Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 13

The set of all extreme probability measures on a sample space of n points
has cardinality n (the probability measuresδωthat are concentrated in one
point).

1.2.2 Dictionary ‘Classical↔Quantum’

We summarize the basic terminology in classical and quantum
probability, following Parthasarathy,cf.[Par03]. For simplicity, we
shall restrict ourselves to ‘finite’ probability spaces, that is, assume
thatΩis a finite set andHhas finite dimension.

Classical Quantum

Sample space A setΩ={ω 1 ,... ,ωn} A Hilbert spaceH=Cn

Events Subsets ofΩthat Orthogonal projections

form aσ-algebra inH, they form a lattice,

(also a Boolean algebra) which is not Boolean (or

distributive), for example, in

generalE∧(F 1 ∨F 2 ) 6 =

(E∧F 1 )∨(E∧F 2 )

Random Measurable functions Self-adjoint operators

variables/ f:Ω→R, X:H→H,X∗=X,

observables they form a comm. they span a non-comm.

(von Neumann) algebra, (von Neumann) algebra,

to each eventE∈F events are observables

we get a r.v. (^1) E. with values in{0, 1}.
Note thatEλ= (^1) {λ}(X).
Probability An additive function A density matrix, that is, a
distribution/ P:F →[0, 1] pos. operator with
state determined bynpos. real tr(ρ) = 1
numberspk=P({ωk}) P(X=λ) =tr(ρEλ),
s.t.∑nk= 1 pk= 1 P(X∈E) =tr(ρ (^1) E(X)),
P(E) =∑ω∈EP({ω}) (^1) E(X) =∑λ∈E∩σ(X)Eλ).
Expectation E(f) =
∫
ΩfdP E(X) =tr(ρX)
=∑nk= 1 f(ω)P({ω})
Variance Var(f) =E(f^2 )−E(f)^2 Var(X) =E(X^2 )−E(X)^2
=tr(ρX^2 )−
(
tr(ρX)
) 2
.
cont...