A Purification of a Density Matrix

[Pete5876]

I'm trying to find the purification of this density matrix
$$\rho=\cos^2\theta \ket{0}\bra{0} + \frac{\sin^2\theta}{2} \left(\ket{1}\bra{1} + \ket{2}\bra{2} \right)
$$

So I think the state (the purification) we're looking for is such Psi that
$$
\ket{\Psi}\bra{\Psi}=\rho
$$

But I'm not confident this is right because this would involve considering a generic state Psi, multiplying it with its bra and equating the coefficients which is too complicated to be right.

How do you "purify" a mixed state expressed as a density matrix?

 

[Hyperfine]

There is a substantial body of literature on this. Have you consulted that literature and if so what conclusions have you drawn?

 

[Pete5876]

I did and as you pointed out there is a substantial body of literature. I'm a slow reader and an even slower learner. We don't go by any textbook at uni and I have no idea what purification might possibly entail.

After all, we're not tensor-crossing with any other space so tracing one space out of another can't even be applied. What could they possibly mean by "purification"?

 

[Hyperfine]

What could they possibly mean by "purification"?

You might start with:

Introduction to quantum information processing: Partial Trace and State Purification (PDF auto-download)

Purification (Online video lecture notes from TUDelft)

​

 

[vanhees71]

First of all you should check whether $\hat{\rho}$ is a pure state to begin with. It's a pure state if and only if $\hat{\rho}^2=\hat{\rho}$!