The set of quantum states {|ϕn⟩} in the definition of the density operator ρ=∑npn|ϕn⟩⟨ϕn|
Example 1 In the Stern-Gerlach (SG) set-up, the state of the silver atoms coming out of the oven and before passing through the magnetic field, is imperfectly known because Sz remained unmeasured. Therefore, on the ignorance ground, such an ensemble will be represented by ρ=12(|↑⟩⟨↑|+|↓⟩⟨↓|).
Example 2 Consider an unpolarized light moving in the z-direction so that its polarization must be in the xy-plane. Since we do not know the state vector, it is described by the density operator ρ=12(|x⟩⟨x|+|y⟩⟨y|)
Question Can someone suggest an example of a mixed ensemble where the states {|ϕn⟩} need not be orthonormal and need not form a basis? I'm not looking for the trivial example where the desity operator describes a pure state.
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