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In quantum information theory, the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable. In other words, the reduction criterion is a ''separability criterion''. It was first proved in 〔 and independently formulated in.〔 Violation of the reduction criterion is closely related to the distillability of the state in question.〔 ==Details== Let ''H''1 and ''H''2 be Hilbert spaces of finite dimensions ''n'' and ''m'' respectively. ''L''(''Hi'') will denote the space of linear operators acting on ''Hi''. Consider a bipartite quantum system whose state space is the tensor product : An (un-normalized) mixed state ''ρ'' is a positive linear operator (density matrix) acting on ''H''. A linear map Φ: ''L''(''H''2) → ''L''(''H''1) is said to be positive if it preserves the cone of positive elements, i.e. ''A'' is positive implied ''Φ''(''A'') is also. From the one-to-one correspondence between positive maps and entanglement witnesses, we have that a state ''ρ'' is entangled if and only if there exists a positive map ''Φ'' such that : is not positive. Therefore, if ''ρ'' is separable, then for all positive map Φ, : Thus every positive, but not completely positive, map Φ gives rise to a necessary condition for separability in this way. The reduction criterion is a particular example of this. Suppose ''H''1 = ''H''2. Define the positive map Φ: ''L''(''H''2) → ''L''(''H''1) by : It is known that Φ is positive but not completely positive. So a mixed state ''ρ'' being separable implies : Direct calculation shows that the above expression is the same as : where ''ρ''1 is the partial trace of ''ρ'' with respect to the second system. The dual relation : is obtained in the analogous fashion. The reduction criterion consists of the above two inequalities. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「reduction criterion」の詳細全文を読む スポンサード リンク
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