FOUNDATIONS OF QUANTUM MECHANICS

III. 2. PURE AND MIXED STATES 47
Each mapping which maps a measurable subset of Γ to a number in the interval [0, 1], thereby
satisfying (III. 23) and (III. 24), is called a probability measure. It is not difficult to see that every
probability distribution ρ corresponds univocally to a probability measure and vice versa, this is even
true for δ - functions. Therefore we can also represent a state, in the extended meaning, by a probability
measure on Γ.
Analogous to this reasoning we now aim to let the physical states in quantum mechanics correspond
to probability measures on H. Since we want to preserve the structure of H, we do not consider
arbitrary subsets of H, instead we look at the set P(H) of all subspaces of H generated by orthogonal
projectors, or, equivalently, at the projectors projecting on those subspaces. What we are thus looking
for is a probability measure on P (H), i.e. a mapping
µ : P (H) → [0, 1], (III. 25)
which is additive in the relevant manner; if P 1 , P 2 , . . . , P N is a set of pairwise orthogonal projectors,
P i ⊥ P j for i ≠ j, the following holds,
( ∑
µ
j
P j
)
= ∑ j
and the mapping satisfies
µ(P j ), (III. 26)
µ(0 ) = 0 and µ(11) = 1. (III. 27)
In 1957 A.M. Gleason proved the following theorem.
GLEASON’S THEOREM:
Every probability measure µ on P (H) can, under the condition that dim H > 2, be
written as
µ(P ) = Tr P W, (III. 28)
for a certain operator W satisfying the following requirements: 1
(i) W = W † ,
(ii) ⟨ψ | W | ψ⟩ 0 ∀ |ψ⟩ ∈ H,
(iii) Tr W = 1. (III. 29)
The original proof of Gleason’s theorem is extraordinarily difficult. In the appendix of these
lecture notes, p. 183, ff, we prove a simplified version of this theorem for the interested reader.
1 Conditions (i) and (ii) of (III. 29) are not mutually independent in the complex Hilbert space of the formalism of
quantum mechanics, in this space (i) is in fact superfluous. In a complex Hilbert space all positive operators are self -
adjoint, and an operator A is uniquely defined by all matrix elements of the form ⟨ψ | A | ψ⟩. This is, however, not the case
in a real space, where Gleason’s theorem is also valid. In that case (i) and (ii) are independent.