Density Operator / Purification
Hello, I'm new here. I'm a member over at mathhelpforum, but thought you guys would be better to ask a couple of questions I have.
I've been working through a question on density operators and polarisation vectors. The very last part of the question asks to show that:
$\displaystyle \rho_{\beta} = \frac{1}{2} (1 + \tanh{\beta} \simga_3)$ is a density operator on$\displaystyle \mathbb{C}^2 $, which I can do. It then asks if this defines a pure state, which it doesn't. It then asks to find the purification of $\displaystyle \rho_{\beta}$.
This is where I'm lost. We haven't mentioned ''purification'' in lectures (granted I missed one lecture last week, but the online notes do not mention purification)
I looked on line, and couldn't find an explanation of purification that I can understand (we haven't done partial traces yet so that is probably not the way we're expected to go about it)
I'm guessing that it has something to do with adjusting the polarisation vector to lie on the surface of the Bloch sphere.
Earlier in the question we were asked to show that in a spin half system, that the dispersion of $\displaystyle \textbf{\sigma \ldot A}$ with respect to a density operator vanishes iff the state is pure and the polarisation is parallel to A , which I could do.
Does this condition have any relevance to finding the purification of $\displaystyle \rho_{\beta}$ ?
Any help would be much appreciated, I'm very confused.
