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| In [[quantum information theory]], the '''reduction criterion''' is a necessary condition a [[Mixed state (physics)|mixed state]] must satisfy in order for it to be [[separable state|separable]]. In other words, the reduction criterion is a ''separability criterion''. It was fist proved in <ref name="HH99" />
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| and independently formulated in.<ref name="Cerf99" /> Violation of the reduction criterion is closely related to the [[entanglement distillation|distillability]] of the state in question.<ref name="HH99" />
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| ==Details==
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| Let ''H''<sub>1</sub> and ''H''<sub>2</sub> be Hilbert spaces of finite dimensions ''n'' and ''m'' respectively. ''L''(''H<sub>i</sub>'') will denote the space of linear operators acting on ''H<sub>i</sub>''. Consider a bipartite quantum system whose state space is the tensor product
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| :<math> H = H_1 \otimes H_2.</math>
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| An (un-normalized) mixed state ''ρ'' is a positive linear operator (density matrix) acting on ''H''.
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| A linear map Φ: ''L''(''H''<sub>2</sub>) → ''L''(''H''<sub>1</sub>) is said to be positive if it preserves the cone of positive elements, i.e. ''A'' is positive implied ''Φ''(''A'') is also.
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| From the one-to-one correspondence between positive maps and [[entanglement witness]]es, we have that a state ''ρ'' is entangled if and only if there exists a positive map ''Φ'' such that
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| :<math>(I \otimes \Phi)(\rho)</math>
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| is not positive. Therefore, if ''ρ'' is separable, then for all positive map Φ,
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| :<math>(I \otimes \Phi)(\rho) \geq 0.</math>
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| Thus every positive, but not [[completely positive map|completely positive]], map Φ gives rise to a necessary condition for separability in this way. The reduction criterion is a particular example of this.
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| Suppose ''H''<sub>1</sub> = ''H''<sub>2</sub>. Define the positive map Φ: ''L''(''H''<sub>2</sub>) → ''L''(''H''<sub>1</sub>) by
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| :<math>\Phi(A) = \operatorname{Tr}A - A.</math>
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| It is known that Φ is positive but not completely positive. So a mixed state ''ρ'' being separable implies
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| :<math>(I \otimes \Phi) (\rho) \geq 0.</math>
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| Direct calculation shows that the above expression is the same as
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| :<math>I \otimes \rho_1 - \rho \geq 0</math> | |
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| where ''ρ''<sub>1</sub> is the [[partial trace]] of ''ρ'' with respect to the second system. The dual relation
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| :<math>\rho_2 \otimes I - \rho \geq 0</math>
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| is obtained in the analogous fashion. The reduction criterion consists of the above two inequalities. The reduction criterion is
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| ==References==
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| <references>
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| <ref name="HH99">{{cite journal|journal=Phys. Rev. A|volume='''59'''|pages= 4206|year=1999|author=M. Horodecki and P. Horodecki|title=Reduction criterion of separability and limits for a class of distillation protocols|doi=10.1103/PhysRevA.59.4206|arxiv=quant-ph/9708015}}</ref>
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| <ref name="Cerf99">{{cite journal|journal=Phys. Rev. A|volume='''60'''|pages= 898|year=1999|author=N. Cerf ''et al.''|title=Reduction criterion for separability|doi=10.1103/PhysRevA.60.898|arxiv=quant-ph/9710001}}</ref>
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| </references>
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| {{DEFAULTSORT:Reduction Criterion}}
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| [[Category:Quantum information science]]
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Hi there. Let me begin by introducing the writer, her name is Myrtle Cleary. His wife doesn't like it the way he does but what he really likes performing is to do aerobics and he's been doing it for quite a while. Puerto Rico is where he and his spouse live. Hiring is her day occupation now and she will not alter it whenever soon.
My website; http://www.sddch.org/?document_srl=345265