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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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