Noncommutative Mathematics for Quantum Systems

30 Noncommutative Mathematics for Quantum Systems

random variables (jt)t∈I. For a quantum random variable
j:B →Awe will callφj=Φ◦jitsdistributionin the stateΦ. For a
quantum stochastic process(jt)t∈I the functionalsφt = Φ◦jt :
B → C are called marginal distributions. The joint distribution
Φ◦(‰t∈Ijt)of a quantum stochastic process is a functional on the
free product‰t∈IB, see Section 1.8.
Two quantum stochastic processes
(
j(t^1 ):B →A 1

)

t∈I

and

(

jt(^2 ):B →A 2

)

t∈I

on the same ∗-algebra B over possibly different quantum
probability spaces(A 1 ,Φ 1 ) and(A 2 ,Φ 2 )are calledequivalent, if
their joint distributions coincide. This is the case, if and only if all
their moments agree, that is, if

Φ 1

(

jt( 11 )(b 1 )···j(tn^1 )(bn)

)

=Φ 2

(

j(t 12 )(b 1 )···j(tn^2 )(bn)

)

holds for alln∈N,t 1 ,... ,tn∈Iand allb 1 ,... ,bn∈B.
In Example 1.2.5 we defined quantum random variables as
elements of quantum probability spaces. This can be viewed as a
special case of the definition we just gave, by associating toX∈ A
the unique unital *-algebra homomorphismjX:C〈a,a∗〉→Asuch
thatjX(a) =X, whereC〈a,a∗〉denotes the free unital algebra over
{a,a∗}with the obvious involution.
Similarly, the term ‘quantum stochastic process’ is sometimes
also used for an indexed family(Xt)t∈Iof operators on a Hilbert
space or more generally of elements of a quantum probability
space. We will reserve the name operator process for this. An
operator process(Xt)t∈I⊆A(whereAis a∗-algebra of operators)
always defines a quantum stochastic process(jt:C〈a,a∗〉→A)t∈I
on the free∗-algebra with one generator, if we setjt(a) =Xtand
extend jt as a ∗-algebra homomorphism. On the other hand,
operator processes can be obtained from quantum stochastic
processes(jt:B → A)t∈Iby choosing an elementxof the algebra
Band settingXt=jt(x).
The notion of independence we use for Levy processes on ́
involutive bialgebras is the so-called tensor or boson
independence. In Section 1.8 we will see that other interesting
notions of independence exist.