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| {{Quantum mechanics|cTopic=Fundamental concepts}}
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| '''Quantum entanglement''' is a physical phenomenon that occurs when pairs (or groups) of [[particle]]s are generated or interact in ways such that the [[quantum state]] of each member must subsequently be described relative to the other.
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| Quantum entanglement is a product of [[quantum superposition]]. However, the state of each member is indefinite in terms of physical properties such as [[position (vector)|position]],<ref>"Wave functions could describe combinations of different states, so-called superpositions. For example, an electron could be in a superposition of several different locations", from {{cite journal|author1=Max Tegmark|author2=John Archibald Wheeler|title=100 Years of the Quantum|year=2001|pages=68–75|volume=284|issue=2003|journal=Sci.Am.:,; ''Spektrum Wiss. Dossier N1:6-14''|arxiv=quant-ph/0101077}}</ref> [[momentum]], [[spin (physics)|spin]], [[Polarization (waves)|polarization]], etc. in a manner distinct from the intrinsic uncertainty of quantum superposition. When a [[Measurement#Quantum mechanics|measurement]] is made on one member of an entangled pair and the outcome is thus known (e.g., clockwise spin), the other member of the pair is at any subsequent time<ref>Brian Greene, ''The Fabric of the Cosmos'', p. 11 speaks of "an instantaneous bond between what happens at widely separated locations."</ref> always found (when measured) to have taken the appropriately correlated value (e.g., counterclockwise spin). There is thus a correlation between the results of measurements performed on entangled pairs, and this correlation is observed even though the entangled pair may be separated by arbitrarily large distances.<ref>"Decoherence was worked out in great detail by Los Alamos scientist Wojciech Zurek, Zeh and others over the following decades. They found that coherent quantum superpositions persist only as long as they remain secret from the rest of the world." from {{cite journal|author1=Max Tegmark|author2=John Archibald Wheeler|title=100 Years of the Quantum|year=2001|pages=68–75|volume=284|issue=2003|journal=Scientific American |arxiv=quant-ph/0101077|doi=10.1038/scientificamerican0201-68}}</ref> Repeated experiments have verified that this works even when the measurements are performed more quickly than light could travel between the sites of measurement: there is no [[speed of light|lightspeed]] or slower influence that can pass between the entangled particles.<ref>Francis, Matthew. [http://arstechnica.com/science/2012/10/quantum-entanglement-shows-that-reality-cant-be-local/ Quantum entanglement shows that reality can't be local], ''Ars Technica'', 30 October 2012</ref> Recent experiments have measured entangled particles within less than one part in 10,000 of the light travel time between them;<ref>{{cite journal
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| | author =Juan Yin, et al.
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| | title =Bounding the speed of `spooky action at a distance
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| | journal =Phys. Rev. Lett. 110, 260407
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| | year =2013
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| | url =http://arxiv.org/abs/1303.0614
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| }}</ref> according to the formalism of quantum theory, the effect of measurement happens instantly.<ref>Matson, John. [http://www.nature.com/news/quantum-teleportation-achieved-over-record-distances-1.11163 Quantum teleportation achieved over record distances], ''Nature'', 13 August 2012</ref><ref name=Griffiths2004>{{citation | author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn= 0-13-111892-7}}</ref>
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| This behavior is consistent with quantum theory, and has been demonstrated experimentally with [[photon]]s, [[electron]]s, [[molecule]]s the size of [[buckyballs]],<ref>[http://www.nature.com/nature/journal/v401/n6754/abs/401680a0.html Nature: Wave–particle duality of C<sub>60</sub> molecules, 14 October 1999]. Abstract, subscription needed for full text</ref><ref>[[Olaf Nairz]], [[Markus Arndt]], and [[Anton Zeilinger]], "Quantum interference experiments with large molecules", American Journal of Physics, 71 (April 2003) 319-325.</ref> and even small diamonds.<ref>{{cite journal |journal=Science |date=2 December 2011 |volume=334 |issue=6060 |pages=1253–1256 |doi=10.1126/science.1211914 |url=http://www.sciencemag.org/content/334/6060/1253.full |title=Entangling macroscopic diamonds at room temperature |authors=K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, I. A. Walmsley |laysummary=http://www.newscientist.com/article/dn21235-entangled-diamonds-blur-quantumclassical-divide.html}}</ref><ref>[http://www.sciencemag.org/content/334/6060/1253/suppl/DC1 sciencemag.org], supplementary materials</ref> It is an area of extremely active research by the physics community. However, there is some heated debate<ref>Physicist John Bell depicts the Einstein camp in this debate in his article entitled "Bertlmann's socks and the nature of reality", p. 142 of ''Speakable and unspeakable in quantum mechanics'': "For EPR that would be an unthinkable 'spooky action at a distance'. To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story." And again on p. 144 Bell says: "Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other." Downloaded 5 July 2011 from http://philosophyfaculty.ucsd.edu/faculty/wuthrich/GSSPP09/Files/BellJohnS1981Speakable_BertlmannsSocks.pdf</ref> about whether a possible classical [[hidden variable theory|underlying mechanism]] could explain entanglement. The difference in opinion derives from espousal of various [[interpretations of quantum mechanics]].
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| Research into quantum entanglement was initiated by a 1935 paper by [[Albert Einstein]], [[Boris Podolsky]], and [[Nathan Rosen]] describing the [[EPR paradox]]<ref name="Einstein1935">{{cite journal |author=Einstein A, Podolsky B, Rosen N |title=Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? |journal=Phys. Rev. |volume=47 |issue=10 |pages=777–780 |year=1935 |doi=10.1103/PhysRev.47.777|bibcode = 1935PhRv...47..777E }}</ref> and several papers by [[Erwin Schrödinger]] shortly thereafter.<ref name="Schrödinger1935">{{cite journal |author=Schrödinger E |title=Discussion of probability relations between separated systems |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=31 |issue=4 |pages=555–563 |year=1935 |doi=10.1017/S0305004100013554 |last2=Born |first2=M.}}</ref><ref name="Schrödinger1936">{{cite journal |author=Schrödinger E |title=Probability relations between separated systems |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=32 |issue=3 |pages=446–452 |year=1936 |doi=10.1017/S0305004100019137 |last2=Dirac |first2=P. A. M.}}</ref> Although these first studies focused on the counterintuitive properties of entanglement, with the aim of criticizing quantum mechanics, eventually entanglement was verified experimentally,<ref>[http://www.sciencenews.org/view/feature/id/65093/title/75_years_of_entanglement 75 years of entanglement | Science News]</ref> and recognized as a valid, fundamental feature of quantum mechanics. The focus of the research has now changed to its utilization as a resource for communication and computation.
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| {{toclimit|3}}
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| == History ==
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| The counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen.<ref name="Einstein1935"/> In this study, they formulated the [[EPR paradox]] (Einstein, Podolsky, Rosen paradox), a thought experiment that attempted to show that [[quantum mechanics|quantum mechanical theory]] was [[Incompleteness of quantum physics|incomplete]]. They wrote: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."<ref name="Einstein1935"/>
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| However, they did not coin the word ''entanglement,'' nor did they generalize the special properties of the state they considered. Following the EPR paper, [[Erwin Schrödinger]] wrote a letter (in German) to Einstein in which he used the word ''Verschränkung'' (translated by himself as ''entanglement'') "to describe the correlations between two particles that interact and then separate, as in the EPR experiment."<ref name=MK>Kumar, M., ''Quantum'', Icon Books, 2009, p. 313.</ref> He shortly thereafter published a seminal paper defining and discussing the notion, and terming it "entanglement." In the paper he recognized the importance of the concept, and stated:<ref name="Schrödinger1935"/> "I would not call [entanglement] ''one'' but rather ''the'' characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."
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| Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the [[theory of relativity]].<ref>Alisa Bokulich, Gregg Jaeger, ''Philosophy of Quantum Information and Entanglement'', Cambridge University Press, 2010, xv.</ref> Einstein later famously derided entanglement as "''spukhafte Fernwirkung''"<ref name="spukhafte">Letter from Einstein to Max Born, 3 March 1947; ''The Born-Einstein Letters; Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955'', Walker, New York, 1971. (cited in {{cite web | title = Quantum Entanglement and Communication Complexity (1998) | id = {{citeseerx|10.1.1.20.8324}} | author = M. P. Hobson | coauthors = ''et al.''|pages=1/13}})</ref> or "spooky [[Action at a distance (physics)|action at a distance]]."
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| The EPR paper generated significant interest among physicists and inspired much discussion about the foundations of quantum mechanics (perhaps most famously [[De Broglie–Bohm theory|Bohm's interpretation]] of quantum mechanics), but relatively little other published work. So, despite the interest, the flaw in EPR's argument was not discovered until 1964, when [[John Stewart Bell]] proved that one of their key assumptions, the [[principle of locality]], was not consistent with the hidden variables interpretation of quantum theory that EPR purported to establish. Specifically, he demonstrated an upper limit, seen in [[Bell's inequality]], regarding the strength of correlations that can be produced in any theory obeying [[local realism]], and he showed that quantum theory predicts violations of this limit for certain entangled systems.<ref>{{cite journal |author = J. S. Bell |title = On the Einstein- Poldolsky-Rosen paradox |journal = Physics |year = 1964}}</ref> His inequality is experimentally testable, and there have been numerous [[Bell test experiments|relevant experiments]], starting with the pioneering work of Freedman and Clauser in 1972<ref name="Clauser">{{cite journal|doi=10.1103/PhysRevLett.28.938|last1=Freedman|first1=Stuart J.|last2=Clauser|first2=John F.|title=Experimental Test of Local Hidden-Variable Theories|journal=Physical Review Letters |volume=28 |issue=14 |pages=938–941|year=1972 |bibcode=1972PhRvL..28..938F}}</ref> and Aspect's experiments in 1982.<ref>{{cite journal |author = A. Aspect, P. Grangier, and G. Roger |title = Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities |journal = Physical Review Letters |volume = 49 |issue = 2 |pages = 91–94 |year = 1982 |doi = 10.1103/PhysRevLett.49.91 |bibcode=1982PhRvL..49...91A}}</ref> They have all shown agreement with quantum mechanics rather than the principle of local realism. However, the issue is not finally settled, for each of these experimental tests has left open at least one [[Loopholes in Bell test experiments|loophole]] by which it is possible to question the validity of the results.
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| The work of Bell raised the possibility of using these super strong correlations as a resource for communication. It led to the discovery of [[quantum key distribution]] protocols, most famously [[BB84]] by Bennet and Brassard and [[E91 protocol|E91]] by Artur Ekert. Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.
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| [[David Kaiser]] of MIT mentioned in his book, ''How the Hippies Saved Physics'', that the possibilities of instantaneous long-range communication derived from [[Bell's theorem]] stirred interest among [[hippie]]s, [[psychic]]s, and even the CIA, with the counter-culture playing a critical role in its development toward practical use.<ref>{{cite book| url=http://books.google.com/books?id=i59PHb9XhJcC&printsec=frontcover&dq=How+the+Hippies+Saved+Physics:+Science,+Counterculture+and+the+Quantum+Revival.#v=onepage&q&f=false| title=How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival| publisher= W. W. Norton| year= 2011| isbn= 978-0-393-07636-3}}</ref>
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| == Concept ==
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| {{Tone|section|date=October 2013}}
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| [[Quantum mechanics|Quantum systems]] can become entangled through various types of interactions (see section on methods below). Quantum entanglement is a product of quantum superposition, i.e., of the fundamental aspect of quantum mechanics where the complete state of a system is expressed as a sum of basis states, or [[eigenstates]] of some [[observable]](s). Though it is common to speak of single quantum systems as existing in superpositions of basis states, the same is also valid for the quantum state of a pair or group of quantum systems. If the quantum state of a pair of particles is in a definite superposition, and that superposition cannot be factored out into the product of two states (one for each particle), then that pair is entangled. If entangled, one constituent cannot be fully described without considering the other(s). They remain entangled until a measurement is made and they decohere through interaction with the environment (i.e. measurement device).<ref name="Peres1993">Asher Peres, ''Quantum Theory, Concepts and Methods'', Kluwer, 1993; ISBN 0-7923-2549-4 p. 115.</ref>
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| An example of entanglement occurs when a [[subatomic particle]] [[Particle decay|decays]] into a pair of other particles. These decay events obey the various [[conservation laws]], and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a spin-zero particle could decay into a pair of spin-1/2 particles. Since the total spin before and after this decay must be zero (conservation of angular momentum), whenever the first particle is measured to be [[Spin (physics)#Spin direction|spin up]], the other when measured is always found to be [[Spin (physics)#Spin direction|spin down]]. This type of entangled pair, where the particles always have opposite spin, is known as the ''spin anti-correlated'' case, and if the probabilities for measuring each spin are equal, the pair is said to be in the [[singlet state]].
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| For example, assume that each of two hypothetical experimenters, [[Alice and Bob]], measures the spin of one of a pair of entangled particles (with zero total spin). If Alice makes the measurement first, then her result will be entirely unpredictable, with a 50% probability of the spin being up or down. If Bob subsequently measures the spin of his particle, the measurement will be entirely predictable―always opposite to Alice's, hence perfectly anti-correlated.
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| The correlation seen with aligned measurements (i.e., up and down only) can be simulated classically. To make an analogous experiment, a coin might be sliced along the circumference into two half-coins, in such a way that each half-coin is either [[Coin flipping|"heads" or "tails"]], and each half-coin put in a separate envelope and distributed respectively to Alice and to Bob, randomly. If Alice then "measures" her half-coin, by opening her envelope, for her the measurement will be unpredictable, with a 50% probability of her half-coin being "heads" or "tails", and Bob's "measurement" of his half-coin will always be opposite, hence perfectly anti-correlated.
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| If Alice and Bob measure the spin of their respective particles in directions other than just along the same axis, and check whether or not the results follow [[Bell's inequality]], they would find that the entangled bipartite pure system always violates Bell's inequality, while the classical system must always satisfy Bell's inequality. Bell's inequality is a very sensitive tool for judging quantum entanglement. The correlation of quantum entanglement can not be explained simply using the concepts of classical physics.
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| The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are [[Incompatible observables|incompatible]] in the sense that these measurements' maximum simultaneous precision is constrained by the uncertainty principle. In classical physics this does not make sense, since any number of properties can be measured simultaneously with arbitrary accuracy. Bell's theorem implies, and it has been proven mathematically, that [[Incompatible observables|compatible]] measurements cannot show Bell-like correlations,<ref>{{cite journal|last1=Cirel'son|first1=B. S.|title=Quantum generalizations of Bell's inequality|journal=Letters in Mathematical Physics|volume=4|issue=2|pages=93–100| year=1980|doi=10.1007/BF00417500|bibcode=1980LMaPh...4...93C}}</ref> and thus entanglement is a fundamentally non-classical phenomenon.
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| Entanglement is required to preserve the [[Uncertainty principle]], as seen in the EPR paradox. For example, say that a high energy photon decays into an electron / positron pair, and the position of the electron and the momentum of the positron are then measured. If we don't allow entanglement in the physical description of the pair, the position and momentum of each particle can still be deduced by reference to the conservation of momentum, violating the Uncertainty principle. Alternatively, if we require the uncertainty principle to hold true, and still disallow entanglement in the physical description of the pair, the uncertainty principle would allow violations in the law of conservation of momentum, because strong correlation in both position and momentum would be impossible (i.e. one would not be able to effectively deduce the position and momentum of the electron because they could not be highly correlated with both the position and momentum of the positron).
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| Even when measurements of the entangled particles are made in moving [[special relativity|relativistic]] reference frames, in which each respective measurement occurs before the other, the measurement results remain correlated.<ref>{{cite journal |author = H. Zbinden, et al. |title = Experimental test of nonlocal quantum correlations in relativistic configurations |journal = Phys. Rev. A |doi = 10.1103/PhysRevA.63.022111|year = 2001}}</ref><ref name=LG>Some of the history of both referenced Zbinden, et al. experiments is provided in Gilder, L., ''The Age of Entanglement'', Vintage Books, 2008, pp. 321-324.</ref>
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| In a 2012 experiment, "delayed-choice [[Quantum teleportation#Entanglement swapping|entanglement swapping]]" was used to decide whether two particles were entangled or not after they had already been measured.<ref name="Xiao-song2012">{{cite journal |author=Xiao-song Ma, Stefan Zotter, Johannes Kofler, Rupert Ursin, Thomas Jennewein, Časlav Brukner & Anton Zeilinger |title=Experimental delayed-choice entanglement swapping |journal=Nature Physics |date=26 April 2012 |doi=10.1038/nphys2294}}</ref>
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| In a 2013 experiment, entanglement swapping has been used to create entanglement between photons that never coexisted in time, thus demonstrating that "the nonlocality of quantum mechanics, as manifested by entanglement, does not apply only to particles with spacelike separation, but also to particles with timelike [i.e., temporal] separation".<ref>E. Megidish, A. Halevy, T. Shacham, T. Dvir, L. Dovrat, and H. S. Eisenberg, "Entanglement Swapping between Photons that have Never Coexisted", ''Physical Review Letters'', Volume 110, Issue 21, 22 May 2013.</ref>
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| In three independent experiments it was shown that classical separable states can carry entangled states.<ref>http://physicsworld.com/cws/article/news/2013/dec/11/classical-carrier-could-create-entanglement - Classical carrier could create entanglement</ref>
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| == Non-locality and hidden variables ==
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| There is much confusion about the meaning of entanglement, [[quantum nonlocality|non-locality]] and [[hidden variable theory|hidden variables]] and how they relate to each other. As described above, entanglement is an experimentally verified and accepted property of nature, which has critical implications for the [[interpretations of quantum mechanics]]. The question becomes, "How can one account for something that was at one point indefinite with regard to its spin (or whatever is in this case the subject of investigation) suddenly becoming definite in that regard even though no physical interaction with the second object occurred, and, if the two objects are sufficiently far separated, could not even have had the time needed for such an interaction to proceed from the first to the second object?"<ref>The Stanford encyclopedia <http://plato.stanford.edu/entries/qt-epr/> says that Niels Bohr distinguished between "mechanical disturbances" and "an influence on the very conditions which define the possible types of predictions regarding the future behavior of [the other half of an entangled] system."</ref> The latter question involves the issue of [[Principle of locality|locality]], i.e., whether for a change to occur in something the agent of change has to be in physical contact (at least via some intermediary such as a [[Field (physics)|field force]]) with the thing that changes. Study of entanglement brings into sharp focus the dilemma between locality and the completeness or lack of completeness of quantum mechanics.
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| Bell's theorem and related results rule out a local realistic explanation for quantum mechanics (one which obeys the principle of locality while also ascribing definite values to quantum observables). However, in other interpretations, the experiments that demonstrate the apparent non-locality can also be described in local terms: If each distant observer regards the other as a quantum system, communication between the two must then be treated as a measurement process, and this communication is strictly local.<ref>Sidney Coleman: Quantum Mechanics in Your Face – A lecture given by Sidney Coleman at the New England sectional meeting of the American Physical Society (Apr. 9, 1994). http://media.physics.harvard.edu/video/?id=SidneyColeman_QMIYF</ref> In particular, in the many worlds interpretation, the underlying description is fully local.<ref>Locality in the Everett Interpretation of Heisenberg-Picture Quantum Mechanics http://arxiv.org/abs/quant-ph/0103079</ref> More generally, the question of locality in quantum physics is extraordinarily subtle and sometimes hinges on precisely how it is defined.
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| In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While it is true that a bipartite quantum state must be entangled in order for it to produce non-local correlations, there exist entangled states that do not produce such correlations. A well-known example of this is the [[Werner state]] that is entangled for certain values of <math>p_{sym}</math>, but can always be described using local hidden variables.<ref name=werner1989>{{cite journal | last = Werner| first = R.F. | title = Quantum States with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model | journal = [[Physical Review A]] | volume = 40| pages = 4277–4281 | year = 1989 |doi=10.1103/PhysRevA.40.4277 | pmid=9902666 | issue=8}}</ref> In short, entanglement of a two-party state is necessary but not sufficient for that state to be non-local. It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to non-locality as well as to [[quantum teleportation]] and to [[superdense coding]], whereas non-locality is defined according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.
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| == Quantum mechanical framework ==
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| The following subsections are for those with a good working knowledge of the formal, mathematical description of [[quantum mechanics]], including familiarity with the formalism and theoretical framework developed in the articles: [[bra-ket notation]] and [[mathematical formulation of quantum mechanics]].
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| === Pure states ===
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| Consider two noninteracting systems <math>A</math> and <math>B</math>, with respective [[Hilbert space]]s <math>H_A</math> and <math>H_B</math>. The Hilbert space of the composite system is the [[tensor product]]
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| : <math> H_A \otimes H_B .</math>
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| If the first system is in state <math>\scriptstyle| \psi \rangle_A</math> and the second in state <math>\scriptstyle| \phi \rangle_B</math>, the state of the composite system is
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| : <math>|\psi\rangle_A \otimes |\phi\rangle_B.</math>
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| States of the composite system which can be represented in this form are called ''[[separable state]]s'', or (in the simplest case) ''[[product state]]s''.
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| Not all states are separable states (and thus product states). Fix a [[basis (linear algebra)|basis]] <math>\scriptstyle \{|i \rangle_A\}</math> for <math>H_A</math> and a basis <math>\scriptstyle \{|j \rangle_B\}</math> for <math>H_B</math>. The most general state in <math>\scriptstyle H_A \otimes H_B</math> is of the form
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| : <math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.
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| This state is separable if there exist <math>c^A_i,c^B_j</math> so that <math>\scriptstyle c_{ij}= c^A_ic^B_j,</math> yielding <math>\scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math>\scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for all <math>c^A_i,c^B_j</math> we have <math>\scriptstyle c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an ''entangled state''.
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| For example, given two basis vectors <math>\scriptstyle \{|0\rangle_A, |1\rangle_A\}</math> of <math>H_A</math> and two basis vectors <math>\scriptstyle \{|0\rangle_B, |1\rangle_B\}</math> of <math>H_B</math>, the following is an entangled state:
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| : <math>{1 \over \sqrt{2}} \bigg( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \bigg)</math>.
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| If the composite system is in this state, it is impossible to attribute to either system <math>A</math> or system <math>B</math> a definite [[pure state]]. Another way to say this is that while the [[von Neumann entropy]] of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry.<ref name="JaegerEtAl95">{{cite journal |author=Jaeger G, Shimony A, Vaidman L |title=Two Interferometric Complementarities |journal=Phys. Rev. |volume=51 |issue=1 |pages=54–67 |year=1995 |doi=10.1103/PhysRevA.51.54|bibcode = 1995PhRvA..51...54J }}</ref> It is worthwhile to note that the above example is one of four [[Bell states]], which are (maximally) entangled pure states (pure states of the <math> H_A \otimes H_B </math> space, but which cannot be separated into pure states of each <math> H_A </math> and <math> H_B </math>).
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| Now suppose Alice is an observer for system <math>A</math>, and Bob is an observer for system <math>B</math>. If in the entangled state given above Alice makes a measurement in the <math>\scriptstyle \{|0\rangle, |1\rangle\}</math> eigenbasis of A, there are two possible outcomes, occurring with equal probability:<ref name=nielchuang>
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| {{cite book
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| | last = Nielsen | first = Michael A.
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| | coauthors = Chuang, Isaac L.
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| | year = 2000
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| | title = Quantum Computation and Quantum Information
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| | publisher = [[Cambridge University Press]]
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| | pages = 112–113
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| | isbn = 0-521-63503-9
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| }}</ref>
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| # Alice measures 0, and the state of the system collapses to <math>\scriptstyle |0\rangle_A |1\rangle_B</math>.
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| # Alice measures 1, and the state of the system collapses to <math>\scriptstyle |1\rangle_A |0\rangle_B</math>.
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| If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system ''B'' has been altered by Alice performing a local measurement on system ''A''. This remains true even if the systems ''A'' and ''B'' are spatially separated. This is the foundation of the [[EPR paradox]].
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| The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see [[no-communication theorem]].
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| === Ensembles ===
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| As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has a large number of copies of the same system, then the state of this ''ensemble'' is described by a [[density matrix]], which is a [[positive matrix]], or a [[trace class]] when the state space is infinite dimensional, and has trace 1. Again, by the [[spectral theorem]], such a matrix takes the general form:
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| : <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>
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| where the positive valued <math>w_i</math>'s sum up to 1, and in the infinite dimensional case, we would take the closure of such states in the trace norm. We can interpret <math>\rho</math> as representing an ensemble where <math>w_i</math> is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a ''pure ensemble''. When there is less than total information about the state of a quantum system we need [[#Reduced density matrices|density matrices]] to represent the state.
| |
| | |
| Following the definition in previous section, for a bipartite composite system, mixed states are just density matrices on <math>H_A \otimes H_B</math>.
| |
| Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as<ref name=Laloe>{{citation|last=Laloe|first=Franck|title=Do We Really Understand Quantum Mechanics| publisher=Cambridge University Press|year=2012| isbn = 978-1-107-02501-1}}</ref>{{rp|131–132}}
| |
| | |
| : <math>\rho = \sum_i p_i \rho_i^A \otimes \rho_i^B, </math>
| |
| | |
| where <math>p_i</math>'s are positive valued probabilities, <math>\rho_i^A</math>'s and <math>\rho_i^B</math>'s are themselves states on the subsystems ''A'' and ''B'' respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. We can assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are pure ensembles. A state is then said to be ''entangled'' if it is not separable. In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be [[NP-hard]].<ref>{{cite journal |author=Gurvits L |title=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |journal=Proceedings of the thirty-fifth annual ACM symposium on Theory of computing |year=2003 |doi=10.1145/780542.780545 |page=10 |isbn=1-58113-674-9}}</ref> For the <math> 2 \times 2</math> and <math> 2 \times 3</math> cases, a necessary and sufficient criterion for separability is given by the famous [[Peres-Horodecki criterion|Positive Partial Transpose (PPT)]] condition.<ref>{{cite journal |author=Horodecki M, Horodecki P, Horodecki R |title=Separability of mixed states: necessary and sufficient conditions |journal=Physics Letters A |volume=223 |page=210 |year=1996 |doi=10.1016/S0375-9601(96)00706-2 |bibcode=1996PhLA..223....1H|arxiv = quant-ph/9605038 }}</ref>
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| | |
| Experimentally, a mixed ensemble might be realized as follows. Consider a "black-box" apparatus that spits [[electron]]s towards an observer. The electrons' Hilbert spaces are [[identical particles|identical]]. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with [[spin (physics)|spins]] aligned in the positive <math>\mathbf{z}</math> direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative <math>\mathbf{y}</math> direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.
| |
| | |
| === Reduced density matrices ===
| |
| | |
| The idea of a reduced density matrix was introduced by [[Paul Dirac]] in 1930.<ref>{{cite journal|doi=10.1017/S0305004100016108|title=Note on Exchange Phenomena in the Thomas Atom|year=2008|last1=Dirac|first1=P. A. M.|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=26|issue=3|page=376|bibcode=1930PCPS...26..376D}}</ref> Consider as above systems <math>A</math> and <math>B</math> each with a Hilbert space <math>H_A</math>, <math>H_B</math>. Let the state of the composite system be
| |
| | |
| : <math> |\Psi \rangle \in H_A \otimes H_B. </math>
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| | |
| As indicated above, in general there is no way to associate a pure state to the component system <math>A</math>. However, it still is possible to associate a density matrix. Let
| |
| | |
| : <math>\rho_T = |\Psi\rangle \; \langle\Psi|</math>.
| |
| | |
| which is the [[projection operator]] onto this state. The state of <math>A</math> is the [[partial trace]] of <math>\rho_T</math> over the basis of system <math>B</math>:
| |
| | |
| : <math>\rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox{Tr}_B \; \rho_T </math>.
| |
| | |
| <math>\rho_A</math> is sometimes called the reduced density matrix of <math>\rho</math> on subsystem ''A''. Colloquially, we "trace out" system ''B'' to obtain the reduced density matrix on ''A''.
| |
| | |
| For example, the reduced density matrix of <math>A</math> for the entangled state
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| <math>\scriptstyle ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B ) / \sqrt{2}</math>
| |
| discussed above is
| |
| | |
| : <math>\rho_A = (1/2) \bigg( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \bigg)</math>
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| | |
| This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of <math>A</math> for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math>
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| discussed above is
| |
| | |
| : <math>\rho_A = |\psi\rangle_A \langle\psi|_A .</math>
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| | |
| In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure. Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one dimensional [[AKLT Model|AKLT spin chain]]:<ref name='Fan2004'>{{cite journal | doi = 10.1103/PhysRevLett.93.227203 | title = Entanglement in a Valence-Bond Solid State | journal = Physical Review Letters | date = 2004-11-26 | first = H | last = Fan | page = 227203 | coauthors = Korepin V, Roychowdhury V | volume = 93 | issue = 22 | pmid = 15601113 |arxiv=quant-ph/0406067 | bibcode=2004PhRvL..93v7203F}}</ref> the ground state can be divided into a block and an environment. The reduced density matrix of the block is [[Proportionality (mathematics)|proportional]] to a projector to a degenerate ground state of another Hamiltonian.
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| The reduced density matrix also was evaluated for [[Heisenberg model (quantum)|XY spin chains]], where it has full rank. It was proved that in the thermodynamic limit, the spectrum of the reduced density matrix of a large block of spins is an exact geometric sequence<ref>{{cite journal|doi=10.1007/s11128-010-0197-7|arxiv=1002.2931|title=Spectrum of the density matrix of a large ''block of'' spins of the XY model in one dimension|year=2010|last1=Franchini|first1=F.|last2=Its|first2=A. R.|last3=Korepin|first3=V. E.|last4=Takhtajan|first4=L. A.|journal=Quantum Information Processing|volume=10|issue=3|pages=325–341}}</ref> in this case.
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| === Entropy ===
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| In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.
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| ==== Definition ====
| |
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| [[File:Von Neumann entropy for bipartite system plot.svg|right|thumb|200px|The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.]]
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| In classical information theory, the [[Shannon entropy]], <math>H</math> is associated to a probability distribution,<math>p_1, \cdots, p_n</math>, in the following way:<ref name="SE">{{cite web |url=http://authors.library.caltech.edu/5516/1/CERpra97b.pdf#page=10 |title=Information-theoretic interpretation of quantum error-correcting codes |first1=Nicolas J. |last1=Cerf |first2=Richard |last2=Cleve }}</ref>
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| : <math>H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i</math>.
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| Since a mixed state ρ is a probability distribution over an ensemble, this leads naturally to the definition of the [[von Neumann entropy]]:
| |
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| : <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right),</math>.
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| | |
| In general, one uses the [[Borel functional calculus]] to calculate <math>\; \log \rho</math>. If ρ acts on a finite dimensional Hilbert space and has eigenvalues
| |
| <math>\lambda_1, \cdots, \lambda_n</math>, the Shannon entropy is recovered:
| |
| | |
| : <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i</math>.
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| Since an event of probability 0 should not contribute to the entropy, and given that <math> \lim_{p \to 0} p \log p \;=\; 0</math>, the convention is adopted that <math>0 \log 0 \; = 0</math>. This extends to the infinite dimensional case as well: if ρ has [[projection-valued measure|spectral resolution]] <math> \rho = \int \lambda d P_{\lambda}</math>, assume the same convention when calculating
| |
| | |
| : <math> \rho \log_2 \rho = \int \lambda \log_2 \lambda d P_{\lambda} .</math>
| |
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| As in [[entropy|statistical mechanics]], the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is <math>\log 2</math> (which can be shown to be the maximum entropy for <math>2 \times 2</math> mixed states).
| |
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| ==== As a measure of entanglement ====
| |
| | |
| Entropy provides one tool which can be used to quantify entanglement, although other entanglement measures exist.<ref name="arxiv.org">{{cite journal|author1=Plenio|title=An introduction to entanglement measures|year=2007|pages=1–51|volume=1|journal=Quant. Inf. Comp. |arxiv=quant-ph/0504163|bibcode=2005quant.ph..4163P|last2=Virmani}}</ref> If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.
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| For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.
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| It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/''n'',...,1/''n''}. Therefore, a bipartite pure state
| |
| | |
| : <math>\rho \in H \otimes H</math>
| |
| | |
| is said to be a '''maximally entangled state''' if the reduced state of ''ρ'' is the diagonal matrix
| |
| | |
| : <math>\begin{bmatrix} \frac{1}{n}& \; & \; \\ \; & \ddots & \; \\ \; & \; & \frac{1}{n}\end{bmatrix}.</math>
| |
| | |
| For mixed states, the reduced von Neumann entropy is not the unique entanglement measure.
| |
| | |
| As an aside, the information-theoretic definition is closely related to [[entropy (statistical views)|entropy]] in the sense of statistical mechanics{{Citation needed|date=January 2009}} (comparing the two definitions, we note that, in the present context, it is customary to set the [[Boltzmann constant]] <math>k = 1</math>). For example, by properties of the [[Borel functional calculus]], we see that for any [[unitary operator]] ''U'',
| |
| | |
| : <math>S(\rho) \; = S(U \rho U^*).</math>
| |
| | |
| Indeed, without the above property, the von Neumann entropy would not be well-defined. In particular, ''U'' could be the time evolution operator of the system, i.e.
| |
| | |
| : <math>U(t) \; = \exp \left(\frac{-i H t }{\hbar}\right)</math>
| |
| | |
| where ''H'' is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system. This associates the reversibility of a process with its resulting entropy change, i.e., a process is reversible if, and only if, it leaves the entropy of the system invariant. This provides a connection between [[quantum information theory]] and [[thermodynamics]]. [[Rényi entropy]] also can be used as a measure of entanglement.
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| === Quantum field theory ===
| |
| | |
| The [[Reeh-Schlieder theorem]] of [[quantum field theory]] is sometimes seen as an analogue of quantum entanglement.
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| == Applications ==
| |
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| Entanglement has many applications in [[quantum information theory]]. With the aid of entanglement, otherwise impossible tasks may be achieved.
| |
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| Among the best-known applications of entanglement are [[superdense coding]] and [[quantum teleportation]].<ref>{{cite journal |last1=Bouwmeester |first1=Dik |last2=Pan |first2=Jian-Wei|last3=Mattle |first3=Klaus|last4=Eibl |first4=Manfred |last5=Weinfurter |first5=Harald|last6=Zeilinger |first6=Anton|year=1997 |title=Experimental Quantum Teleportation |journal=Nature |volume=390 |pages=575–579 |format=PDF |lastauthoramp=6 |url=http://qudev.ethz.ch/content/courses/QSIT06/pdfs/Bouwmeester97.pdf }}</ref>
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| | |
| Most researchers believe that entanglement is necessary to realize [[quantum computer|quantum computing]] (although this is disputed by some<ref name="jozsa02">{{cite journal|author1=Richard Jozsa|author2=Noah Linden|doi=10.1098/rspa.2002.1097|title=On the role of entanglement in quantum computational speed-up|year=2002|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=459|issue=2036|pages=2011–2032|arxiv=quant-ph/0201143|bibcode = 2003RSPSA.459.2011J }}</ref>).
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| Entanglement is used in some protocols of [[quantum cryptography]].<ref name="ekert91">{{cite journal |doi=10.1103/PhysRevLett.67.661 |title=Quantum cryptography based on Bell's theorem |year=1991 |last1=Ekert |first1=Artur K. |journal=Physical Review Letters |volume=67 |issue=6 |pages=661–663 |pmid=10044956}}</ref><ref name="horodecki10">{{cite arxiv|eprint=1006.0468|author1=Karol Horodecki|author2=Michal Horodecki|author3=Pawel Horodecki|author4=Ryszard Horodecki|author5=Marcin Pawlowski|author6=Mohamed Bourennane|title=Contextuality offers device-independent security|class=quant-ph|year=2010}}</ref> This is because the "shared noise" of entanglement makes for an excellent [[one-time pad]]. Moreover, since measurement of either member of an entangled pair destroys the entanglement they share, entanglement-based quantum cryptography allows the sender and receiver to more easily detect the presence of an interceptor.
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| In [[interferometry]], entanglement is necessary for surpassing the [[standard quantum limit]] and achieving the [[Heisenberg limit]].
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| === Entangled States ===
| |
| | |
| There are several canonical entangled states that appear often in theory and experiments.
| |
| | |
| For two [[qubits]], the [[Bell state]]s are
| |
| | |
| : <math>|\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B)</math>
| |
| : <math>|\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B)</math>.
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| These four pure states are all maximally entangled (according to the [[entropy of entanglement]]) and form an [[orthonormal]] [[basis (linear algebra)]] of the Hilbert space of the two qubits. They play a fundamental role in [[Bell's theorem]].
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| For M>2 qubits, the [[Greenberger–Horne–Zeilinger state|GHZ state]] is
| |
| | |
| : <math>|\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}},</math>
| |
| | |
| which reduces to the Bell state <math>|\Phi^+\rangle</math> for <math>M=2</math>. The traditional GHZ state was defined for <math>M=3</math>. GHZ states are occasionally extended to ''[[Qudit|qudits]]'', i.e. systems of ''d'' rather than 2 dimensions.
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| | |
| Also for M>2 qubits, there are [[spin squeezed states]].<ref>[http://qwiki.stanford.edu/index.php/Spin_Squeezed_State Database error - Qwiki]</ref> Spin squeezed states are a class of states satisfying certain restrictions on the uncertainty of spin measurements, and are necessarily entangled.<ref>Masahiro Kitagawa and Masahito Ueda, "[http://pra.aps.org/abstract/PRA/v47/i6/p5138_1 Squeezed Spin States]", Phys. Rev. A 47, 5138–5143 (1993).</ref>
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| | |
| For two [[boson]]ic modes, a [[NOON state]] is
| |
| | |
| : <math>|\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}, \, </math>
| |
| | |
| This is like a Bell state <math>|\Phi^+\rangle</math> except the basis kets 0 and 1 have been replaced with "the ''N'' photons are in one mode" and "the ''N'' photons are in the other mode".
| |
| | |
| Finally, there also exist [[twin Fock states]] for bosonic modes, which can be created by feeding a [[Fock state]] into two arms leading to a beam-splitter. They are the sum of multiple of NOON states, and can used to achieve the Heisenberg limit.<ref>[http://prl.aps.org/abstract/PRL/v71/i9/p1355_1 Phys. Rev. Lett. 71, 1355 (1993): Interferometric detection of optical phase shifts at the Heisenberg limit]</ref>
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| | |
| For the appropriately chosen measure of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to prepare experimentally.
| |
| | |
| === Methods of creating entanglement ===
| |
| | |
| Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is [[spontaneous parametric down-conversion]] to generate a pair of photons entangled in polarisation.<ref name="horodecki2007">{{cite journal |author=Horodecki R, Horodecki P, Horodecki M, Horodecki K |title=Quantum entanglement |journal=Rev. Mod. Phys. |arxiv=quant-ph/0702225 |doi =10.1103/RevModPhys.81.865 |year=2007|pages=865–942 |bibcode=2009RvMP...81..865H |volume=81 |issue=2}}</ref> Other methods include the use of a [[fiber coupler]] to confine and mix photons, the use of [[quantum dot]]s to trap electrons until decay occurs, the use of the [[Hong-Ou-Mandel effect]], etc. In the earliest tests of Bell's theorem, the entangled particles were generated using [[atomic cascade]]s.
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| | |
| It is also possible to create entanglement between quantum systems that never directly interacted, through the use of [[Quantum teleportation#Entanglement swapping|entanglement swapping]].
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| | |
| == Wormholes ==
| |
| | |
| [[File:LorentzianWormhole.jpg|thumb|300px|Exact mathematical plot of a Lorentzian wormhole (Schwarzschild wormhole)|alt=A double-ended funnel, with the narrow ends stuck to each other]]
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| | |
| Entangling two black holes, then pulling them apart, forms a [[wormhole]]—essentially a "shortcut"—connecting them. Similarly looked at in terms of [[string theory]], entangling two [[quarks]] does the same.<ref name=scid1312>{{cite web|url=http://www.sciencedaily.com/releases/2013/12/131205142218.htm |title=You can't get entangled without a wormhole: Physicist finds entanglement instantly gives rise to a wormhole |doi=10.1103/PhysRevLett.111.211603 |publisher=Sciencedaily.com |date=2013-12-05 |accessdate=2013-12-09}}</ref><ref>{{cite journal |last=Sonner |first=Julian |year=2013 |title=Holographic Schwinger effect and the geometry of entanglement |url=http://arxiv.org/pdf/1307.6850.pdf |journal=Physical Review Letters |volume=111 |issue=21 |doi=10.1103/PhysRevLett.111.211603 |format=PDF}}</ref>
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| | |
| These theoretical results support the theory that the laws of [[gravity]] are not fundamental, but instead arise from entanglement. While quantum mechanics correctly describes interactions at a microscopic level, it has not been able to explain gravity. A theory of [[quantum gravity]] would show that classical gravity is not fundamental, as [[Albert Einstein]] proposed, but rather emerges from a more basic, quantum-based phenomenon.<ref name=scid1312/>
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| The [[Schwinger effect]] creates two particles from a vacuum. Under an electric field, the particles can be "caught" before they disappear back into the vacuum. Once extracted, these particles are entangled. The entangled particles can be mapped in space-time, a four-dimensional space. In contrast, gravity is thought to exist in the fifth dimension as, according to Einstein's laws, it acts to "bend" and shape space-time.<ref name=scid1312/>{{Dubious|date=February 2014}}
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| | |
| [[Holographic duality]] is the principle says that all events in the fifth dimension are translatable into events in the other four.<ref>{{cite web|url=http://www.wired.com/wiredscience/2013/07/hidden-particle-world/ |title='Holographic Duality' Hints at Hidden Subatomic World - Wired Science |publisher=Wired.com |date=2013-07-03 |accessdate=2013-12-09}}</ref> It reveals that a [[wormhole]] is created along with the particles. More fundamentally, the results suggest that gravity and its ability to bend space-time emerge from entanglement.<ref name=scid1312/>
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| == See also ==
| |
| | |
| {{columns-list|3|
| |
| | |
| * [[Concurrence (quantum computing)]]
| |
| * [[Entanglement distillation]]
| |
| * [[Entanglement witness]]
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| * [[Ghirardi–Rimini–Weber theory]]
| |
| * [[Multipartite entanglement]]
| |
| * [[Observer effect (physics)]]
| |
| * [[Photon entanglement]]
| |
| * [[Quantum coherence]]
| |
| * [[Quantum phase transition]]
| |
| * [[Quantum pseudo-telepathy]]
| |
| * [[Retrocausality]]
| |
| * [[Separable state]]
| |
| * [[Squashed entanglement]]
| |
| * [[John Clive Ward|Ward's probability amplitude]]
| |
| * [[Wheeler-Feynman absorber theory]]
| |
| * [[Ansible]]
| |
| * [[Superluminal communication|Faster-than-light communication]]
| |
| | |
| }}
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| | |
| == References ==
| |
| {{Reflist|30em}}
| |
| | |
| == Further reading ==
| |
| {{refbegin}}
| |
| | |
| * {{cite book |author=Bengtsson I, [[Karol Życzkowski|Życzkowski K]] |chapter=Geometry of Quantum States |title=An Introduction to Quantum Entanglement |publisher=Cambridge University Press |location=Cambridge |year=2006}}
| |
| * {{cite book |author=Steward EG |title=Quantum Mechanics: Its Early Development and the Road to Entanglement |publisher=Imperial College Press |year=2008 |isbn=978-1-86094-978-4}}
| |
| * {{cite journal |author=Horodecki R, Horodecki P, Horodecki M, Horodecki K |title=Quantum entanglement |journal=Rev. Mod. Phys. |arxiv=quant-ph/0702225 |doi =10.1103/RevModPhys.81.865 |year=2009 |bibcode=2009RvMP...81..865H |volume=81 |issue=2 |pages=865–942}}
| |
| * {{cite book |author=Jaeger G |year=2009 |title=Entanglement, Information, and the Interpretation of Quantum Mechanics |location=Heildelberg |publisher=Springer |isbn=978-3-540-92127-1}}
| |
| * {{cite journal |author=Plenio MB, Virmani S |title=An introduction to entanglement measures |journal=Quant. Inf. Comp. |volume=1 |issue=7 |page=151 |arxiv=quant-ph/0504163 |year=2007 |bibcode=2005quant.ph..4163P |last2=Virmani}}
| |
| * {{cite journal |author=Shadbolt PJ, Verde MR, Peruzzo A, Politi A, Laing A, Lobino M, Matthews JCF, Thompson MG, O'Brien JL |title=Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit |journal=Nature Photonics |arxiv=/1108.3309 |doi=10.1038/nphoton.2011.283 |year=2012 |volume=6 |pages=45–59 }}
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| {{refend}}
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| == External links ==
| |
| {{div col|cols=2}}
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| | |
| * [http://prola.aps.org/abstract/PR/v47/i10/p777_1 The original EPR paper]
| |
| * [http://plato.stanford.edu/entries/qt-entangle/ Quantum Entanglement at Stanford Encyclopedia of Philosophy]
| |
| * [http://physicsworldarchive.iop.org/index.cfm?action=summary&doc=11%2F3%2Fphwv11i3a29%40pwa-xml&qt= How to entangle photons experimentally (subscription required)]
| |
| * [http://www.physicaltv.com.au/DanceFilmEntanglementTheoryrichardJamesAllenkarenPearlmangaryHayesmixedRealityLiveActionsecondLifeMachinima_619_1307_3_0.html A creative interpretation of Quantum Entanglement]
| |
| * [http://www.science20.com/hammock_physicist/einstein_got_it_wrong_can_you_do_better-85544 Albert's chest: entanglement for lay persons]
| |
| * [http://davidjarvis.ca/entanglement/ How Quantum Entanglement Works]
| |
| * [http://www.youtube.com/watch?v=xM3GOXaci7w Explanatory video by Scientific American magazine]
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| | |
| {{Use dmy dates|date=July 2011}}
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| * [http://news.yahoo.com/two-diamonds-linked-strange-quantum-entanglement-190805281.html Two Diamonds Linked by Strange Quantum Entanglement]
| |
| * [http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/index.html Entanglement experiment with photon pairs - interactive]
| |
| * [http://www.physorg.com/news63037231.html Multiple entanglement and quantum repeating]
| |
| * [http://www.mathpages.com/home/kmath521/kmath521.htm Quantum Entanglement and Bell's Theorem at MathPages]
| |
| * Audio - Cain/Gay (2009) [http://www.astronomycast.com/physics/ep-140-entanglement/ Astronomy Cast] Entanglement
| |
| * [http://www.imperial.ac.uk/quantuminformation Recorded research seminars at Imperial College relating to quantum entanglement]
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| * [http://www.osa.org/meetings/topicalmeetings/ICQI/default.aspx Quantum Entanglement and Decoherence: 3rd International Conference on Quantum Information (ICQI)]
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| * [http://www.npl.co.uk/server.php?show=ConWebDoc.433 Ion trapping quantum information processing]
| |
| * [http://www.spectrum.ieee.org/aug07/5378/1 IEEE Spectrum On-line: ''The trap technique'']
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| * [http://www.sciam.com/article.cfm?id=was-einstein-wrong-about-relativity Was Einstein Wrong?: A Quantum Threat to Special Relativity]
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| * [http://en.citizendium.org/wiki/Entanglement_(physics) Citizendium: Entanglement]
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| * [http://www.youtube.com/watch?gl=IT&hl=it&v=ta09WXiUqcQ Spooky Actions At A Distance?: Oppenheimer Lecture, Prof. David Mermin (Cornell University) Univ. California, Berkeley, 2008. Non-mathematical popular lecture on YouTube, posted Mar 2008)]
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| {{div col end}}
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| {{DEFAULTSORT:Quantum Entanglement}}
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| [[Category:Quantum information science]]
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| [[Category:Quantum mechanics]]
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| [[Category:Faster-than-light communication]]
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