A quantum stochastic Lie-Trotter product formula

A QUANTUM STOCHASTIC LIE-TROTTER PRODUCT FORMULA

J. Martin Lindsay^∗ and Kalyan B. Sinha^∗∗

∗Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK

e-mail: j.m.lindsay@lancaster.ac.uk

∗∗Jawaharlal Nehru Center for Advanced Scientific Research, and

Indian Institute of Science, Bangalore, India e-mail: kbsjaya@yahoo.co.in

Abstract A Trotter product formula is established for unitary quantum stochas- tic processes governed by quantum stochastic differential equations with constant bounded coefficients.

Key words Trotter product formula, quantum stochastic cocycle, one-parameter semigroup, random unitary evolution, noncommutative probability.

1. Introduction

The aim of this paper is to establish a quantum probabilistic counterpart to the well- known Trotter product formula for one-parameter unitary groups and contraction semigroups [25] and its forerunner, the Lie product formula for one-parameter sub- groups of Lie groups (see [4, 21, 22]). Some years ago K. R. Parthasarathy and the second-named author obtained a stochastic Trotter product formula for unitary- operator valued evolutions constituted from independent increments of indepen- denta classical Brownian motions [18]. This predated the founding of quantum stochastic calculus by Hudson and Parthasarathy [8]. In this paper Brownian incre- ments are replaced by the fundamental quantum martingales, namely the creation, preservation and annihilation processes of quantum stochastic calculus [2, 6, 9, 15, 17, 23], and we prove a Lie-Trotter type product formula for unitary quantum stochastic processes on a Hilbert space which satisfy a quantum stochastic differ- ential equation with constant bounded coefficients. The case of quantum stochastic differential equations with unbounded coefficients, and more general kinds of quan- tum stochastic cocycle on operator spaces andC^∗-algebras, will be addressed in the forthcoming paper [11].

2. Unitary Quantum Stochastic Cocycles

In this section we fix our notations and recall the essential facts about quantum stochastic differential equations and unitary quantum stochastic cocycles that we need here.

Let k be a complex Hilbert space, with fixed countable orthonormal basis, which we refer to as thenoise dimension space. WriteFkfor the symmetric Fock space over the Hilbert spaceK := L²(R₊;k)and(f)for the normalised expo- nential vectorexp(−f²/2)ε(f),f ∈K. WhenR₊is replaced by[s, t[, we write K_[s,t[andF_k,[s,t[instead; the continuous tensor decomposition

F_k =F_k,[0,s[⊗ F_k,[s,t[⊗ F_k,[t,∞[,

corresponding to the direct sum decompositionK =K_[0,s[⊕K_[s,t[⊕K_[t,∞[, is in constant use below. For a start, a bounded quantum stochastic process on aninitial Hilbert spacehwith noise dimension spacekis a family of operators(Xt)t≥0 on h⊗ F_k satisfying the adaptedness condition

Xt∈B(h⊗ F_k,[0,t[)⊗I_Fk,[t,∞[=B(h)⊗B(F_k,[0,t[)⊗I_Fk,[t,∞[, (2.1) for all t ∈ R+. For f ∈ K, f_[s,t[ denotes the function equal to f on [s, t[ and zero elsewhere; c_[s,t[ is defined similarly, for c ∈ k. Let S_k andS_k denote the subspaces ofKconsisting of step functions, respectively step functions which have their discontinuities in the dyadic set D := {j2⁻ⁿ : j, n ∈ Z+}, and let E_k and E_k be the (dense) subspaces Lin{ε(f) : f ∈ S_k} and Lin{ε(f) : f ∈ S_k} of F_k. For evaluation purposes, we always take theright-continuous versionsof step functions. Theorderof a functionf ∈ S_kis the least nonnegative integerN such thatf is constant on all intervals of the form[j2^−N,(j+ 1)2^−N[forj∈Z₊.

Thetime-shiftsemigroup(Θ^kt)t≥0of unital *-monomorphisms ofB(F_k)is de- fined by

Θ^kt(X) =I_Fk,[0,t[⊗Γ(θ^k_t)XΓ(θ^k_t)^∗, t∈R₊,X∈B(F_k),

whereΓ(θ^k_t) : F_k → F_k,[t,∞[ is the unitary (second quantisation) operator deter- mined by

Γ(θ_t^k)(f) =(θ_t^kf) where (θ^k_tf)(s) =f(s−t) fors∈[t,∞[.

Let {Λ^μν : μ, ν ≥ 0} denote the fundamental quantum semimartingales for the noise dimension spacek, with respect to its fixed orthonormal basis. Then the quantum stochastic (QS) integral equation

Ut=I_h⊗Fk+ t

0 UsF_ν^μΛ^νμ(ds) (2.2) (where summation over the repeated greek indices is understood), has a unique strongly continuous solution, consisting of unitary operators onh⊗ F_k, provided

that the matrix of bounded operators[Fν^μ]on the initial spacehsatisfies the follow- ing structural relations [8]. It must have the block matrix structure

K [Mk] [L^j] [W_k^j−δ^j_k]

.

of an operator F ∈ B(h⊕ (h ⊗k)), where [L^j] is the block column matrix of an arbitrary operator L ∈ B(h;h⊗k), [W_k^j] is the block matrix form of a unitary operatorW ∈ B(h⊗k), [Mk] is the block row matrix of the operator M = −L^∗W ∈ B(h⊗k;h), and K = iH − 1

2L^∗L for a selfadjoint operator H ∈B(h), so that

Mk =−

j≥1

(L^j)^∗W_k^j, k≥0,andK=iH−1 2

j≥1

(L^j)^∗L^j.

These structure relations may equivalently be expressed by the following two identities, for allv= (v^μ)μ≥0inh⊕(h⊗k) =

μ≥0h:

μ,ν≥0

v^μ,((F_μ^ν)^∗+F_ν^μ+

j≥1(F_μ^j)^∗F_ν^j

v^ν = 0, (2.3a)

μ,ν≥0

v^μ,

(F_μ^ν)^∗+F_ν^μ+

j≥1F_j^μ(F_j^ν)^∗

v^ν = 0; (2.3b) the first corresponds to isometry and the second to coisometry.

A contractive quantum stochastic process(Ut)t≥0satisfying

Us+t=UsΘs(Ut), U₀=I_h⊗F, s, t≥0, (2.4) where

Θt:= idB(h)⊗Θ^kt

t≥0, is called aquantum stochastic contraction cocycle.

If(Ut)t≥0is a QS contraction cocycle then the operators onhdefined by u, Ptv=u⊗(0), Utv⊗(0) u, v∈h, t∈R₊,

define a contraction semigroup(Pt)t≥0 on hknown as the(vacuum) expectation semigroup of the cocycle, and the cocycle(Ut)t≥0 is calledMarkov-regularif its expectation semigroup is norm-continuous.

Theorem 2.1. [12]Let(Ut)t≥0be a unitary quantum stochastic process onhwith noise dimension spacek. Then the following are equivalent:

1. (Ut)t≥0satisfies (2.2), for a matrix of bounded operators[Fν^μ]; 2. (Ut)t≥0is a Markov-regular quantum stochastic cocycle.

The implication (i)⇒(ii) follows from the form that solutions of such QS dif- ferential equations take, by virtue of the time-homogeneity of the quantum noises:

I_Fk,[0,t[⊗Γ(θ^k_t)Λ^μν[a, b]Γ(θ^k_t)^∗ = Λ^μν[a+t, b+t],

and the time-independence of the coefficients of the QS differential equation.

The converse implication (ii)⇒(i) may be deduced from the Quantum Martin- gale Representation Theorem [19] applied to the regular quantum martingale

Ut− t

0

UsK ds

t≥0

in which the operatorK is the generator of the expectation semigroup of(Ut)t≥0

(see [7]). However the more powerful method of proof in [12] goes via the follow- ing intermediate characterisation which is of considerable use itself, as we shall see below:

(iii) there are semigroups{(P^c,d)t≥0 : c, d∈ k}such that, for allf, g ∈ S_k and t∈R₊,

u⊗(f_[0,t[), Vtv⊗(g_[0,t[)=u, P_t^f₁⁽₋^t0_t0^),g(t0)· · · P_t^f_m+1^(tm₋^),g_tm^(tm)v, (2.5) wheret₀ = 0,tm+1 =tand{t₁ <· · · < tm} ⊂ Dis the (possibly empty) union of the sets of discontinuity off andgin the open interval]0, t[.

Remarks: The matrix of bounded operators[Fν^μ]necessarily satisfies the structural relations required for unitarity (2.3).

The identity (2.5) is known as thesemigroup decompositionand the collection {(P_t^c,d)t≥0 : c, d ∈ k} as theassociated semigroups of the cocycle. Clearly the associated semigroups are determined by

u, P_t^c,dv=u⊗(c_[0,t[), Utv⊗(d_[0,t[), u, v∈h, (2.6) and(P^0,0)t≥0is the expectation semigroup of the cocycle.

In fact, each associated semigroup(P^c,d)t≥0is itself the expectation semigroup of another unitary QS cocycle, namely the cocycle

U_t^c,d:= (I_h⊗W_t^c)^∗Ut(I_h⊗W_t^d)

t≥0, where theWeyl cocyclesare defined by

W_t^c(f) =e^{−iImc[0,t[,f}(f+c_[0,t[), f ∈S_k, c∈k, t∈R₊.

Markov-regularity for a QS contraction cocycle actually implies that all of its associated semigroups are norm-continuous. In fact, in terms of the block matrix form of[Fν^μ], the semigroup(P^c,d)t≥0has bounded generator

G_c,d:=K+L^c+M_d+W_d^c−1

2(||c||²+||d||²)I_h, (2.7) where, in terms of basis expansions ofc andd, the operators here are defined as follows:

L^c=

j≥1

c^jL^j, Md=

k≥1

d^kMkandW_d^c=

j,k≥1

c^jd^kW_k^j,

the convergence here being in the strong operator topology (see [13]).

Given a unitary QS cocycle(Ut)t≥0, the family(Us,t := Θs(U_(t−s)))_0≤s≤t is atime-homogeneous adapted unitary evolution, that is: for alla≥0and0≤r ≤ s≤t:

1. Us+a,t+a= Θa(Us,t);

2. Us,t ∈B(h)⊗I_Fk,[0,t[⊗B(F_k,[s,t[)⊗I_Fk,[t,∞[; 3. Ur,t=Ur,sUs,t.

Conversely, if(Us,t)0≤s≤tis such an evolution then(Ut := U₀,t)t≥0 defines a unitary QS cocycle, and it is easily seen that the passages between QS cocycle and adapted time-homogeneous evolution are mutually inverse.

The corresponding QS integral equation satisfied by(Us,t)0≤s≤tis Ur,t=I_h⊗Fk+

t r

Ur,sF_ν^μΛ^νμ(ds).

Adapted evolutions that are not time-homogeneous arise as solutions of QS differential equations with time-dependent coefficients[Fν^μ].

3. Trotter Product of Quantum Stochastic Cocycles

Let(U_t¹)t≥0 and(U_t²)t≥0 be two unitary QS cocycles on the same initial spaceh, with noise dimension spacesk₁ andk₂ having fixed countable orthonormal bases.

Suppose that they are both Markov-regular, equivalently that they satisfy QS dif- ferential equations

dU_t^l =U_s^l(l)F_ν^μ_l^lΛ^νμ^l_l(dt), U₀^l =I_h⊗F(l), (3.1) l = 1,2, for matrices of bounded operators [⁽¹⁾Fν^μ₁¹] and [⁽²⁾Fν^μ₂²] satisfying the structural relations which guarantee unitarity of the processes. HereF⁽¹⁾andF⁽²⁾ denote the Fock spacesF_k1 andF_k2 respectively.

Our aim is to obtain a unitary cocycle(Ut)t≥0 as a Lie-Trotter type product of the cocycles(U_t¹)t≥0and(U_t²)t≥0, in the same spirit as that of [18]. To this end let kbe the noise dimension spacek₁⊕k₂, setF =F_k, and, by ‘concatenating’ the orthonormal bases fork1andk2 to form an orthonormal basis ofk, let[Fν^μ]be the matrix of bounded operators onhhaving block matrix form

K [Mk] [L^j] [W_k^j−δ_k^jI_h]

=

⎡

⎣

(1)K+⁽²⁾K ⁽¹⁾M ⁽²⁾M

(1)L ⁽¹⁾W −⁽¹⁾I 0

(2)L 0 ⁽²⁾W −⁽²⁾I

⎤

⎦. (3.2)

Here^(l)I :=I_h⊗kland [^(l)F_ν^μl

l] =

(l)K [^(l)Mk_l] [^(l)L^jl] [^(l)W_k^jl

l −δ^j_k^l

lI_h]

is the block matrix decomposition of^(l)F, in which

(l)K=iHl−1 2

j_l≥1

(^(l)L^jl)^∗(l)L^jl and(Hl)^∗=Hl,

for l = 1,2. (We are slightly cheating in terms of indices since if the noise di- mension spacek₁ is infinite dimensional then we cannotexactly count 1,2,· · · , dimk1,dimk1+ 1,· · ·. However allisjustified by a proper indexing, or alterna- tively by working coordinate-free as in [11]) Thus, settingH=H₁+H₂,

K =iH−1 2

j≥1

(L^j)^∗L^j

=iH₁+iH₂−1 2

j₁≥1

(⁽¹⁾L^j1)^∗(⁽¹⁾L^j1)−1 2

j₂≥1

(⁽²⁾L^j2)^∗(⁽²⁾L^j2), and

[Mk] =

−⁽¹⁾L^∗(1)W −⁽²⁾L^∗(2)W

=

⎡

⎣−

j≥1

(L^j)^∗W_k^j

⎤

⎦.

Thus[Fν^μ]satisfies the structure relations (2.3) for unitarity of the solution of the QS differential equation (2.2) to be unitary.

Forc^l, d^l∈kllet(^(l)P_t^cl,dl)t≥0denote the corresponding associated semigroup of the cocycle (U_t^l)t≥0 (l = 1,2). For each n ∈ N define a unitary process (Un^(1,2)(t))t≥0as follows:

U_n^(1,2)(t) :=

U₀⁽¹_,2^,_−n²⁾ U₂⁽¹_−n^,2)_,2·2−n· · ·U_t⁽¹n^,2)

−1,tⁿ₀

U_t⁽¹n^,2)

0,t , t∈R₊, where, with[·]denoting the integer part,

tⁿ_k := 2⁻ⁿ

[2ⁿt] +k

fork∈Z,≥ −[≈], (3.3) and, lettingΣ₂,1denote the tensor flipB(h⊗F⁽²⁾⊗F⁽¹⁾)→B(h⊗F⁽¹⁾⊗F⁽²⁾) = B(h⊗ F),

U_s,t^(1,2) := Θs(U_t⁽¹₋^,_s²⁾), 0≤s≤t, (3.4) where

U_t^(1,2):=

U_t¹⊗I⁽²⁾ Σ₂,1

U_t²⊗I⁽¹⁾

, t∈R+.

HereI^(l)is the identity operator onF^(l)(l= 1,2), and we are using the natural isometric isomorphismF⁽¹⁾⊗ F⁽²⁾ =F. Also define a family of contractions on hby

u,^(1,2)P_t^c,dv=u⊗(c_[0,t[), U_t^(1,2)v⊗(d_[0,t[), u, v∈h, forc, d∈kandt∈R₊(cf. (2.6)).

Remarks: For eacht∈R+,n∈Nandkas in (3.3),

tⁿ₀ ≤tⁿ₀⁺¹≤t < tⁿ₁⁺¹< tⁿ₁ and|tⁿ_k+1−tⁿ_k|= 2⁻ⁿ.

In particular the sequence(tⁿ₁)decreases totand the sequence(tⁿ₀)is nondecreas- ing and converges tot.

In general, neither(U_t^(1,2))t≥0nor(Un^(1,2)(t))t≥0are cocycles themselves. How- ever they are both unitary QS processes and the two-parameter process(U_s,t^(1,2))_0≤s≤t

enjoys the factorisations

U_s,t^(1,2)∈B(h)⊗I_[0,s[⊗B(F_[s,t[)⊗I_[t,∞[ (3.5) in whichI_[0,s[andI_[t,∞[denote the identity operators onF_[0,s[andF_[t,∞[. By the same token,(^(1,2)P^c,d)t≥0is typically not a semigroup.

Lemma 3.1. Let (U_t¹)t≥0 and (U_t²)t≥0 be unitary QS cocycles on h with noise dimension spacesk₁ andk₂ respectively. Setk:= k₁⊕k₂ and let(U_t^(1,2))t≥0 be as defined above. Lett∈R₊, then forc=_c1

c²

,d=_d1

d²

∈k=k₁⊕k₂,

(1,2)P_t^(c,d)=⁽¹⁾P_t^c1,d1(2)P_t^c2,d2,

and, forf, g∈S_kandngreater than the orders of bothfandg, u⊗(f_[0,t[), U_n^(1,2)(t)v⊗(g_[0,t[)=

u,_(1,2)

P₂^f_−n^{(0),g(0)(1,2)}P₂^f_−n^{(2−n),g(2−n)}· · · ^(1,2)P₂^f_−n^{(tn−1),g(tn−1)}_(1,2)

P₍^f_t⁽₋^tn0_tn^),g(tn0)

0) v

.

Proof: These both follow from factorisations; the first from u⊗(c_[0,t[)(U_t^(1,2)v⊗(d_[0,t[)

=u⊗(c¹_[0,t[)U_t¹

(E^∗U_t²F v)⊗(d¹_[0,t[) ,

whereE andF are the isometric operatorsh→ h⊗ F⁽²⁾defined respectively by v → v⊗(c²_[0,t[)andv → v⊗(d²_[0,t[); in turn, the second from the semigroup decomposition (3.5) and(h) =(h_[0,s[)⊗(h_[s,t[)⊗(h_[t,∞[), forh=f, g.2 We now come to our quantum stochastic product formula. For its proof we use the following version of the classical Lie product formula. For bounded operators Z₁andZ₂onh,

e^hZ1e^hZ2_[t/h]→e^t(Z1+Z2)ash→0, (3.6) in operator norm, uniformly on bounded time intervals (see e.g. Theorem VIII. 29 of [21], where the proof is obviously valid for operators).

Theorem 3.2.Let(U_t¹)t≥0and(U_t²)t≥0be unitary QS cocycles onhwith noise di- mension spacesk₁andk₂, satisfying the quantum stochastic differential equations

(3.1), and let(Ut)t≥0 be the unitary QS cocycle onhwith noise dimension space k := k₁ ⊕k₂ satisfying the QS differential equation (2.2) where[Fν^μ]is given by (3.2). Then,

U_n^(1,2)(t)→Utasn→ ∞, (3.7) in the strong operator topology onB(h⊗ F), for eacht≥0.

Proof: Lett∈R₊. First note that, sinceUtis unitary and eachUn^(1,2)(t)is unitary and so a contraction, it suffices to prove thatUn^(1,2)(t) → Utin the weak operator topology. Also, the uniform boundedness of the operatorsUn^(1,2)(t) means that it suffices to fixu, v∈handf =_f1

f²

, g =_g1

g²

∈S_k ⊂L(R₊;k=k⊕k), and prove the following:

u⊗(f_[0,t[)U_n^(1,2)(t)v⊗(g_[0,t[) → u⊗(f_[0,t[)Utv⊗(g_[0,t[). (3.8) By the semigroup representation (2.5),

R.H.S. of (3.8) =u, P_t^f(t0),g(t0)

1−t₀ · · · P_t^f(tm),g(tm)

m+1−t_m v, (3.9) wheret₀ = 0,tm+1 =tand{t₁ <· · ·< tm} ⊂ Dare the points in]0, t[(if any) wheref orghas a discontinuity. Forngreater than the orders of the step functions f andg, Lemma 3.1 implies that the L.H.S. of (3.8) equals

u,(⁽¹⁾P₂^f_−n^{1(t0),g1(t0)(2)}P₂^f_−n^2(t0),g2(t0))^{[2n(t1−t0)]}· · ·

· · ·(⁽¹⁾P₂^f_−n^{1(tm),g1(tm)(2)}P₂^f_−n^2(tm),g2(tm))^{[2n(tm+1−tm)]}

(⁽¹⁾P₍^f_t¹₋⁽_t^t_n^m),g1(tm)

0) (2)P₍^f_t²₋⁽_t^t_n^m),g2(tm)

0) )v

. The Lie product formula (3.6) and the joint continuity of operator composition on bounded sets, therefore implies that

nlim→∞( L.H.S.of (3.8)) =u, Q^f_t1⁽₋^t0_t⁾₀^,g(t0)· · · Q^f_tm+1^(tm)₋^,g_t⁽_m^tm)v, (3.10) where(Q^c,d_t )t≥0is the semigroup generated byG⁽¹⁾_c1,d1 +G⁽²⁾_c2,d2. Now

G⁽¹⁾_c1,d1 +G⁽²⁾_c2,d2 = ⁽¹⁾K+⁽¹⁾L^c1 +⁽¹⁾M_d1 +⁽¹⁾W_d^c₁¹−¹₂(c¹²+d¹²)I_h +⁽²⁾K+⁽²⁾L^c2 +⁽²⁾M_d2+⁽²⁾W_d^c₂² −¹₂(c²²+d²²)I_h

= K+L^c+Md+W_d^c− ¹₂(c²+d²)I_h,

which, by (2.7), is the generator of the semigroup(P_t^c,d)t≥0, for eachc, d∈k. The

result therefore follows from (3.10) and (3.9). 2

Remark: The joint continuity of operator composition on bounded sets also gives a straightforward extension of this result to time-homogeneous adapted unitary evo- lutions(Us,t)_0≤s≤t:

U_n^(1,2)(s, t)→Us,t asn→ ∞,

in the strong operator topology, for all0≤s≤t, where U_n^(1,2)(s, t) :=U_s,s^(1,n²⁾

1

U_s⁽¹n^,2) 1,sⁿ₂U_s⁽¹n^,2)

2,sⁿ₃ · · ·U_t⁽¹n^,2)

−1,tⁿ₀

U_t⁽¹n^,2) 0,t .

4. Extensions and an Example

The quantum stochastic product formula also holds for Markov-regular QS con- traction cocycles, with the same proof, since these are equally characterised as contraction processes which satisfy a QS differential equation of the form (2.2), in other words Theorem 2.1 still holds; contractivity of the cocycle corresponds precisely to the matrix of coefficients of the QS differential equation satisfying the inequality

μ,ν≥0

v^μ,

(F_μ^ν)^∗+F_ν^μ+

j≥1(F_μ^j)^∗F_ν^j

v^ν ≤0,

equivalently,

μ,ν≥0

v^μ,

(F_μ^ν)^∗+F_ν^μ+

j≥1F_j^μ(F_j^ν)^∗

v^ν ≤0,

for allv = (v^μ)^μ≥0 ∈h⊗(C⊕k) =

μ≥h[5, 16], cf. the equalities (2.3) for the unitary case. However in this case the convergence of the Trotter products is only assured in the weak operator topology (or rather in the hybrid normF_k-weak operator topology, see [14]).

Using an extension of the standard Trotter product formula to products of sev- eral semigroups, our QS product formula extends to cover a finite number of QS unitary (or contraction) cocycles(U_t¹)t≥0, ... ,(U_t^p)t≥0. The coefficient matrix for the QS differential equation of the resulting QS cocycle will then have the block matrix form:

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

(1)K+· · ·+^(p)K ⁽¹⁾M ⁽²⁾M · · · ^(p)M

(1)L ⁽¹⁾W −⁽¹⁾I 0 · · · 0

(2)L 0 ⁽²⁾W −⁽²⁾I . .. ...

... ... . .. . .. 0

(p)L 0 · · · 0 ^(p)W −^(p)I

⎤

⎥⎥

⎥⎥

⎥⎥

⎦ .

Example: The cocycles considered in [18] are the random unitaries defined by U^l(s, t, ω^l) =e^{i(ωl(t)−ωl(s))Hl}, 0≤s≤t,

for l = 1,2, where ω¹ and ω² are paths of two independent classical Brownian motions(B_t¹)t≥0and(B_t²)t≥0, andH₁andH₂are selfadjoint operators on a Hilbert spaceh. Recall the notation (3.3). By viewingω:= (ω¹, ω²)as a path of the two- dimensional Brownian motion((B_t¹, B_t²))t≥0with probability spaceΩ, and

U^(1,2)(s, t, ω) :=e^{i(ω1(t)−ω1(s))H1}e^{i(ω2(t)−ω2(s))H2}, 0≤s≤t,

as multiplication operators onL²(Ω;h), it is shown—under the assumption that the nonnegative symmetric operator(H₁)²+ (H₂)² is selfadjoint—that the sequence (Un^(1,2)(s, t, ω))n≥1of unitary operators:

U^(1,2)(s, sⁿ₁, ω)

U^(1,2)(sⁿ₁, sⁿ₂, ω)· · ·U^(1,2)(tⁿ₋₁, tⁿ₀, ω)

U^(1,2)(tⁿ₀, t, ω) converges in the weak operator topology to the unique contraction-operator valued process satisfying the classical stochastic differential equations

dtU(s, t, ω)v=i U(s, t, ω)H₁v dB_t¹(ω) +i U(s, t, ω)H₂v dB_t²(ω)

−¹₂U(s, t, ω)

(H₁)²+ (H₂)² v dt forv ∈Dom (H₁)² + (H₂)², and that if the process(U(s, t, ω))_0≤s≤tis unitary- valued then the convergence is strong.

Remark: Under the assumption of selfadjointness ofd

l=1(H_l)², the correspond- ing result is shown to hold for any finite number of such unitary cocycles (U^l(s, t, ω))t≥0,l= 1, . . . , d.

This may be recast in our quantum stochastic setting by identifying the Brown- ian motion(B_t^l)t≥0 with the quantum stochastic process(Q^l_t:= (A^l_t^∗+A^l_t)⁻)t≥0

onF^(l)(where the bar denotes operator closure), and setting U_t^l =e^iHl(t), t≥0,

whereH_l(t)is the selfadjoint operatorH_l⊗Q⁽_t^l)onh⊗F^(l), forl= 1,· · · , d. Here however the coefficients of the corresponding differential equation are unbounded, with coefficients having block matrix form

(l)F =

−¹₂(Hl)² iHl

iHl 0

, l= 1,· · ·, d,

and

F =

⎡

⎢⎢

⎢⎣

−¹₂K iH₁ · · · iH_d iH₁ 0 · · · 0

... ... . .. ... iHd 0 · · · 0

⎤

⎥⎥

⎥⎦whereK = (H₁)²+· · ·(Hd)².

This class of example is discussed in more detail in [11].

5. Concluding Remarks

The methods of this paper extend to more general QS cocycles. Firstly, quantum stochastic Trotter product formulae may be obtained for completely contractive QS cocycles on operators spaces and completely positive QS cocycles onC^∗-algebras.

Secondly,strongly continuous(as opposed to Markov-regular) QS cocycles may be shown to satisfy the QS Trotter product formula developed here. Conversely, the formula may be used to construct QS cocycles from simpler cocycles with lower

dimensional noises. This yields potential applications to multidimensional diffu- sions. The basic conditions under which Trotter products converge is that the sum of sufficiently many pairs of associated semigroup generators are pregenerators of contraction semigroups. Here the assumption of analyticity of the expectation semigroups of the constituent cocycles helps ([10]). As in the Markov-regular case, strong (as opposed to weak) operator convergence holds for Trotter products of iso- metric QS cocycles if and only if the limiting cocycle is isometric. Coisometry, on the other hand, is equivalent to isometry of thedualcocycle (see [9]). Unitarity for strongly continuous contraction cocycles is assured when the cocycle satisfies a QS differential differential whose coefficients satisfyFeller conditions(see [23]).

All these extensions are treated in [11]. They are facilitated by characterisa- tions of QS cocycles in terms of (a small number of) their associated semigroups ([1, 14]). Here Skeide’s multidimensional generalisation ([24]) of a theorem of Parthasarathy and Sunder ([20]) plays a key role. The homomorphic property of Trotter product limits of Evans-Hudson type cocycles on operator algebras is tack- led in [3].

Acknowledgement

Both authors acknowledge support from the UK-India Education and Research Ini- tiative (UKIERI). KBS is grateful for the support of a Bhatnagar Fellowship from CSIR, India, and JML is grateful for the hospitality of the Delhi Centre of the In- dian Statistical Institute, where this collaboration began, and the J.N. Centre for Advanced Research, Bangalore.

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