Is entanglement the only way to get mixed state that is consistent with the Schrödinger equation?

Question

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.

Answer 1

1. Purely mathematically, this is certainly true, since any mixed state in a Hilbert space $H$ can be "purified", i.e. we can exhibit a pure state in $H\otimes H$ whose partial trace is the mixed state.

2. In "physical" terms, a mixed state doesn't need to always arise as the partial trace of a larger entangled system: If I have an electron source and I tell you that half the electrons it spits out are spin-up (with respect to some spin operator) and the other half is spin down, then you will likely model what you know about an electron in the beam by assigning it a mixed state of 50% spin-up and 50% spin-down. So this mixed state models incomplete knowledge about the pure state of the individual electrons, but it didn't arise from any sort of entanglement or larger pure state - whether or not my source internally uses entanglement to achieve this outcome is completely irrelevant for your model.