FOUNDATIONS OF QUANTUM MECHANICS

50 CHAPTER III. THE POSTULATES
Now we have to show that the 1 - dimensional projectors are the only extreme elements. A state
operator is self - adjoint and has, according to the spectral theorem, p. 26, a complete orthonormal
set of eigenstates |w i , j⟩, where j is the degeneracy, j = 1, . . . , n i , and which has M ∈ N +
different w i . We can write an arbitrary W ∈ S (H) as
W =
M∑
∑n i
i=1 j=1
w i W i,j , (III. 41)
where
W i,j := |w i , j⟩ ⟨w i , j|, and
M∑
n i = dim H. (III. 42)
i=1
For w i it holds that
M∑
n i w i = 1 and 0 < w i < 1 (III. 43)
i=1
because, according to (III. 29) (ii) and (III. 29) (iii),
w i = ⟨w i , j | W | w i , j⟩ 0 and Tr W =
M∑
n i w i = 1. (III. 44)
i=1
Thus we se that the sum (III. 41) is a convex decomposition of W .
A convex decomposition W = w 1 W 1 + w 2 W 2 can always be decomposed further through
expansion of W 1 and W 2 . In case of a bounded convex set the expansion ends on extreme
elements. Therefore, if W is an extreme element, the sum has to reduce to one term. In that case
W is a 1 - dimensional projector, and we see that all extreme elements of S(H) are 1 - dimensional
projectors. □
Physical states which are represented by 1 - dimensional projectors are called pure states, where
states which can be divided non - trivially are called mixed states or mixtures. To see that pure states
correspond to the vector states of H, consider W to be the 1 - dimensional projector P ψ projecting
on the vector |ψ⟩. The state defined by this state operator through (III. 28) behaves exactly like the
vector state |ψ⟩; for arbitrary |ϕ⟩ it holds that
µ W (P ϕ ) = Tr P ϕ W = Tr P ϕ P ψ = ⟨ψ | P ϕ | ψ⟩ = |⟨ψ | ϕ⟩| 2 , (III. 45)
which means that the probability to find the state |ϕ⟩ in the state |ψ⟩ is equal to (III. 6). 2 It holds
especially that µ(P ψ ) = 1, and if |ϕ⟩ ⊥ |ψ⟩, then µ(P ϕ ) = 0. We see that the state P ψ assigns a
2 ‘The probability to find the state |ϕ⟩’ is shorthand for the probability to find, upon measurement of the quantity corresponding
to the projector |ϕ⟩ ⟨ϕ|, the value 1.