follows:

There is a probability space Λ and the observed
outcomes by both Alice and Bob result by random sampling of the parameter .

The values observed by Alice or Bob are functions of the local detector
settings and the hidden parameter only. Thus

Value observed by Alice with detector setting a is A(a,λ)

Value observed by Bob with detector setting b is B(b,λ)

Implicit in assumption 1) above, the hidden parameter space Λ has a probability measure ρ
and the expectation of a random variable X on Λ with respect to ρ is
written

where for accessibility of notation we assume that the probability measure
has a density.

Bell's inequality. The CHSH inequality (1) holds under the hidden
variables assumptions above.

For simplicity, let us first assume the observed values are +1 or −1; we
remove this assumption in Remark 1 below.

Let .
Then at least one of

is 0. Thus

and therefore

Remark 1. The correlation inequality (1) still holds if the variables
A(a,λ), B(b,λ) are allowed to take on any real values
between −1 and +1. Indeed, the relevant idea is that each summand in the above
average is bounded above by 2. This is easily seen to be true in the more
general case:

To justify the upper bound 2 asserted in the last inequality, without loss of
generality, we can assume that

In that case

Remark 2. Though the important component of the hidden parameter λ in Bell's original proof is associated with the source
and is shared by Alice and Bob, there may be others that are associated with the
separate detectors, these others being independent. This argument was used by
Bell in 1971, and again by Clauser and Horne in 1974,[11]
to justify a generalisation of the theorem forced on them by the real
experiments, in which detectors were never 100% efficient. The derivations were
given in terms of the averages of the outcomes over the local detector
variables. The formalisation of local realism was thus effectively changed,
replacing A and B by averages and retaining the symbol λ but with a slightly different meaning. It was henceforth
restricted (in most theoretical work) to mean only those components that were
associated with the source.

However, with the extension proved in Remark 1, CHSH inequality still holds
even if the instruments themselves contain hidden variables. In that case,
averaging over the instrument hidden variables gives new variables:

on Λ which still have values in the range
[−1, +1] to which we can apply the previous result.

[edit]
Bell inequalities are violated by quantum mechanical
predictions

In the usual quantum
mechanical formalism, the observables X and Y are represented
as self-adjoint
operators on a Hilbert space. To compute
the correlation, assume that X and Y are represented by matrices in a
finite dimensional space and that X and Y commute; this special
case suffices for our purposes below. The von Neumann measurement postulate
states: a series of measurements
of an observable X on a series of identical systems in state φ produces a distribution of real values. By the assumption
that observables are finite matrices, this distribution is discrete. The
probability of observing λ is non-zero if and only if λ is an eigenvalue of the
matrix X and moreover the probability is

where EX (λ) is the projector corresponding to the
eigenvalue λ. The system state immediately after the measurement
is

From this, we can show that the correlation of commuting observables X
and Y in a pure state ψ is

We apply this fact in the context of the EPR paradox. The
measurements performed by Alice and Bob are spin measurements on electrons.
Alice can choose between two detector settings labelled a and a′;
these settings correspond to measurement of spin along the z or the
x axis. Bob can choose between two detector settings labelled b
and b′; these correspond to measurement of spin along the z′ or
x′ axis, where the x′ – z′ coordinate system is rotated
135° relative to the x – z coordinate system. The spin observables
are represented by the 2 × 2 self-adjoint matrices:

These are the Pauli spin matrices
normalized so that the corresponding eigenvalues are +1, −1. As is customary, we denote
the eigenvectors of Sx by

Let φ be the spin singlet state for a pair of
electrons discussed in the EPR paradox. This is a specially constructed state
described by the following vector in the tensor product

Now let us apply the CHSH formalism to the measurements that can be performed
by Alice and Bob.

Illustration of Bell test for spin 1/2 particles.
Source produces spin singlet pairs, one particle of each pair is sent to Alice
and the other to Bob. Each performs one of the two spin
measurements.

The operators B(b'), B(b) correspond to Bob's spin measurements
along x′ and z′. Note that the A operators commute with the
B operators, so we can apply our calculation for the correlation. In this
case, we can show that the CHSH inequality fails. In fact, a straightforward
calculation shows that

and

so that

Bell's Theorem: If the quantum mechanical formalism is correct, then the
system consisting of a pair of entangled electrons cannot satisfy the principle
of local realism. Note that
is indeed the upper bound for quantum mechanics called Tsirelson's bound.
The operators giving this maximal value are always isomorphic
to the Pauli matrices.

[edit]
Practical experiments testing Bell's theorem

Main article: Bell test
experiments

Scheme of a "two-channel" Bell test
The
source S produces pairs of "photons", sent in opposite directions. Each photon
encounters a two-channel polariser whose orientation (a or b) can be set by the
experimenter. Emerging signals from each channel are detected and coincidences
of four types (++, −−, +− and −+) counted by the coincidence
monitor.

Experimental tests can determine whether the Bell inequalities required by
local realism hold up to the empirical evidence.

Bell's inequalities are tested by "coincidence counts" from a Bell test
experiment such as the optical one shown in the diagram. Pairs of particles are
emitted as a result of a quantum process, analysed with respect to some key
property such as polarisation direction, then detected. The setting
(orientations) of the analysers are selected by the experimenter.

Bell test experiments to date overwhelmingly violate Bell's inequality.
Indeed, a table of Bell test experiments performed prior to 1986 is given in 4.5
of Redhead, 1987.[12]
Of the thirteen experiments listed, only two reached results contradictory to
quantum mechanics; moreover, according to the same source, when the experiments
were repeated, "the discrepancies with QM could not be reproduced".

Nevertheless, the issue is not conclusively settled. According to Shimony's
2004 Stanford Encyclopedia overview article:[1]

Most of the dozens of experiments performed so far have favored Quantum
Mechanics, but not decisively because of the 'detection loopholes' or the
'communication loophole.' The latter has been nearly decisively blocked by a
recent experiment and there is a good prospect for blocking the
former.

To explore the 'detection loophole', one must distinguish the classes of
homogeneous and inhomogeneous Bell inequality.

The standard assumption in Quantum Optics is that "all photons of given
frequency, direction and polarization are identical" so that photodetectors
treat all incident photons on an equal basis. Such a fair sampling
assumption generally goes unacknowledged, yet it effectively limits the range of
local theories to those which conceive of the light field as corpuscular. The
assumption excludes a large family of local realist theories, in particular, Max
Planck's description. We must remember the cautionary words of Albert
Einstein[13]
shortly before he died: "Nowadays every Tom, Dick and Harry ('jeder Kerl' in
German original) thinks he knows what a photon is, but he is mistaken".

Objective physical properties for Bell’s analysis (local realist
theories) include the wave amplitude of a light signal. Those who maintain
the concept of duality, or simply of light being a wave, recognize the
possibility or actuality that the emitted atomic light signals have a range of
amplitudes and, furthermore, that the amplitudes are modified when the signal
passes through analyzing devices such as polarizers and beam splitters. It
follows that not all signals have the same detection probability[14].

[edit]
Two classes of Bell
inequalities

The fair sampling problem was faced openly in the 1970s. In early
designs of their 1973 experiment, Freedman and Clauser[15]
used fair sampling in the form of the Clauser-Horne-Shimony-Holt
(CHSH[10])
hypothesis. However, shortly afterwards Clauser and Horne[11]
made the important distinction between inhomogeneous (IBI) and homogeneous (HBI)
Bell inequalities. Testing an IBI requires that we compare certain coincidence
rates in two separated detectors with the singles rates of the two detectors.
Nobody needed to perform the experiment, because singles rates with all
detectors in the 1970s were at least ten times all the coincidence rates. So,
taking into account this low detector efficiency, the QM prediction actually
satisfied the IBI. To arrive at an experimental design in which the QM
prediction violates IBI we require detectors whose efficiency exceeds 82% for
singlet states, but have very low dark rate and short dead and resolving times.
This is well above the 30% achievable[16]
so Shimony’s optimism in the Stanford Encyclopedia, quoted in the preceding
section, appears over-stated.

[edit]
Practical challenges

Main article: Loopholes
in Bell test experiments

Because detectors don't detect a large fraction of all photons, Clauser and
Horne[11]
recognized that testing Bell's inequality requires some extra assumptions. They
introduced the No Enhancement Hypothesis (NEH):

a light signal, originating in an atomic cascade for example, has a certain
probability of activating a detector. Then, if a polarizer is interposed
between the cascade and the detector, the detection probability cannot
increase.

Given this assumption, there is a Bell inequality between the coincidence
rates with polarizers and coincidence rates without polarizers.

The experiment was performed by Freedman and Clauser[15],
who found that the Bell's inequality was violated. So the no-enhancement
hypothesis cannot be true in a local hidden variables model. The
Freedman-Clauser experiment reveals that local hidden variables imply the new
phenomenon of signal enhancement:

In the total set of signals from an atomic cascade there is a subset whose
detection probability increases as a result of passing through a linear
polarizer.

This is perhaps not surprising, as it is known that adding noise to data can,
in the presence of a threshold, help reveal hidden signals (this property is
known as stochastic
resonance[17]).
One cannot conclude that this is the only local-realist alternative to Quantum
Optics, but it does show that the word loophole is biased. Moreover, the
analysis leads us to recognize that the Bell-inequality experiments, rather than
showing a breakdown of realism or locality, are capable of revealing important
new phenomena.

[edit]
Theoretical
challenges

Most advocates of the hidden variables idea believe that experiments have
ruled out local hidden variables. They are ready to give up locality, explaining
the violation of Bell's inequality by means of a "non-local" hidden variable
theory, in which the particles exchange information about their states. This
is the basis of the Bohm interpretation
of quantum mechanics, which requires that all particles in the universe be able
to instantaneously exchange information with all others. A recent experiment
ruled out a large class of non-Bohmian "non-local" hidden variable theories.[18]

If the hidden variables can communicate with each other faster than light,
Bell's inequality can easily be violated. Once one particle is measured, it can
communicate the necessary correlations to the other particle. Since in
relativity the notion of simultaneity is not absolute, this is unattractive. One
idea is to replace instantaneous communication with a process which travels
backwards in time along the past Light cone. This is the idea
behind a transactional
interpretation of quantum mechanics, which interprets the statistical
emergence of a quantum history as a gradual coming to agreement between
histories that go both forward and backward in time[19].

A few advocates of deterministic models have not given up on local hidden
variables. E.g., Gerard 't Hooft has argued that the superdeterminism
loophole cannot be dismissed[20].

The quantum mechanical wavefunction can also provide a local realistic
description, if the wavefunction values are interpreted as the fundamental
quantities that describe reality. Such an approach is called a many-worlds
interpretation of quantum mechanics. In this view, two distant observers
both split into superpositions when measuring a spin. The Bell inequality
violations are no longer counterintuitive, because it is not clear which copy of
the observer B observer A will see when going to compare notes. If reality
includes all the different outcomes, locality in physical space (not outcome
space) places no restrictions on how the split observers can meet up.

This implies that there is a subtle assumption in the argument that realism
is incompatible with quantum mechanics and locality. The assumption, in its
weakest form, is called counterfactual
definiteness. This states that if the results of an experiment are always
observed to be definite, there is a quantity which determines what the outcome
would have been even if you don't do the experiment.

Many worlds interpretations are not only counterfactually indefinite, they
are factually indefinite. The results of all experiments, even ones that have
been performed, are not uniquely determined.

[edit]
Final remarks

This section does not cite any
references or
sources.
Please help improve this article by adding citations to reliable
sources. Unsourced material may be challenged
and removed.
(November 2009)

The phenomenon of quantum entanglement that is behind violation of Bell's
inequality is just one element of quantum physics which cannot be represented by
any classical picture of physics; other non-classical elements are complementarity
and wavefunction
collapse. The problem of interpretation
of quantum mechanics is intended to provide a satisfactory picture of these
non-classical elements of quantum physics.

The EPR paper "pinpointed" the unusual properties of the entangled
states, e.g. the above-mentioned singlet state, which is the foundation for
present-day applications of quantum physics, such as quantum
cryptography. This strange non-locality was originally supposed to be a Reductio ad
absurdum, because the standard interpretation could easily do away with
action-at-a-distance by simply assigning to each particle definite spin-states.
Bell's theorem showed that the "entangledness" prediction of quantum mechanics
have a degree of non-locality that cannot be explained away by any local
theory.

In well-defined Bell experiments (see the paragraph on "test
experiments") one can now falsify either quantum mechanics or Einstein's
quasi-classical assumptions: currently many experiments of this kind have been
performed, and the experimental results support quantum mechanics, though some
believe that detectors give a biased sample of photons, so that until nearly
every photon pair generated is observed there will be loopholes.

What is powerful about Bell's theorem is that it doesn't come from any
particular physical theory. What makes Bell's theorem unique and powerful is
that it relies only on the general properties of quantum mechanics. Physical
theories which assume a deterministic variable inside the particle can account
for the experimental results only by assuming that this variable cannot causally
change other variables far away.

[edit]
See also

Bell test
experiments

CHSH Bell test

Counterfactual
definiteness

Leggett–Garg
inequality

Local hidden
variable theory

Mott problem

Quantum
entanglement

Quantum
mechanical Bell test prediction

Measurement
in quantum mechanics

Renninger
negative-result experiment

GHZ
State

[edit]
Notes

^ a
b
c
d
Article on Bell's Theorem by Abner Shimony in the Stanford
Encyclopedia of Philosophy, (2004).

^
Griffiths, David J. Introduction to Quantum Mechanics: Second Edition.
Pearson Prentice Hall, 1998. p. 423.

^
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[edit]
References

A. Aspect et al., Experimental Tests of Realistic Local Theories via
Bell's Theorem, Phys. Rev. Lett. 47, 460 (1981)

A. Aspect et al., Experimental Realization of
Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's
Inequalities, Phys. Rev. Lett. 49, 91 (1982).

A. Aspect et al., Experimental Test of Bell's Inequalities Using
Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982).

A. Aspect and P. Grangier, About resonant scattering and other
hypothetical effects in the Orsay atomic-cascade experiment tests of Bell
inequalities: a discussion and some new experimental data, Lettere al
Nuovo Cimento 43, 345 (1985)

B. D'Espagnat, The
Quantum Theory and Reality, Scientific American, 241, 158
(1979)

J. S. Bell, On the problem of hidden variables in quantum
mechanics, Rev. Mod. Phys. 38, 447 (1966)

J. S. Bell, Introduction to the hidden variable question,
Proceedings of the International School of Physics 'Enrico Fermi', Course IL,
Foundations of Quantum Mechanics (1971) 171–81

J. S. Bell, Bertlmann’s socks and the nature of reality, Journal de
Physique, Colloque C2, suppl. au numero 3, Tome 42 (1981) pp C2
41–61

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics
(Cambridge University Press 1987) [A collection of Bell's papers, including
all of the above.]

J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and
implications, Reports on Progress in Physics 41, 1881 (1978)

J. F. Clauser and M. A. Horne, Phys. Rev D 10, 526–535 (1974)

E. S. Fry, T. Walther and S. Li, Proposal for a loophole-free test of
the Bell inequalities, Phys. Rev. A 52, 4381 (1995)

E. S. Fry, and T. Walther, Atom based tests of the Bell Inequalities —
the legacy of John Bell continues, pp 103–117 of Quantum
[Un]speakables, R.A. Bertlmann and A. Zeilinger (eds.) (Springer,
Berlin-Heidelberg-New York, 2002)

R. B. Griffiths, Consistent Quantum Theory', Cambridge University
Press (2002).

L. Hardy, Nonlocality for 2 particles without inequalities for almost
all entangled states. Physical Review Letters 71 (11) 1665–1668
(1993)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum
Information, Cambridge University Press (2000)

P. Pearle, Hidden-Variable Example Based upon Data Rejection,
Physical Review D 2, 1418–25 (1970)

A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht,
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P. Pluch, Theory of Quantum Probability, PhD Thesis, University of
Klagenfurt, 2006.

B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991.

M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe,
and D.J. Wineland, Experimental violation of Bell's inequalities with
efficient detection,(Nature, 409, 791–794, 2001).

S. Sulcs, The Nature of Light and Twentieth Century Experimental
Physics, Foundations of Science 8, 365–391 (2003)

S. Gröblacher et al., An experimental test of non-local
realism,(Nature, 446, 871–875, 2007).

D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe,
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The comic Dilbert, by Scott
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[edit]
Further reading

The following are intended for general audiences.

Amir D. Aczel, Entanglement: The greatest mystery in physics (Four
Walls Eight Windows, New York, 2001).

A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox
(Plenum Press, New York and London, 1999)

J. Baggott, The Meaning of Quantum Theory (Oxford University Press,
1992)

N. David Mermin, "Is the moon there when nobody looks? Reality and the
quantum theory", in Physics Today, April 1985, pp. 38–47.

Louisa Gilder, The Age of Entanglement: When Quantum Physics Was
Reborn (New York: Alfred A. Knopf, 2008)

Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN
0-375-72720-5)

Nick Herbert, Quantum Reality: Beyond the New Physics (Anchor,
1987, ISBN
0-385-23569-0)

D. Wick, The infamous boundary: seven decades of controversy in quantum
physics (Birkhauser, Boston 1995)

R. Anton Wilson, Prometheus Rising (New Falcon Publications, 1997,
ISBN
1-56184-056-4)

Gary Zukav "The Dancing Wu Li Masters" (Perennial Classics, 2001, ISBN
0-06-095968-1)

[edit]
External links

An explanation of Bell's Theorem, based on N. D. Mermin's
article, "Bringing Home the Atomic World: Quantum Mysteries for
Anybody," Am. J. of Phys. 49 (10), 940 (October 1981)

Quantum Entanglement Includes a simple explanation of Bell's
Inequality.

Bell's theorem on arXiv.org

Disproofs of Bell, GHZ, and Hardy Type Theorems and the Illusion
of Entanglement Claimed Disproof of Bell's Theorem

Disproof of "Disproof of Bell's Theorem by Clifford Algebra
Valued Local Variables" Claims to refute "Disproof of Bell's
Theorem by Clifford Algebra Valued Local Variables"

"Disproof of Bell's Theorem" : more critics Claims to
refute "Disproof of Bell's Theorem by Clifford Algebra Valued Local
Variables"

Interactive experiments with single photons: entanglement and
Bell´s theorem

Retrieved from "http://en.wikipedia.org/wiki/Bell%27s_theorem"

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