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Revision as of 06:13, 16 June 2006 by Mct mht (talk | contribs) (typo. copyedit intro.)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)In quantum mechanics, especially quantum information, purification refers to the fact that every mixed state acting on finite dimensional Hilbert spaces can be viewed as the reduced state of some pure state.
Statement
Let ρ be a density matrix acting on a Hilbert space of finite dimension n, then there exist a Hilbert space and a pure state such that the partial trace of with respect to
Proof
A density matrix is by definition positive semidefinite. So ρ has square root factorization . Let be another copy of the n-dimensional Hilbert space with any orthonormal basis . Define by
Direct calculation gives
This proves the claim.
Note
- The vectorial pure state is in the form specified by the Schmidt decomposition.
- Since square root decompositions of a positive semidefinite matrix is not unique in general, neither are purifications.
An application: Stinespring's theorem
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By combining Choi's theorem on completely positive maps and purification of a mixed state, we can recover the Stinespring dilation theorem for the finite dimensional case.
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