FOUNDATIONS OF QUANTUM MECHANICS

184 APPENDIX A. GLEASON’S THEOREM
GLEASON’S THEOREM FOR PURE STATES:
Under the condition that dim H > 2, a 1 - dimensional projector P 0 ∈ P (H) exists on
which the measure µ : P (H) → [0, 1] is concentrated, such that
µ(P ) = Tr P 0 P (A. 4)
for all P ∈ P (H).
The original proof by A.M. Gleason uses sophisticated mathematical methods and is rather
opaque. Several authors have undertaken attempts a to give a more simple proof, particularly
C. Piron (1976), J. Dorling (unpublished) and R. Cooke, M. Keane and B. Moran (1985), where
the commentaries on the ‘elementary proof’ of Cooke, Keane and Moran by R.I.G. Hughes (1989)
are clarifying.
The following proof is a mixture of all this work. It exists of four steps, which, for that matter, do
not coincide with the sections.
A. 2 CONVERSION TO A 3 - DIMENSIONAL REAL PROBLEM
Before taking the first step, we discuss a number of simple observations. First, the probability
measure of Gleason’s theorem has to be continuous in P . In section III. 1, p. 48, we showed that
discontinuous probability measures exist for dim H = 2. Therefore, the requirement dim H > 2
holds without further mentioning throughout this appendix. Second, since the trace of a projector P
is equal to the dimension of the subspace onto which it projects, the trace of a 1 - dimensional projector
is 1, which yields for µ being concentrated on P 0
µ(11) = Tr P 0 11 = 1, (A. 5)
in accordance with (A. 2) and (A. 4). Third, every measure is entirely determined by giving its values
on the 1 - dimensional projectors, and, since every higher - dimensional projector P is the sum
of orthogonal 1 - dimensional projectors P i we can, with (A. 1), determine µ (P ) from the values
of µ(P i ). Fourth, for every Hilbert space, (A. 4) at the same time defines an extreme measure µ on H
which is concentrated on P 0 , and as of now we will indicate this measure by µ 0 ,
µ 0 (P ) := Tr P 0 P. (A. 6)
Using the idempotence of P 0 we have
µ 0 (P 0 ) = Tr P 0 2 = Tr P 0 = 1, (A. 7)
from which we see that (A. 4) holds for µ = µ 0 and P = P 0 . Since this measure, being concentrated
on P 0 , assigns the value 0 to all projectors orthonormal to P 0 , it can also easily be verified that this
measure satisfies the requirements (A. 1) and (A. 2).
The foregoing observations lead to the conclusion that to prove Gleason’s theorem for pure states
we have to prove that µ = µ 0 for all P ∈ P (H). Now we will take the first step.