Purification of quantum state

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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.

In purely linear algebraic terms, it can be viewed as a statement about positive-semidefinite matrices.

Statement[edit]

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

We say that is the purification of .

Proof[edit]

A density matrix is by definition positive semidefinite. So ρ can be diagonalized and written as for some basis . Let be another copy of the n-dimensional Hilbert space with an orthonormal basis . Define by

Direct calculation gives

This proves the claim.

Note[edit]

  • The vectorial pure state is in the form specified by the Schmidt decomposition.
  • Since square root decompositions of a positive semidefinite matrix are not unique, neither are purifications.
  • 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 of a positive map remains a positive map.

An application: Stinespring's theorem[edit]

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.