Superposition & Mixture : Preparation and Representation of

[Swamp Thing]

I have been reading this explanation about superpositions and mixtures. The author takes the example of two non-overlapping regions in space, each covered by a gaussian wavefunction. He goes on to compare the superposition and the mixture made up of those two gaussian functions, based on their different representations in terms of their density matrices.

My question is, how would one actually prepare a mixture exactly like the example discussed there?

 

[DrDu]

The article states that a mixture cannot be represented by a wavefunction. This isn't completely correct. E.g. introducing additional states $|+\rangle$ and $|-\rangle$, which are supposed to be orthogonal, you can write the mixed state as $p_1|\psi_1\rangle |+\rangle+p_2 |\psi_2\rangle |-\rangle$.
This is especially important in symmetry breaking, where (sub)spaces with different symmetry become orthogonal.

 

[Swamp Thing]

What would $|+\rangle$ and $|-\rangle$ be in the example that he talks about?

 

[DrDu]

Just two states from an auxillary space.