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Early on and influential critique leveled against quantum mechanics

The Einstein–Podolsky–Rosen paradox (EPR paradox) is a idea experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR), with which they argued that the description of physical reality provided past quantum mechanics was incomplete.^[1] In a 1935 paper titled "Can Breakthrough-Mechanical Description of Concrete Reality be Considered Complete?", they argued for the being of "elements of reality" that were not role of quantum theory, and speculated that it should be possible to construct a theory containing them. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

The thought experiment involves a pair of particles prepared in what later authors would refer to equally an entangled state. Einstein, Podolsky, and Rosen pointed out that, in this state, if the position of the first particle were measured, the result of measuring the position of the second particle could be predicted. If instead the momentum of the first particle were measured, then the outcome of measuring the momentum of the second particle could be predicted. They argued that no activity taken on the first particle could instantaneously affect the other, since this would involve information being transmitted faster than light, which is forbidden by the theory of relativity. They invoked a principle, afterward known as the "EPR benchmark of reality", positing that, "If, without in any way disturbing a arrangement, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, and so there exists an element of reality corresponding to that quantity." From this, they inferred that the 2nd particle must have a definite value of both position and of momentum prior to either being measured. But in quantum mechanics these 2 observables are incompatible and information technology therefore does not acquaintance simultaneous values for both to any arrangement. Therefore, Einstein, Podolsky, and Rosen concluded that breakthrough theory did not provide a complete description of reality.^[two]

History [edit]

The work was done at the Plant for Advanced Study in 1934, which Einstein had joined the previous year, later on he had fled the rise of Nazi Frg.^{[3] [4]} The resulting paper was written by Podolsky, and while Einstein was listed as an writer he did non experience it properly represented his view.^[five] The publication of the newspaper prompted a response by Niels Bohr, which he published in the same periodical, in the aforementioned yr, using the aforementioned championship.^[6] This exchange was only one chapter in a prolonged debate between Bohr and Einstein about the fundamental nature of reality.

Einstein struggled unsuccessfully for the rest of his life to find a theory that could better comply with his thought of locality. Since his decease, experiments analogous to the one described in the EPR paper have been carried out (notably past the group of Alain Aspect in the 1980s) that have confirmed that physical probabilities, as predicted by quantum theory, do exhibit the phenomena of Bell-inequality violations that are considered to invalidate EPR's preferred "local hidden-variables" type of explanation for the correlations to which EPR first drew attention.^{[7] [8]}

Paradox [edit]

The original paper^[nine] purports to describe what must happen to "two systems I and Ii, which we allow to interact", and subsequently some time "we suppose that there is no longer any interaction between the two parts." The EPR clarification involves "two particles, A and B, [which] interact briefly and and then motility off in contrary directions."^[10] According to Heisenberg'south doubt principle, information technology is impossible to measure out both the momentum and the position of particle B exactly; even so, it is possible to measure out the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B tin be known. Alternatively, the exact momentum of particle A tin can be measured, so the exact momentum of particle B can be worked out. Equally Manjit Kumar writes, "EPR argued that they had proved that ... [particle] B can have simultaneously verbal values of position and momentum. ... Particle B has a position that is real and a momentum that is real. EPR appeared to have contrived a means to establish the exact values of either the momentum or the position of B due to measurements made on particle A, without the slightest possibility of particle B being physically disturbed."^[x]

EPR tried to set a paradox to question the range of true awarding of quantum mechanics: Quantum theory predicts that both values cannot exist known for a particle, and however the EPR thought experiment purports to show that they must all have determinate values. The EPR paper says: "Nosotros are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not consummate."^[10] The EPR newspaper ends by saying: "While we accept thus shown that the wave part does not provide a complete clarification of the physical reality, nosotros left open the question of whether or non such a clarification exists. Nosotros believe, all the same, that such a theory is possible." The 1935 EPR paper condensed the philosophical word into a physical statement. The authors claim that given a specific experiment, in which the outcome of a measurement is known before the measurement takes place, there must exist something in the real world, an "element of reality", that determines the measurement outcome. They postulate that these elements of reality are, in modernistic terminology, local, in the sense that each belongs to a certain betoken in spacetime. Each element may, again in modern terminology, only exist influenced by events which are located in the backward light cone of its betoken in spacetime, i.e. the past). These claims are founded on assumptions near nature that constitute what is now known equally local realism.

Article headline regarding the EPR paradox paper in the May 4, 1935, issue of The New York Times.

Though the EPR newspaper has often been taken as an exact expression of Einstein's views, information technology was primarily authored past Podolsky, based on discussions at the Plant for Advanced Study with Einstein and Rosen. Einstein later on expressed to Erwin Schrödinger that, "information technology did not come out likewise as I had originally wanted; rather, the essential matter was, so to speak, smothered by the ceremonial."^[11] Einstein would afterward keep to nowadays an individual account of his local realist ideas.^[12] Shortly before the EPR paper appeared in the Physical Review, The New York Times ran a news story about it, under the headline "Einstein Attacks Quantum Theory".^[13] The story, which quoted Podolsky, irritated Einstein, who wrote to the Times, "Any data upon which the article 'Einstein Attacks Breakthrough Theory' in your issue of May 4 is based was given to you without authority. It is my invariable exercise to discuss scientific matters only in the advisable forum and I deprecate advance publication of whatsoever announcement in regard to such matters in the secular press."^{[14] : 189}

The Times story also sought out comment from physicist Edward Condon, who said, "Of course, a cracking deal of the argument hinges on just what meaning is to be fastened to the word 'reality' in physics."^{[xiv] : 189} The physicist and historian Max Jammer later noted, "[I]t remains a historical fact that the earliest criticism of the EPR paper — moreover, a criticism which correctly saw in Einstein's conception of concrete reality the key problem of the whole issue — appeared in a daily newspaper prior to the publication of the criticized newspaper itself."^{[14] : 190}

Bohr's reply [edit]

Bohr's response to the EPR paper was published in the Concrete Review later in 1935.^[6] He argued that EPR had reasoned fallaciously. Because measurements of position and of momentum are complementary, making the pick to mensurate one excludes the possibility of measuring the other. Consequently, a fact deduced regarding one arrangement of laboratory apparatus could not be combined with a fact deduced by means of the other, and so, the inference of predetermined position and momentum values for the 2nd particle was not valid. Bohr concluded that EPR's "arguments practice not justify their conclusion that the quantum clarification turns out to be essentially incomplete."

Einstein's ain argument [edit]

In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.^{[5] [15] [xvi] [17] : 83ff} He explicitly de-emphasized EPR's attribution of "elements of reality" to the position and momentum of particle B, saying that "I couldn't care less" whether the resulting states of particle B allowed one to predict the position and momentum with certainty.^[a]

For Einstein, the crucial role of the statement was the demonstration of nonlocality, that the choice of measurement washed in particle A, either position or momentum, would lead to ii different quantum states of particle B. He argued that, because of locality, the real state of particle B could not depend on which kind of measurement was done in A and that the quantum states therefore cannot be in ane-to-one correspondence with the real states.^[5]

Later developments [edit]

Bohm's variant [edit]

In 1951, David Bohm proposed a variant of the EPR thought experiment in which the measurements have discrete ranges of possible outcomes, dissimilar the position and momentum measurements considered by EPR.^{[18] [19] [20]} The EPR–Bohm idea experiment tin can exist explained using electron–positron pairs. Suppose we have a source that emits electron–positron pairs, with the electron sent to destination A, where in that location is an observer named Alice, and the positron sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source and then that each emitted pair occupies a breakthrough country called a spin singlet. The particles are thus said to be entangled. This tin can be viewed as a quantum superposition of two states, which we telephone call state I and land Two. In state I, the electron has spin pointing upwardly along the z-axis (+z) and the positron has spin pointing down along the z-centrality (−z). In state II, the electron has spin −z and the positron has spin +z. Because it is in a superposition of states, it is impossible without measuring to know the definite country of spin of either particle in the spin singlet.^{[21] : 421–422}

The EPR idea experiment, performed with electron–positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to Bob (right), who can perform spin measurements.

Alice now measures the spin forth the z-centrality. She can obtain one of two possible outcomes: +z or −z. Suppose she gets +z. Informally speaking, the quantum land of the system collapses into state I. The quantum land determines the likely outcomes of whatsoever measurement performed on the system. In this instance, if Bob later on measures spin along the z-axis, at that place is 100% probability that he will obtain −z. Similarly, if Alice gets −z, Bob will go +z. In that location is nothing special about choosing the z-centrality: according to quantum mechanics the spin singlet land may equally well be expressed every bit a superposition of spin states pointing in the x direction.^{[22] : 318} Suppose that Alice and Bob had decided to mensurate spin along the ten-axis. We'll telephone call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob'due south positron has spin −x. In country IIa, Alice'due south electron has spin −10 and Bob'south positron has spin +10. Therefore, if Alice measures +x, the system 'collapses' into land Ia, and Bob volition get −x. If Alice measures −x, the system collapses into state IIa, and Bob volition get +x.

Whatever axis their spins are measured along, they are always constitute to be opposite. In quantum mechanics, the ten-spin and z-spin are "incompatible observables", meaning the Heisenberg uncertainty principle applies to alternating measurements of them: a quantum state cannot possess a definite value for both of these variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. At present, instead of measuring the z-spin as well, Bob measures the x-spin. Co-ordinate to quantum mechanics, when the organisation is in state I, Bob's x-spin measurement will have a fifty% probability of producing +10 and a l% probability of -x. It is incommunicable to predict which effect will appear until Bob really performs the measurement. Therefore, Bob's positron volition have a definite spin when measured along the same axis as Alice's electron, just when measured in the perpendicular axis its spin volition be uniformly random. It seems as if information has propagated (faster than light) from Alice's apparatus to make Bob'south positron assume a definite spin in the appropriate axis.

Bell's theorem [edit]

In 1964, John Stewart Bell published a paper^[7] investigating the puzzling situation at that time: on i hand, the EPR paradox purportedly showed that quantum mechanics was nonlocal, and suggested that a hidden-variable theory could heal this nonlocality. On the other manus, David Bohm had recently developed the first successful hidden-variable theory, but it had a grossly nonlocal character.^{[23] [24]} Bell fix out to investigate whether it was indeed possible to solve the nonlocality problem with subconscious variables, and constitute out that first, the correlations shown in both EPR's and Bohm'due south versions of the paradox could indeed exist explained in a local way with subconscious variables, and second, that the correlations shown in his own variant of the paradox couldn't be explained by any local subconscious-variable theory. This second effect became known every bit the Bell theorem.

To empathise the first result, consider the following toy hidden-variable theory introduced later by J.J. Sakurai:^{[25] : 239–240} in information technology, quantum spin-singlet states emitted by the source are really estimate descriptions for "truthful" physical states possessing definite values for the z-spin and 10-spin. In these "true" states, the positron going to Bob always has spin values opposite to the electron going to Alice, only the values are otherwise completely random. For example, the outset pair emitted by the source might exist "(+z, −x) to Alice and (−z, +10) to Bob", the next pair "(−z, −ten) to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement centrality is aligned with Alice'south, he will necessarily go the opposite of whatever Alice gets; otherwise, he volition become "+" and "−" with equal probability.

Bell showed, however, that such models can simply reproduce the singlet correlations when Alice and Bob brand measurements on the same axis or on perpendicular axes. As soon as other angles betwixt their axes are allowed, local subconscious-variable theories become unable to reproduce the quantum mechanical correlations. This difference, expressed using inequalities known as "Bell's inequalities", is in principle experimentally testable. After the publication of Bell's newspaper, a variety of experiments to test Bell's inequalities were devised. All experiments conducted to date accept found beliefs in line with the predictions of quantum mechanics.^[8] The present view of the situation is that quantum mechanics flatly contradicts Einstein'due south philosophical postulate that any adequate physical theory must fulfill "local realism". The fact that breakthrough mechanics violates Bell inequalities indicates that any hidden-variable theory underlying breakthrough mechanics must be not-local; whether this should exist taken to imply that quantum mechanics itself is non-local is a matter of debate.^{[26] [27]}

Steering [edit]

Inspired by Schrödinger'due south handling of the EPR paradox back in 1935,^{[28] [29]} Wiseman et al. formalised it in 2007 as the phenomenon of quantum steering.^[xxx] They defined steering as the situation where Alice's measurements on a part of an entangled state steer Bob's office of the land. That is, Bob'due south observations cannot be explained by a local hidden land model, where Bob would accept a fixed breakthrough land in his side, that is classically correlated merely otherwise independent of Alice's.

Locality in the EPR paradox [edit]

Locality has several different meanings in physics. EPR depict the principle of locality as asserting that physical processes occurring at one place should have no immediate consequence on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to exist a result of special relativity, which states that energy tin can never be transmitted faster than the speed of calorie-free without violating causality;^{[21] : 427–428 [31]} however, information technology turns out that the usual rules for combining quantum mechanical and classical descriptions violate EPR's principle of locality without violating special relativity or causality.^{[21] : 427–428 [31]} Causality is preserved because at that place is no mode for Alice to transmit messages (i.e., information) to Bob by manipulating her measurement centrality. Whichever axis she uses, she has a 50% probability of obtaining "+" and fifty% probability of obtaining "−", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what upshot she gets. Furthermore, Bob is but able to perform his measurement once: in that location is a fundamental holding of quantum mechanics, the no-cloning theorem, which makes it impossible for him to brand an capricious number of copies of the electron he receives, perform a spin measurement on each, and await at the statistical distribution of the results. Therefore, in the 1 measurement he is allowed to brand, at that place is a 50% probability of getting "+" and fifty% of getting "−", regardless of whether or not his axis is aligned with Alice's.

As a summary, the results of the EPR thought experiment do not contradict the predictions of special relativity. Neither the EPR paradox nor whatsoever quantum experiment demonstrates that superluminal signaling is possible; however, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky activity at a distance".^[b] The conclusion they drew was that quantum mechanics is not a complete theory.^[33]

Mathematical formulation [edit]

Bohm's variant of the EPR paradox can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional complex vector infinite Five, with each breakthrough land corresponding to a vector in that infinite. The operators respective to the spin along the 10, y, and z management, denoted S_x , S_y , and Due south_z respectively, tin can be represented using the Pauli matrices:^{[25] : 9}

$${\displaystyle S_{x}={\frac {\hbar }{two}}{\brainstorm{bmatrix}0&1\\1&0\end{bmatrix}},\quad S_{y}={\frac {\hbar }{2}}{\brainstorm{bmatrix}0&-i\\i&0\end{bmatrix}},\quad S_{z}={\frac {\hbar }{2}}{\begin{bmatrix}ane&0\\0&-1\terminate{bmatrix}}}$$

where ${\displaystyle \hbar }$ is the reduced Planck constant (or the Planck abiding divided by 2π).

The eigenstates of Southward_z are represented as

$${\displaystyle \left|+z\right\rangle \leftrightarrow {\begin{bmatrix}ane\\0\terminate{bmatrix}},\quad \left|-z\correct\rangle \leftrightarrow {\begin{bmatrix}0\\ane\end{bmatrix}}}$$

and the eigenstates of S_x are represented as

$${\displaystyle \left|+x\right\rangle \leftrightarrow {\frac {1}{\sqrt {ii}}}{\begin{bmatrix}i\\one\end{bmatrix}},\quad \left|-ten\right\rangle \leftrightarrow {\frac {1}{\sqrt {2}}}{\begin{bmatrix}i\\-1\end{bmatrix}}}$$

The vector infinite of the electron-positron pair is ${\displaystyle V\otimes V}$ , the tensor product of the electron's and positron's vector spaces. The spin singlet land is

$${\displaystyle \left|\psi \correct\rangle ={\frac {ane}{\sqrt {2}}}{\biggl (}\left|+z\right\rangle \otimes \left|-z\correct\rangle -\left|-z\correct\rangle \otimes \left|+z\right\rangle {\biggr )}}$$

where the two terms on the right paw side are what we accept referred to as land I and country 2 above.

From the above equations, it can be shown that the spin singlet can besides be written as

$${\displaystyle \left|\psi \right\rangle =-{\frac {one}{\sqrt {2}}}{\biggl (}\left|+10\right\rangle \otimes \left|-10\correct\rangle -\left|-x\right\rangle \otimes \left|+x\right\rangle {\biggr )}}$$

where the terms on the right hand side are what we have referred to as state Ia and state IIa.

To illustrate the paradox, we demand to bear witness that later Alice's measurement of S_z (or S_ten ), Bob's value of S_z (or Southward_x ) is uniquely adamant and Bob's value of S_x (or S_z ) is uniformly random. This follows from the principles of measurement in breakthrough mechanics. When South _z is measured, the system country ${\displaystyle |\psi \rangle }$ collapses into an eigenvector of S _z. If the measurement result is +z, this means that immediately afterward measurement the organization state collapses to

$${\displaystyle \left|+z\right\rangle \otimes \left|-z\correct\rangle =\left|+z\correct\rangle \otimes {\frac {\left|+ten\right\rangle -\left|-x\right\rangle }{\sqrt {2}}}}$$

Similarly, if Alice's measurement consequence is −z, the state collapses to

$${\displaystyle \left|-z\right\rangle \otimes \left|+z\right\rangle =\left|-z\correct\rangle \otimes {\frac {\left|+x\right\rangle +\left|-x\right\rangle }{\sqrt {two}}}}$$

The left manus side of both equations show that the measurement of Southward _z on Bob'southward positron is now adamant, it will exist −z in the first case or +z in the second case. The right hand side of the equations show that the measurement of S _x on Bob'due south positron will return, in both cases, +10 or -ten with probability 1/2 each.

See also [edit]

• Bohr-Einstein debates: The argument of EPR
• CHSH Bell test
• Coherence
• Correlation does non imply causation
• ER=EPR
• GHZ experiment
• Measurement trouble
• Philosophy of information
• Philosophy of physics
• Popper's experiment
• Superdeterminism
• Quantum entanglement
• Quantum information
• Quantum pseudo-telepathy
• Quantum teleportation
• Quantum Zeno effect
• Synchronicity
• Ward's probability amplitude

Notes [edit]

References [edit]

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2. ^ Peres, Asher (2002). Breakthrough Theory: Concepts and Methods. Kluwer. p. 149.
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4. ^ Levenson, Thomas (nine June 1917). "The Scientist and the Fascist". The Atlantic . Retrieved 28 June 2021.
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15. ^ ^{a b} Howard, D. (1985). "Einstein on locality and separability". Studies in History and Philosophy of Science Part A. 16 (3): 171–201. Bibcode:1985SHPSA..16..171H. doi:10.1016/0039-3681(85)90001-9.
16. ^ Sauer, Tilman (2007-12-01). "An Einstein manuscript on the EPR paradox for spin observables". Studies in History and Philosophy of Scientific discipline Role B: Studies in History and Philosophy of Modern Physics. 38 (4): 879–887. Bibcode:2007SHPMP..38..879S. CiteSeerX10.1.one.571.6089. doi:x.1016/j.shpsb.2007.03.002. ISSN 1355-2198.
17. ^ Einstein, Albert (1949). "Autobiographical Notes". In Schilpp, Paul Arthur (ed.). Albert Einstein: Philosopher-Scientist. Open Court Publishing Company.
18. ^ Bohm, D. (1951). Quantum Theory, Prentice-Hall, Englewood Cliffs, folio 29, and Affiliate 5 department 3, and Chapter 22 Department 19.
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20. ^ Reid, M. D.; Drummond, P. D.; Bowen, W. P.; Cavalcanti, Eastward. G.; Lam, P. Chiliad.; Bachor, H. A.; Andersen, U. L.; Leuchs, G. (2009-12-10). "Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications". Reviews of Modern Physics. 81 (4): 1727–1751. arXiv:0806.0270. Bibcode:2009RvMP...81.1727R. doi:10.1103/RevModPhys.81.1727. S2CID 53407634.
21. ^ ^{a b c} Griffiths, David J. (2004). Introduction to Breakthrough Mechanics (2d ed.). Prentice Hall. ISBN978-0-13-111892-8.
22. ^ Laloe, Franck (2012). "Do We Really Sympathise Quantum Mechanics". American Journal of Physics. 69 (6): 655–701. arXiv:quant-ph/0209123. Bibcode:2001AmJPh..69..655L. doi:10.1119/i.1356698. (Erratum: doi:10.1119/1.1466818)
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25. ^ ^{a b} Sakurai, J. J.; Napolitano, Jim (2010). Modern Quantum Mechanics (second ed.). Addison-Wesley. ISBN978-0805382914.
26. ^ Werner, R. F. (2014). "Comment on 'What Bell did'". Journal of Physics A. 47 (42): 424011. Bibcode:2014JPhA...47P4011W. doi:10.1088/1751-8113/47/42/424011. S2CID 122180759.
27. ^ Żukowski, M.; Brukner, Č. (2014). "Quantum non-locality—it ain't necessarily then...". Journal of Physics A. 47 (42): 424009. arXiv:1501.04618. Bibcode:2014JPhA...47P4009Z. doi:x.1088/1751-8113/47/42/424009. S2CID 119220867.
28. ^ Schrödinger, Due east. (October 1936). "Probability relations betwixt separated systems". Mathematical Proceedings of the Cambridge Philosophical Club. 32 (three): 446–452. Bibcode:1936PCPS...32..446S. doi:10.1017/s0305004100019137. ISSN 0305-0041. S2CID 122822435.
29. ^ Schrödinger, E. (Oct 1935). "Give-and-take of Probability Relations between Separated Systems". Mathematical Proceedings of the Cambridge Philosophical Society. 31 (4): 555–563. Bibcode:1935PCPS...31..555S. doi:10.1017/s0305004100013554. ISSN 0305-0041. S2CID 121278681.
30. ^ Wiseman, H. M.; Jones, S. J.; Doherty, A. C. (2007). "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox". Concrete Review Letters. 98 (14): 140402. arXiv:quant-ph/0612147. Bibcode:2007PhRvL..98n0402W. doi:x.1103/PhysRevLett.98.140402. ISSN 0031-9007. PMID 17501251. S2CID 30078867.
31. ^ ^{a b} Blaylock, Guy (Jan 2010). "The EPR paradox, Bell's inequality, and the question of locality". American Journal of Physics. 78 (1): 111–120. arXiv:0902.3827. Bibcode:2010AmJPh..78..111B. doi:10.1119/1.3243279. S2CID 118520639.
32. ^ Albert Einstein Max Born, Briefwechsel 1916-1955 (in German) (3 ed.). München: Langen Müller. 2005. p. 254.
33. ^ Bell, John (1981). "Bertlmann's socks and the nature of reality". J. Physique Colloques. C22: 41–62. Bibcode:1988nbpw.conf..245B.

Selected papers [edit]

• Eberhard, P. H. (1977). "Bell's theorem without subconscious variables". Il Nuovo Cimento B. Series eleven. 38 (1): 75–80. Bibcode:1977NCimB..38...75E. doi:10.1007/bf02726212. ISSN 1826-9877. S2CID 51759163.
• Eberhard, P. H. (1978). "Bell's theorem and the dissimilar concepts of locality". Il Nuovo Cimento B. Serial 11. 46 (two): 392–419. Bibcode:1978NCimB..46..392E. doi:10.1007/bf02728628. ISSN 1826-9877. S2CID 118836806.
• Einstein, A.; Podolsky, B.; Rosen, Due north. (1935-05-15). "Can Breakthrough-Mechanical Description of Physical Reality Be Considered Complete?" (PDF). Concrete Review. 47 (x): 777–780. Bibcode:1935PhRv...47..777E. doi:ten.1103/physrev.47.777. ISSN 0031-899X.
• Fine, Arthur (1982-02-01). "Hidden Variables, Joint Probability, and the Bell Inequalities". Physical Review Letters. 48 (5): 291–295. Bibcode:1982PhRvL..48..291F. doi:x.1103/physrevlett.48.291. ISSN 0031-9007.
• A. Fine, Do Correlations demand to exist explained?, in Philosophical Consequences of Breakthrough Theory: Reflections on Bell'due south Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986).
• Hardy, Lucien (1993-09-13). "Nonlocality for two particles without inequalities for almost all entangled states". Physical Review Messages. 71 (11): 1665–1668. Bibcode:1993PhRvL..71.1665H. doi:10.1103/physrevlett.71.1665. ISSN 0031-9007. PMID 10054467.
• 1000. Mizuki, A classical interpretation of Bell'southward inequality. Annales de la Fondation Louis de Broglie 26 683 (2001)
• Peres, Asher (2005). "Einstein, Podolsky, Rosen, and Shannon". Foundations of Physics. 35 (3): 511–514. arXiv:quant-ph/0310010. Bibcode:2005FoPh...35..511P. doi:10.1007/s10701-004-1986-vi. ISSN 0015-9018. S2CID 119556878.
• P. Pluch, "Theory for Breakthrough Probability", PhD Thesis Academy of Klagenfurt (2006)
• Rowe, K. A.; Kielpinski, D.; Meyer, 5.; Sackett, C. A.; Itano, W. 1000.; Monroe, C.; Wineland, D. J. (2001). "Experimental violation of a Bell'due south inequality with efficient detection". Nature. 409 (6822): 791–794. Bibcode:2001Natur.409..791R. doi:10.1038/35057215. hdl:2027.42/62731. ISSN 0028-0836. PMID 11236986. S2CID 205014115.
• Smerlak, Matteo; Rovelli, Carlo (2007-02-03). "Relational EPR". Foundations of Physics. 37 (3): 427–445. arXiv:quant-ph/0604064. Bibcode:2007FoPh...37..427S. doi:ten.1007/s10701-007-9105-0. ISSN 0015-9018. S2CID 11816650.

Books [edit]

• Bong, John S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN 0-521-36869-3.
• Fine, Arthur (1996). The Shaky Game: Einstein, Realism and the Quantum Theory. 2nd ed. Univ. of Chicago Printing.
• Gribbin, John (1984). In Search of Schrödinger'due south Cat. Black Swan. ISBN 978-0-552-12555-0
• Leaderman, Leon; Teresi, Dick (1993). The God Particle: If the Universe Is the Answer, What Is the Question? Houghton Mifflin Company, pp. 21, 187–189.
• Selleri, Franco (1988). Quantum Mechanics Versus Local Realism: The Einstein–Podolsky–Rosen Paradox. New York: Plenum Press. ISBN 0-306-42739-7.

External links [edit]

• Stanford Encyclopedia of Philosophy: The Einstein–Podolsky–Rosen Argument in Quantum Theory; 1.2 The argument in the text
• Internet Encyclopedia of Philosophy: "The Einstein-Podolsky-Rosen Argument and the Bong Inequalities"
• Stanford Encyclopedia of Philosophy: Abner Shimony (2019) "Bell's Theorem"
• EPR, Bell & Aspect: The Original References
• Does Bell's Inequality Principle rule out local theories of quantum mechanics? from the Usenet Physics FAQ
• Theoretical utilize of EPR in teleportation
• Effective use of EPR in cryptography
• EPR experiment with single photons interactive
• Spooky Actions At A Altitude?: Oppenheimer Lecture past Prof. Mermin
• Original paper

What Is Signaling Site Criterion,

Source: https://en.wikipedia.org/wiki/EPR_paradox

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