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 H_A of finite dimension n. Then there exist a Hilbert space H_B and a pure state | \psi \rangle \in H_A \otimes H_B such that the partial trace of | \psi \rangle \langle \psi | with respect to H_B

\operatorname{tr_B} \left( | \psi \rangle \langle \psi | \right )= \rho.

We say that | \psi \rangle is the purification of \rho.

Proof[edit]

A density matrix is by definition positive semidefinite. So ρ can be diagonalized and written as \rho = \sum_{i =1} ^n p_i | i \rangle \langle i | for some basis \{ | i \rangle \}. Let H_B be another copy of the n-dimensional Hilbert space with any orthonormal basis \{ | i' \rangle \}. Define | \psi \rangle \in H_A \otimes H_B by

| \psi \rangle = \sum_{i} \sqrt{p_i} |i \rangle \otimes | i' \rangle.

Direct calculation gives


\operatorname{tr_B} \left( | \psi \rangle \langle \psi | \right )= 
\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.

This proves the claim.

Note[edit]

  • 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 is 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.