The Einstein–Podolsky–Rosen (EPR) paradox is a
thought experiment
A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...
proposed by physicists
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
,
Boris Podolsky and
Nathan Rosen
Nathan Rosen (; March 22, 1909 – December 18, 1995) was an American and Israeli physicist noted for his study on the structure of the hydrogen molecule and his collaboration with Albert Einstein and Boris Podolsky on entangled wave functions and ...
, which argues that the description of physical reality provided by
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
is incomplete.
In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these
hidden variables. Resolutions of the paradox have important implications for the
interpretation of quantum mechanics.
The thought experiment involves a pair of particles prepared in what would later become known as an
entangled state
State most commonly refers to:
* State (polity), a centralized political organization that regulates law and society within a territory
**Sovereign state, a sovereign polity in international law, commonly referred to as a country
**Nation state, a ...
. 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 result of measuring the momentum of the second particle could be predicted. They argued that no action taken on the first particle could instantaneously affect the other, since this would involve information being transmitted faster than light, which is impossible according to the
theory of relativity
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical ph ...
. They invoked a principle, later known as the "EPR criterion of reality", which posited that: "If, without in any way disturbing a system, we can predict with certainty (i.e., with
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity." From this, they inferred that the second particle must have a definite value of both position and of momentum prior to either quantity being measured. But quantum mechanics considers these two observables
incompatible and thus does not associate simultaneous values for both to any system. Einstein, Podolsky, and Rosen therefore concluded that quantum theory does not provide a complete description of reality.
The "Paradox" paper
The term "Einstein–Podolsky–Rosen paradox" or "EPR" arose from a paper written in 1934 after Einstein joined the
Institute for Advanced Study
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
, having
fled the rise of Nazi Germany.
The original paper purports to describe what must happen to "two systems I and II, which we permit to interact", and after some time "we suppose that there is no longer any interaction between the two parts." The EPR description involves "two particles, A and B,
hich
Ij () is a village in Golabar Rural District of the Central District in Ijrud County, Zanjan province, Iran
Iran, officially the Islamic Republic of Iran (IRI) and also known as Persia, is a country in West Asia. It borders Iraq ...
interact briefly and then move off in opposite directions."
According to
Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly; however, it is possible to measure the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Alternatively, the exact momentum of particle A can be measured, so the exact momentum of particle B can be worked out. As
Manjit Kumar writes, "EPR argued that they had proved that ...
articleB can have simultaneously exact 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."
[
EPR tried to set up a paradox to question the range of true application of quantum mechanics: quantum theory predicts that both values cannot be known for a particle, and yet the EPR thought experiment purports to show that they must both have determinate values. The EPR paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."][ The EPR paper ends by saying: "While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible." The 1935 EPR paper condensed the philosophical discussion into a physical argument. 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 modern terminology, ]local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
Arts, entertainment, and media
* ''Local'' (comics), a limited series comic book by Bria ...
, in the sense that each belongs to a certain point in spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. Each element may, again in modern terminology, only be influenced by events that are located in the backward light cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single Event (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...
of its point in spacetime (i.e. in the past). These claims are thus founded on assumptions about nature that constitute what is now known as local realism.
Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored by Podolsky, based on discussions at the Institute for Advanced Study
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
with Einstein and Rosen. Einstein later expressed to Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger ( ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was an Austrian-Irish theoretical physicist who developed fundamental results in quantum field theory, quantum theory. In particul ...
that, "it did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by the formalism." Einstein would later go on to present an individual account of his local realist ideas. Shortly before the EPR paper appeared in the ''Physical Review
''Physical Review'' is a peer-reviewed scientific journal. The journal was established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the Ame ...
,'' ''The New York Times
''The New York Times'' (''NYT'') is an American daily newspaper based in New York City. ''The New York Times'' covers domestic, national, and international news, and publishes opinion pieces, investigative reports, and reviews. As one of ...
'' ran a news story about it, under the headline "Einstein Attacks Quantum Theory". The story, which quoted Podolsky, irritated Einstein, who wrote to the ''Times,'' "Any information upon which the article 'Einstein Attacks Quantum Theory' in your issue of May 4 is based was given to you without authority. It is my invariable practice to discuss scientific matters only in the appropriate forum and I deprecate advance publication of any announcement in regard to such matters in the secular press."
The ''Times'' story also sought out comment from physicist Edward Condon
Edward Uhler Condon (March 2, 1902 – March 26, 1974) was an American nuclear physicist, a pioneer in quantum mechanics, and a participant during World War II in the development of radar and, very briefly, of nuclear weapons as part of the Ma ...
, who said, "Of course, a great deal of the argument hinges on just what meaning is to be attached to the word 'reality' in physics." The physicist and historian Max Jammer
Max Jammer (; born Moshe Jammer, ; 13 April 1915 – 18 December 2010), was an Israeli physicist and philosophy of physics, philosopher of physics. He was born in Berlin, Germany. He was Rector and Acting President at Bar-Ilan University from 19 ...
later noted, " remains a historical fact that the earliest criticism of the EPR paper – moreover, a criticism that correctly saw in Einstein's conception of physical reality the key problem of the whole issue – appeared in a daily newspaper prior to the publication of the criticized paper itself."
Bohr's reply
The publication of the paper prompted a response by Niels Bohr
Niels Henrik David Bohr (, ; ; 7 October 1885 – 18 November 1962) was a Danish theoretical physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the No ...
, which he published in the same journal (''Physical Review
''Physical Review'' is a peer-reviewed scientific journal. The journal was established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the Ame ...
''), in the same year, using the same title. (This exchange was only one chapter in a prolonged debate between Bohr and Einstein about the nature of quantum reality.)
He argued that EPR had reasoned fallaciously. Bohr said measurements of position and of momentum are complementary, meaning the choice to measure 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 second particle was not valid. Bohr concluded that EPR's "arguments do not justify their conclusion that the quantum description turns out to be essentially incomplete."
Einstein's own argument
In his own publications and correspondence, Einstein indicated that he was not satisfied with the EPR paper and that Podolsky had authored most of it. He later used a different argument to insist that quantum mechanics is an incomplete theory. 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.
For Einstein, the crucial part of the argument was the demonstration of nonlocality, that the choice of measurement done in particle A, either position or momentum, would lead to ''two 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 one-to-one correspondence with the real states. Einstein struggled unsuccessfully for the rest of his life to find a theory that could better comply with his idea of locality.
Later developments
Bohm's variant
In 1951, David Bohm
David Joseph Bohm (; 20 December 1917 – 27 October 1992) was an American scientist who has been described as one of the most significant Theoretical physics, theoretical physicists of the 20th centuryDavid Peat Who's Afraid of Schrödinger' ...
proposed a variant of the EPR thought experiment in which the measurements have discrete ranges of possible outcomes, unlike the position and momentum measurements considered by EPR. The EPR–Bohm thought experiment can be explained using electron–positron
The positron or antielectron is the particle with an electric charge of +1''elementary charge, e'', a Spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest mass, mass as an electron. It is the antiparticle (antimatt ...
pairs. Suppose we have a source that emits electron–positron pairs, with the electron sent to destination ''A'', where there is an observer named Alice
Alice may refer to:
* Alice (name), most often a feminine given name, but also used as a surname
Literature
* Alice (''Alice's Adventures in Wonderland''), a character in books by Lewis Carroll
* ''Alice'' series, children's and teen books by ...
, and the positron sent to destination ''B'', where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. The particles are thus said to be entangled. This can be viewed as a quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödi ...
of two states, which we call state I and state II. In state I, the electron has spin pointing upward along the ''z''-axis (+''z'') and the positron has spin pointing downward along the ''z''-axis (−''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 state of spin of either particle in the spin singlet.
Alice now measures the spin along the ''z''-axis. She can obtain one of two possible outcomes: +''z'' or −''z''. Suppose she gets +''z''. Informally speaking, the quantum state of the system collapses into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the ''z''-axis, there is 100% probability that he will obtain −''z''. Similarly, if Alice gets −''z'', Bob will get +''z''. There is nothing special about choosing the ''z''-axis: according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the ''x'' direction.[ ]
Whatever axis their spins are measured along, they are always found to be opposite. In quantum mechanics, the ''x''-spin and ''z''-spin are "incompatible observables", meaning the Heisenberg uncertainty principle
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
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. Now, instead of measuring the ''z''-spin as well, Bob measures the ''x''-spin. According to quantum mechanics, when the system is in state I, Bob's ''x''-spin measurement will have a 50% probability of producing +''x'' and a 50% probability of -''x''. It is impossible to predict which outcome will appear until Bob actually ''performs'' the measurement. Therefore, Bob's positron will have a definite spin when measured along the same axis as Alice's electron, but when measured in the perpendicular axis its spin will be uniformly random. It seems as if information has propagated (faster than light) from Alice's apparatus to make Bob's positron assume a definite spin in the appropriate axis.
Bell's theorem
In 1964, John Stewart Bell
John Stewart Bell (28 July 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum mechanics, quantum physics regarding hidden-variable theory, hidden-variable theor ...
published a paper investigating the puzzling situation at that time: on one 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 hand, David Bohm had recently developed the first successful hidden-variable theory, but it had a grossly nonlocal character. Bell set out to investigate whether it was indeed possible to solve the nonlocality problem with hidden variables, and found out that first, the correlations shown in both EPR's and Bohm's versions of the paradox could indeed be explained in a local way with hidden variables, and second, that the correlations shown in his own variant of the paradox couldn't be explained by ''any'' local hidden-variable theory. This second result became known as the Bell theorem.
To understand the first result, consider the following toy hidden-variable theory introduced later by J.J. Sakurai: in it, quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the ''z''-spin and ''x''-spin. In these "true" states, the positron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+''z'', −''x'') to Alice and (−''z'', +''x'') to Bob", the next pair "(−''z'', −''x'') to Alice and (+''z'', +''x'') to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "−" with equal probability.
Bell showed, however, that such models can only reproduce the singlet correlations when Alice and Bob make measurements on the same axis or on perpendicular axes. As soon as other angles between their axes are allowed, local hidden-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 paper, a variety of experiments to test Bell's inequalities were carried out, notably by the group of Alain Aspect
Alain Aspect (; born 15 June 1947) is a French physicist noted for his experimental work on quantum entanglement.
Aspect was awarded the 2022 Nobel Prize in Physics, jointly with John Clauser and Anton Zeilinger, "for experiments with Quantum e ...
in the 1980s; all experiments conducted to date have found behavior in line with the predictions of quantum mechanics. The present view of the situation is that quantum mechanics flatly contradicts Einstein's philosophical postulate that any acceptable physical theory must fulfill "local realism". The fact that quantum mechanics violates Bell inequalities indicates that any hidden-variable theory underlying quantum mechanics must be non-local; whether this should be taken to imply that quantum mechanics ''itself'' is non-local is a matter of continuing debate.
Steering
Inspired by Schrödinger's treatment of the EPR paradox back in 1935, Howard M. Wiseman et al. formalised it in 2007 as the phenomenon of quantum steering. They defined steering as the situation where Alice's measurements on a part of an entangled state ''steer'' Bob's part of the state. That is, Bob's observations cannot be explained by a ''local hidden state'' model, where Bob would have a fixed quantum state in his side, which is classically correlated but otherwise independent of Alice's.
Locality
'' Locality'' has several different meanings in physics. EPR describe the principle of locality as asserting that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
, which states that energy can never be transmitted faster than the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
without violating causality; however, it 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.[ Causality is preserved because there is no way for Alice to transmit messages (i.e., information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "−", completely at ]random
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is able to perform his measurement only ''once'': there is a fundamental property of quantum mechanics, the no-cloning theorem
In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computer, quantum computing among ...
, which makes it impossible for him to make an arbitrary number of copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% 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 any 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 action at a distance". The conclusion they drew was that quantum mechanics is not a complete theory.
Mathematical formulation
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 space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'', with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the ''x'', ''y'', and ''z'' direction, denoted ''Sx'', ''Sy'', and ''Sz'' respectively, can be represented using the Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
:
where is the reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
(or the Planck constant divided by 2π).
The eigenstate
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s of ''Sz'' are represented as
and the eigenstates of ''Sx'' are represented as
The vector space of the electron-positron pair is , the tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of the electron's and positron's vector spaces. The spin singlet state is
where the two terms on the right hand side are what we have referred to as state I and state II above.
From the above equations, it can be shown that the spin singlet can also be written as
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 need to show that after Alice's measurement of ''Sz'' (or ''Sx''), Bob's value of ''Sz'' (or ''Sx'') is uniquely determined and Bob's value of ''Sx'' (or ''Sz'') is uniformly random. This follows from the principles of measurement in quantum mechanics
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability ...
. When ''Sz'' is measured, the system state collapses into an eigenvector of ''S''z. If the measurement result is +''z'', this means that immediately after measurement the system state collapses to
Similarly, if Alice's measurement result is −''z'', the state collapses to
The left hand side of both equations show that the measurement of ''S''z on Bob's positron is now determined, it will be −''z'' in the first case or +''z'' in the second case. The right hand side of the equations show that the measurement of ''Sx'' on Bob's positron will return, in both cases, +''x'' or −''x'' with probability 1/2 each.
See also
* Aspect's experiment
* Bohr-Einstein debates: The argument of EPR
* Coherence
* Correlation does not imply causation
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...
* CHSH inequality
* ER = EPR
* GHZ experiment
* Measurement problem
In quantum mechanics, the measurement problem is the ''problem of definite outcomes:'' quantum systems have superpositions but quantum measurements only give one definite result.
The wave function in quantum mechanics evolves deterministically ...
* Philosophy of information
* Philosophy of physics
In philosophy, the philosophy of physics deals with conceptual and interpretational issues in physics, many of which overlap with research done by certain kinds of theoretical physicists. Historically, philosophers of physics have engaged with ...
* Popper's experiment
* Superdeterminism
* Quantum entanglement
Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic o ...
* Quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
* Quantum pseudo-telepathy
Quantum pseudo-telepathy describes the use of quantum entanglement to eliminate the need for classical communications.
A nonlocal game is said to display quantum pseudo-telepathy if players who can use entanglement can win it with certainty whi ...
* Quantum teleportation
Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from on ...
* Quantum Zeno effect
In quantum mechanics, frequent measurements cause the quantum Zeno effect, a reduction in transitions away from the systems initial state, slowing a systems time evolution.
Sometimes this effect is interpreted as "a system cannot change while you ...
* Synchronicity
Synchronicity () is a concept introduced by Carl Jung, founder of analytical psychology, to describe events that coincide in time and appear meaningfully related, yet lack a discoverable causal connection. Jung held that this was a healthy fu ...
* Ward's probability amplitude
Notes
References
Selected papers
*
*
*
*
* A. Fine, ''Do Correlations need to be explained?'', in ''Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem'', edited by Cushing & McMullin (University of Notre Dame Press, 1986).
*
* M. Mizuki, ''A classical interpretation of Bell's inequality''. Annales de la Fondation Louis de Broglie 26 683 (2001)
*
* P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006)
*
*
Books
* Bell, John S. (1987). ''Speakable and Unspeakable in Quantum Mechanics''. Cambridge University Press. .
* Fine, Arthur (1996). ''The Shaky Game: Einstein, Realism and the Quantum Theory''. 2nd ed. Univ. of Chicago Press.
* Gribbin, John (1984). ''In Search of Schrödinger's Cat''. Black Swan.
* Lederman, 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. .
External links
* Stanford Encyclopedia of Philosophy: The Einstein–Podolsky–Rosen Argument in Quantum Theory
1.2 The argument in the text
* ''Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia with around 900 articles about philosophy, philosophers, and related topics. The IEP publishes only peer review, peer-reviewed and blind-refereed original p ...
'':
The Einstein-Podolsky-Rosen Argument and the Bell Inequalities
* ''Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
'': Abner Shimony (2019)
Bell's Theorem
from the Usenet Physics FAQ
EPR experiment with single photons interactive
Spooky Actions At A Distance?: Oppenheimer Lecture by Prof. Mermin
Original paper
{{DEFAULTSORT:Einstein-Podolsky-Rosen paradox
EPR paradox
EPR may refer to:
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* EPR (nuclear reactor), European Pressurised-Water Reactor
* EPR paradox (Einstein–Podolsky–Rosen paradox), in physics
* Earth potential rise, in electrical engineering
* East Pacific Rise, a mid-ocea ...
Physical paradoxes
Quantum measurement
Thought experiments in quantum mechanics