Quantum stochastic calculus on Boolean Fock space
22
* † * †
Transcript of Quantum stochastic calculus on Boolean Fock space
Quantum stochastic calculus on Boolean Fock space
Anis BEN GHORBAL∗ Michael SCHÜRMANN†
Institut für Mathematik und Informatik,Ernst-Moritz-Arndt-Universität Greifswald,
Friedrich-Ludwig-Jahn-Straÿe 15a,D-17487 Greifswald, GERMANY.
Abstract
In this paper we establish a theory of stochastic integration with respect to the basiceld operator processes in the Boolean case. This leads to a Boolean version of quantumItô's product formula and has applications to the theory of dilations of quantum dynamicalsemi-groups.
1 Introduction
A quantum stochastic calculus can be developed in dierent frameworks: R.L. Hudson andK.R. Parthasarathy, see [9], initiated a theory of stochastic integration on Bose Fock spacewith respect to the basic eld operator processes by using the Bosonic canonical commutationrelations and the Riemann-Lebesgue integral. They replace Brownian motion by annihilationand creation operator processes on Boson Fock space and include the Poisson process byintroducing the preservation operator process. Moreover they give a non-commutative Itôformula and a realization of the classical Poisson process in Bose Fock space. Similar resultshold for a stochastic calculus on Fermi Fock space, see [2]. For the free case B. Kümmererand R. Speicher [12] presented the stochastic calculus for the creation and annihilationprocesses on full Fock space. In [19] R. Speicher included the preservation process and gavea stochastic integration theory on full Fock space by using a kernel representation for alarge class of bounded operators on full Fock space. After giving the denition of processesand stochastic integrals Speicher proves an Itô formula and uses it (under necessary andsucient conditions on the coecients) to show the uniqueness of the solution of linearstochastic dierential equations. There exists also a unied approach [1].
∗[email protected] Supported by the European Network Training Research QuantumProbability with Applications to Physics, Information Theory and Biology
According to [17, 15, 4] there are exactly three notions of non-commutative independencesatisfying a set of natural axioms; see also [14] for a more general treatment which leads to venotions independence. To each notion of stochastic independence in the non-commutativecase there corresponds a Fock space which comes together with a notion of adaptedness foroperator processes on this Fock space. Like in the classical theory of stochastic integration(where the notion of stochastic independence is unique) adaptedness is the fundamentalproperty for the integrands of stochastic integration. In Bose, Fermi and free calculus,adaptedness is given by the symmetric anti-symmetric and the Free Fock space respectively.The Bose and Fermi case both are associated with tensor independence where as the freecase comes from free independence.
The third notion of non-commutative independence is the so-called Boolean independencewhich already (without calling it Boolean) appeared in a paper by W. von Waldenfels on thepressure broadening of spectral lines [21]; see also [20] for analysis of Boolean convolution.It plays an important rôle in the theory of free groups [6], in the unifying approach of R.Lenczewski [13] and in the theory of quantum stochastic calculi using Hilbert modules by M.Skeide [18]. Moreover, there is an interesting relation to results of R.L. Hudson, P.D.F. Ion,K.R. Parthasarathy [8] and B. Kümmerer, W. Schröder [10]; see also [5]. In this paper weestablish a theory of stochastic integration with respect to the basic eld operator processesin the Boolean case. This leads to a Boolean version of quantum Itô's product formulaand has applications to the theory of dilations of quantum dynamical semi-groups; see also[7].
This paper is organized as follows. In Section 2 we give the denition of Boolean Fockspace and of the basic integration processes which are built by creation, preservation andannihilation operators. We introduce Boolean independence and the Boolean version ofadaptedness for integrand processes. In Section 3 we develop the theory of stochastic integralsof adapted processes by using Riemann sums with Itô's choice of intermediate points andwith respect the basic integrator processes. In Section 4 we establish an Itô formula. As anapplication of the previous sections we solve a quantum stochastic dierential equation inSection 5. In fact we give an explicit form of the solution which is not possible in the Bose,Fermi and free case. We investigate also when the solution is unitary. In Section 6 we usethis for constructing dilations of completely positive semi-groups.
2 Preliminaries
For a given Hilbert space H denote by Γ (H) the Hilbert space
Γ (H) = C⊕H
which is called Boolean Fock space over H. The vector (1,0) is called the vacuum vector andis denoted by Ω.
The vacuum expectation E0 : B (Γ (H)) → C is the state dened by
E0 (A) := 〈Ω, AΩ〉Γ(H) .
For any U ∈ H and T ∈ B (H) we dene the
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• creation operator A∗ (U) : Γ (H) −→ Γ (H),
• annihilation operator A (U) : Γ (H) −→ Γ (H),
• preservation operator Λ (T ) : Γ (H) −→ Γ (H),
on Boolean Fock space as follows:
A∗ (U)
(αV
)=
(0
α U
), A (U)
(αV
)=
(〈U, V 〉H
0
), Λ (T )
(αV
)=
(0
T (V )
),
where α ∈ C and V ∈ H. We have that A (U) and A∗ (U) are adjoint to each other on Γ (H)and that
A (U) A∗ (V ) = 〈U , V 〉HPΩ,
where PΩ is the projection on CΩ.
Remark 2.1 Let U , V ∈ H, then we have
A∗ (U) A (V ) = Λ (|U〉 〈V |) ,
where |U〉 〈V | is the bounded operator on H dened by
|U〉 〈V | (W ) = 〈V , W 〉H U,
for all W ∈ H.
We have also
(Λ (T ))∗
= Λ (T ∗) .
Now let h be a Hilbert space and dene the Hilbert space L2 (R+; h) by
L2 (R+; h) :=
f : R+ −→ h measurable ;
∫ ∞
0
‖f (t)‖2h dt < ∞
.
If we suppose that h is separable then we have the following identication
L2 (R+; h) ∼= L2 (R+)⊗ h,
as the tensor product of two Hilbert spaces.
The Hilbert space L2 (R+) can be looked upon as a continuous sum of Hilbert spaces, thusfor each t ∈ R+ we have the following orthogonal decomposition
L2 (R+) ∼= L2 ([0, t[)⊕ L2 ([t,∞[) ,
and more generally if 0 < t1 < . . . < tn we obtain
L2 (R+) ∼= L2 ([0, t1[)⊕ L2 ([t1, t2[)⊕ . . .⊕ L2 ([tn,∞[) ,
where for a Borel subset B of R+ we use the notation ΓB (h) for the subspace Γ (L2 (B; h))of Γ (L2 (R+; h)).
For a given u ∈ h and r ∈ B (h) we put:
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• ut = χ[0,t[ ⊗ u as an element of L2 (R+; h) dened by ut (s) := χ[0,t[ (s) u ∈ h,
• rt = χ[0,t[ ⊗ r as bounded operator on L2 (R+; h) where now χ[0,t[ is identied with thecorresponding multiplication operator on L2 (R+; h), i.e.[
rt (f)](s) = χ[0,t[ (s) f (s) ,
for each f ∈ L2 (R+; h).
Using the above notations we now dene the following operators on Γ (L2 (R+; h))
A∗t (u) = A∗ (ut
), At(u) = A
(ut), Λt(r) = Λ
(rt).
which are called creation, preservation and annihilation operator respectively.
Let h be a separable Hilbert space. We consider Boolean Fock space ΓR+ (h). A boundedoperator F on ΓR+ (h) is given by
F =
(α 〈ξ || ζ〉 r
),
where
• α ∈ C,
• 〈ξ | ∈ B (L2 (R+; h) ; C) with ξ ∈ L2 (R+; h) given by 〈ξ | (f) := 〈ξ, f〉L2(R+;h),
• | ζ〉 ∈ B (C; L2 (R+; h)) with ζ ∈ L2 (R+; h) given by | ζ〉 (λ) := λ ζ,
• r is a bounded operator on L2 (R+; h).
The following remark motivates for the independence associated with Boolean Fock space.
Remark 2.2 For a given element B of the Borel σ-algebra B (R+) we consider the Hilbertsubspace ΓB (h) of the Boolean Fock space. We have the canonical isomorphism
ΓR+ (h) ∼= ΓB (h)⊕ L2 (Bc; h) .
where Bc is the complement of B on R+. With this denition B (ΓB (h)) becomes a ∗-sub-algebra of A = B
(ΓR+ (h)
).
Let there be given a Borel decomposition B1, B2, . . . , Bn of R+, i.e.
R+ =
n.⋃
p=1
Bp, where Bi ∩Bj = ∅ for each i 6= j.
Consider the following ∗-sub-algebras A1 = B (ΓB1 (h)) , . . . ,An = B (ΓBn (h)) of A. Weprove that they are independent under the Boolean product (see [20, 4]).
Let a1, . . . , am be a family of elements where ak ∈ Aik with ik ∈ 1, . . . , n and ik+1 6= ik.Then we have
[am−1am] (Ω) =
(αm−1 〈ξm−1||ζm−1〉 rm−1
)(αm
ζm
)=
(αm−1 αm
αm ζm−1
),
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where we used
〈ξm−1 | ζm〉L2(R+;h) = 0 = rm−1 (ζm) .
Then by the same arguments we obtain
[a1a2 . . . am] (Ω) =
(α1 α2 . . . αm−1 αm
−−−−−−−
).
Thus we have,
ΦΩ (a1a2 . . . am−1am) = 〈Ω , [a1a2 . . . am−1am] (Ω)〉= α1 α2 . . . αm−1 αm
= ΦΩ (a1) ΦΩ (a2) . . . ΦΩ (am−1) ΦΩ (am) .
A family of bounded operators F = (Ft)t>0 on Boolean Fock space ΓR+ (h) is an adaptedprocess if for arbitrary t > 0, Ft is the ampliation to Γ[0,t[ (h)⊕L2 ([t, +∞[; h) of an operatorin Γ[0,t[ (h), i.e.
Ft
(λf
)= Ft
(λ
fχ[0,t[
), for all
(λf
)∈ Γ.
The adapted processes form a left ideal of B(ΓR+ (h)
)which we denote by L (h). F ∈ L (h)
is said to be continuous if the map t 7→ Ft is norm-continuous. We denote by Lc (h) thesubspace of L (h) of continuous processes. For Ft ∈ L (h) we
Ft =
(αt 〈ξt || ζt〉 rt
).
Then ξt, ζt ∈ L2 (R+; h) and rt ∈ B ((R+; h)).
Let g be a separable Hilbert space. A mapping γ : R+ → g is said to be locally squareintegrable if t 7→ γ (t) is a measurable mapping and for each t ∈ R+ we have(∫ t
0
‖γ (s)‖2g ds
) 12
= ‖γ‖(t) < ∞.
We denote a such space by L2loc (R+; g) u
⋃t
L2 ([0, t[; g) .
Denition 2.3 An element F = (Ft)t>0 ∈ L (h) is said to be locally square integrable if foreach U ∈ ΓR+ (h) the map t 7→ Ft (U) is measurable and satises
• t 7−→ αt ∈ L2loc (R+; C),
• t 7−→ ζt ∈ L2loc (R+; L2 (R+; h)),
• t 7−→ ξt ∈ L2loc (R+; L2 (R+; h)).
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We denote by L2 (h) the subspace of L (h) of locally square integrable processes. It is clearthat Lc (h) ⊂ L2 (h).
Proposition 2.4 A (u) = (At (u))t>0, Λ (r) = (Λt (r))t>0 and A∗ (u) = (A∗t (u))t>0 are
adapted and continuous processes.
Proof: The adeptness comes from the denition of the operators. Let 0 6 s 6 t. Then it iseasily seen that ∥∥(A∗
t (u)− A∗s (u))
∥∥2
Γ=∥∥(At (u)− As (u))
∥∥2
Γ= (t− s) ‖u‖2
H∥∥(Λt (r)− Λs (r))∥∥2
Γ= ‖r‖2 (t− s) .
Now let h and g be two separable Hilbert spaces, u ∈ h, v ∈ g, γ ∈ L2loc (R+; g) and
f ∈ L2 (R+; h). Then we have∣∣∣∣∫ t
0
〈v, γ (s)〉g 〈u, f (s)〉h ds∣∣∣∣ 6
∫ t
0
∣∣∣〈v, γ (s)〉g 〈u, f(s)〉h∣∣∣ ds
6 ‖v‖g ‖u‖h
∫ t
0
‖γ (s)‖g ‖f(s)‖h ds
6 ‖v‖g ‖u‖h ‖γ‖(t) ‖f‖L2(R+;h) . (2.1)
Then we dene the element∫ t
0
〈u, f (s)〉h γ (s) ds of g by the equation
⟨v,
∫ t
0
〈u, f (s)〉h γ (s) ds
⟩g
:=
∫ t
0
〈u, f (s)〉h 〈v, γ (s)〉g ds, (2.2)
Lemma 2.5 Then s 7→ γ (s) 〈u , f (s)〉h ∈ L1loc (R+; g) and∥∥∥∥∫ t
0
〈u, f (s)〉h γ (s) ds
∥∥∥∥ 6 ‖γ‖(t) ‖u‖h ‖f‖L2(R+;h) .
Proof: Direct consequence of (2.1).
We put
Lt,u,γ (f) :=
∫ t
0
〈u , f (s)〉h γ (s) ds,
for each f ∈ L2 (R+, h). It follows from Lemma 2.5 that Lt,u,γ : L2 (R+, h) → g is a boundedoperator with ‖Lt,u,γ‖ 6 ‖γ‖(t) ‖u‖h. Equation (2.2) gives also∫ t
0
〈u, f (s)〉h 〈v, γ (s)〉g ds =
∫ t
0
〈〈γ (s) , v〉g u, f (s)〉hds,
Moreover, the adjoint operator is given by
L∗t,u,γ (v) = 〈γ (·) , v〉g χ[0,t[ (·) u =∣∣χ[0,t[ (·) u
⟩〈γ (·)| (v)
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for each v ∈ g. In particular, if γ1 ∈ L2loc(R+; L2 (R+; h)) and γ2 ∈ L2
loc (R+; C) we have
L∗t,u,γ1(g) = 〈γ2 (·) , g〉L2(R+;h) χ[0,t[ (·) u,
L∗t,u,γ2(λ) = λ γ2 (·) χ[0,t[ (·) u,
for each g ∈ L2 (R+; h) and λ ∈ C.
3 Stochastic integrals
Let F = (Ft)t>0 ∈ L2 (h). By Lemma 2.5 Lt,u,α : L2 (R+; h) −→ C and Lt,u,ζ : L2 (R+; h) −→L2 (R+; h) given by
Lt,u,α (f) :=
∫ t
0
αs 〈u, f (s)〉h ds and Lt,u,ζ (f) :=
∫ t
0
ζs 〈u, f (s)〉h ds
are bounded operators. We put∫ t
0
Fs dAs (u) :=
(0 Lt,u,α
0 Lt,u,ζ
)= Xt. (3.3)
Then by Lemma 2.5 ∥∥∥∥∫ t
0
Fs dAs (u)
∥∥∥∥ 6(‖ζ‖(t) + ‖α‖(t)
)‖u‖ .
Theorem 3.1 Let F = (Ft)t>0 ∈ L2 (h) be such that α is continuous and ζ is weaklycontinuous, and u ∈ h. Let
Zn = 0 = t0,n < t1,n < · · · < tkn,n = t
be a sequence of partitions of the interval [0, t] such that limn→∞ |Zn| = 0.
Then the limit of the Itô-Riemann sums exists
Sn =n∑
j=0
Ftj,n
(Atj+1,n
(u)− Atj,n(u))−−−→n→∞
Xt
in the weak topology.
Proof: Let
(λf
)∈ ΓR+ (h), then we have
Sn
(λf
)=
n∑j=0
Ftj,n
(10
)∫ tj+1,n
tj,n
〈u, f (s)〉h ds=n∑
j=0
(αtj,n
ζtj,n
)∫ tj+1,n
tj,n
〈u, f (s)〉H ds.
We denote by F u a the primitive function of s 7→ 〈u, f (s)〉h. Then we obtain
Sn
(λf
)=
n∑j=0
(αtj,n
ζtj,n
)(F u (tj+1,n)− F u (tj,n)) .
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Therefore for λ′ ∈ C and f ′ ∈ L2 (R+; h) we obtain
⟨(λ′
f ′
), Sn
(λf
)⟩Γ
−−−→n→∞
⟨(λ′
f ′
),
∫ t
0
αs dFu (s)∫ t
0
ζs dFu (s)
⟩
Γ
,
where we used the continuity of α and of 〈f ′, ζt〉 and∫ t
0
αs dFu (s)∫ t
0
ζsdFu (s)
=
∫ t
0
αs 〈u, f (s)〉h ds∫ t
0
〈u, f (s)〉h ζs ds
.
Proposition 3.2 For F ∈ L2 (h) and u ∈ h the mapping
t 7−→∫ t
0
FsdAs (u)
from R+ to B(ΓR+ (h)
)is adapted and continuous in norm.
Proof: The adaptedness is clear. Now for the norm-continuity we have∥∥∥∥∫ t
0
FdA (u)−∫ s
0
FdA (u)
∥∥∥∥ 6(‖ζ‖(s,t) + ‖α‖(s,t)
)‖u‖
where again we used Lemma 2.5.
Next we dene for F ∈ L2 (h) and u ∈ h∫ t
0
dA∗s (u) Fs :=
(∫ t
0
F ∗s dAs (u)
)∗:=
(0 0
L∗t,u,α L∗t,u,ξ
)= Yt. (3.4)
Proposition 3.3 Let F ∈ L2 (h) such that α is continuous and ξ is weakly continuous, andu ∈ h. Let
Zn = 0 = t0,n < t1,n < · · · < tkn,n = t
be a sequence of partitions of the interval [0, t] such that limn→∞ |Zn| = 0.
Then the limit of the Itô-Riemann sum exists
S ′n =n∑
j=0
(A∗
tj+1,n(u)− A∗
tj,n(u))
Ftj,n−−−−→n→+∞
Yt
in the weak topology.
Moreover, for F ∈ L2 (h) and u ∈ h the mapping
t 7−→∫ t
0
dA∗s (u) Fs
from R+ to B(ΓR+ (h)
)is adapted and continuous in norm.
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Proof: Let
(λf
)∈ ΓR+ (h) then we have
S ′n
(λf
)=
n∑j=0
(A∗
tj+1,n(u)− A∗
tj,n(u))( λ αtj,n
+ 〈ξtj,n, f〉
λ ζtj,n+ rtj,n
(f)
)
=n∑
j=0
(0[
λ αtj,n+⟨ξtj,n
, f⟩
L2(R+;h)
]χ[tj,n,tj+1,n[ (·) u
)Using the continuity of α and the weak continuity of ξ, we obtain⟨(
λ′
f ′
), S ′n
(λf
)⟩−−−→n→∞
⟨(λ′
f ′
),
(0
λ α. u χ (·) + 〈ξ., f〉L2(R+;h) u χ (·)
)⟩Γ
which gives the proof of the rst assertion. For the second assertion we use (3.4) and Lemma2.5.
Let (Zn)n∈N be a sequence of partitions of the interval [0, t] as in Proposition 3.1 and 3.3such that lim
n→+∞|Zn| = 0. Let F ∈ L2 (h) and u ∈ h. Then we have
n∑j=0
Ftj,n
(A∗
tj+1,n(u)− A∗
tj,n(u))( λ
f
)=
n∑j=0
Ftj,n
(0
λ χ[tj,n,tj+1,n[ (·) u
)= 0,
for each λ∈C and f ∈L2(R+; h) where we used that Ftj,nmust vanish on Γ[tj,n,tj+1,n[ (h) (i.e.
F is adapted). Thus we put, for F ∈ L2 (H)∫ t
0
Fs dA∗s (u) = 0.
Also we put, for F ∈ L2 (h) and r ∈ B (h)∫ t
0
dAs (u) Fs =
∫ t
0
Fs dΛs (r) =
∫ t
0
dΛs (r) Fs = 0,
by similar arguments.
Remark 3.4 If F ∈ L1 (h), i.e. α ∈ L1loc (R+; C), ζ, ξ ∈ L1
loc (R+; L2 (R+; h)) and r ∈L1
loc (R+;B (L2 (R+; h))) we put
∫ t
0
Fs ds :=
∫ t
0
αs ds
⟨∫ t
0
ξs ds
∣∣∣∣∣∣∣∣∫ t
0
ζs ds
⟩ ∫ t
0
rs ds
(3.5)
where ⟨∫ t
0
ξs ds
∣∣∣∣ (f) :=
∫ t
0
〈ξs, f〉L2(R+;h) ds[∫ t
0
rs ds
](f) :=
∫ t
0
rs (f) ds,
for f ∈ l2 (R+; h).
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The process of constant equal to 1 (the identity operator on the Boolean Fock space) is notadapted, but we put∫ t
0
dAs (u) = At (u) ,
∫ t
0
dA∗s (u) = At (u) ,
∫ t
0
dΛs (r) = Λt (r) .
Denote by L21 (h) (resp. L1
1 (h)) the linear span of 1 and L2 (h) (resp. L1 (h)).
4 Quantum Itô's formula
We need the following result:
Lemma 4.1 Let F ∈ L2 (h) and let X and Y be two elements of L2 (h) continuous in normgiven by
Xt =
(0 〈γt|0 lt
), Yt =
(0 0
|βt〉 lt
).
Then FX and Y F are in L2 (h). Furthermore, if G ∈ L2 (h) then FtPΩGt ∈ L1 (h).
Proof: We have
FtXt =
(αt 〈ξt||ζt〉 rt
)(0 〈γt|0 lt
)=
(0 〈αt γt + l∗t (ξt)|0 |ζt〉 〈ξt|+ rt lt
).
Thus by using the continuity in norm of X we obtain∫ t
0
‖αs γs + l∗s (ξs)‖ ds 6∫ t
0
‖αs γs‖ ds +
∫ t
0
‖l∗s (ξs)‖ ds
6 C1
∫ t
0
|αs| ds + C2
∫ t
0
‖ξs‖ ds.
Thus FX ∈ L2 (h).
For Y F we have
YtFt =
(0 0
|βt〉 lt
)(αt 〈ξt||ζt〉 rt
)=
(0 0∣∣∣αt βt + lt (ζt)
⟩|βt〉 〈ξt|+ lt rt
)(4.6)
and we obtain ∫ t
0
∥∥∥αs βs + ls (ζs)∥∥∥ ds 6 C ′
1
∫ t
0
‖αs‖ ds + C ′2
∫ t
0
‖ζs‖ ds.
Now let G ∈ L2 (h) with
Gt =
αt
⟨ξt
∣∣∣∣∣∣ ζt
⟩rt
.
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Then we have
Ft PΩ Gt = Ft
(αt
⟨ξt
∣∣∣0 0
)=
αt αt
⟨αt ξt
∣∣∣| αt ζt〉
∣∣∣ζt
⟩⟨ξt
∣∣∣ . (4.7)
which yields the last assertion of the lemma.
Theorem 4.2 Let F , G ∈ L2 (h) and u, v ∈ h. We put
Xt =
∫ t
0
Fs dAs (u) and Yt =
∫ t
0
dA∗s (v) Gs.
Then
XtYt = 〈u, v〉h∫ t
0
Fs PΩ Gs ds
and
YtXt =
∫ t
0
dA∗s (v) GsXs +
∫ t
0
YsFs dAs (u) .
Proof: Let F , G ∈ L2 (h) with
Ft =
(αt 〈ξt|| ζt〉 rt
), Gt =
αt
⟨ξt
∣∣∣∣∣∣ ζt
⟩rt
,
for t > 0 and let
(λf
)∈ ΓR+ (h). Then, on the one hand, we have
XtYt =
(Lt,u,α L∗
t,v,αLt,u,α L∗
t,v,ξ
Lt,u,ζ L∗t,v,α
Lt,u,ζ L∗t,v,ξ
),
where for λ ∈ C and f ∈ L2 (R+; h)
Lt,u,α L∗t,v,α
(λ) = λ
∫ t
0
αs αs 〈u, v〉h ds,
Lt,u,α L∗t,v,ξ
(f) =
∫ t
0
αs
⟨ξs, f
⟩L2(R+;h)
〈u, v〉h ds,
Lt,u,ζ L∗t,v,α
(λ) = λ
∫ t
0
αs 〈u, v〉h ζs ds,
Lt,u,ζ L∗t,v,ξ
(f) =
∫ t
0
⟨ξs, f
⟩L2(R+;h)
〈u, v〉h ζs ds.
On the other hand, from 4.7 we have
Fs PΩ Gs =
αs αs
⟨αs ξs
∣∣∣| αs ζs〉
∣∣∣ζs
⟩⟨ξs
∣∣∣ .
11
The rst equality follows from a simple computation.
For the second assertion observe that
YtXt =
0 0
0 L∗t,v,α Lt,u,α + L∗t,v,ξ
Lt,u,ζ
.
Applying 4.6 to our situation this gives
YtFt =
(0 0
αt L∗t,v,α +
∣∣∣L∗t,v,ξ
(ζt)⟩
L∗t,v,α 〈ξt|+ L∗t,v,ξ
rt
).
Notice that
αt L∗t,v,α =
∣∣αt α. v χ[0,t[ (·)⟩.
We put
γ1t = αt α. v χ[0,t[ (·) and γ2
t = L∗t,v,ξ
(ζt) .
Then, by combining Lemma 4.1 and the previous notations we obtain∫ t
0
Ys FsdAs (u) =
(0 0
0 Lt,u,γ1+γ2
)
where for each f ∈ L2 (R+; h) we have
Lt,u,γ1+γ2 (f) =
∫ t
0
αs α. χ[0,s[ (·) 〈u, f (s)〉h ds v +
∫ t
0
L∗s,v,ξ
(ζs) 〈u, f (s)〉 ds
=
∫ t
.
αs α. 〈u, f (s)〉h ds χ[0,t[ (·) v +
∫ t
0
⟨ξ., ζs
⟩L2(R+;h)
χ[0,s[ (·) 〈u, f (s)〉h ds v
=
∫ t
.
αs α. 〈u, f (s)〉h ds χ[0,t[ (·) v +
∫ t
.
⟨ξ., ζs
⟩L2(R+;h)
〈u, f (s)〉h ds χ[0,t[ (·) v
For the second part we have
GtXt =
0 αt Lt,u,α +⟨L∗t,u,ζ
(ξt
)∣∣∣0∣∣∣ζt
⟩ Lt,u,α + rt Lt,u,ζ
.
Now we put
γ4t = L∗t,u,ζ
(ξt
),
and we remark that
αt Lt,u,α (f) = αt
∫ t
0
αs 〈u, f (s)〉h ds =⟨γ3
t
∣∣ (f) ,
12
where now we put
γ3t = αt α. χ[0,t[ (·) u.
We obtain ∫ t
0
dA∗s (v) Ys Gs =
(0 0
0 L∗t,v,γ3+γ4
),
where now we have for f ∈ L2 (R+; h)
L∗t,v,γ3+γ4 (f) =
∫ ∞
0
⟨α. αsχ[0,.[ (s) u, f (s)
⟩hds χ[0,t[ (·) v +
⟨L∗.,u,ζ
(ξ.
), f⟩
L2(R+;h)χ[0,t[ (·) v
=
∫ .
0
αs α. 〈u, f (s)〉 ds χ[0,t[ (·) v +
∫ .
0
⟨ξ., ζs
⟩〈u, f (s)〉 ds χ[0,t[ (·) v.
The second equality follows.
Remark 4.3 Under the same hypothesis as in the previous theorem and for r ∈ B (h) wehave
XtΛt (r) =
∫ t
0
FsdAs (r∗u) , Λt (r) Xt =
∫ t
0
Λs (r) FsdAs (u) ,
YtΛt (r) =
∫ t
0
dA∗s (u) GsΛs (r) , Λt (r) Yt =
∫ t
0
dA∗s (ru) Gs.
Given u ∈ h and r ∈ B (h), the dierentials dAt (u), dΛt (r), dA∗t (v) and dt called elementary.
The formal multiplication rule for two elementary dierentials is given by the Itô table
dAs (u1) dΛs (r1) dA∗s (v1) ds
dAs (u2) 0 dAs (r∗1u2) 〈u2, v1〉H PΩ ds 0
dΛs (r2) 0 dΛs (r2r1) dA∗s (r2v1) 0
dA∗s (v2) 0 0 0 0
ds 0 0 0 0
Let dMt, dNt be one of the elementary dierentials and FM , GM , FN , GN ∈ L21 (h). Then
the product of two expressions of the form FMt dMt G
Mt and FN
t dNt GNt with elementary
dierentials dMt and dNt is computed according to the Itô table and to the rule
(dMt) (α1 + Ft) (dNt) = α (dMt) (dNt) , (4.8)
for α ∈ C and F in L2 (h) (resp. in L1 (h)).
Theorem 4.4 (Integration by parts) Let FX , GX , F Y , GY ∈ L21 (h), HX , HY ∈ L1
1 (h),αX , βX , αY , βY ∈ C and uX , vX , uY , vY ∈ h, rX , rY ∈ B (h). We put
dXt = FXt dAt
(uX)
+ dΛt
(rX)
+ dA∗t
(vX)
GXt + HX
t dt
X0 = αXPΩ + βX1
13
and dYt = F Y
t dAt
(uY)
+ dΛt
(rY)
+ dA∗t (v
Y ) GYt + HY
t dt
Y0 = αY PΩ + βY 1
Then d (XtYt) = (dXt) Yt + Xt (dYt) + (dXt) (dYt)
X0 Y0 =(αXαY + αXβY + βXαY
)PΩ + βXβY 1,
where (dXt) (dYt) is computed using distributivity, rule (4.8) and the Itô table.
Proof: By using Theorem 4.2 and Remark 4.3 we obtain an explicit form of XtYt. Now bya simple identication the result follows.
There is another formulation of the previous result. In fact let u ∈ h and r ∈ B (h), we putthe following notations for the elementary dierentials
dLp,s (u, r) =
dAs (u) if p = 1dΛs (r) if p = 2dA∗
s (u) if p = 3ds if p = 4 .
Let (Ft)t>0 and (Gt)t>0 be two families of operator processes in L21 (h) and p ∈ 1, 2, 3, 4 we
set
Xp,t =
∫ t
0
Fs dLp,s (u, r) , Yp,t =
∫ t
0
dLp,s (u, r) Gs.
We denote by Int (L2) the sub-space of the processes of stochastic integrals
Int(L2)
:=
(Xp,t)t>0 , (Yp,t)t>0 with (Ft)t>0 , (Gt)t>0 ∈ L2
1 (h) for p = 1, 2, 3 and (Ft)t>0 ,
(Gt)t>0 ∈ L11 (h) for p = 4
.
We use the dierential notations
dXp,t = Ft dLp,t (u, r) , dYp,t = dLp,t (u, r) Gt. (4.9)
We denote now by Int (L2) the linear span of Int (L2).
Corollary 4.5 Int (L2) is a ∗-algebra.Proof: The proof is a simple deduction of Theorem 4.4, the Itô table and the rules (4.8).In fact let (Xp,t)t>0 , (Yq,t)t>0 ∈ Int (L2) with
dXp,t = Ft dLp,t (u1, r1) , dYq,t = dLq,t (u2, r2) Gt
where ui ∈ h, ri ∈ B (h) and (Ft)t>0, (Gt)t>0 ∈ L21 (h). We may assume also that Xp,0 = 0 =
Yq,0. Then by Theorem 4.4 we obtain
d (Xp,tYq,t) = (dXp,t) Yq,t + Xp,t (dYq,t) + (dXp,t) (dYq,t)
= Fs dLp,s (u1, r1) Yq,t + Xp,t dLq,s (u2, r2) Gs
+ Fs dLp,s (u1, r1) dLq,s (u2, r2) Gs
Now by combining the Itô table and the denitions of stochastic integrals we obtain X Y ∈Int (L2) then the result follows.
14
5 Quantum stochastic dierential equations
Let h be a Hilbert space, u, v ∈ h, β ∈ C and w ∈ B (h). Our aim is to construct anoperator-valued process U = (Ut) on the Boolean Fock space ΓR+ (h) satisfying
dUt = Ut (dAt (u) + dΛt (w) + dA∗t (v) + β PΩdt)
U0 = idΓR+(h) = 1
(5.10)
and to nd conditions on u, v, w and β such that U is a unitary operator. We begin bysolving equation (5.10) and by proving unicity. Then we give the condition for unitarity. Tosimplify thinks we put
dMt = dAt (u) + dΛt (w) + dA∗t (v) + β PΩdt.
If we put Ft = Ut − 1 the quantum dierential equation (Q.D.E.) (5.10) becomesdFt = FtdMt + dMt
F0 = 0(5.11)
Notice that we want to solve equation (5.11) on Boolean Fock space. The solution (if itexists!) has the form
Ft =
(αt 〈ξt||ζt〉 rt
)where F is an adapted operator-valued process. Thus by applying stochastic integration,the Q.S.D.E. becomes
dFt = dMt + FtdMt
⇐⇒
(αt 〈ξt||ζt〉 rt
)=
(β t 〈ut (·)|∣∣vχ[t,0[ (·)
⟩wt
)+
(0 Lt,u,α
0 Lt,u,ζ
)+
β
∫ t
0
αs ds 0∣∣∣∣β ∫ t
0
ζs ds
⟩0
.
By a simple identication we obtain
αt = βt + β
∫ t
0
αs ds
〈ξt| =⟨χ[0,t[ (·) v
∣∣+ Lt,u,α
ζt = χ[0,t[ (·) u + β
∫ t
0
ζs ds
rt = wt + Lt,u,ζ
The rst equation gives
αt = β
∫ t
0
αs ds + β t ⇐⇒ αt = exp (βt)− 1.
15
For the second equation we have for each f ∈ L2 (R+; h)
〈ξt, f〉 =⟨χ[0,t[ (·) u, f
⟩+⟨α. χ[0,t[ (·) u, f
⟩which yields
ξt = γ1χ[0,t[ (·) u.
where γ1 is the function on R+ given by
γ1 : s 7−→ exp (βs) (5.12)
For the third equation we recall that F must be adapted. We have ζt (x) = 0, for x > t.Then for x 6 t, the equation becomes
ζt (x) = β
∫ t
0
ζs (x) ds + v = β
∫ t
x
ζs (x) ds + v. (5.13)
Therefore
ζt (x) = Cx exp (βt)
where Cx satises equation (5.13). Thus we obtain
ζt (x) = exp (β (t− x)) v for x 6 t,
and where ζt (x) vanishes for x > t. So we have
ζt (x) = exp (β (t− x)) χ[0,t[ (x) v, i.e. ζt = exp (β (t− .)) χ[0,t[ (·) v
Finally, we have for each t > 0,
Ft =
(exp (βt)− 1 Lt,u,γ1
L∗t,v,γ2Lt,u,ζ + wt
)
where γ1 is given by (5.12) and γ2 by
γ2 : s 7−→ exp (β (t− s)) . (5.14)
Let us now prove that such a solution is unique. Let G be another solution of (5.11), .i.e.dG = GdM + dM
G0 = 0
Thus by combining the above equation and (5.11) for F , we obtain the following dierentialequation
d (F −G) = (F −G) dM
F0 −G0 = 0(5.15)
16
By using the same arguments as above and the initial condition, we conclude that a solutionof (5.15) is 0. So we have uniqueness of the solution of equation (5.11) and so of (5.10). Theunique solution of (5.10) is
Ut =
(exp (βt) Lt,u,γ1
L∗t,v,γ2Lt,u,ζ + wt + idL2(R+;h)
)(5.16)
where γ1 and γ2 are given by (5.12) and (5.14) respectively. It is not dicult to see that foreach t ∈ R+, Ut is a bounded operator and that its adjoint is given by
U∗t =
(exp
(βt)
Lt,v,γ2
L∗t,u,γ1L∗t,u,ζ + w∗M[0,t[ + idL2(R+;h)
)Now we come to the basic theorem.
Proposition 5.1 Let u, v ∈ h, w ∈ B (h) and β ∈ C. Then the quantum stochasticdierential equation
dUt = Ut (dAt (u) + dΛt (w) + dA∗t (v) + β PΩdt)
with initial condition U0 = 1 has a unique solution U = (Ut)t>0 given by (5.16).
The operator U is a unitary process if and only ifβ = −1
2‖u‖2 + α i
v = −w (u)− u
w + 1 is a unitary operator
(5.17)
where α is a real number.
Proof: Suppose that β, u, v and w are given as in (5.17). Then it is easy to see that U is aunitary process.
We investigate when the process U is unitary. If this is the case, U and U∗ satisfy
dU = U (dM)
dU∗ = (dM∗) U∗
respectively where dM∗ = dA∗ (u) + dΛ (w∗) + dA (v) + β PΩ dt. Now by combining theunitarity condition U∗U = 1 = UU∗ and Itô's formula, one obtains
d (U∗U) = 0 = d (UU∗) ⇐⇒
(dU∗) U + U∗ (dU) + (dU∗) (dU) = 0
(dU) U∗ + U (dU∗) + (dU) (dU∗) = 0
For the rst equality we have
(dU∗) U + U∗ (dU) + (dU∗) (dU) = 0
⇐⇒ dM∗ + dM + (dM∗) (dM) = 0
⇐⇒ (dA∗ (u) + dΛ (w∗) + dA (v) + β PΩ dt)
+(dA (u) + dΛ (w1) + dA∗ (v) + β PΩ dt)
+ (dA (w∗ (v)) + dΛ (w∗ w) + dA∗ (w∗ (v)) + ‖v‖2 PΩdt) = 0.
17
Thus we obtain
v + u + w∗ (v) = 0 (5.18)
w∗ + w + w∗ w = 0 (5.19)
β + β + ‖v‖2 = 0 (5.20)
For the second equality we multiply on the left by U∗ and on the right by U to obtain
dM + dM∗ + (dM) (dM∗) = 0
⇐⇒ (dA (u) + dΛ (w) + dA∗ (v) + β PΩ dt) + (dA∗ (u) + dΛ (w∗) + dA (v) + β PΩ dt)
+(dA (w (u)) + dΛ (w w∗) + dA∗ (w (u)) + ‖u‖2 PΩ dt) =
We have
u + v + w (u) = 0 (5.21)
w + w∗ + w w∗ = 0 (5.22)
β + β + ‖u‖2 = 0 (5.23)
Equations (5.19) and (5.22) imply that w + 1 is a unitary operator on h. Thus equations(5.18) and (5.21) become equivalent and we have
v = −w (u)− u.
We have also that (5.20) and (5.23) are equivalent and
β = iα− 1
2‖u‖2 where α ∈ R.
6 Dilations
In the same way as in the Bosonic and free cases [9, 12, 2, 19], see also [3], we will nowconstruct dilations of special semigroups of completely positive operators. For the generaldenitions of dilations see [11].
Let h be a nite-dimensional Hilbert space; i.e. h = Cm for some m ∈ N. We consider thequantum dynamical semi-group
Tt = exp (tG) : B (h) −→ B (h) ,
t > 0, with Lindblad generator G given by
G : B (h) −→ B (h)
X 7−→ L X L∗ − 12L L∗, X+ i [H, X] ,
where L, H ∈ B (h), H is a self-adjoint operator, [ , ] denotes the commutator and , isthe anti-commutator.
18
For a given X ∈ B (h) we identify X with X ⊗ PΩ ∈ B (h) ⊗ B (Γ (L2 (R))) that is we usethe embedding
ι : B (h) −→ B(h⊗ Γ (L2 (R)))X 7−→ X ⊗PΩ
(6.24)
Let U be the solution of the Q.S.D.E.dUt = Ut
(LdAt + (W − 1) dΛt −WL∗dA∗
t +(iH − 1
2LL∗
)PΩdt
)U0 = 1
, (6.25)
where L, W , H ∈ B (h), W is a unitary operator, H is a selfadjoint operator (where weidentify X ∈ B (h) with ι (X)). Ut is nothing but a special case of a Lévy process on K 〈m〉;see [16] for the Bose case. Thus the solution of (6.25) exists and is unique.
For the rest of this section it will be convenient to consider Boolean Fock space Γ (L2 (R))on all of R rather than on R+. This causes no real problems. Stochastic calculus can beextended in the obvious way. The advantage is that the shift process becomes unitary.
We dene the time shift, see [12], St : Γ (L2 (R)) → Γ (L2 (R)) on Boolean Fock space by
St
(λf
):=
(λ
stf
),
where st is the shift on L2 (R) given by
[stf ] (x) := f (x + t) .
Lemma 6.1 If Ut is the solution of the Q.S.D.E. (6.25), then
Ur+t = Ut S∗t Ur St, r, t ∈ R+,
which is called the cocycle property of Ut.
Proof: This can either be seen directly from (5.16) or by a uniqueness argument usingBoolean quantum stochastic calculus.
For t ∈ R+ consider the following mapping
Tt : B(h⊗ Γ (L2 (R))) −→ B(h⊗ Γ (L2 (R)))Y 7−→ Ut S
∗t Y St U
∗t
(6.26)
Let Y ∈ B(h⊗ Γ (L2 (R))) then one obtains
TrTt (Y ) = Tr (Ut S∗t Y St U
∗t )
= Ur S∗r Ut S∗t Y St U
∗t Sr U∗
r
= Ur+t S∗r+t Y Sr+t U
∗r+t
= Tr+t (Y ) ,
where we used Lemma 6.1. Therefore Tt denes a semigroup of automorphisms of B (h) ⊗B (Γ (L2 (R))).
19
Theorem 6.2 Let Tt be dened by (6.26). Then (B (h)⊗ B (Γ (L2 (R))) , Tt) is a dilationof (B (h) , Tt) , i.e. the following diagram
B (h)
ι
Tt // B (h)
B(h⊗ Γ (L2 (R)))Tt
// B(h⊗ Γ (L2 (R)))
Ec
OO
commutes where Ec is the conditional expectation given by
〈h1, Ec [X] h2〉h = 〈h1 ⊗ Ω, X (h2 ⊗ Ω)〉h⊗Γ(L2(R)) ,
for all h1, h2 ∈ h, X ∈ B(h⊗ Γ (L2 (R))).
Proof: We have to prove that
Tt (X) = [Ec Tt ι] (X) .
for each X ∈ B (h) and t > 0. Recall that we have
[Ec Tt ι] (X) = Ec (Ut S∗t ι (X) St U
∗t ) = Ec (Ut ι (X) U∗
t ) .
For X ∈ B (h) put
Xt = Ut ι (X) U∗t .
Then by using Theorem 4.4, one obtains
dXt = (dUt) ι (X) U∗t + Ut ι (X) (dU∗
t ) + (dUt) ι (X) (dU∗t )
= Ut
(− [L, ι (X)] W ∗ dAt + (W ι (X) W ∗ − ι (X)) dΛt −W [L∗, ι (X)] dA∗
t
+
i [H, ι (X)]− 1
2(L L∗ ι (X)− 2L ι (X) L∗ + ι (X) L L∗)
PΩ dt
)U∗
t .
Using ι (Y )PΩ = ι (Y ) for X ∈ B (h), we obtain, for h1, h2 ∈ h,
〈h1, [Tt (X)−X] h2〉h =
∫ t
0
〈h1, Ts (G (X)) h2〉h ds.
Thus Tt = exp (tG).
The general case of quantum dynamical semi-groups on B (h), h nite-dimensional, can betreated similarly. If the generator is given by
L (A) =k∑
r=1
(LrAL∗r −
1
2LrL
∗r, A
)+ i [H, A] ,
the corresponding Q.S.D.E. is of the form
dUt = Ut
(k∑
r=1
(LrdA
(r)t +(Wr−1) dΛ
(r)t −WrL
∗rdA
∗ (r)t
)+
(iH− 1
2
k∑r=1
LrL∗r
)PΩ dt
),
with A(r)t , Λ
(r)t , A
∗ (r)t , r = 1, . . . , k, k-dimensional annihilation, preservation and creation
processes.
20
References
[1] Accardi, L.; Fagnola, F.; Quaegebeur, J. : A representation free quantum stochasticcalculus, J. Funct. Anal. 104, 149-197, 1992.
[2] Applebaum, D.B.; Hudson, R.L. : Fermion Ito's formula and stochastic evolutions,Commun. Math. Phys. 96, 473-496, 1984.
[3] Bhat, B.V.R.; Skeide, M. : Tensor product systems of Hilbert modules and dilationsof completely positive semigroups, Preprint, Revised version, Rome, 2000.
[4] Ben Ghorbal, A.; Schürmann, M. : Non-commutative notions of stochastic indepen-dence, Math. Proc. Cambridge Philos. Soc. 133, no. 3, 531-561, 2002.
[5] Ben Ghorbal, A. : Feynman Kac formula by the Boolean stochastic calculus, inpreparation.
[6] Bo»ejko, M.: Uniformly bounded representations of free groups, J. Reine Angew.Math. 377, 170-186, 1987.
[7] Dogan, H. : Quantenstochastische Integration für Boole-adaptierte Prozesse, Diplo-marbeit University of Heidelberg, 2000.
[8] Hudson, R.L.; Ion, P.D.F.; Parthasarathy K.R. : Time-orthogonal unitary dilationsand noncommutative Feynman-Kac formulae, Commun. Math. Phys. 83, 261-280,1982.
[9] Hudson, R. L.; Parthasarathy, K. R. : Quantum Ito's formula and stochastic evo-lutions, Commun. Math. Phys., 93, 301-323, 1984.
[10] Kümmerer, B.; Schröder, W. : A new construction of unitary dilations: singularcoupling to white noise, Quantum probability and applications, II (Heidelberg, 1984),Lecture Notes in Math., 1136, 332-347, 1985.
[11] Kümmerer, B. : Survey on a theory of non-commutative stationary Markov pro-cesses, Quantum probability and applications III, (Proc. Conf., Oberwolfach/Germany1987), Lect. Notes Math. 1303, 154-182, 1988.
[12] Kümmerer, B.; Speicher, R. : Stochastic integration on the Cuntz algebra O∞, J.Funct. Anal. 103, No.2, 372-408, 1992.
[13] Lenczewski, R. : Stochastic calculus on multiple symmetric Fock spaces, Inn. Di-mens. Anal. Quantum Probab. Relat. Top. 4, no. 3, 309-346. 2001.
[14] Muraki, N. : The Five Independences as Natural Products, EMAU GreifswaldPreprint-Reihe Mathematik 3/2002, 2002.
[15] Speicher, R. : On universal product, Fields Inst. Commun. 12, 257-266, 1997.
[16] Schürmann, M. : White noise on bialgebras, Lect. Notes Math. 1544, 1993.
21
[17] Schürmann, M. : Direct sums of tensor products and non-commutative indepen-dence, J. Funct. Anal. 133, No.1, 1-9, 1995.
[18] Skeide, M. : Quantum stochastic calculus on full Fock modules, J. Funct. Anal. 173,no. 2, 401-452, 2000.
[19] Speicher, R. : Stochastic integration on the full Fock space with the help of a kernelcalculus, Publ. Res. Inst. Math. Sci. 27, No.1, 149-184, 1991.
[20] Speicher, R.; Woroudi, R.: Boolean convolution, Fields Inst. Commun. 12, 267-279,1997.
[21] Waldenfels, W. von : An approach to the theory of pressure broadening of spectrallines, In: Behara, M, Krickeberg, K.; Wolfowitz, J. (eds.), Probability and InformationTheory II, Lect. Notes Math. 296, 19-69, 1973.
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