Colloquium: The Einstein-Podolsky-Rosen paradox:
From concepts to applications
M. D. Reid and P. D. Drummond
ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and
Ultrafast Spectroscopy, Swinburne University of Technology, P.O. Box 218, Melbourne,
Victoria 3122 Australia
W. P. Bowen
School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072,
Australia
E. G. Cavalcanti
Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia
P. K. Lam and H. A. Bachor
ARC Centre of Excellence for Quantum-Atom Optics, Building 38, The Australian National
University, Canberra, Australian Capital Territory 0200, Australia
U. L. Andersen
Department of Physics, Technical University of Denmark, Building 309, 2800 Lyngby,
Denmark
G. Leuchs
Max-Planck Institute for the Science of Light and Department of Physics, Universität
Erlangen-Nürnberg, D-91058 Erlangen, Germany
共Published 10 December 2009兲
This Colloquium examines the field of the Einstein, Podolsky, and Rosen 共EPR兲 gedanken
experiment, from the original paper of Einstein, Podolsky, and Rosen, through to modern theoretical
proposals of how to realize both the continuous-variable and discrete versions of the EPR paradox.
The relationship with entanglement and Bell’s theorem are analyzed, and the progress to date towards
experimental confirmation of the EPR paradox is summarized, with a detailed treatment of the
continuous-variable paradox in laser-based experiments. Practical techniques covered include
continuous-wave parametric amplifier and optical fiber quantum soliton experiments. Current
proposals for extending EPR experiments to massive-particle systems are discussed, including spin
squeezing, atomic position entanglement, and quadrature entanglement in ultracold atoms. Finally,
applications of this technology to quantum key distribution, quantum teleportation, and entanglement
swapping are examined.
DOI: 10.1103/RevModPhys.81.1727 PACS number共s兲: 03.65.Ud, 03.67.Bg, 03.75.Gg, 42.50.Xa
CONTENTS
I. Introduction 1728
II. The Continuous Variable EPR Paradox 1729
A. The 1935 argument: EPR’s elements of reality 1729
B. Schrödinger’s response: entanglement and
separability 1730
III. Discrete Spin Variables and Bell’s Theorem 1731
A. The EPR-Bohm paradox: Early EPR experiments 1731
B. Bell’s theorem 1731
C. Experimental tests of Bell’s theorem 1732
IV. EPR Argument for Real Particles and Fields 1732
A. Inferred Heisenberg inequality: Continuous variable
case 1732
B. Criteria for the discrete EPR paradox 1734
C. A practical linear-estimate criterion for EPR 1734
D. Experimental criteria for demonstrating the paradox 1735
V. Theoretical Model for a Continuous Variable EPR
Experiment 1735
A. Two-mode squeezed states 1735
B. Measurement techniques 1736
C. Effects of loss and imperfect detectors 1737
VI. EPR, Entanglement, and Bell Criteria 1738
A. Steering 1738
B. Symmetric EPR paradox 1738
C. EPR as a special type of entanglement 1738
D. EPR and Bell’s nonlocality 1739
VII. Continuous-Wave EPR Experiments 1740
A. Parametric oscillator experiments 1740
B. Experimental results 1741
VIII. Pulsed EPR Experiments 1742
A. Optical fiber experiment 1742
B. Parametric amplifier experiment 1743
REVIEWS OF MODERN PHYSICS, VOLUME 81, OCTOBER–DECEMBER 2009
0034-6861/2009/81共4兲/1727共25兲 ©2009 The American Physical Society1727
IX. Spin EPR and Atoms 1744
A. Transfer of optical entanglement to atomic
ensembles 1744
B. Conditional atom ensemble entanglement 1745
X. Application of EPR Entanglement 1745
A. Entanglement-based quantum key distribution 1745
B. Quantum teleportation and entanglement swapping 1746
XI. Outlook 1747
Acknowledgments 1748
References 1748
I. INTRODUCTION
In 1935, Einstein, Podolsky, and Rosen 共EPR兲 origi-
nated the famous “EPR paradox” 共Einstein et al., 1935兲.
This argument concerns two spatially separated particles
which have both perfectly correlated positions and mo-
menta, as is predicted possible by quantum mechanics.
The EPR paper spurred investigations into the nonlocal-
ity of quantum mechanics, leading to a direct challenge
of the philosophies taken for granted by most physicists.
Furthermore, the EPR paradox brought into sharp focus
the concept of entanglement, now considered to be the
underpinning of quantum technology.
Despite its large significance, relatively little has been
done to directly realize the original EPR gedanken ex-
periment. Most published discussion has centred around
the testing of theorems by Bell 共1964兲, whose work was
derived from that of EPR, but proposed more stringent
tests dealing with a different set of measurements. The
purpose of this Colloquium is to give a different per-
spective. We go back to EPR’s original paper, and ana-
lyze the current theoretical and experimental status and
implications of the EPR paradox itself, as an indepen-
dent body of work.
A paradox is “a seemingly absurd or self-
contradictory statement or proposition that may in fact
be true.”
1
The EPR conclusion was based on the as-
sumption of local realism, and thus the EPR argument
pinpoints a contradiction between local realism and the
completeness of quantum mechanics. The argument was
therefore termed a “paradox” by Schrödinger 共1935b兲,
Bohm 共1951兲, Bohm and Aharonov 共1957兲, and Bell
共1964兲. EPR took the prevailing view of their era that
local realism must be valid. They argued from this
premise that quantum mechanics must be incomplete.
With the insight later provided by Bell 共1964兲, the EPR
argument is best viewed as the first demonstration of
problems arising from the premise of local realism.
The intention of EPR was to motivate the search for a
theory “better” than quantum mechanics. However,
EPR never questioned the correctness of quantum me-
chanics, only its completeness. They showed that if a set
of assumptions, which we now call local realism, is up-
held, then quantum mechanics must be incomplete. Ow-
ing to the subsequent work of Bell, we now know what
EPR did not know: local realism, the “realistic philoso-
phy of most working scientists” 共Clauser and Shimony,
1978兲, is itself in question. Thus, an experimental real-
ization of the EPR proposal provides a way to demon-
strate a type of entanglement inextricably connected
with quantum nonlocality.
In the sense that the local realistic theory envisaged
by them cannot exist, EPR were “wrong.” What EPR
did reveal in their paper, however, was an inconsistency
between local realism and the completeness of quantum
mechanics. Hence, we must abandon at least one of
these premises. Their analysis was clever, insightful, and
correct. The EPR paper therefore provides a way to dis-
tinguish quantum mechanics as a complete theory from
classical reality, in a quantitative sense.
The conclusions of the EPR argument can only be
drawn if certain correlations between the positions and
momenta of the particles can be confirmed experimen-
tally. The work of EPR, like that of Bell, requires experi-
mental demonstration, since it could be supposed that
the quantum states in question are not physically acces-
sible, or that quantum mechanics itself is wrong. It is not
feasible to prepare the perfect correlations of the origi-
nal EPR proposal. Instead, we show that the violation of
an inferred Heisenberg uncertainty principle—an “EPR
inequality”—is eminently practical. These EPR in-
equalities provide a way to test the incompatibility of
local realism, as generalized to a nondeterministic situa-
tion, with the completeness of quantum mechanics. Vio-
lating an EPR inequality is a demonstration of the EPR
paradox.
In a nutshell, EPR experiments provide an important
complement to those proposed by Bell. While the con-
clusions of Bell’s theorem are stronger, the EPR ap-
proach is applicable to a greater variety of physical sys-
tems. Most Bell tests have been confined to single
photon counting measurements with discrete outcomes,
whereas recent EPR experiments have involved con-
tinuous variable outcomes and high detection efficien-
cies. This leads to possibilities for tests of quantum non-
locality in new regimes involving massive particles and
macroscopic systems. Significantly, new applications in
the field of quantum information are feasible.
In this Colloquium, we outline the theory of EPR’s
seminal paper, and also provide an overview of more
recent theoretical and experimental achievements. We
discuss the development of the EPR inequalities, and
how they can be applied to quantify the EPR paradox
for both spin and amplitude measurements. A limiting
factor for the early spin EPR experiments of Wu and
Shaknov 共1950兲, Freedman and Clauser 共1972兲, Aspect,
Grangier, and Roger 共1981兲, and others was the low de-
tection efficiencies, which meant probabilities were sur-
mised using a postselected ensemble of counts. In con-
trast, the more recent EPR experiments report an
amplitude correlation measured over the whole en-
semble, to produce unconditionally, on demand, states
that give the entanglement of the EPR paradox. How-
ever, causal separation has not yet been achieved. We
explain the methodology and development of these ex-
1
Compact Oxford English Dictionary, 2006,
www.askoxford.com
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periments, first performed by Ou, Pereira, Kimble, and
Peng 共1992兲.
An experimental realization of the EPR proposal will
always imply entanglement, and we analyze the relation-
ship between entanglement, the EPR paradox, and
Bell’s theorem. In looking to the future, we review re-
cent experiments and proposals involving massive par-
ticles, ranging from room-temperature spin-squeezing
experiments to proposals for the EPR-entanglement of
quadratures of ultracold Bose-Einstein condensates. A
number of possible applications of these EPR experi-
ments have already been proposed, for example, in the
areas of quantum cryptography and quantum teleporta-
tion. Finally, we discuss these, with emphasis on those
applications that use the form of entanglement closely
associated with the EPR paradox.
II. THE CONTINUOUS VARIABLE EPR PARADOX
Einstein, Podolsky, and Rosen 共1935兲 focused atten-
tion on the nonlocality of quantum mechanics by consid-
ering the case of two spatially separated quantum par-
ticles that have both maximally correlated momenta and
maximally anticorrelated positions. In their paper en-
titled “Can Quantum-Mechanical Description of Physi-
cal Reality Be Considered Complete?,” they pointed out
an apparent inconsistency between such states and the
premise of local realism, arguing that this inconsistency
could only be resolved through a completion of quan-
tum mechanics. Presumably EPR had in mind to supple-
ment quantum theory with a hidden variable theory,
consistent with the “elements of reality” defined in their
paper.
After Bohm 共1952兲 demonstrated that a 共nonlocal兲
hidden variable theory was feasible, subsequent work by
Bell 共1964兲 proved the impossibility of completing quan-
tum mechanics with local hidden variable theories. This
resolves the paradox by pointing to a failure of local
realism itself—at least at the microscopic level. The
EPR argument nevertheless remains significant. It re-
veals the necessity of either rejecting local realism or
completing quantum mechanics 共or both兲.
A. The 1935 argument: EPR’s elements of reality
The EPR argument is based on the premises that are
now generally referred to as local realism 共quotes are
from the original paper兲:
• “If, without disturbing a system, we can predict with
certainty the value of a physical quantity,” then
“there exists an element of physical reality corre-
sponding to this physical quantity.” The element of
reality represents the predetermined value for the
physical quantity.
• The locality assumption postulates no action at a dis-
tance, so that measurements at a location B cannot
immediately “disturb” the system at a spatially sepa-
rated location A.
EPR treated the case of a nonfactorizable pure state
兩
典, which describes the results for measurements per-
formed on two spatially separated systems at A and B
共Fig. 1兲. “Nonfactorizable” means entangled, that is, we
cannot express 兩
典 as a simple product 兩
典=兩
典
A
兩
典
B
,
where 兩
典
A
and 兩
典
B
are quantum states for the results of
measurements at A and B, respectively.
In the first part of their paper, EPR pointed out in a
general way the puzzling aspects of such entangled
states. The key issue is that one can expand 兩
典 in terms
of more than one basis, which correspond to different
experimental settings, parametrized by
. Consider the
state
兩
典 =
冕
dx兩
x
典
,A
兩u
x
典
,B
. 共1兲
Here the eigenvalue x could be continuous or discrete.
The parameter setting
at the detector B is used to
define a particular orthogonal measurement basis
兩u
x
典
,B
. On measurement at B, this projects out a wave
function 兩
x
典
,A
at A, the process called “reduction of
the wave packet.” The puzzling issue is that different
choices of measurements
at B will cause reduction of
the wave packet at A in more than one possible way.
EPR state that, “as a consequence of two different mea-
surements” at B, the “second system may be left in
states with two different wave functions.” Yet, “no real
change can take place in the second system in conse-
quence of anything that may be done to the first sys-
tem.” Schrödinger 共1935b, 1936兲 studied this case as
well, referring to the apparent influence by B on the
remote system A as “steering.” Despite the apparently
acausal nature of state collapse 共Herbert, 1982兲, the lin-
earity or “no-cloning” property of quantum mechanics
rules out superluminal communication 共Dieks, 1982;
Wootters and Zurek, 1982兲.
The problem was crystallized by EPR with a specific
example, shown in Fig. 1. EPR considered two spatially
separated subsystems, at A and B, each with two observ-
ables x
ˆ
and p
ˆ
where x
ˆ
and p
ˆ
are noncommuting quan-
tum operators, with commutator 关x
ˆ
,p
ˆ
兴=x
ˆ
p
ˆ
−p
ˆ
x
ˆ
=2C⫽0.
The results of the measurements x
ˆ
and p
ˆ
are denoted x
and p, respectively, and we follow this convention
throughout the paper. We note that EPR assumed a con-
tinuous variable spectrum, but this is not crucial to the
concepts they raised. In our treatment we scale the ob-
servables so that C = i, for simplicity, which gives rise to
the Heisenberg uncertainty relation
FIG. 1. 共Color online兲 The original EPR gedanken experi-
ment. Two particles move from a source S into spatially sepa-
rated regions A and B, and yet continue to have maximally
correlated positions and anticorrelated momenta. This means
one may make an instant prediction, with 100% accuracy, of
either the position or momentum of particle A by performing a
measurement at B.
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⌬x⌬p 艌 1, 共2兲
where ⌬x and ⌬p are the standard deviations in the re-
sults x and p, respectively.
EPR considered the quantum wave function
defined
in a position representation
共x,x
B
兲 =
冕
e
共ip/ប兲共x−x
B
−x
0
兲
dp, 共3兲
where x
0
is a constant implying spacelike separation.
Here the pairs x and p refer to the results for position
and momentum measurements at A, while x
B
and p
B
refer to position and momentum at B. We leave off the
superscript for system A, to emphasize the inherent
asymmetry that exists in the EPR argument, where one
system 共A兲 is “steered” by the other 共B兲.
According to quantum mechanics, one can “predict
with certainty” that a measurement x
ˆ
will give result
x
B
+x
0
, if a measurement x
ˆ
B
, with result x
B
, was already
performed at B. One may also predict with certainty the
result of measurement p
ˆ
, for a different choice of mea-
surement at B. If the momentum at B is measured to be
p, then the result for p
ˆ
is −p. These predictions are made
“without disturbing the second system” at A, based on
the assumption, implicit in the original EPR paper, of
“locality.” The locality assumption can be strengthened
if the measurement events at A and B are causally sepa-
rated 共such that no signal can travel from one event to
the other, unless faster than the speed of light兲.
The remainder of the EPR argument may be summa-
rized as follows 共Clauser and Shimony, 1978兲. Assuming
local realism, one deduces that both the measurement
outcomes, for x and p at A, are predetermined. The per-
fect correlation of x with x
B
+x
0
implies the existence of
an element of reality for the measurement x
ˆ
. Similarly,
the correlation of p with −p
B
implies an element of re-
ality for p
ˆ
. Although not mentioned by EPR, it will
prove useful to mathematically represent the elements
of reality for x
ˆ
and p
ˆ
by the respective variables
x
A
and
p
A
, whose “possible values are the predicted results of
the measurement” 共Mermin, 1990兲.
To continue the argument, local realism implies the
existence of two elements of reality,
x
A
and
p
A
, that
simultaneously predetermine, with absolute definiteness,
the results for either measurement x or p at A. These
elements of reality for the localized subsystem A are not
themselves consistent with quantum mechanics. Simulta-
neous determinacy for both the position and momentum
is not possible for any quantum state. Hence, assuming
the validity of local realism, one concludes quantum me-
chanics to be incomplete. Bohr’s early reply 共Bohr, 1935兲
to EPR was essentially a defense of quantum mechanics
and a questioning of the relevance of local realism.
B. Schrödinger’s response: entanglement and separability
It was soon realized that the paradox was intimately
related to the structure of the wavefunction in quantum
mechanics, and the opposite ideas of entanglement and
separability. Schrödinger 共1935a兲 pointed out that the
EPR two-particle wave function in Eq. 共3兲 was
verschränkten—which he later translated as entangled
共Schrödinger, 1935b兲—i.e., not of the separable form
A
B
. Both he and Furry 共1936兲 considered as a possible
resolution of the paradox that this entanglement de-
grades as the particles separate spatially, so that EPR
correlations would not be physically realizable. Experi-
ments considered in this Colloquium show this reso-
lution to be untenable microscopically, but the proposal
led to later theories which only modify quantum me-
chanics macroscopically 共Ghirardi et al., 1986; Bell, 1988;
Bassi and Ghirardi, 2003兲.
Quantum inseparability 共entanglement兲 for a general
mixed quantum state is defined as the failure of
ˆ
=
冕
dP共兲
ˆ
A
丢
ˆ
B
, 共4兲
where 兰dP共兲= 1 and
ˆ
is the density operator.
2
Here
is a discrete or continuous label for component states,
and
ˆ
A,B
correspond to density operators that are re-
stricted to the Hilbert spaces A, B respectively.
The definition of inseparability extends beyond that of
the EPR situation, in that one considers a whole spec-
trum of measurement choices, parametrized by
for
those performed on system A, and by
for those per-
formed on B. We introduce the new notation x
ˆ
A
and
x
ˆ
B
to describe all measurements at A and B. Denoting
the eigenstates of x
ˆ
A
by 兩x
A
典, we define P
Q
共x
A
兩
,兲
=具x
A
兩
ˆ
A
兩x
A
典 and P
Q
共x
B
兩
,兲= 具x
B
兩
ˆ
B
兩x
B
典, which are the
localized probabilities for observing results x
A
and x
B
,
respectively. The separability condition 共4兲 then implies
that joint probabilities P共x
A
,x
B
兲 are given as
P共x
A
,x
B
兲 =
冕
dP共兲P
Q
共x
A
兩兲P
Q
共x
B
兩兲. 共5兲
We note the restriction that, for example,
⌬
2
共x
A
兩兲⌬
2
共p
A
兩兲艌 1, where ⌬
2
共x
A
兩兲 and ⌬
2
共p
A
兩兲 are
the variances of P
Q
共x
A
兩
,兲 for the choices
corre-
sponding to position x
A
and momentum p
A
, respectively.
The original EPR state of Eq. 共3兲 is not separable.
The most precise signatures of entanglement rely on
entropic or more general information-theoretic mea-
sures. This can be seen in its simplest form when
ˆ
is a
pure state so that Tr
ˆ
2
=1. Under these conditions, it
follows that
ˆ
is entangled if and only if the von Neu-
mann entropy measure of either reduced density matrix
ˆ
A
=Tr
B
ˆ
or
ˆ
B
=Tr
A
ˆ
is positive. Here the entropy is
defined as
2
Here we use entanglement in the simplest sense to mean a
state for a composite system which is nonseparable, so that Eq.
共4兲 fails. The issues of the EPR paradox that make entangle-
ment interesting demand that the systems A and B can be
spatially separated, and these are the types of systems ad-
dressed here. The relation, between a quantum correlation and
entanglement, is discussed by Shore 共2008兲.
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S关
ˆ
兴 =−Tr
ˆ
ln
ˆ
. 共6兲
When
ˆ
is a mixed state, one must turn to variational
measures like the entanglement of formation to obtain
necessary and sufficient conditions 共Bennett et al., 1996兲.
The entanglement of formation leads to the popular
concurrence measure for two qubits 共Wootters, 1998兲.A
necessary but not sufficient condition for entanglement
is the positive partial transpose criterion of Peres 共1996兲.
III. DISCRETE SPIN VARIABLES AND BELL’S
THEOREM
A. The EPR-Bohm paradox: Early EPR experiments
As the continuous variable EPR proposal was not ex-
perimentally realizable at the time, much of the early
work relied on an adaptation of the EPR paradox to
spin measurements by Bohm 共1951兲, as depicted in Fig.
2.
Specifically, Bohm considered two spatially separated
spin-1/2 particles at A and B produced in an entangled
singlet state 共often referred to as the EPR-Bohm state or
the Bell state兲,
兩
典 =
1
冑
2
冉
冏
1
2
冔
A
冏
−
1
2
冔
B
−
冏
−
1
2
冔
A
冏
1
2
冔
B
冊
. 共7兲
Here 兩±
1
2
典
A
are eigenstates of the spin operator J
ˆ
z
A
, and
we use J
ˆ
z
A
, J
ˆ
x
A
, J
ˆ
y
A
to define the spin components mea-
sured at location A. The spin eigenstates and measure-
ments at B are defined similarly. By considering differ-
ent quantization axes, one obtains different but
equivalent expansions of 兩
典 in Eq. 共1兲, just as EPR sug-
gested.
Bohm’s reasoning is based on the existence, for Eq.
共7兲, of a maximum anticorrelation between not only J
ˆ
z
A
and J
ˆ
z
B
, but J
ˆ
y
A
and J
ˆ
y
B
, and also J
ˆ
x
A
and J
ˆ
x
B
. An assump-
tion of local realism would lead to the conclusion that
the three spin components of particle A were simulta-
neously predetermined, with absolute definiteness. Since
no such quantum description exists, this is the situation
of an EPR paradox. A simple explanation of the
discrete-variable EPR paradox has been presented by
Mermin 共1990兲 in relation to the three-particle
Greenberger-Horne-Zeilinger correlation 共Greenberger
et al., 1989兲.
An early attempt to realize EPR-Bohm correlations
for discrete 共spin兲 variables came from Bleuler and
Bradt 共1948兲, who examined the gamma radiation emit-
ted from positron annihilation. These are spin-one par-
ticles which form an entangled singlet. Here correlations
were measured between the polarizations of emitted
photons, but with very inefficient Compton-scattering
polarizers and detectors, and no control of causal sepa-
ration. Several further experiments were performed
along similar lines 共Wu and Shaknov, 1950兲, as well as
with correlated protons 共Lamehi-Rachti and Mittig,
1976兲. While these are sometimes regarded as demon-
strating the EPR paradox 共Bohm and Aharonov, 1957兲,
the fact that they involved extremely inefficient detec-
tors, with postselection of coincidence counts, makes
this interpretation debatable.
B. Bell’s theorem
The EPR paper concludes by referring to theories
that might complete quantum mechanics: “…we have
left open the question of whether or not such a descrip-
tion exists. We believe, however, that such a theory is pos-
sible.” The seminal works of Bell 共1964, 1988兲 and
Clauser et al. 共1969兲 共CHSH兲 clarified this issue, to show
that this speculation was wrong. Bell showed that the
predictions of local hidden variable 共LHV兲 theories dif-
fer from those of quantum mechanics, for the Bell state,
Eq. 共7兲.
Bell-CHSH considered theories for two spatially sepa-
rated subsystems A and B. As with separable states,
Eqs. 共4兲 and 共5兲, it is assumed there exist parameters
that are shared between the subsystems and which de-
note localized—though not necessarily quantum—states
for each. Measurements can be performed on A and B,
and the measurement choice is parametrized by
and
,
respectively. Thus, for example,
may be chosen to be
either position and momentum, as in the original EPR
gedanken experiment, or an analyzer angle as in the
Bohm-EPR gedanken experiment. We denote the result
of the measurement labelled
at A as x
A
, and use simi-
lar notation for outcomes at B. The assumption of Bell’s
locality is that the probability P共x
A
兩兲 for x
A
depends on
and
, but is independent of
; and similarly for
P共x
B
兩兲. The local hidden variable assumption of Bell
and CHSH then implies the joint probability P共x
A
,x
B
兲
to be
P共x
A
,x
B
兲 =
冕
dP共兲P共x
A
兩兲P共x
B
兩兲, 共8兲
where P共兲 is the distribution for the . This assumption,
which we call Bell-CHSH local realism, differs from Eq.
共5兲 for separability, in that the probabilities P共x
A
兩兲 and
P共x
B
兩兲 do not arise from localized quantum states.
From the assumption Eq. 共8兲 of LHV theories, Bell and
CHSH derived constraints, referred to as Bell’s inequali-
ties. They showed that quantum mechanics predicts a
FIG. 2. 共Color online兲 The Bohm gedanken EPR experiment.
Two spin-
1
2
particles prepared in a singlet state move from the
source into spatially separated regions A and B, and give an-
ticorrelated outcomes for J
A
and J
B
, where
is x, y,orz.
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Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009
