Editing Quantum entanglement (section)

==== As a measure of entanglement ====
Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.<ref name="Plenio">{{cite journal|last1=Plenio |first1=Martin B. |first2=Shashank |last2=Virmani|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}}</ref><ref name="Vedral2002">{{cite journal | last = Vedral | first = Vlatko |author-link = Vlatko Vedral | doi = 10.1103/RevModPhys.74.197 | arxiv = quant-ph/0102094 | bibcode=2002RvMP...74..197V | volume=74 | issue = 1 | title=The role of relative entropy in quantum information theory | year=2002 | journal=Reviews of Modern Physics | pages=197–234 | s2cid = 6370982 }}</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. 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.<ref>{{cite journal |last1=Hill |first1=S |last2=Wootters |first2=W. K. |title=Entanglement of a Pair of Quantum Bits |journal=Phys. Rev. Lett. |arxiv=quant-ph/9703041 |doi =10.1103/PhysRevLett.78.5022 |year=1997 |volume=78 |issue=26 |pages=5022–5025 |bibcode=1997PhRvL..78.5022H |s2cid=9173232 }}</ref>

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {{mset|1/''n'', ..., 1/''n''}}.<ref name="Nielsen-2010"/>{{rp|505}} Therefore, a bipartite pure state {{math|''ρ'' ∈ ''H''<sub>A</sub> ⊗ ''H''<sub>B</sub>}} is said to be a ''maximally entangled state'' if the reduced state of each subsystem of {{mvar|ρ}} is the diagonal matrix<ref>{{Cite journal |last1=Enríquez |first1=M. |last2=Wintrowicz |first2=I. |last3=Życzkowski |first3=K. |author-link3=Karol Życzkowski |date=March 2016 |title=Maximally Entangled Multipartite States: A Brief Survey |journal=Journal of Physics: Conference Series |volume=698 |issue=1 |pages=012003 |doi=10.1088/1742-6596/698/1/012003 |bibcode=2016JPhCS.698a2003E |issn=1742-6588|doi-access=free }}</ref>
: <math>\begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}.</math>

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.<ref name="Bengtsson2017"/>{{rp|471}}

[[Rényi entropy]] also can be used as a measure of entanglement.<ref name="Bengtsson2017"/>{{rp|447,480}}<ref>{{cite journal |last1=Wang |first1=Yu-Xin |last2=Mu |first2=Liang-Zhu |last3=Vedral |first3=Vlatko |last4=Fan |first4=Heng |date=17 February 2016 |title=Entanglement Rényi α entropy |url=https://link.aps.org/doi/10.1103/PhysRevA.93.022324 |journal=Physical Review A |language=en |volume=93 |issue=2 |page=022324 |arxiv=1504.03909 |doi=10.1103/PhysRevA.93.022324 |bibcode=2016PhRvA..93b2324W |issn=2469-9926}}</ref>