14 Noncommutative Mathematics for Quantum Systems
continued
Extreme The set of all probab. The extreme points of the
points distributions onΩis a setS(H)of states onH
compact convex set are exactly the one-dim.
withnextreme points projections onto the rays
δωk,k=1,... ,n. Cu,u∈Ha unit vector.
IfP=δωk,then the Ifρ=Puthen Var(X) =
distribution of any r.v. =‖(X−〈u,Xu〉)u‖^2 ,
is concentrated in one thusVar(X) = 0 iff
point. uis an eigenvector ofX.In quantum probability or quantum physics,degeneracy of the
state does not kill the uncertainty of the observables. The gives an
indication that the nature or origin of the randomness in quantum
mechanics is fundamentally different from the generally accepted
description in classical probability. One might call extreme states,
that is, density matrices that are rank one operators,degenerate, by
analogy with the situation in classical probability where they
correspond to Dirac measures. However, in quantum probability
this situation is very different from the one in classical probability.
Even if we have maximal information on the state, that is, if the
state is pure and therefore an extreme point of the set of states, the
Classical Quantum
Product Given two systems Given two systems
spaces described by(Ωi,Fi,Pi), described by(Hi,ρi),
systems i=1, 2, i=1, 2,
(Ω 1 ×Ω 2 ,F 1 ⊗F 2 ,P 1 ⊗P 2 ) (H 1 ⊗H 2 ,ρ 1 ⊗ρ 2 )
describes both independent describes both indep.
systems as a single system systems as a single
system
→independence
→entanglement
Reversible is modeled by is modeled by unitary operators
dynamics bijective (measurable) U:H→H
mapsT:Ω→Ω in the Heisenberg
for r.v.:f7→f◦T picture:X7→U∗XU
in the Schrodinger ̈
for prob. measures ρ7→UρU∗
P7→P◦T−^1 orψ7→Uψ.