Editing Quantum entanglement (section)

=== Testing a system for entanglement ===

A density matrix ''ρ'' is called separable if it can be written as a convex sum of product states, namely
<math display="block">{\rho=\sum_j p_j \rho_j^{(A)}\otimes\rho_j^{(B)}}</math>
with <math>0\le p_j\le 1</math> probabilities. By definition, a state is entangled if it is not separable.

For 2-qubit and qubit-qutrit systems (2&nbsp;×&nbsp;2 and 2&nbsp;×&nbsp;3 respectively) the simple [[Peres–Horodecki criterion]] provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized.<ref name="NP-hard1">{{cite book|last=Gurvits |first=L. |chapter=Classical deterministic complexity of Edmonds' problem and quantum entanglement |title=Proceedings of the 35th ACM Symposium on Theory of Computing |publisher=ACM Press |location=New York |year=2003 |pages=10–19 |doi=10.1145/780542.780545|isbn=1-58113-674-9 }}</ref><ref name="NP-hard2">{{cite journal |author=Gharibian |first=Sevag |year=2010 |title=Strong NP-Hardness of the Quantum Separability Problem |journal=Quantum Information and Computation |volume=10 |pages=343–360 |arxiv=0810.4507 |doi=10.26421/QIC10.3-4-11 |s2cid=621887 |number=3&4}}</ref> Other separability criteria include (but not limited to) the [[range criterion]], [[reduction criterion]], and those based on uncertainty relations.<ref>{{cite journal |last1=Hofmann |first1=Holger F. |last2=Takeuchi |first2=Shigeki |title=Violation of local uncertainty relations as a signature of entanglement |journal=Physical Review A |date=22 September 2003 |volume=68 |issue=3 |page=032103 |doi=10.1103/PhysRevA.68.032103 |arxiv=quant-ph/0212090 |bibcode=2003PhRvA..68c2103H |s2cid=54893300 }}</ref><ref>{{cite journal |last1=Gühne |first1=Otfried |title=Characterizing Entanglement via Uncertainty Relations |journal=Physical Review Letters |date=18 March 2004 |volume=92 |issue=11 |page=117903 |doi=10.1103/PhysRevLett.92.117903|pmid=15089173 |arxiv=quant-ph/0306194 |bibcode=2004PhRvL..92k7903G |s2cid=5696147 }}</ref><ref>{{cite journal |last1=Gühne |first1=Otfried |last2=Lewenstein |first2=Maciej |title=Entropic uncertainty relations and entanglement |journal=Physical Review A |date=24 August 2004 |volume=70 |issue=2 |page=022316 |arxiv=quant-ph/0403219 |doi=10.1103/PhysRevA.70.022316 |bibcode=2004PhRvA..70b2316G |s2cid=118952931}}</ref><ref>{{cite journal |last1=Huang |first1=Yichen |title=Entanglement criteria via concave-function uncertainty relations |journal=Physical Review A |date=29 July 2010 |volume=82 |issue=1 |page=012335 |doi=10.1103/PhysRevA.82.012335 |bibcode=2010PhRvA..82a2335H }}</ref> See Ref.<ref>{{cite journal|last1=Gühne|first1=Otfried|last2=Tóth|first2=Géza| title=Entanglement detection |journal=Physics Reports|volume=474|issue=1–6|pages=1–75|doi=10.1016/j.physrep.2009.02.004|arxiv=0811.2803 |bibcode=2009PhR...474....1G |year=2009|s2cid=119288569}}</ref> for a review of separability criteria in discrete-variable systems and Ref.<ref name=FriisEtAl2019entanglement>{{cite journal|last1= Friis |first1=Nicolai |last2= Vitagliano |first2=Giuseppe |last3=Malik |first3=Mehul |last4=Huber |first4=Marcus |date=2019 |title=Entanglement certification from theory to experiment |journal=Nature Reviews Physics |language=en|volume=1|issue=|pages=72–87|doi=10.1038/s42254-018-0003-5 |issn=2522-5820 |arxiv=1906.10929 |s2cid=125658647}}</ref> for a review on techniques and challenges in experimental entanglement certification in discrete-variable systems.

A numerical approach to the problem is suggested by [[Jon Magne Leinaas]], [[Jan Myrheim]] and [[Eirik Ovrum]] in their paper "Geometrical aspects of entanglement".<ref>{{cite journal |last1=Leinaas| first1=Jon Magne| last2=Myrheim| first2=Jan| last3=Ovrum| first3=Eirik| year=2006| title=Geometrical aspects of entanglement| journal=Physical Review A| volume=74| issue=1| page=012313| s2cid=119443360| doi=10.1103/PhysRevA.74.012313| arxiv=quant-ph/0605079| bibcode=2006PhRvA..74a2313L}}</ref> Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached.

In continuous variable systems, the Peres–Horodecki criterion also applies. Specifically, Simon<ref>{{cite journal|last1=Simon|first1=R.|title=Peres–Horodecki Separability Criterion for Continuous Variable Systems |journal=Physical Review Letters|volume=84|issue=12|pages=2726–2729|pmid=11017310 |doi=10.1103/PhysRevLett.84.2726|arxiv=quant-ph/9909044|bibcode=2000PhRvL..84.2726S|s2cid=11664720 |year=2000}}</ref> formulated a particular version of the Peres–Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for <math> 1\oplus1 </math>-mode Gaussian states (see Ref.<ref>{{cite journal|last1=Duan|first1=Lu-Ming |last2=Giedke|first2=G.|last3=Cirac|first3=J. I.|last4=Zoller|first4=P.|title=Inseparability Criterion for Continuous Variable Systems|journal=Physical Review Letters|volume=84|issue=12 |pages=2722–2725 |doi=10.1103/PhysRevLett.84.2722|pmid=11017309|arxiv=quant-ph/9908056|bibcode=2000PhRvL..84.2722D |year=2000|s2cid=9948874}}</ref> for a seemingly different but essentially equivalent approach). It was later found<ref>{{cite journal|last1=Werner|first1=R. F.|last2=Wolf|first2=M. M.|title=Bound Entangled Gaussian States|journal=Physical Review Letters|volume=86|issue=16|pages=3658–3661|pmid=11328047 |arxiv=quant-ph/0009118 |doi=10.1103/PhysRevLett.86.3658|bibcode=2001PhRvL..86.3658W|year=2001 |s2cid=20897950}}</ref> that Simon's condition is also necessary and sufficient for <math> 1\oplus n </math>-mode Gaussian states, but no longer sufficient for <math> 2\oplus2 </math>-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators<ref>{{cite journal|last1=Shchukin |first1=E.|last2=Vogel |first2=W. |title=Inseparability Criteria for Continuous Bipartite Quantum States |journal=Physical Review Letters|volume=95|issue=23 |pages=230502|doi=10.1103/PhysRevLett.95.230502|pmid=16384285|bibcode=2005PhRvL..95w0502S |arxiv=quant-ph/0508132|year=2005|s2cid=28595936}}</ref><ref>{{cite journal| last1=Hillery|first1=Mark|last2=Zubairy |first2=M.Suhail|title=Entanglement Conditions for Two-Mode States|journal=Physical Review Letters |volume=96|issue=5|page=050503|year=2006|doi=10.1103/PhysRevLett.96.050503|arxiv=quant-ph/0507168 |bibcode=2006PhRvL..96e0503H|pmid=16486912|s2cid=43756465}}</ref> or by using entropic measures.<ref>{{cite journal| last1=Walborn|first1=S.|last2=Taketani|first2=B.|last3=Salles|first3=A.|last4=Toscano |first4=F.|last5=de Matos Filho|first5=R.|title=Entropic Entanglement Criteria for Continuous Variables |journal=Physical Review Letters |volume=103|issue=16|doi=10.1103/PhysRevLett.103.160505|arxiv=0909.0147 |bibcode=2009PhRvL.103p0505W|pmid=19905682|page=160505 |year=2009 |s2cid=10523704}}</ref><ref>{{cite journal |last1=Huang |first1=Yichen |date=October 2013 |title=Entanglement Detection: Complexity and Shannon Entropic Criteria |journal=IEEE Transactions on Information Theory |volume=59 |issue=10 |pages=6774–6778 |doi=10.1109/TIT.2013.2257936 |s2cid=7149863}}</ref>