Pure mixed entangled confusion

[kop442000]

I have read quite a bit on here and elsewhere, but am still coming to terms with pure, mixed, entangled and product states.

I know a state is either mixed or pure, but then is engtaglement a sub-class of being mixed? Or a sub-class of being pure? Or is entangled another name for pure? Or something different altogether?

Also is a product state another way to say mixed? Is it the opposite to entanged?

As you can see, I am very confused!

Thanks in advance of replies posted.
Kop442000.

 

[Fredrik]

Pure state = a projection operator for a 1-dimensional subspace, i.e. an operator of the form $|\psi\rangle\langle\psi|$
Mixed state = a density operator (a linear operator satisfying 2.6-2.8) that isn't a pure state

One way to see if a density operator $\rho$ is pure or not is to compute $\operatorname{Tr}\rho^2$. The result is =1 if and only if the state is pure. (It's <1 for mixed states).

Entangled state = a state of a composite system that can't be expressed as a tensor product $|\psi_1\rangle\otimes\cdots\otimes|\psi_n\rangle$ (It would therefore have to be expressed as a linear combination of such tensor products).