If we treat our entire system (say an electron and a bunch of atoms) quantum mechanically then all possible interactions will be unitary transformations. Thus any state that I describe will always be a pure state.
But if I observe only a subspace of my system (just the spin of the electron, say) I need to trace out rest of the space and I end up with a density matrix.
If my states were separable to begin with then my density matrix will correspond to a pure state. The only way to get a mixed state would be if the spin of my electron was entangled with the rest of the system. Right?
In other words, is a mixed state always an entangled state in a higher dimension?
Edit: My question is not about purification. I do not care if I can find a state in my complete Hilbert space by purification. Rather, is entanglement the only way to go from a pure state to a mixed state. Thus it isn't a duplicate of this.