Purification of quantum state

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.

Let ρ be a density matrix acting on a Hilbert space ${\displaystyle H_{A}}$ of finite dimension n. Then there exist a Hilbert space ${\displaystyle H_{B}}$ and a pure state ${\displaystyle |\psi \rangle \in H_{A}\otimes H_{B}}$ such that the partial trace of ${\displaystyle |\psi \rangle \langle \psi |}$ with respect to ${\displaystyle H_{B}}$

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