electrons identical spins Hilbert spaces spin
Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits electrons towards an observer. The electrons' Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state 




|


z

+
⟩


{\displaystyle |\mathbf {z} +\rangle }

 with spins aligned in the positive z direction, and the other with state 




|


y

−
⟩


{\displaystyle |\mathbf {y} -\rangle }

 with spins aligned in the negative y direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.
