Noncommutative Mathematics for Quantum Systems

20 Noncommutative Mathematics for Quantum Systems

see for example the discussion in [Gil98]. However, in such a
model the random variable that describes the spin measured by
Alice depends on the choice of the direction of Bob’s spin
measurement. As these experiments may be carried out
simultaneously in two places that are far away, they should not
influence each other.

1.3.2 Gleason’s theorem

Gleason’s theorem shows that for Hilbert spaces with dimH≥ 3
states onB(H)are in one-to-one correspondence with probability
measures onP(H)(to be defined below). This is not the case for
dimH = 2; there exist many additive probability measures on
P(C^2 )that do not extend to states onB(C^2 ).
LetHbe a separable Hilbert space and denoted byP(H)the set
of orthogonal projections onH.

Definition 1.3.1 A mapP :P(H) → [0, 1]is called anadditive
probability measureonP(H)ifP(idH) =1 and

P

(

n

∑

k= 1

Ek

)

=

n

∑

k= 1

P(Ek)

for alln ≥ 1 and all familiesE 1 ,... ,Enof pairwise orthogonal
projections.

If the additivity condition holds also for countable families, then
we say thatPis aσ-additive probability measureonP(H).

Gleason’s theorem says that for a Hilbert space of dimension
bigger than or equal to three all probability measures onP(H)
arise by restriction from states onB(H).

Theorem 1.3.2(Gleason’s theorem, [nlab Gl], [Par92, Corollary 8.10],
[Ara09, Theorem 2.3]) LetdimH≥ 3. Then each additive measure on
P(H) can be uniquely extended to a state on B(H). Conversely, the
restriction of every state toP(H)is an additive measure onP(H).
The same holds forσ-additive probability measures and normal states:
Everyσ-additive probability measure can be extended to a normal state
and every normal state restricts to aσ-additive probability measure.

For a proof of Gleason’s theorem see, for example, [Par92].