From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 143

Theorem 2.3.26(Bell-Kochen-Specker theorem). LetHbe a complex Hilbert
space of finite dimension=2, or infinite dimensional and separable. There is no
quantum stateρ:L(H)→[0,1], in the sense of Def. 2.3.21, such thatρ(L(H)) =
{ 0 , 1 }

Proof. DefineS:={x∈H|||x||=1}endowed with the topology induced byH,
and letT∈B 1 (H) be the representative ofρusing Gleason’s theorem. The map
fρ:Sx→〈x, T x〉=ρ(〈x, 〉x)∈Cis continuous becauseT is bounded. We
havefρ(S)⊂{ 0 , 1 },where{ 0 , 1 }is equipped with the topology induced byC.
SinceSis connected its image must beconnected also. So eitherfρ(S)={ 0 }or
fρ(S)={ 1 }. In the first caseT= 0 which is impossible becausetrT=1,inthe
second casetr T= 2 which is similarly impossible.

This negative result produces no-go theorems in some attempts to explain QM in
terms of CM introducinghidden variables[ 6 ].

Remark 2.3.27.In view of Proposition 2.3.23 and Theorem 2.3.24, assuming
thatHhas finite dimension or is separable, we henceforth identifyS(H)with the
subset ofB 1 (H)of positive operators with unit trace. We simply disregard the
states inHwith dimension 2 which are not of this form especially taking (b) in
remark 2.3.25 into account.

We are in a position to state some definitions of interest for physicists, especially
the distinction between pure and mixed states, so we proceed to analyse the struc-
ture of the space of the states. To this end, we remind the reader that, ifCis a
convex set in a vector space,e∈Cis calledextremeif it cannot be written as
e=λx+(1−λ)y, withλ∈(0,1),x, y∈C{e}.
We have the following simple result whose proof can be found in[ 5 ].

Proposition 2.3.28.LetHbe a complex separable Hilbert space.
(a)S(H)is a convex closed subset inB 1 (H)whose extreme points are those
of the form:ρψ:=〈ψ, 〉ψfor every vectorψ∈Hwith||ψ||=1.(Thissetsupa
bijection between extreme states and elements ofPH.)
(b)A stateρ∈S(H)is extreme if and only ifρρ=ρ. (All the elements of
S(H)however satisfy〈x, ρρx〉≤〈x, ρx〉for allx∈H.)
(c)Any stateρ∈S(H)is a linear combination of extreme states, including
infinite combinations in the strong operator topology. In particular there is always
a decomposition

ρ=

∑

φ∈N

pφ〈φ,〉φ,