Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications

TL;DR: In this article, the authors examined the field of the EPR 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 EPR paradox.

Abstract: This Colloquium examines the field of the Einstein, Podolsky, and Rosen (EPR) 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.

Summary

I. INTRODUCTION

• The EPR conclusion was based on the assumption of local realism, and thus the EPR argument pinpoints a contradiction between local realism and the completeness of quantum mechanics.
• In the sense that the local realistic theory envisaged by them cannot exist, EPR were "wrong.".
• It is not feasible to prepare the perfect correlations of the original EPR proposal.

II. THE CONTINUOUS VARIABLE EPR PARADOX

• Einstein, Podolsky, and Rosen ͑1935͒ focused attention on the nonlocality of quantum mechanics by considering the case of two spatially separated quantum particles that have both maximally correlated momenta and maximally anticorrelated positions.
• Presumably EPR had in mind to supplement quantum theory with a hidden variable theory, consistent with the "elements of reality" defined in their paper.
• After Bohm ͑1952͒ demonstrated that a ͑nonlocal͒ hidden variable theory was feasible, subsequent work by Bell ͑1964͒ proved the impossibility of completing quantum mechanics with local hidden variable theories.
• It reveals the necessity of either rejecting local realism or completing quantum mechanics ͑or both͒.

IV. EPR ARGUMENT FOR REAL PARTICLES AND FIELDS

• To recreate the precise gedanken proposal of EPR, one needs perfect correlations be-tween the positions of two separated particles, and also between their momenta.
• In order to demonstrate the existence of EPR correlations for real experiments, one therefore needs to minimally extend the EPR argument, in particular their definition of local realism, to situations where there is less than perfect correlation.
• This definition is the meaning of local realism in the text below.
• As considered by Furry ͑1936͒ and Reid ͑1989͒, this allows the derivation of an inequality whose violation indicates the EPR paradox.
• Like EPR, the authors assume causal separation of the observations and the validity of quantum mechanics.

P͑x

• The element of reality and hidden variable have similar meanings, except that the element of reality is a special hidden variable following from the EPR logic.
• This indicates an inconsistency of local realism with the completeness of quantum mechanics.
• The assumption that the state depicted by a particular pair x A , p A has an equivalent quantum description demands that the conditional probabilities satisfy the same relations as the probabilities for a quantum state.
• One can in principle use any quantum uncertainty constraint ͑Cavalcanti and Reid, 2007͒.

B. Criteria for the discrete EPR paradox

• The discrete variant of the EPR paradox was treated in Sec. III.
• Conclusive experimental realization of this paradox needs to account for imperfect sources and detectors, just as in the continuous variable case.
• Here the correlation is described in terms of Stokes operators for the polarization of the fields.
• ͑1981͒ used two-channel analyzers to demonstrate a perfect spin-EPR correlation but were constrained by weak photon detection efficiency.
• This is lower than the 58% threshold given above.

C. A practical linear-estimate criterion for EPR

• Nevertheless, an inference variance, which is the variance of the conditional distribution, has been measured for twin beam intensity distributions by Zhang, Kasai, and Watanabe ͑2003͒, who achieved ⌬ inf 2 x = 0.62.
• This was proposed by Reid ͑1989͒ as a practical procedure for measuring EPR correlations.
• There is also an analogous optimum for the value of gЈ.
• The linear approach thus gives the minimum possible ⌬ inf x in the Gaussian case.
• This was first experimentally achieved by Ou, Pereira, Kimble, and Peng ͑1992͒.

D. Experimental criteria for demonstrating the paradox

• The authors now summarize experimental criteria sufficient to realize the EPR paradox.
• To achieve this, one must have two spatially separated subsystems at A and B. ͑1͒.
• This point has been extensively discussed in literature on Bell's inequalities and is needed to justify the locality assumption, given that EPR assumed idealized instantaneous measurements.
• The EPR correlation is demonstrated when the product of the average errors in the inferred results x est and p est for x ând p ˆat A falls below a bound determined by the corresponding Heisenberg uncertainty principle.

V. THEORETICAL MODEL FOR A CONTINUOUS VARIABLE EPR EXPERIMENT

• As a physically realizable example of the original continuous variable EPR proposal, suppose the two systems A and B are localized modes of the electromagnetic field, with frequencies A,B and boson operators a ˆand b ˆ, respectively.
• As a result, these modes become correlated.
• The parameter r is called the squeezing parameter.
• The Schmidt decomposition, which is not unique, is a useful tool for identifying the pairs of EPR observables ͑Huang and Eberly, 1993; Ekert and Knight, 1995; Law et al., 2000͒.
• Giovannetti et al. ͑2001͒ presented an exciting scheme for demonstrating the EPR paradox for massive objects using radiation pressure acting on an oscillating mirror.

B. Measurement techniques

• Quadrature phase amplitudes can be measured using homodyne detection techniques developed for the detection of squeezed light fields.
• In the experimental proposal of Drummond and Reid ͑1990͒, carried out by Ou, Pereira, Kimble, and Peng ͑1992͒, an intracavity nondegenerate downconversion scheme was used.
• Single timedomain modes are obtained through spectral filtering of the photocurrent.
• These behave effectively as described in the simple model given above, together with corrections for cavity detuning and nonlinearity that are negligible near resonance, and not too close to the critical threshold ͑Dechoum et al., 2004͒.
• The field quadrature amplitudes are symbolized Y and X.

C. Effects of loss and imperfect detectors

• Crucial to the validity of the EPR experiment is the accurate calibration of the correlation relative to the vacuum limit.
• In optical experiments, this limit is the vacuum noise level as defined within quantum theory.
• To provide a simple but accurate model of detection inefficiencies, the authors consider an imaginary beam splitter ͑Fig. 3͒ placed before the photodetector at each location and A/B gives the fractional homodyne efficiency due to optical transmission, mode matching and photodetector losses at A and B, respectively.
• Details of the modeling of the detection losses were also discussed by Ou, Pereira, and Kimble ͑1992͒.
• Since the loss model is linear, the final state, although no longer pure, is Gaussian, Eq. ͑27͒.

VI. EPR, ENTANGLEMENT, AND BELL CRITERIA

• The authors have understood a "demonstration of the EPR paradox" to be a procedure that closely follows the original EPR gedanken experiment.
• Most generally, the EPR paradox is demonstrated when one can confirm the inconsistency between local realism and the completeness of quantum mechanics, since this was the underlying EPR objective.
• The authors point out in this section that the inconsistency can be shown in more ways than one.
• There are many uncertainty relations or constraints placed on the statistics of a quantum state, and for each such relation there is an EPR criterion.
• This has been discussed for the case of entanglement by Gühne ͑2004͒, and for EPR by Cavalcanti and Reid ͑2007͒.

B. Symmetric EPR paradox

• Where the authors violate the condition ͑5͒ for separability, to demonstrate entanglement, it is necessarily the case that the parameters for each localized system cannot be represented as a quantum state.
• The demonstration of entanglement, for sufficient spatial separations, gives inconsistency of Bell's local realism with completeness of quantum mechanics, and the authors provide an explicit link between entanglement and the EPR paradox.

C. EPR as a special type of entanglement

• While generalizations of the paradox have been presented, the authors propose to reserve the title "EPR paradox" for those experiments that minimally extend the original EPR argument, so that criteria given in Sec. IV are satisfied.
• That an EPR paradox implies entanglement is most readily seen by noting that a separable ͑nonentangled͒ source, as given by Eq. ͑4͒, represents a local realistic description in which the localized systems A and B are described as quantum states ˆ.
• This was first carried out experimentally by Ou, Pereira, Kimble, and Peng ͑1992͒.
• Further criteria sufficient to prove entanglement for continuous variable measurements were presented by Simon ͑2000͒ and Duan et al. ͑2001͒, who adapted the positive partial transpose ͑PPT͒ criterion of Peres ͑1996͒.

A. Parametric oscillator experiments

• The first continuous variable test of the EPR paradox was performed by Ou, Pereira, Kimble, and Peng ͑1992͒.
• This is well above the 50% efficiency threshold required for EPR.
• This issue, combined with the restricted detector separations used to date, means that a true causally separated EPR experiment is yet to be carried out, although this is certainly not impossible.
• This proposal uses cavities which are single mode in the vicinity of each of the resonant frequencies, so modes must be spatially separated after output from the cavity.
• These are in an approximate two-mode squeezed state, with the quadrature operators as given by Eq. ͑26͒.

B. Experimental results

• In reality, the authors are restricted to the physically achievable case where losses do exist, and the high nonlinearities required for extremely high gains are difficult to obtain.
• The pump field for each optical parametric amplifier was produced by frequency doubling an Nd:YAG laser to 532 nm.
• These results were verified by calibrating the loss.
• Dualbeam second-harmonic generation can also theoretically produce EPR correlations ͑Lim and Saffman, 2006͒.
• The EPR paradox was tested for the actual position and momentum of single photons ͑Fedorov et al., 2004 , 2006; Guo and Guo, 2006͒ by Howell et al. ͑2004͒ to realize an experiment more in direct analogy with the original EPR paradox.

VIII. PULSED EPR EXPERIMENTS

• One solution is to place the nonlinear medium inside a cavity, as discussed above, and another one, which will be discussed in this section, is to use high power pump laser pulses.
• Using such a source the effective interaction length can be dramatically shortened.
• The high finesse cavity conditions can be relaxed or for extreme high peak power pulses, the use of a cavity can be completely avoided.
• Broadband entanglement is of particular importance for the field of quantum information science, where, for example it allows for fast communication of quantum states by means of quantum teleportation ͑Sec. X͒.
• This may also allow truly causal EPR experiments, which are yet to be carried out.

A. Optical fiber experiment

• The first experimental realization of pulsed EPR entanglement, shown in Fig. 7 , was based on the approach of mixing two squeezed beams on a 50/ 50 beam splitter as outlined above for continuous wave light.
• 1987; Rosenbluh and Shelby, 1991͒ along two orthogonal polarization axes of the same polarization maintaining fiber ͑Silberhorn et al., 2001͒.
• ͑2006͔͒ to generate two spatially separated EPR modes possessing quantum correlations between the amplitude quadratures and the phase quadratures.
• Which together with the Kerr effect enable soliton formation at a certain threshold pulse energy, thereby ensuring a constant peak power level of the pulses along the fiber.
• This fact renders the verification procedure of proving EPR entanglement somewhat more difficult since standard homodyne detectors cannot be used.

B. Parametric amplifier experiment

• An alternative approach, which does not involve GAWBS, is the use of pulsed downconversion.
• The output of the parametric amplifier was then a pulsed two-mode squeezed vacuum state with a pulse duration of 150 fs and a repetition rate of 780 kHz.
• Without correcting for detector inefficiencies the authors deduce that the EPR paradox was not demonstrated in this experiment since the product of the conditional variances amounts to 2 = 1.06.
• In the experiment by Silberhorn et al. ͑2001͒, measurements were performed in the frequency domain similar to the previously discussed coutinuous wave experiments:.
• The frequency bandwidth of the detection system was too small to resolve successive pulses, which arrived at the detector with a frequency of 163 MHz.

IX. SPIN EPR AND ATOMS

• Experimental realizations of the paradox with massive particles are important, both due to their closeness in spirit with the original EPR proposal and because such massive entities could reasonably be considered more closely bound to the concept of local realism than fields.
• Here the authors focus on experiments based on atomic ensembles, which have shown the most promise for tests of the EPR paradox.
• These include the use of buffer gases ͑Phillips et al., 2001͒ and paraffin coatings ͑Julsgaard et al., 2001͒ in room temperature vapor cells to minimize collisions between atoms and the effect of wall collisions, respectively, and the use of cold atoms in magneto-optic traps ͑Geremia et al., 2004͒.
• These techniques have led to long decoherence times of the order of 1 ms for the collective spin states.

A. Transfer of optical entanglement to atomic ensembles

• Polzik ͑1999͒ showed that the optical entanglement generated by a parametric oscillator, as described in Sec. VII, could be transferred to the collective spin state of a pair of distant atomic ensembles.
• In both cases, however, at least 50% loss was introduced due to spontaneous emission.
• As discussed in Sec. V, the EPR paradox cannot be tested when symmetric losses exceed 50%.
• The first experimental demonstration of quantum state transfer from the polarization state of an optical field to the collective spin state of an atomic ensemble was performed by Hald et al. ͑1999͒.
• The extension of these results to pairs of spatially separated entangled ensembles has yet to be performed experimentally.

B. Conditional atom ensemble entanglement

• The other approach to experimental demonstration of collective spin entanglement in atomic ensembles is to rely on conditioning measurements to prepare the state ͑Julsgaard et al., 2004; Chou et al., 2005͒.
• This approach has the advantage of not requiring any nonclassical optical resources.
• A subsequent experiment along these lines by Geremia et al. ͑2004͒ utilized control techniques to further enhance the generation of QND-based collective spin squeezing.
• This conditioned the state of the atomic ensembles into a collective entangled state of the type required to test the EPR paradox.
• The principle of the experiment by van der Wal et al. ͑2003͒ was the same.

X. APPLICATION OF EPR ENTANGLEMENT

• A review of continuous variable quantum information protocols has been given by Braunstein and van Loock ͑2005͒.
• The authors focus on three continuous variable quantum information protocols that utilize shared EPR entanglement between two parties.
• They are entanglement-based quantum key distribution, quantum teleportation, and entanglement swapping.

A. Entanglement-based quantum key distribution

• In quantum key distribution ͑QKD͒, a sender ͑Alice͒ wants to communicate with a receiver ͑Bob͒ in secrecy.
• They achieve this by first cooperatively finding a method to generate a secret key that is uniquely shared between the two of them.
• Figure 3 shows that the EPR paradox can be demonstrated when Alice and Bob get together to perform conditional variance measurements of the quadrature amplitudes of a pair of entangled beams.
• Alice keeps one of the entangled beams and transmits the other to Bob.
• Since the results of measurements between Alice and Bob are never perfectly identical, Alice and Bob are required to reconcile the results of their measurements.

B. Quantum teleportation and entanglement swapping

• Figure 8 gives the schematic of the protocol.
• A well-accepted measure of teleportation efficacy is the overlap of the wave function of the output state with the original input state.
• Victor verifies the efficacy of entanglement swapping using conditional variance measurements of his entangled beam with Bob's teleportation output beam.
• Measurements of fidelity have to be averaged over a significant region of the quadrature amplitude phase space before the suggested bounds are valid classical and no-cloning limits.

XI. OUTLOOK

• The Einstein-Podolsky-Rosen gedanken experiment has been realized through a series of important developments, both theoretical and technological.
• Experiments have measured violation of the inferred Heisenberg uncertainty principle, thus confirming EPR-entanglement.
• A question often arising is the utility of such measurements, given that Bell inequality violations are a more powerful indication of the failure of local realism.
• The beauty of the EPR approach is its simplicity, from both a theoretical and a practical point of view.

Figures

FIG. 4. Effect of detector efficiencies A and B on the EPR paradox. The inference error product = infXA infYA is plotted for a two-mode squeezed state with r=2: A= B= solid line ; fixed A=1 but varying = B dashed line ; fixed B=1 but varying = A dash-dotted line . The EPR paradox is sensitive to the losses B of the steering system B, but insensitive to A, those of the inferred system A. No paradox is possible for B 0.5, regardless of A, but a paradox is always possible with B 0.5, provided only A 0.

FIG. 6. EPR paradox measure and entanglement measures: a 2 Eqs. 23 , 20 , and 33 and b normalized entanglement measure D of Duan et al. Eq. 34 vs total efficiency . The dashed lines are theoretical predictions for 2 and D. The points are experimental data with error bars. It is more difficult to realize the EPR paradox than to demonstrate entanglement. From Bowen, Schnabel, et al., 2003.

FIG. 9. A history of continuous variable EPR experiments: a EPR paradox measure 2 Eq. 23 and b inseparability measure D Eq. 34 . Where D 0.5, one can infer an EPR paradox, using 2D Sec. VI . The grey labels in a indicate that 2 has not been measured directly, but is inferred by the authors. From b we see that an EPR paradox could have been inferred in other experiments as well. i Ou, Pereira, Kimble, and Peng 1992 , ii Zhang et al. 2000 inferred from a variance product measurement , iii Silberhorn et al. 2001 , iv Julsgaard et al. 2001 , v Schori et al. 2002 , vi Bowen, Treps, et al. 2002 , vii Bowen, Schnabel, et al. 2003 , viii Glöckl et al. 2003 , ix Josse et al. 2004 , x Hayasaka et al. 2004 , xi Takei et al. 2005 , xii Laurat et al. 2005 , xiii Wenger et al. 2005 , xiv Huntington et al. 2005 , xv Villar et al. 2005 , xvi Nandan et al. 2006 , xvii Jing et al. 2006 , xviii Takei et al. 2006 , xix Yoshino et al. 2007 , xx Zhang et al. 2007 , xxi Dong et al. 2007 , xxii Keller et al. 2008 , xxiii Grosse et al. 2008 , xxiv Wagner et al. 2008 , and xxv Boyer et al. 2008 . Inseparability has also been verified using other measures, such as negativity Ourjoumtsev et al., 2007 .

FIG. 3. Color online Schematic diagram of the measurement of the EPR paradox using field quadrature phase amplitudes. Spatially separated fields A and B radiate outwards from the EPR source, usually given by Eq. 24 . The field quadrature amplitudes are symbolized Y and X. The fields combine with an intense local oscillator LO field, at beam splitters BS. The outputs of each BS are detected by photodiodes and their difference current is proportional to the amplitude Y or X, depending on the phase shift . A gain g is introduced to read out the final conditional variances, Eq. 30 . Here A and B are the nonideal efficiencies that model losses, defined in Sec. V.

FIG. 5. The original EPR parametric downconversion experiment using an intracavity nonlinear crystal and homodyne detection, following the procedure depicted in Fig. 3. Figure reprinted from Ou, Pereira, Kimble, and Peng, 1992, with permission.

FIG. 1. Color online The original EPR gedanken experiment. Two particles move from a source S into spatially separated 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.

FIG. 8. Color online Schematic of quantum teleportation and entanglement swapping. In teleportation, Alice and Bob share a pair of entangled beams. in is the input state Alice teleports to Bob. The use of electro-optic feedforward on both the amplitude and phase quadrature on Bob’s entangled beam produces an output state out , which he measures using optical homodyne detection, as in Fig. 3. In entanglement swapping, Alice and Victor also share a pair of entangled beams. Alice uses her share of this pair as the input state in . The teleportation protocol is again performed. Victor verifies the efficacy of entanglement swapping using conditional variance measurements of his entangled beam with Bob’s teleportation output beam. The elements are: beam splitters BS, local oscillator LO, phase shift , difference/sum currents / . Semicircles are photodiodes, while triangles show electronic gain.

FIG. 2. Color online The Bohm gedanken EPR experiment. Two spin-12 particles prepared in a singlet state move from the source into spatially separated regions A and B, and give anticorrelated outcomes for J A and J B, where is x, y, or z.

FIG. 7. The original demonstration of pulsed EPR entanglement. The soliton experiment uses orthogonal polarization modes in a fiber Sagnac interferometer and a Mach-Zehnder interferometer for fiber-birefringence compensation. Notation: /2 is half-wave plate; G is a gradient index len; 50/50 is beam splitter of 50% reflectivity; ŝ and p̂ are two amplitude squeezed beams from the respective polarization states; â and b̂ are EPR entangled beams. From Silberhorn et al., 2001.

Full text

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
1728
Reid et al.: Colloquium: The Einstein-Podolsky-Rosen …
Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009

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
d␭P共␭兲
␳
ˆ
␭
A
丢
␳
ˆ
␭
B
, 共4兲
where 兰d␭P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
d␭P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
␭
d␭P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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Frequently asked questions

Q1. What have the authors contributed in "Colloquium: the einstein-podolsky-rosen paradox: from concepts to applications" ?

This Colloquium examines the field of the Einstein, Podolsky, and Rosen EPR 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. 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. 

Q2. What is the squeezing parameter in steady-state continuous-wave experiments?

In these steady-state continuous-wave experiments, however, the squeezing parameter r is time independent, and given by the inputoutput parametric gain G, such that G=e2r. 

Q3. Why was the use of a cavity circumvented?

Due to the high peak powers of the frequency doubled pulses as well as the particular choice of a highly nonlinear optical material KNBO3 , the use of a cavity was circumvented despite the fact that a very thin 100 m crystal was employed. 

Q4. How can the authors test quantum theory for increasingly macroscopic systems?

Using fieldquadrature measurements and multiparticle states, it is likely that quantum theory and its alternatives can be tested for increasingly macroscopic systems Marshall et al., 2003 using the EPR paradox. 

Q5. What was the recent use of a traveling-wave OPA?

A degenerate waveguide technique, together with a beam splitter, was recently used to demonstrate pulsed entanglement using a traveling-wave OPA Zhang et al., 2007 . 

Q6. What is the decoherence rate of atomic ensembles?

One might expect that since spin-squeezed and entangled atomic ensembles contain a large number N of atoms, the decoherence rate of such systems would scale as N , where is the single-atom decay rate. 

Q7. What is the critical feature of the collective spin states?

a critical feature of these collective spin states is that excitation due to interaction with light is distributed symmetrically amongst all of the atoms. 

Q8. What is the way to measure teleportation efficacy?

A well-accepted measure of teleportation efficacy is the overlap of the wave function of the output state with the original input state. 

Q9. What is the logical choice for labeling the element of reality by the outcomes?

Since the set of predicted distributions are the conditionals P x xB , one for each value of xB, the logical choice is to label the element of reality by the outcomes xB, but bearing in mind the set of predetermined results is not the set xB , but is the set of associated conditional distributions P x xB . 

Q10. How many fidelitys can be obtained using classical measure and regenerate strategies?

For a Gaussian distribution of coherent states, with mean photon number n̄, the average fidelity using classical measure and regenerate strategies is limited to F n̄+1 / 2n̄+1 

Q11. What frequency band was used to characterize the quantum noise properties?

The quantum noise properties were characterized at a specific Fourier component within a narrow frequency band, typically in the range 100–300 kHz. 

Q12. What is the first experimental realization of pulsed EPR entanglement?

The first experimental realization of pulsed EPR entanglement, shown in Fig. 7, was based on the approach of mixing two squeezed beams on a 50/50 beam splitteras outlined above for continuous wave light. 

Q13. How many entangled beams did Silberhorn et al. find a degree?

The symmetry of the entangled beams allowed one to infer from this number the degree of EPR violation, which was found to be 2=0.64±0.08. 

Q14. What are the techniques that can be used to test the EPR paradox?

These techniques have significant potential for quantum information networks Duan et al., 2001 and are also capable of generating a collective entangled state of the form required to test the EPR paradox. 

Q15. What is the other approach to demonstrating collective spin entanglement in atomic ensembles?

The other approach to experimental demonstration of collective spin entanglement in atomic ensembles is to rely on conditioning measurements to prepare the state Julsgaard et al., 2004; Chou et al., 2005 . 

Q16. What was the optimum level of EPR paradox achieved to date?

The optimum level of EPR paradox achieved to date was by Bowen, Schnabel, et al. 2003 using a pair of type The authoroptical parametric oscillators. 

Q17. Who performed the first experimental demonstration of quantum state transfer from the polarization state of an optical?

The first experimental demonstration of quantum state transfer from the polarization state of an optical field to the collective spin state of an atomic ensemble was performed by Hald et al.