Noncommutative Mathematics for Quantum Systems

104 Noncommutative Mathematics for Quantum Systems

Like for bialgebras, the opposite comultiplication∆op=F◦∆of a
dual semigroup(B,∆,ε)defines a new dual semigroup(B,∆op,ε).
In the following we restrict the index set to a finite interval[0,T],
with some fixedT >0. Then we can define time reversal as the
map[0,T] 3 t7→T−t∈[0,T]. It is straight forward to adapt the
definition of Levy processes to finite time intervals. ́

Proposition 1.9.6 Let{jst : B → (A,Φ)} 0 ≤s≤t≤T be a quantum
stochastic process on a dual semigroup (B,∆,ε) and define its
time-reversed process{jopst} 0 ≤s≤t≤Tby

jopst =jT−t,T−s

for 0 ≤s≤t≤∞.

(i) The process{jst} 0 ≤s≤t≤T is a tensor (boolean, free, respectively)

L ́evy process on the dual semigroup(B,∆,ε) if and only if the

time-reversed process {jopst} 0 ≤s≤t≤T is a tensor (boolean, free,

respectively) L ́evy process on the dual semigroup(B,∆op,ε).

(ii) The process{jst} 0 ≤s≤t≤T is a monotone (anti-monotone, resp.)

L ́evy process on the dual semigroup(B,∆,ε) if and only if the

time-reversed process {jopst} 0 ≤s≤t≤T is an anti-monotone

(monotone, resp.) L ́evy process on the dual semigroup(B,∆op,ε).

Proof The equivalence of the stationarity and continuity
property for the quantum stochastic processes{jst} 0 ≤s≤t≤T and

{j

op
st}^0 ≤s≤t≤Tis clear.
The increment property for{jst} 0 ≤s≤t≤T with respect to∆ is
equivalent to the increment property of{jstop} 0 ≤s≤t≤Twith respect
to∆op, since

mA◦

(

jopst ‰joptu

)

◦∆op = mA◦

(

jT−t,T−s‰jT−u,T−t

)

◦F◦∆

= mA◦γA,A◦

(

jT−u,T−t‰jT−t,T−s

)

◦∆

= mA◦

(

jT−u,T−t‰jT−t,T−s

)

◦∆

for all 0≤s≤t≤u≤T.
If the process {jst} 0 ≤s≤t≤T has monotonically independent
increments, that is, if the n-tuples (js 1 t 2 ,... ,jsntn) are
monotonically independent for alln∈Nand all 0≤s 1 ≤t 1 ≤
s 2 ≤ ··· ≤ tn, then the n-tuples(jsntn,... ,js 1 t 1 ) = (jopT−tn,T−sn,

... ,jTop−t 1 ,T−s 1 )are anti-monotonically independent and therefore