Understanding Quantum Purification

[aaaa202]

I am not very good at this, but I really want to understand it.

Definition:
We say lψ>_AB is a purification of ρ_A if Tr_B[lψ><ψl_AB ] = ρ_A.

Note that ρ_A is a density matrix.

My book proceeds to give an example of a purifying system as:

√ρ_A ⊗ 1*lΦ>, where 1 is identity and lΦ> = ∑_ilii> = ∑_ili>⊗li>

How can I see that this is true? I am not sure I even know how to perform the tensor product on the LHS. Is this correct?
√ρ_A ⊗ 1*lΦ> = √ρ_A ⊗ (∑_ili>⊗li>) = (∑_i√ρ_Ali>⊗√ρ_Ali>)

 

[Strilanc]

That's definitely not correct. The matrix sizes of the left-hand and right-hand sizes can't possibly match up because you took $\rho_A$ and suddenly put it on both sides of the tensor product. I think you're confused about what your book is doing. The starting expression $\sqrt{\rho_A} \otimes \left| \Phi \right\rangle$ already makes no sense in context. Like... it's tensor-producting a matrix against a column-vector.

To do a purification on paper you generally:

- Perform an eigendecomposition of the density matrix $\rho$, giving you $\rho = \sum p_i \left| v_i \right\rangle \left\langle v_i \right|$ where each $p_i$ is a probability and each $\left| v_i \right\rangle$ is some pure state (the $p_i$'s are the eigenvalues and the $\left| v_i \right\rangle$'s are the eigenvectors).
- Introduce a secondary pure state which has a state for each eigenvector and amplitudes that give probabilities that will match the eigenvalues. Perhaps $\sum \sqrt{p_i} \left| i \right\rangle$.
- Match the secondary pure state's components against the density matrix' eigenvectors, inside the sum. $\psi_{pure} = \sum \sqrt{p_i} \left| i \right\rangle \left| v_i \right\rangle$.
- The new overall density matrix is $\rho_{pure} = \sum \sum \sqrt {p_i} \sqrt{p_j} \left| i \right\rangle \left| v_i \right\rangle \left\langle v_j \right| \left\langle j \right|$, and tracing over the introduced secondary state's space will result in the original density matrix.

This is exactly what the wikipedia page on quantum purification does.