Editing Quantum entanglement (section)

=== Entanglement as a resource ===
In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations.<ref name="Chitambar2019">
{{cite journal
 | last1 = Chitambar | first1 = Eric
 | last2 = Gour | first2 = Gilad
 | title = Quantum resource theories
 | journal = Reviews of Modern Physics
 | volume = 91
 | number = 2
 | pages = 025001
 | doi = 10.1103/RevModPhys.91.025001
 | arxiv = 1806.06107
 | year = 2019
 | bibcode = 2019RvMP...91b5001C
 | s2cid = 119194947
}}</ref><ref name="GG-2022">
{{cite journal
 | last1 = Georgiev
 | first1 = Danko D.
 | last2 = Gudder
 | first2 = Stanley P.
 | title = Sensitivity of entanglement measures in bipartite pure quantum states
 | journal = Modern Physics Letters B
 | volume = 36
 | number = 22
 | pages = 2250101–2250255
 | doi = 10.1142/S0217984922501019
 | arxiv = 2206.13180
 | year = 2022
 | bibcode = 2022MPLB...3650101G
 | s2cid = 250072286
}}</ref> The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitrary [[quantum operation]]s can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called [[LOCC]] (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. 

If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state <math>|\Psi\rangle</math> she has in her lab. Alice performs a joint measurement on <math>|\Psi\rangle</math> together with her half of the entangled state  and tells Bob the results. Using Alice's results Bob operates on his half of the entangled state to make it equal to <math>|\Psi\rangle</math>. Since Alice's measurement necessarily erases the quantum state of the system in her lab, the state <math>|\Psi\rangle</math> is not copied, but transferred: it is said to be "[[quantum teleportation|teleported]]" to Bob's laboratory through this protocol.<ref name="Nielsen-2010">{{cite book |last1=Nielsen |first1=Michael A. |title=Quantum Computation and Quantum Information |title-link=Quantum Computation and Quantum Information |last2=Chuang |first2=Isaac L. |publisher=Cambridge Univ. Press |year=2010 |isbn=978-0-521-63503-5 |edition=10th anniversary|location=Cambridge}}</ref>{{rp|27}}<ref name="horodecki2007"/>{{rp|875}}<ref>{{cite journal|arxiv=1505.07831 |title=Advances in Quantum Teleportation |first1=S. |last1=Pirandola |first2=J. |last2=Eisert |first3=C. |last3=Weedbrook |first4=A. |last4=Furusawa |first5=S. L. |last5=Braunstein |journal=Nature Photonics |volume=9 |pages=641–652 |year=2015 |issue=10 |doi=10.1038/nphoton.2015.154|bibcode=2015NaPho...9..641P }}</ref>

[[File:Entanglement swapping.svg|thumb|Entanglement of states from independent sources can be swapped through Bell state measurement.<ref name="GuoReview2023">{{cite journal |last1=Hu |first1=Xiao-Min |last2=Guo |first2=Yu |last3=Liu |first3=Bi-Heng |last4=Li |first4=Chuan-Feng |last5=Guo |first5=Guang-Can |date=June 2023 |title=Progress in quantum teleportation |url=https://www.nature.com/articles/s42254-023-00588-x |journal=Nature Reviews Physics |language=en |volume=5 |issue=6 |pages=339–353 |doi=10.1038/s42254-023-00588-x |bibcode=2023NatRP...5..339H |issn=2522-5820}}</ref>{{rp|341}}]]
[[Entanglement swapping]] is variant of teleportation that allows two parties that have never interacted to share an entangled state. The swapping protocol begins with two EPR sources. One source emits an entangled pair of particles A and B, while the other emits a second entangled pair of particles C and D. Particles B and C are subjected to a measurement in the basis of Bell states. The state of the remaining particles, A and D, collapses to a Bell state, leaving them entangled despite never having interacted with each other.<ref name="horodecki2007"/><ref name="Pan1998">{{Cite journal |last1=Pan |first1=J.-W. |last2=Bouwmeester |first2=D. |last3=Weinfurter |first3=H. |last4=Zeilinger |first4=A. |author-link4=Anton Zeilinger |year=1998 |title=Experimental entanglement swapping: Entangling photons that never interacted |journal=[[Physical Review Letters]] |volume=80 |number=18 |pages=3891–3894 |doi=10.1103/PhysRevLett.80.3891 |bibcode=1998PhRvL..80.3891P }}</ref>

An interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.<ref name="horodecki2007" />