Editing Quantum entanglement (section)

== Concept ==

=== Meaning of entanglement ===
Just as [[energy]] is a resource that facilitates mechanical operations, entanglement is a resource that facilitates performing tasks that involve communication and computation.<ref name="Nielsen-2010"/>{{rp|106}}<ref name="Rieffel2011"/>{{rp|218}}<ref name="Bengtsson2017">{{cite book|first1=Ingemar |last1=Bengtsson |first2=Karol |last2=Życzkowski |author-link2=Karol Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |year=2017 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-02625-4}}</ref>{{rp|435}}<ref name="Bub2023">{{cite SEP|url-id=qt-entangle |author-first=Jeffrey |author-last=Bub |author-link=Jeffrey Bub |title=Quantum Entanglement and Information |date=2023-05-02}}</ref> The mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system.<ref name="Rau2021">{{cite book|first=Jochen |last=Rau |title=Quantum Theory: An Information Processing Approach |publisher=Oxford University Press |year=2021 |isbn=978-0-19-289630-8}}</ref> If the quantum state that describes a pair of particles is entangled, then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other. However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life. Instead, entanglement can be thought of as ''potential'' correlation that can be used to generate actual correlation in an appropriate experiment.<ref name="Fuchs2011">{{cite book |first=Christopher A. |last=Fuchs |title=Coming of Age with Quantum Information |date=6 January 2011 |publisher=Cambridge University Press |isbn=978-0-521-19926-1 }}</ref>{{rp|130}} The correlations generated from an entangled quantum state cannot in general be replicated by classical probability.<ref name="Holevo2001">{{cite book|first=Alexander S. |last=Holevo |author-link=Alexander Holevo |title=Statistical Structure of Quantum Theory |publisher=Springer |series=[[Lecture Notes in Physics|Lecture Notes in Physics. Monographs]] |year=2001 |isbn=3-540-42082-7}}</ref>{{rp|33}}

An example of entanglement is a [[subatomic particle]] that [[Particle decay|decays]] into an entangled pair of other particles. The 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 (physics)|spin]]-zero particle could decay into a pair of spin-1/2 particles. If there is no orbital angular momentum, the total spin angular momentum after this decay must be zero (by the [[conservation of angular momentum]]). Whenever the first particle is measured to be [[Spin (physics)#Direction|spin up]] on some axis, the other, when measured on the same axis, is always found to be [[Spin (physics)#Direction|spin down]]. This is called the spin anti-correlated case and the pair is said to be in the [[singlet state]]. Perfect anti-correlations like this could be explained by "hidden variables" within the particles. For example, we could hypothesize that the particles are made in pairs such that one carries a value of "up" while the other carries a value of "down". Then, knowing the result of the spin measurement upon one particle, we could predict that the other will have the opposite value. Bell illustrated this with a story about a colleague, Bertlmann, who always wore socks with mismatching colors. "Which colour he will have on a given foot on a given day is quite unpredictable," Bell wrote, but upon observing "that the first sock is pink you can be already sure that the second sock will not be pink."<ref>{{cite journal|first=J. |last=Bell |title=Bertlmann's Socks and the Nature of Reality |journal=Journal de Physique Colloques |year=1981 |volume=42 (C2) |pages=41–62 |doi=10.1051/jphyscol:1981202 |url=https://hal.science/jpa-00220688v1}}</ref> Revealing the remarkable features of quantum entanglement requires considering multiple distinct experiments, such as spin measurements along different axes, and comparing the correlations obtained in these different configurations.<ref name="Zwiebach2022">{{cite book|first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8}}</ref>{{rp|§18.8}}

Quantum [[physical system|systems]] can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below on [[#Methods of creating entanglement|methods]]. Entanglement is broken when the entangled particles [[quantum decoherence|decohere]] through interaction with the environment; for example, when a measurement is made. In more detail, this process involves the particles becoming entangled with the environment, as a consequence of which, the quantum state describing the particles themselves is no longer entangled.<ref name="Peres1993">{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |publisher=Kluwer |year=1993 |isbn=0-7923-2549-4 }}</ref>{{rp|369}}<ref>{{cite journal|doi=10.1016/j.physrep.2019.10.001 |first=Max |last=Schlosshauer |title=Quantum decoherence |journal=Physics Reports |volume=831 |date=25 October 2019 |pages=1–57 |arxiv=1911.06282|bibcode=2019PhR...831....1S }}</ref>

Mathematically, an entangled system can be defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. When entanglement is present, one constituent cannot be fully described without considering the other(s).<ref name="Mermin2007">{{cite book|first=N. David |last=Mermin |author-link=N. David Mermin |title=Quantum Computer Science: An Introduction |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-87658-2}}</ref>{{rp|18–19}}<ref name="Zwiebach2022"/>{{rp|§1.5}} The state of a composite system is always expressible as a sum, or [[quantum superposition|superposition]], of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.<ref name="Rieffel2011">{{Cite book |last1=Rieffel |first1=Eleanor |author-link1=Eleanor Rieffel |title=Quantum Computing: A Gentle Introduction |title-link=Quantum Computing: A Gentle Introduction |last2=Polak |first2=Wolfgang |date=2011 |publisher=MIT Press |isbn=978-0-262-01506-6 |series=Scientific and engineering computation |location=Cambridge, Mass}}</ref>{{Rp|page=39}}

=== Paradox ===
{{main|EPR paradox}}
The singlet state described above is the basis for one version of the EPR paradox. In this variant, introduced by [[David Bohm]], a source emits particles and sends them in opposite directions. The state describing each pair is entangled.<ref>{{cite book|first=David |last=Bohm |author-link=David Bohm |title=Quantum Theory |orig-year=1951 |year=1989 |publisher=Dover |edition=reprint |isbn=0-486-65969-0 |pages=611–622}}</ref> In the standard textbook presentation of quantum mechanics, performing a spin measurement on one of the particles causes the wave function for the whole pair to [[wave function collapse|collapse]] into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is 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 name="Zwiebach2022"/>{{rp|§18.8}}<ref name="Griffiths">{{cite book|first1=David J. |last1=Griffiths |author-link1=David J. Griffiths |first2=Darrell F. |last2=Schroeter |title=Introduction to Quantum Mechanics |title-link=Introduction to Quantum Mechanics (book) |edition=3rd |year=2018 |publisher=Cambridge University Press |isbn=978-1-107-18963-8 }}</ref>{{rp|447–448}}

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''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} there are [[inertial frame]]s in which {{math|''x''<sub>1</sub>}} is first and others in which {{math|''x''<sub>2</sub>}} 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.<ref>{{cite journal|first=Asher |last=Peres |author-link=Asher Peres |doi=10.1103/PhysRevA.61.022117 |title=Classical interventions in quantum systems. II. Relativistic invariance |journal=Physical Review A |volume=61 |pages=022117 |date=2000-01-18|issue=2 |arxiv=quant-ph/9906034 |bibcode=2000PhRvA..61b2117P }}</ref>

=== Failure of local hidden-variable theories ===
A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "[[hidden-variables theory|hidden variables]]".<ref name="Gibney2017">
{{cite journal
 | last = Gibney
 | first = Elizabeth
 | title = Cosmic Test Bolsters Einstein's "Spooky Action at a Distance"
 | journal = Scientific American
 | url = https://www.scientificamerican.com/article/cosmic-test-bolsters-einsteins-ldquo-spooky-action-at-a-distance-rdquo/
 | year = 2017
}}</ref> The state of the particles being measured contains some 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.

[[Local hidden-variable theory|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 realist or hidden variables view were correct, the results would always satisfy [[Bell's inequality]]. A [[Bell test|number of experiments]] have shown in practice that Bell's inequality is not satisfied.<ref name = "Clauser"/><ref>{{cite journal|last1=Dehlinger |first1=Dietrich |first2=M. W. |last2=Mitchell |title=Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory |journal=American Journal of Physics |volume=70 |number=9 |year=2002 |pages=903–910 |arxiv=quant-ph/0205171 |doi=10.1119/1.1498860|bibcode=2002AmJPh..70..903D }}</ref><ref>{{cite journal|date=May 2018|title=Challenging local realism with human choices |journal=Nature |volume=557 |issue=7704 |pages=212–216 |doi=10.1038/s41586-018-0085-3 |bibcode=2018Natur.557..212B |author1=BIG Bell Test Collaboration |pmid=29743691 |arxiv=1805.04431 }}</ref><ref>{{cite journal|title=Cosmic Bell Test Using Random Measurement Settings from High-Redshift Quasars|date=20 August 2018 |journal=Physical Review Letters |volume=121 |number=8 |pages=080403 |doi=10.1103/PhysRevLett.121.080403 |last1 = Rauch |first1 = Dominik |pmid=30192604 |display-authors=etal |arxiv=1808.05966|bibcode=2018PhRvL.121h0403R }}</ref> Moreover, 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=Zbinden |first=H. |author2=Gisin |author3=Tittel |display-authors=1 |year=2001 |title=Experimental test of nonlocal quantum correlations in relativistic configurations |url=http://archive-ouverte.unige.ch/unige:37034 |journal=Physical Review A |volume=63 |issue=2 |pages=22111 |arxiv=quant-ph/0007009 |bibcode=2001PhRvA..63b2111Z |doi=10.1103/PhysRevA.63.022111 |s2cid=44611890}}</ref><ref name=Gilder2009/>{{rp|321–324}}

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.