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Based on AdS/CFT correspondence, Mark Van Raamsdonk suggested that spacetime arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the space-time. Induced gravity can emerge from the entanglement first law.

Based on AdS/CFT correspondence, Mark Van Raamsdonk suggested that spacetime arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the space-time. Induced gravity can emerge from the entanglement first law.

基于 AdS/CFT对偶的理论，Mark Van Raamsdonk 提出时空是作为量子自由度的一种涌现现象而产生的，这种量子自由度是纠缠在一起的，生活在时空的边界上。诱导引力可以产生于纠缠第一定律。

基于AdS/CFT对应关系， Mark Van Raamsdonk提出时空是量子自由度的一种涌现现象，量子自由度纠缠在时空的边界上。诱导引力可以从纠缠第一定律中产生。

The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel [[faster than light]]) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the [[Copenhagen interpretation]], the result of a spin measurement on one of the particles is a collapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured.<ref>{{cite book|last1=Rupert W.|first1=Anderson|title=The Cosmic Compendium: Interstellar Travel|date=28 March 2015|publisher=The Cosmic Compendium|isbn=9781329022027|page=100|edition=First|url=https://books.google.com/books?id=JxauCQAAQBAJ&pg=PA100&lpg=PA100&dq=The+outcome+is+taken+to+be+random,+with+each+possibility+having+a+probability+of+50%25.+However,+if+both+spins+are+measured+along+the+same+axis,+they+are+found+to+be+anti-correlated.+This+means+that+the+random+outcome+of+the+measurement+made+on+one+particle+seems+to+have+been+transmitted+to+the+other,+so+that+it+can+make+the+%22right+choice%22+when+it+too+is+measured#v=onepage}}</ref>

The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel [[faster than light]]) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the [[Copenhagen interpretation]], the result of a spin measurement on one of the particles is a collapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured.<ref>{{cite book|last1=Rupert W.|first1=Anderson|title=The Cosmic Compendium: Interstellar Travel|date=28 March 2015|publisher=The Cosmic Compendium|isbn=9781329022027|page=100|edition=First|url=https://books.google.com/books?id=JxauCQAAQBAJ&pg=PA100&lpg=PA100&dq=The+outcome+is+taken+to+be+random,+with+each+possibility+having+a+probability+of+50%25.+However,+if+both+spins+are+measured+along+the+same+axis,+they+are+found+to+be+anti-correlated.+This+means+that+the+random+outcome+of+the+measurement+made+on+one+particle+seems+to+have+been+transmitted+to+the+other,+so+that+it+can+make+the+%22right+choice%22+when+it+too+is+measured#v=onepage}}</ref>

 

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements [[spacelike]], hence, any causal effect connecting the events would have to travel faster than light. According to the principles of [[special relativity]], it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events {{math|''x''₁}} and {{math|''x''₂}} there are [[inertial frame]]s in which {{math|''x''₁}} is first and others in which {{math|''x''₂}} is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements [[spacelike]], hence, any causal effect connecting the events would have to travel faster than light. According to the principles of [[special relativity]], it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events {{math|''x''₁}} and {{math|''x''₂}} there are [[inertial frame]]s in which {{math|''x''₁}} is first and others in which {{math|''x''₂}} is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.

In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations. A well-known example is the Werner states that are entangled for certain values of <math>p_{sym}</math>, but can always be described using local hidden variables. Moreover, it was shown that, for arbitrary numbers of parties, there exist states that are genuinely entangled but admit a local model.

In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations. A well-known example is the Werner states that are entangled for certain values of <math>p_{sym}</math>, but can always be described using local hidden variables. Moreover, it was shown that, for arbitrary numbers of parties, there exist states that are genuinely entangled but admit a local model.

在媒体和流行科学中，量子非定域性经常被描述为等价于纠缠。虽然这对于纯二体量子态来说是正确的，但是一般来说纠缠只对于非局域关联是必要的，但是存在混合纠缠态，不产生这样的关联。一个众所周知的例子是 Werner 状态，它纠缠于 < math > p _ { sym } </math > 的某些值，但总是可以使用局部隐变量来描述。此外，研究还表明，对于任意数目的当事人，存在真正纠缠但承认局部模型的状态。

在媒体和大众科学中，量子非定域性常常被描述为与纠缠等价。虽然这对于纯二部量子态是正确的，但一般来说纠缠只对非局域关联是必要的，但是存在不产生这种关联的混合纠缠态。一个著名的例子是沃纳态，它纠缠在<math>p{sym}</math>的某些值上，但总是可以用局部隐藏变量来描述。此外，研究还表明，对于任意数目的当事方，存在真正纠缠但允许局部模型的态。

The mentioned proofs about the existence of local models assume that there is only one copy of the quantum state available at a time. If the parties are allowed to perform local measurements on many copies of such states, then many apparently local states (e.g., the qubit Werner states) can no longer be described by a local model. This is, in particular, true for all distillable states. However, it remains an open question whether all entangled states become non-local given sufficiently many copies.

The mentioned proofs about the existence of local models assume that there is only one copy of the quantum state available at a time. If the parties are allowed to perform local measurements on many copies of such states, then many apparently local states (e.g., the qubit Werner states) can no longer be described by a local model. This is, in particular, true for all distillable states. However, it remains an open question whether all entangled states become non-local given sufficiently many copies.

(In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two widely separated detectors attempting to detect the particle in two different places must instantaneously attain appropriate correlation, so that they do not both detect the particle.)

(In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two widely separated detectors attempting to detect the particle in two different places must instantaneously attain appropriate correlation, so that they do not both detect the particle.)

 

In short, entanglement of a state shared by two parties is necessary but not sufficient for that state to be non-local. It is important to recognize that entanglement is more commonly viewed as an algebraic concept, noted for being a prerequisite 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.

In short, entanglement of a state shared by two parties is necessary but not sufficient for that state to be non-local. It is important to recognize that entanglement is more commonly viewed as an algebraic concept, noted for being a prerequisite 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.

=== Hidden variables theory ===

=== Hidden variables theory 隐藏变量理论===

A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables".<ref>{{Cite news|url=https://www.scientificamerican.com/article/cosmic-test-bolsters-einsteins-ldquo-spooky-action-at-a-distance-rdquo/?WT.mc_id=SA_FB_PHYS_NEWS|title=Cosmic Test Bolsters Einstein's "Spooky Action at a Distance"|last=magazine|first=Elizabeth Gibney, Nature|newspaper=Scientific American|language=en|access-date=2017-02-04}}</ref>  The state of the particles being measured contains some [[hidden-variable theory|hidden variables]], whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed  this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete.

A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables".<ref>{{Cite news|url=https://www.scientificamerican.com/article/cosmic-test-bolsters-einsteins-ldquo-spooky-action-at-a-distance-rdquo/?WT.mc_id=SA_FB_PHYS_NEWS|title=Cosmic Test Bolsters Einstein's "Spooky Action at a Distance"|last=magazine|first=Elizabeth Gibney, Nature|newspaper=Scientific American|language=en|access-date=2017-02-04}}</ref>  The state of the particles being measured contains some [[hidden-variable theory|hidden variables]], whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed  this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete.

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.

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.

=== Violations of Bell's inequality ===

=== Violations of Bell's inequality 贝尔不等式的违反===

Local hidden variable theories fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the [[local realism|local realist]] or hidden variables view were correct, the results would always satisfy [[Bell's inequality]]. A [[Bell test experiments|number of experiments]] have shown in practice that Bell's inequality is not satisfied. However, prior to 2015, all of these had loophole problems that were considered the most important by the community of physicists.<ref>{{citation |author1=I. Gerhardt |author2=Q. Liu |author3=A. Lamas-Linares |author4=J. Skaar |author5=V. Scarani |author6=V. Makarov |author7=C. Kurtsiefer |year=2011 |title=Experimentally faking the violation of Bell's inequalities |journal=Phys. Rev. Lett. |volume=107 |issue=17 |page=170404 |arxiv=1106.3224 |doi=10.1103/PhysRevLett.107.170404 |bibcode=2011PhRvL.107q0404G |pmid=22107491|s2cid=16306493 }}</ref><ref>{{cite journal | last1 = Santos | first1 = E | year = 2004 | title = The failure to perform a loophole-free test of Bell's Inequality supports local realism | url = | journal = Foundations of Physics | volume = 34 | issue = 11| pages = 1643–1673 | doi=10.1007/s10701-004-1308-z|bibcode = 2004FoPh...34.1643S | s2cid = 123642560 }}</ref> When measurements of the entangled particles are made in moving [[special relativity|relativistic]] reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.<ref>{{cite journal |author = H. Zbinden |title = Experimental test of nonlocal quantum correlations in relativistic configurations |journal = Phys. Rev. A |volume = 63 |issue = 2 |pages = 22111 |doi = 10.1103/PhysRevA.63.022111|year = 2001|arxiv = quant-ph/0007009 |bibcode = 2001PhRvA..63b2111Z |display-authors = 1 |last2 = Gisin |last3 = Tittel |s2cid = 44611890 |url = http://archive-ouverte.unige.ch/unige:37034 }}</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>

Local hidden variable theories fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the [[local realism|local realist]] or hidden variables view were correct, the results would always satisfy [[Bell's inequality]]. A [[Bell test experiments|number of experiments]] have shown in practice that Bell's inequality is not satisfied. However, prior to 2015, all of these had loophole problems that were considered the most important by the community of physicists.<ref>{{citation |author1=I. Gerhardt |author2=Q. Liu |author3=A. Lamas-Linares |author4=J. Skaar |author5=V. Scarani |author6=V. Makarov |author7=C. Kurtsiefer |year=2011 |title=Experimentally faking the violation of Bell's inequalities |journal=Phys. Rev. Lett. |volume=107 |issue=17 |page=170404 |arxiv=1106.3224 |doi=10.1103/PhysRevLett.107.170404 |bibcode=2011PhRvL.107q0404G |pmid=22107491|s2cid=16306493 }}</ref><ref>{{cite journal | last1 = Santos | first1 = E | year = 2004 | title = The failure to perform a loophole-free test of Bell's Inequality supports local realism | url = | journal = Foundations of Physics | volume = 34 | issue = 11| pages = 1643–1673 | doi=10.1007/s10701-004-1308-z|bibcode = 2004FoPh...34.1643S | s2cid = 123642560 }}</ref> When measurements of the entangled particles are made in moving [[special relativity|relativistic]] reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.<ref>{{cite journal |author = H. Zbinden |title = Experimental test of nonlocal quantum correlations in relativistic configurations |journal = Phys. Rev. A |volume = 63 |issue = 2 |pages = 22111 |doi = 10.1103/PhysRevA.63.022111|year = 2001|arxiv = quant-ph/0007009 |bibcode = 2001PhRvA..63b2111Z |display-authors = 1 |last2 = Gisin |last3 = Tittel |s2cid = 44611890 |url = http://archive-ouverte.unige.ch/unige:37034 }}</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>

Consider two arbitrary quantum systems  and , with respective Hilbert spaces  and . The Hilbert space of the composite system is the tensor product

Consider two arbitrary quantum systems  and , with respective Hilbert spaces  and . The Hilbert space of the composite system is the tensor product

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]]. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating 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|s2cid=120680226}}</ref> and thus entanglement is a fundamentally non-classical phenomenon.

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]]. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating 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|s2cid=120680226}}</ref> and thus entanglement is a fundamentally non-classical phenomenon.

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).-->

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).-->

 

如果第一个系统处于状态 < math > scriptstyle | psi rangle _ a </math > ，而第二个系统处于状态 < math > scriptstyle | phi rangle _ b </math > ，则复合系统的状态为

如果第一个系统处于状态 < math > scriptstyle | psi rangle _ a </math > ，而第二个系统处于状态 < math > scriptstyle | phi rangle _ b </math > ，则复合系统的状态为

=== Other types of experiments ===

=== Other types of experiments其他类型的试验 ===

In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time.<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 |volume=8 |issue=6 |pages=480–485 |date=26 April 2012 |doi=10.1038/nphys2294|arxiv = 1203.4834 |bibcode = 2012NatPh...8..480M |last2=Zotter |last3=Kofler |last4=Ursin |last5=Jennewein |last6=Brukner |last7=Zeilinger |s2cid=119208488 }}</ref><ref>{{cite journal | last1 = Megidish | first1 = E. | last2 = Halevy | first2 = A. | last3 = Shacham | first3 = T. | last4 = Dvir | first4 = T. | last5 = Dovrat | first5 = L. | last6 = Eisenberg | first6 = H. S. | year = 2013 | title = Entanglement Swapping between Photons that have Never Coexisted | url = | journal = Physical Review Letters | volume = 110 | issue = 21| page = 210403| doi=10.1103/physrevlett.110.210403|arxiv = 1209.4191 |bibcode = 2013PhRvL.110u0403M | pmid=23745845| s2cid = 30063749 }}</ref> The authors claimed that this result was achieved by [[Quantum teleportation#Entanglement swapping|entanglement swapping]] between two pairs of entangled photons after measuring the polarization of one photon of the early pair, and that it proves that quantum non-locality applies not only to space but also to time.

In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time.<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 |volume=8 |issue=6 |pages=480–485 |date=26 April 2012 |doi=10.1038/nphys2294|arxiv = 1203.4834 |bibcode = 2012NatPh...8..480M |last2=Zotter |last3=Kofler |last4=Ursin |last5=Jennewein |last6=Brukner |last7=Zeilinger |s2cid=119208488 }}</ref><ref>{{cite journal | last1 = Megidish | first1 = E. | last2 = Halevy | first2 = A. | last3 = Shacham | first3 = T. | last4 = Dvir | first4 = T. | last5 = Dovrat | first5 = L. | last6 = Eisenberg | first6 = H. S. | year = 2013 | title = Entanglement Swapping between Photons that have Never Coexisted | url = | journal = Physical Review Letters | volume = 110 | issue = 21| page = 210403| doi=10.1103/physrevlett.110.210403|arxiv = 1209.4191 |bibcode = 2013PhRvL.110u0403M | pmid=23745845| s2cid = 30063749 }}</ref> The authors claimed that this result was achieved by [[Quantum teleportation#Entanglement swapping|entanglement swapping]] between two pairs of entangled photons after measuring the polarization of one photon of the early pair, and that it proves that quantum non-locality applies not only to space but also to time.