THE EINSTEIN-PODOLSKY-ROSEN PARADOX AND THE NATURE OF REALITY

Transcription

1 ARTICLE DE FOND THE EINSTEIN-PODOLSKY-ROSEN PARADOX AND THE NATURE OF REALITY BY SHOHINI GHOSE In 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 CAP- CASCA 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 <sghose@wlu.ca>, Assistant Professor, Department of Physics and Computer Science, Wilfrid Laurier University, 75 University Ave W, Waterloo, ON N2L 3C5 LA PHYSIQUE AU CANADA / Vol. 65, No. 4 ( oct. à déc ) C 199

2 THE EINSTEIN-PODOLSKY-ROSEN PARADOX... (GHOSE) 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: (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.) Given these seemingly reasonable assumptions, EPR asserted that (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 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-x 0. 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-x 0. 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). 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. THE NATURE OF REALITY 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 Einstein- Podolsky-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 200 C PHYSICS IN CANADA / VOL. 65, NO. 4 ( Oct.-Dec )

3 THE EINSTEIN-PODOLSKY-ROSEN PARADOX... (GHOSE)AA 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 H s V i + V s H i (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 45 o s, 135o i + 135o s 45o i (+45 o polarization of signal and 135 o polarization of idler, or 135 o polarization of signal photon and +45 o polarization of idler). A valid description along any polarization direction would thus be θ s, θ +90 o i + θ +90 o s, θ i. Quantum mechanics does not allow us to predict with certainty 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. 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 0 o or 45 o. Bob randomly chooses to measure the polarization of each photon along 22.5 o or 67.5 o. They then compute the value of the CHSH correlation function from their measurement outcomes. LA PHYSIQUE AU CANADA / Vol. 65, No. 4 ( oct. à déc ) C 201

4 THE EINSTEIN-PODOLSKY-ROSEN PARADOX... (GHOSE) 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] 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 [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 qrst 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- 202 C PHYSICS IN CANADA / VOL. 65, NO. 4 ( Oct.-Dec )

5 THE EINSTEIN-PODOLSKY-ROSEN PARADOX... (GHOSE)AA 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 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 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. REFERENCES 1. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, (1935). 2. J.S. Bell, Physics 1, (1964). 3. J.S. Bell, Rev. Mod. Phys. 38, (1966). 4. S.J. Freedman and J.F. Clauser, Phys. Rev. Lett. 28, (1972). 5. A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47, (1981). LA PHYSIQUE AU CANADA / Vol. 65, No. 4 ( oct. à déc ) C 203

6 THE EINSTEIN-PODOLSKY-ROSEN PARADOX... (GHOSE) 6. A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, (1982). 7. A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, (1982). 8. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000). 9. S. Ghose, N. Sinclair, S. Debnath, P. Rungta and R. Stock, Phys. Rev. Lett. 102, (2009). 10. J.J. Sakurai, Modern Quantum Mechanics, (Addison-Wesley, New York, 1994). 11. J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987). 12. E. Hecht, Optics, (Addison Wesley, 2001). 13. R.W. Boyd, Nonlinear Optics, (Academic Press, 2003). 14. P.G. Kwiat, P.H. Eberhard, A.M. Steinberg, and R.Y. Chiao, Phys. Rev. A 49, (1994). 15. P.G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 75, (1995). 16. J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt, Phys. Rev. Lett. 23, (1969). 17. A. Garg and N.D. Mermin, Phys. Rev. D 35, (1987). 18. J-A. Larsson, Phys. Rev. A 57, (1998). 19. M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, and W.M. Itano, C. Monroe, and D.J. Wineland, Nature 409, (2001). 20. G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, (1998). 21. R. Stock, N.S. Babcock, M.G.. Raizen, and B.C. Sanders, Phys. Rev. A (2008). 22. W. Rosenfeld et al., Advanced Science Letters, to appear (2009). 23. C.H. Bennett, G. Brassard, and N.D. Mermin, Phys Rev Lett. 65, (1992). 24. A.K. Ekert, Phys. Rev. Lett. 67, (1991). 25. D. Gottesman and H.-K. Lo, Physics Today 53, (2000). 26. Gilles Van Assche, Quantum Cryptography and Secret-Key Distillation, (Cambridge University Press, 2006). 27. J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Götzinger, and V. Sandoghdar, Nature 460, (2009). 28. C-F. Lee, D.A. Leigh, R.G. Pritchard, D. Schultz, S.J. Teat, G.A. Timco, and R.E.P. Winpenny, Nature 458, (2009). 29. L. DiCarlo et al., Nature 460, (2009). 30. M. Hofheinz et al., Nature 459, (28 May 2009). 31. R. Blatt and D.J. Wineland, Nature 453, (2008). 32. S. Olmschenk, D. Hayes, D.N. Matsukevich, P. Maunz, D.L. Moehring, and C. Monroe, quant-ph/ (2009). 33. G.-D. Lin, S.-L. Zhu, R. Islam, K. Kim, M.-S. Chang, S. Korenblit, C. Monroe, and L.-M. Duan, Europhysics Letters 86, (2009). 34. R. Raussendorf, D.E. Browne, and H.J. Briegel, Phys. Rev. A 68, (2003). 35. P. Lougovski, S.J. van Enk, arxiv: (2009). 36. C.-E. Bardyn, T.C.H. Liew, S. Massar, M. McKague, and V. Scarani, arxiv: (2009). 37. A. Acín, N. Gisin, and L. Masanes, Phys. Rev. Lett. 97, (2006). 38. W.T. Kim, and E.J. Son, Phys. Rev. A 71, (2005). 39. A. Cabello, A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 102, (2009). 40. N. Gisin in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle Essays in Honour of Abner Shimony, (Springer Netherlands, 2008). 204 C PHYSICS IN CANADA / VOL. 65, NO. 4 ( Oct.-Dec )