Editing Quantum entanglement (section)

=== 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}}