Editing Quantum entanglement (section)

=== 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>

These four pure states are all maximally entangled and form an [[orthonormal]] [[basis (linear algebra)|basis]] of the Hilbert space of the two qubits.<ref name="Rieffel2011"/>{{rp|38–39}}<ref name="Nielsen-2010"/>{{rp|98}} They provide examples of how quantum mechanics can violate [[Bell's theorem|Bell-type inequalities]].<ref name="Rieffel2011"/>{{rp|62}}<ref name="Nielsen-2010"/>{{rp|116}}

For {{nowrap|''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 {{nowrap|1=''M'' = 2}}. The traditional GHZ state was defined for {{nowrap|1=''M'' = 3}}. GHZ states are occasionally extended to [[qudit]]s, i.e., systems of ''d'' rather than 2 dimensions.<ref>{{Cite journal|last1=Caves |first1=Carlton M. |author-link=Carlton M. Caves |last2=Fuchs |first2=Christopher A. |last3=Schack |first3=Rüdiger |date=2002-08-20 |title=Unknown quantum states: The quantum de Finetti representation |journal=[[Journal of Mathematical Physics]] |volume=43 |number=9 |pages=4537–4559 |arxiv=quant-ph/0104088 |doi=10.1063/1.1494475 |quote=Mermin was the first to point out the interesting properties of this three-system state, following the lead of D. M. Greenberger, M. Horne, and A. Zeilinger, "Going beyond Bell's Theorem," in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69, where a similar four-system state was proposed. |bibcode=2002JMP....43.4537C}}</ref><ref>{{cite journal|first1=Yulin |last1=Chi |display-authors=etal |title=A programmable qudit-based quantum processor |journal=Nature Communications |year=2022 |volume=13 |issue=1 |page=1136 |doi=10.1038/s41467-022-28767-x |pmid=35246519 |bibcode=2022NatCo..13.1166C |pmc=8897515 }}</ref>

Also for {{nowrap|''M'' > 2}} qubits, there are [[Spin squeezing|spin squeezed states]], a class of [[squeezed coherent states]] satisfying certain restrictions on the uncertainty of spin measurements, which are necessarily entangled.<ref>{{cite journal |last1=Kitagawa |first1=Masahiro |last2=Ueda |first2=Masahito |year=1993 |title=Squeezed Spin States |url=https://ir.library.osaka-u.ac.jp/repo/ouka/all/77656/PhysRevA_47_06_005138.pdf |journal=Physical Review A |volume=47 |issue=6 |pages=5138–5143 |bibcode=1993PhRvA..47.5138K |doi=10.1103/physreva.47.5138 |pmid=9909547 |hdl-access=free |hdl=11094/77656}}</ref> Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.<ref>{{cite journal |last1=Wineland |first1=D. J. |last2=Bollinger |first2=J. J. |last3=Itano |first3=W. M. |last4=Moore |first4=F. L. |last5=Heinzen |first5=D. J. |year=1992 |title=Spin squeezing and reduced quantum noise in spectroscopy |journal=Physical Review A |volume=46 |issue=11 |pages=R6797–R6800 |bibcode=1992PhRvA..46.6797W |doi=10.1103/PhysRevA.46.R6797 |pmid=9908086}}</ref>

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 the Bell state <math>|\Psi^+\rangle</math> except the basis states <math>|0\rangle</math> and <math>|1\rangle</math> have been replaced with "the ''N'' photons are in one mode" and "the ''N'' photons are in the other mode".<ref name="Kishore2007">{{cite journal|first1=Kishore T. |last1=Kapale |first2=Jonathan P. |last2=Dowling |author-link2=Jonathan Dowling |title=A Bootstrapping Approach for Generating Maximally Path-Entangled Photon States |arxiv=quant-ph/0612196 |journal=Physical Review Letters |volume=99 |page=053602 |year=2007 |issue=5 |doi=10.1103/PhysRevLett.99.053602|pmid=17930751 |bibcode=2007PhRvL..99e3602K }}</ref>

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 NOON states, and can be used to achieve the [[Heisenberg limit]].<ref>{{cite journal |doi = 10.1103/PhysRevLett.71.1355|pmid = 10055519|title = Interferometric detection of optical phase shifts at the Heisenberg limit|journal = Physical Review Letters|volume = 71|issue = 9|pages = 1355–1358|year = 1993|last1 = Holland|first1 = M. J|last2 = Burnett|first2 = K|bibcode = 1993PhRvL..71.1355H}}</ref>

For the appropriately chosen measures of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled.<ref>{{cite journal|doi=10.1126/science.1097522 |year=2004 |volume=304 |journal=Science |first1=Christian F. |last1=Roos |display-authors=etal |title=Control and Measurement of Three-Qubit Entangled States|issue=5676 |pages=1478–1480 |pmid=15178795 }}</ref><ref name="Kishore2007"/><ref>{{cite journal|last1=Pezzè |first1=L. |last2=Smerzi |first2=A. |last3=Oberthaler |first3=M. K. |last4=Schmied |first4=R. |last5=Treutlein |first5=P. |year=2018 |title=Quantum metrology with nonclassical states of atomic ensembles |journal=Reviews of Modern Physics |volume=90 |number=3 |page=035005 |doi=10.1103/revmodphys.90.035005 |arxiv=1609.01609}}</ref>