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| In [[quantum mechanics]], especially [[quantum information]], '''purification''' refers to the fact that every [[Mixed state (physics)|mixed state]] acting on finite dimensional Hilbert spaces can be viewed as the [[partial trace|reduced state]] of some pure state.
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| In purely linear algebraic terms, it can be viewed as a statement about [[positive-semidefinite matrix|positive-semidefinite matrices]].
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| == Statement == | |
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| Let ρ be a density matrix acting on a Hilbert space <math>H_A</math> of finite dimension ''n''. Then there exist a Hilbert space <math>H_B</math> and a pure state <math>| \psi \rangle \in H_A \otimes H_B</math> such that the partial trace of <math>| \psi \rangle \langle \psi |</math> with respect to <math>H_B</math>
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| :<math>\operatorname{tr_B} \left( | \psi \rangle \langle \psi | \right )= \rho.</math>
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| We say that <math>| \psi \rangle</math> is the purification of <math>\rho</math>.
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| === Proof ===
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| A density matrix is by definition positive semidefinite. So ρ can be [[Diagonalizable matrix|diagonalized]] and written as <math>\rho = \sum_{i =1} ^n p_i | i \rangle \langle i |</math> for some basis <math>\{ | i \rangle \}</math>. Let <math>H_B</math> be another copy of the ''n''-dimensional Hilbert space with any orthonormal basis <math>\{ | i' \rangle \}</math>. Define <math>| \psi \rangle \in H_A \otimes H_B</math> by
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| :<math>| \psi \rangle = \sum_{i} \sqrt{p_i} |i \rangle \otimes | i' \rangle.</math>
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| Direct calculation gives
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| :<math>
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| \operatorname{tr_B} \left( | \psi \rangle \langle \psi | \right )=
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| \operatorname{tr_B} \left( \sum_{i, j} \sqrt{p_ip_j} |i \rangle \langle j | \otimes | i' \rangle \langle j'| \right ) = \sum_{i,j} \delta_{i,j} \sqrt{p_i p_j}| i \rangle \langle j | = \rho.
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| </math>
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| This proves the claim.
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| ==== Note ====
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| * The vectorial pure state <math>| \psi \rangle</math> is in the form specified by the [[Schmidt decomposition]].
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| * Since square root decompositions of a positive semidefinite matrix are not unique, neither are purifications.
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| * In linear algebraic terms, a square matrix is positive semidefinite if and only if it can be purified in the above sense. The ''if'' part of the implication follows immediately from the fact that the [[partial trace]] is a [[Choi's theorem on completely positive maps|positive map]].
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| == An application: Stinespring's theorem ==
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| {{Expand section|date=June 2008}}
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| By combining [[Choi's theorem on completely positive maps]] and purification of a mixed state, we can recover the [[Stinespring factorization theorem|Stinespring dilation theorem]] for the finite dimensional case.
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| {{DEFAULTSORT:Purification Of Quantum State}}
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| [[Category:Linear algebra]]
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| [[Category:Quantum information science]]
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