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.

## Contents

* 1 Statement
* 1.1 Proof
* 1.1.1 Note
* 2 An application: Stinespring's theorem

## Statement

Let ρ be a density matrix acting on a Hilbert space HA of finite dimension n. Then it is possible to construct a second Hilbert space HB and a pure state |ψ⟩∈HA⊗HB such that ρ is the partial trace of |ψ⟩⟨ψ| with respect to HB. While the initial Hilbert space HA might correspond to physically meaningful quantities, the second Hilbert space HB needn't have any physical interpretation whatsoever. However, in physics the process of state purification is assumed to be physical, and so the second Hilbert space HB should also correspond to a physical space, such as the environment. The exact form of HB in such cases will depend on the problem. Here is a proof of principle, showing that at very least HB has to have dimensions greater than or equal to HA.

With these statements in mind, if,

trB⁡(|ψ⟩⟨ψ|)=ρ,

we say that |ψ⟩ purifies ρ.

### Proof

A density matrix is by definition positive semidefinite. So ρ can be diagonalized and written as ρ=∑i=1npi|i⟩⟨i| for some basis {|i⟩}. Let HB be another copy of the n-dimensional Hilbert space with an orthonormal basis {|i′⟩}. Define |ψ⟩∈HA⊗HB by

|ψ⟩=∑ipi|i⟩⊗|i′⟩.

Direct calculation gives

trB⁡(|ψ⟩⟨ψ|)=trB⁡[(∑ipi|i⟩⊗|i′⟩)(∑jpj⟨j|⊗⟨j′|)]=trB⁡(∑i,jpipj|i⟩⟨j|⊗|i′⟩⟨j′|)=∑i,jδijpipj|i⟩⟨j|=∑ipi2|i⟩⟨i|=ρ.

This proves the claim.

#### Note

* The purification is not unique, but if during the construction of |ψ⟩ in the proof above HB is generated by only the {|i′⟩} for which pi is non-zero, any other purification |φ⟩ on HA⊗HC induces an isometry V:HB->HC such that |φ⟩=(I⊗V)|ψ⟩.
* The vectorial pure state |ψ⟩ 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

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.

This article does not cite any external source. HandWiki requires at least one external source. See citing external sources. Please help improve this article. Unsourced material may be removed in future.
Find sources: "Purification of quantum state" – news · newspapers · books · scholar · JSTOR (2021) (Learn how and when to remove this template message)

|

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Purification_of_quantum_state&oldid=191544"