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ARTICLE DE FOND
THE EINSTEIN-PODOLSKY-ROSEN PARADOX AND
NATURE OF REALITY
THE
SHOHINI GHOSE
BY
I
n 1935 Einstein, Podolsky and Rosen wrote a seminal
paper about a thought experiment that led them to
question the completeness of the theory of quantum
mechanics [1]. The paradox identified in this paper
arose from the fact that according to quantum mechanics,
measurements of one member of a correlated (entangled)
pair of objects seem to instantaneously affect the other
member, no matter how far away - an effect Einstein
called ‘spooky action at a distance’. The EPR paradox led
to much scientific and philosophical debate regarding the
interpretation and need for modification of quantum
mechanics, as well as the nature of reality. Then in 1964,
John Bell devised a practical means to test the validity of
the key assumptions about reality and locality underlying
the EPR paradox [2,3]. When the test was implemented a
few years later [4-7], it created a stir in the scientific community by proving the EPR assumptions about local realism to be invalid and confirming that the predictions of
quantum mechanics, although bizarre, were indeed correct. Bell’s theorem has subsequently been called one of
the most profound discoveries of science.
These early studies laid the groundwork for the eventual
development of the field of quantum information processing, which focuses on harnessing quantum mechanical
properties such as entanglement to perform useful tasks
such as computing and communication [8]. The EPR paradox and Bell’s inequalities have gained fresh relevance
and practical use in this new era of quantum information
science. Despite the importance of these topics both for a
fundamental understanding of the foundations of quantum
mechanics, as well as for practical quantum information
processing applications, they are not typically discussed in
much detail in current undergraduate quantum mechanics
SUMMARY
This article is based on talks given during
the 2009 CAP-CASCA Lecture Tour for
undergraduates, and provides a conceptual
discussion of the EPR paradox, the formulation of Bell’s inequalities for testing the
premises underlying the paradox, and the
connection of these ideas to the rapidly
developing new field of quantum information
science.
courses. Nevertheless, these ideas generated a lot of interest, excitement and debate among undergraduate students
during a series of lectures given as part of the 2009 CAPCASCA Lecture Tour. This article is inspired by these lectures and the overwhelmingly positive feedback given by
students and faculty. The goal of the article is to present in
a conceptual manner the main ideas of the EPR paradox
and Bell’s inequalities, and how they have led to modern
quantum information science. Unlike other papers on the
topic, this paper does not include mathematically rigorous
derivations, but instead aims to explain in general terms
the underlying logic behind the mathematics. The focus is
on providing the reader with an overview of the important
issues, an understanding of the key results and current
developments, together with a complete list of references
that can be used to look up further details. The article is
organized in the following manner: the next section discusses the EPR paradox as presented in the original 1935
paper. The following section includes a discussion of
Bell’s inequalities and the key challenges for performing a
foolproof test of these inequalities in actual experiments.
The final section provides a brief introduction to quantum
information science, and the use of entanglement (quantum correlations) as a resource for quantum computing
and communication. It also includes a summary of our
recent surprising results on entanglement and Bell's
inequalities in multipartite quantum systems [9], as well as
some of the latest developments in the field.
THE EPR PARADOX
Most cars today come equipped with a GPS (global positioning system) device that shows the position of the car at
any moment as well its velocity, from which one could in
principle, estimate the momentum. On the other hand if
one were to turn off the GPS device, one could reasonably
claim that there still exist specific values of the position
and momentum of the car although we are not measuring
them with the GPS device. These properties are assumed
to exist independent of whether we measure their values or
not. Furthermore, given certain information about the
car’s initial position and momentum, and applying
Newton’s laws, we can precisely predict their values without having to measure them. But quantum mechanics predicts that if the position of a particle is known, its momentum cannot be precisely known at the same time, and vice
versa [10]. This idea has come to be popularly known as the
‘uncertainty principle’ and can be more mathematically
Shohini Ghose
<[email protected]>,
Assistant Professor,
Department of
Physics and
Computer Science,
Wilfrid Laurier
University,
75 University Ave W,
Waterloo, ON N2L
3C5
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Fig.1
Set-up for the EPR thought experiment. Two entangled particles from a source are spatially separated so that, according
to the EPR assumption of locality, Alice’s measurements of
particle A cannot instantaneously affect Bob’s particle B.
Nevertheless quantum entanglement allows Alice to make
precise predictions about the position and momentum of particle B just by measuring particle A.
stated in terms of commutation of observables [10]. According
to some interpretations, one cannot predict the precise values of
both position and momentum at the same time, not just because
one cannot measure them, but because precise values cannot
actually exist for both properties simultaneously.
Einstein and many others were disturbed by the idea of discarding precise objective values or ‘elements of reality’. The EPR
paper [1] thus discussed an argument that was devised to show
that quantum mechanics was an incomplete theory. The argument was based on certain assumptions about what we mean by
reality and locality. The original statements of the assumptions
in the EPR paper are reformulated here for clarity:
does precisely describe the perfect correlation between their
positions and momenta. If the momentum of one of the particles is p then the momentum of the other is –p. Similarly, if the
position of one of the particles is x, then the position of the
other is x-x0. Such quantum correlations are called entanglement [8]. As we will see in the following discussion, this quantum entanglement leads to the contradiction at the heart of the
EPR paradox.
If Alice measures the position of particle A to be x, then knowing that the particles are entangled, she can predict with certainty, without disturbing particle B, that the location of particle B is x-x0. Therefore, using statement (i) above, the position
of particle B is an element of reality. If Alice instead measures
the momentum of particle A to be p, she can predict with certainty without disturbing particle B, that the momentum of particle B is -p. Therefore the momentum of particle B is an element of reality, from statement (i).
In summary, using the correlations (entanglement) between
particles A and B, EPR showed that the position and momentum of particle B are both elements of reality. However, quantum mechanics cannot predict with certainty the position and
momentum of particle B. This therefore leads to one of the following conclusions:
(a) The premises of local realism stated in (i) and (ii) are false.
(b) Quantum mechanics must be an incomplete theory as it violates statement (iii).
(i) If without disturbing a system we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical
quantity.
(ii) Measurements of a particle at location A cannot instantaneously disturb an object at location B far away, since
information cannot travel from A to B faster than the speed
of light. (This is often called Einstein locality.)
Einstein and co-workers concluded with statement (b) in the
EPR paper; i.e., quantum mechanics must be an incomplete
theory B but did not prove that this was the correct conclusion.
It would take almost thirty years before a practical means of
deciding between statements (a) and (b) was devised.
Given these seemingly reasonable assumptions, EPR asserted
that
THE NATURE OF REALITY
(iii) any complete physical theory must be able to predict the
values of all elements of reality.
The original EPR paper analyzed the positions and momenta of
two correlated particles. Since then, the argument has been
reformulated using different examples such as the spins of two
electrons or the polarizations of two photons. Here, the original
discussion based on positions and momenta of two particles is
presented, as these properties are typically more familiar than
spin or polarization to most students. Consider two quantum
particles that have interacted for some time in the past, but are
no longer interacting, as shown in Figure 1. The particles are
now far enough separated from each other that measurements
made by Alice on particle A cannot instantaneously affect particle B during the experiment and vice versa, in accordance
with statement (ii) above. According to quantum mechanics,
for each particle, if the position is known (by measurement)
then the precise value of momentum cannot be know simultaneously, and vice versa. Nevertheless, quantum mechanics
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In 1964 the physicist John Bell formulated an experiment that
could be used to test the assumptions of local realism made in
the EPR analysis [2,3]. In a later paper [11], Bell described the
question that he was studying in the following way: “The
philosopher on the street… is quite unimpressed by EinsteinPodolsky-Rosen correlations. She can point to many examples
of similar correlations in everyday life. The case of
Dr. Bertlmann's socks is often cited. Dr. Bertlmann likes to
wear two socks of different colors. Which color he will have on
given foot on given day is quite unpredictable. But when you
see the first sock is pink you are sure the second sock will not
be pink. Observation of first, and experience with
Dr. Bertlmann, gives immediate information about second.
There is no mystery in such correlations. Isn’t this EPR business just the same sort of thing?”
Bell devised a practical experiment to answer the following
question: Are the premises of local realism that are obviously
true for correlated socks also true for entangled quantum particles? Although Bell’s original argument is generally applicable
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ty the outcomes of polarization measurements on a single photon along different directions (just like position and momentum), but nevertheless can precisely describe the correlations in
the joint state of both photons [8]. Such a description is exactly
analogous to the entangled pairs of particles considered in the
original EPR paper [1]. Here, the entanglement exists between
the polarizations of the two photons along any direction,
instead of their positions and momenta.
Fig. 2: Linearly polarized light. The electric field F oscillates along
a single plane. The direction of polarization is concisely
depicted as shown on the right.
to any pair of entangled particles, it is most clearly understood
by considering in detail a particular experimental setup. Most
such experiments to test Bell’s inequalities use polarizationentangled photons. Consider a beam of light as shown in Fig. 2.
The light is linearly polarized (the electric field F is confined
to oscillate in a single plane relative to the direction of propagation [12]). The direction of polarization can be along any
direction as viewed head-on and is depicted more concisely as
shown in Fig. 2.
When a linearly polarized laser beam is incident on a nonlinear
crystal then occasionally a single incident photon will be converted into a pair of photons (called signal and idler photons)
via a process called parametric downconversion [13,14] which
conserves momentum and energy. The signal and idler photons
are emitted from the crystal in two cones of light that can be
made to intersect by adjusting the angle of the incoming laser
beam [15]. At the intersection of cones in which the signal and
idler photons have equal frequencies (half the incoming laser
frequency), are pairs of photons whose linear polarizations are
perfectly correlated. If one photon of the pair is measured to
have a particular polarization direction, the other photon’s
polarization is always orthogonal to that of the first photon and
vice versa. Thus if we were to measure whether the polarizations of the photons were horizontal or vertical, then prior to
measurement, the joint polarization state of both photons could
be described as HsVi + Vs Hi (horizontally polarized signal
photon and vertically polarized idler, or vertically polarized
signal photon and horizontally polarized idler). Note that
before measurement, according to quantum mechanics, we
cannot predict which of the two photons is horizontally polarized and which one is vertically polarized – either outcome is
equally likely for each photon. However, after measuring one
of the photons to be horizontally (or vertically) polarized, we
know with certainty that the other must be vertically (or horizontally) polarized. On the other hand, if we wanted to measure whether the polarization of the two photons was along +45
degrees or -45 degrees, then an equivalent description would be
45os , 135oi + 135os 45oi (+45o polarization of signal and 135o
polarization of idler, or 135o polarization of signal photon and
+45o polarization of idler). A valid description along any polarization direction would thus be θs, θ + 90oi + θ + 90os , θi .
Quantum mechanics does not allow us to predict with certain-
Given this source of entangled photon pairs, the following
experiment can be performed to test the predictions of local
realism. Alice receives a sequence of photons, each being one
half of an entangled pair, while Bob receives the other half
(Fig. 3). Alice measures the polarization of each photon she
receives. For each photon, she randomly chooses to measure
the polarization along the 0 or 45 degree directions. (Recall that
according to quantum mechanics, she cannot predict with certainty what the polarization of the photon is prior to measurement). If she chooses to measure the polarization along 0
degrees, she records the following values based on her measurement:
Q = 1 if the polarization is 0 degrees
Q = -1 if the polarization is orthogonal (90 degrees)
If she chooses to measure the polarization along 45 degrees,
she records the following values based on her measurement:
R = 1 if the polarization is 45 degrees
R = -1 if the polarization is orthogonal (135 degrees)
Bob receives his sequence of photons and also measures the
polarization of each photon. For each photon, he randomly
chooses to measure the polarization along the directions
22.5 degrees or 67.5 degrees. If he chooses to measure the
polarization along 22.5 degrees, he records the following values based on his measurement:
S = 1 if the polarization is 22.5 degrees
S = -1 if the polarization is orthogonal (112.5 degrees)
Fig. 3: Experimental set-up for testing the predictions of local realism via Bell inequalities. Alice and Bob receive a sequence of
photons, each being a half of a polarization entangled photon
pair created via parametric downconversion in a nonlinear
crystal. Alice randomly chooses to measure the polarization
of each photon along 0o or 45o . Bob randomly chooses to
measure the polarization of each photon along 22.5o or 67.5o.
They then compute the value of the CHSH correlation function from their measurement outcomes.
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If he chooses to measure the polarization along 67.5 degrees,
he records the following values based on his measurement:
T = 1 if the polarization is 67.5 degrees
T = -1 if the polarization is orthogonal (157.5 degrees)
Let’s analyze the experiment using the premises of local realism. Recall that EPR did not consider quantum mechanics to be
incorrect, merely incomplete. Their belief was that some additional hidden quantity or quantities (hidden variables) would be
required to fully characterize all elements of reality. Bell’s
approach was to ask what would we expect to measure in
experiments like the one outlined above, if we were to allow
for the possibility of such hidden variables that determine the
measured values of Q, R, S and T. Notice that the quantity
M = QS + RS + RT - QT = (R+Q)S+(R=Q)T = 2 or -2
If the experiment is repeated many times then we can calculate
the average value of the quantities QS, RS, RT and QT, denoted by +QS,,+RS,, +RT,, +QT,. If before a measurement, the probability of obtaining the values Q = q, R = r, S = s, T = t is
p (q,r,s,t), then the average is
QS + RS + RT − QT = ∑ p( qrst )[qs + rs + rt − qt ]
qrst
It is apparent that even if we allow for local hidden variables
that characterize the complete state of the system and determine the probabilities p, this average value must always lie
between 2 and -2 (for more details of this derivation
see [11], [16]). Thus the prediction of any local realistic model
of the experiment is that the quantity
+QS, + +RS, + +RT, − +QT,
#2
This CHSH inequality (so named after its inventors [16]) is a
modification of the inequality originally derived by Bell and is
the most commonly used Bell-type inequality today. All that
remains to be seen is whether the average values calculated
from actual experimental data satisfy this Bell-type inequality.
The first experiments testing the CHSH inequality were performed in 1981-82 [5-7] and caused shock waves in the scientific community - the average value of M calculated from the
experimental data was found to be 2 2 , which clearly violated the bound of 2 imposed by the premises of local realism.
The implications of this result are profound and stunning.
Nature cannot always be described by a local realistic model.
Quantum correlation or entanglement is not the same as the
correlation between Bertlmann’s socks. Moreover the experimental result agrees with quantum mechanical predictions [2,8,10]. All experimental tests thus far have confirmed that
quantum entanglement leads to a violation of a Bell inequality.
Therefore these correlations cannot be explained by any local
realistic (also known as local hidden-variable) model.
Furthermore, all measurement statistics have agreed with the
predictions of quantum mechanics. The EPR conclusion [1] that
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local realism is a valid assumption for quantum entangled particles was thus proved wrong by these experiments.
The astounding conclusions of the first Bell inequality experiments led many to question the accuracy of the experimental
data. The detectors used in Bell inequalities experiments are
not perfect. Errors in the detection of photons can make the
experimental results invalid by leading to a false violation of
Bell inequalities. A more careful analysis shows that a detection efficiency of at least 83% is required to obtain conclusive
results [17,18]. So far only one experiment performed in 2001
has had high enough efficiency to close this detection loophole.
The key was to perform the experiment using entangled ions
that can be measured with almost perfect efficiency, instead of
photons that are notoriously difficult to measure at the single
photon level [19]. Another practical problem in performing
actual experiments is to enforce strict Einstein locality. Alice
and Bob must be far enough apart and their measurements must
be fast enough so that they cannot influence each other’s measurements in any way. In fact, in actual experiments, Alice and
Bob are detectors that are located in the same lab and are certainly too close to satisfy the locality condition. This locality
loophole has been closed in an experiment that separated photons over a few hundred metres and used nanosecond scale
detectors that could perform measurements faster than the time
required for any signal to be able to travel the distance between
Alice and Bob [20]. Interestingly, forty-five years after Bell proposed his experiment, and seventy-five years after the original
EPR paper, no experiment has closed both loopholes at the
same time as yet. Recently new experiments for loophole-free
tests of Bell inequalities have been proposed based on fast
manipulation and high-precision measurement of atoms [21]
and entanglement-swapping using photons [22]. An implementation of these experiments is imminent and will be a critical
step in verifying the bizarre nonlocal nature of entangled quantum particles.
QUANTUM INFORMATION SCIENCE
The EPR paradox and Bell’s inequality tests were crucial stepping-stones in the road to the development of the new area of
quantum information science. Quantum information science is
the study of how quantum mechanics can be used to efficiently perform information processing tasks such as computing and
communication, as well as to develop new tasks such as teleportation and quantum cryptography [8]. Entanglement lies at
the heart of many of these quantum information processing
protocols. A detailed discussion of all these protocols is beyond
the scope of this article. However a brief discussion of a quantum key distribution (QKD) algorithm [23] is included here as
an example and also because it is very similar to the set-up for
testing Bell’s inequality described earlier. The goal of this protocol is to generate a secret key (a sequence of 0s and 1s) that
only Alice and Bob share and therefore can use for secure communication. As described in the previous section, Alice and
Bob each receive a sequence of photons, each of which is half
of a polarization-entangled pair. Alice and Bob make polarization measurements of each photon along one of two measure-
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ment directions – they independently choose to measure
whether the polarization is horizontal or vertical, or whether
the polarization is +45 degrees or -45 degrees. They record a 0
if they measure horizontal polarization and 1 if they measure
vertical polarization, or if they choose the other measurement
direction, they record a 0 if they measure +45 degree polarization and 1 if they measure -45 degree polarization. Then they
publicly compare their measurement directions for each photon, but NOT the outcomes of their measurements. For cases in
which their measurement directions agree, they know each
other’s measurement outcome due to the perfect correlations
inherent in the entangled photons. They discard all cases in
which their measurement directions do not agree, since in these
instances, they cannot know with certainty each other’s measurement outcomes. They are left with a perfectly correlated
(secret) set of measurement outcomes that they can use as a key
for secure communication. Bell’s inequalities have provided an
important means of testing the security of quantum key distribution (QKD) schemes [24]. Careful analyses of a variety of
attacks by an eavesdropper Eve have been performed to develop completely secure QKD protocols [25,26]. Today, quantum
cryptography has gone from a theoretical protocol to a working
technology that is being refined by over 100 research groups
around the world and already being marketed by several companies whose clients are, of course, secret! Quantum cryptography was also used for communication security in elections
held in Switzerland in 2007.
Whereas much progress has been made in analyzing entanglement in two particles and designing quantum information processing tasks using pairwise entangled particles, not much is
known about correlations between three or more particles. In
recent studies [9], we have found a quantitative relationship
between entanglement and violation of a Bell inequality for
three particles. Our results are surprising and counterintuitive –
unlike the case of two entangled particles, we found that three
entangled particles do not always violate a tripartite Bell
inequality. Our studies are part of the growing body of research
aimed at mapping out multipartite entanglement, which is
important not only for a fundamental understanding of this
important quantum property, but also for designing large-scale
quantum information processing devices involving large numbers of interacting quantum particles.
It is an exciting time to be a researcher in the rapidly growing
field of quantum information science. Although we have come
a long way since the EPR paper was first published in 1935,
many open questions remain to be answered. One important
milestone will be reached in the near future when a loopholefree test of Bell’s inequalities is eventually implemented. On
the experimental front, much effort is being put into developing efficient ways of encoding, storing and manipulating quantum bits of information (qubits). Rapid advances in quantum
photonics have led to the development of a quantum optical
transistor made from a single molecule [27]. Molecular qubits
could also be the building blocks for future non-silicon based
ultrafast quantum computers [28]. On the solid-state front, an
important step towards a working technology was the recent
demonstration of the worlds first working two-qubit superconducting quantum processor [29]. Superconducting quantum circuits have also been used to create interesting quantum states
of light [30]. Trapped atoms or ions interacting with lasers and
magnetic fields have traditionally been promising systems in
which to develop a toolbox of quantum control [31]. Recently
teleportation of an ion state over a distance of one metre was
demonstrated for the first time [32]. Trapped ions also offer the
possibility of building large-scale devices encoding very large
numbers of qubits [33].
Complementing this research on the theoretical side, is an
effort to quantify multipartite entanglement and develop new
protocols for efficient large-scale quantum information processing. An example is the development of one-way quantum
computation [34] in which amazingly, all logic operations are
implemented by performing conditional measurements on a
particular many-particle entangled state. This breakthrough not
only opens up new avenues for practical implementations of
quantum computing but also gives rise to new fundamental
questions about the role of measurements, and highlights the
importance of entanglement and nonlocality as resources for
information processing. We are learning that Bell’s inequalities
are very useful in a variety of contexts including detection of
entanglement [35], quantum state tomography [36], secure quantum cryptography [37], studies of entanglement of relativistic
particles [38], and studies of energy-time entanglement [39].
Many open questions still remain, including questions related
to the construction and experimental implementation of new
inequalities with more than two detector settings and the study
of Bell-like inequalities that allow some finite nonlocal
resources between the two parties [40]. In spite of the numerous
studies and the rapidly growing number of papers in this area,
there are still heated debates about the interpretation of quantum mechanics, and we still have much to learn in order to
address the kinds of questions first raised by Einstein and his
colleagues back in 1935. The future will thus be challenging
but exciting. We have taken the first small but crucial steps that
we hope will help us eventually make giant leaps towards a
fundamental understanding of these bizarre properties called
entanglement and nonlocality and the design of large-scale
quantum information processing devices.
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