quantum mechanics - What is the entropy of a pure state?

Well, zero of course. Because

$S = -\text{tr}(\rho \ln \rho)$ and $\rho$ for a pure state gives zero entropy.

But... all quantum states are really pure states right? A mixed state just describes our ignorance about a particular system. So how can properties like entropy and temperature come out of ignorance of information? That doesn't make sense to me.