Quantum stochastic Weyl operatorsLuigi Accardi

Centro Vito Volterra, Universita di Roma TorVergataVia di TorVergata, 00133 Roma, Italy

volterra@volterra.mat.uniroma2.itAndreas Boukas

Department of Mathematics, American College of GreeceAghia Paraskevi, Athens 15342, Greece

gxk-personnel@ath.forthnet.gr

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Contents

1. Introduction 32. First order and square of white noise Weyl operators 33. Module form of Weyl evolutions 74. Matrix elements 9References 10

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Thanks. The second author wishes to express his gratitude to Professor Luigi Accardi for his support and guid-

ance over the years, as well as for the hospitality of the Centro Vito Volterra of the Universita di RomaTorVergata on many occasions.

Class of Subject: Primary 81S25Abstract. The quantum stochastic differential equation satisfied by the unitary operator process U =

{U(t) = eiE(t) : t ≥ 0} where E(t) = λ t + z B−t + z B+t + kMt, and B−t , B+

t , Mt are the square of whitenoise processes of [AcLuVo99], is obtained in the module form of [AcBou03b].

1. Introduction

Classical (i.e Ito [Ito51]) and quantum (i.e Hudson Parthasarathy [HuPa84]) stochastic calculi were unifiedby Accardi, Lu, and Volovich in [AcLuVo99] in the context of Hida white noise theory [Hida92], [Kuo96]. Thetheory was extended to include the newly discovered “square of white noise” with the use of renormalizationtechniques.

The problem of the unitarity of the solutions of quantum stochastic differential equations (QSDE) drivenby first order white noise (related to the oscillator algebra of the Heisenberg-Weyl Lie algebra) was solvedby Hudson and Parthasarathy in [HuPa84] who proved that a unitary evolution {U(t) : t ≥ 0} defined in thetensor product of a “system” Hilbert space and a ”noise” Boson-Fock space, satisfies a QSDE of the form

dU(t) = U(t)[(iH − 12 L∗L)dt− L∗WdA(t) + LdA†(t) + (W − I)dΛ(t)](1.1)

where H, L, and W are bounded system space operators, with H self-adjoint and W unitary. HeredA(t), dA†(t), and dΛ(t) are the quantum stochastic differentials of the “annihilation”, “creation”, and“conservation” processes respectively.

The corresponding problem for “square of white noise” evolutions (related to the sl(2;R) Lie algebra)was open for several years. Preliminary work was done by Accardi, Hida, Boukas, and Kuo in [AcBou01a-g],[AcBou00a], and [AcHiKu01]. The subject was brought to a close by Accardi and Boukas in [AcBou03b]where it was shown that square of white noise unitary evolutions satisfy QSDE of the type

dUt = ((− 12 (D−|D−) + iH) dt+ dAt(D−) + dA†t(−l(W )D−) + dLt(W − I))Ut(1.2)

formulated on the module B(HS) ⊗ Γ(K), where HS is a system Hilbert space, K = l2(N) and Γ(K)denotes the Fock space over K (see section 3 below and [AcBou03b] for notation and details).

Applications of quantum stochastic calculus to the control of quantum evolution and Langevin equationscan be found in [AcBou03a], [AcBou02a-b], [Bou94a-b], [Bou93], [Bou96].

In this paper we take a closer look at the “Weyl” unitary operator process U = {U(t) = eiE(t) : t ≥ 0}where E(t) is a linear combination of time and the basic noise processes in both the first power and squareof white noise cases. In the square of white noise case, we put the corresponding QSDE in the module formof [AcBou03b] mentioned above.

We also find explicit formulas for the “matrix elements” of E(t).

2. First order and square of white noise Weyl operators

Let U(sl(2;R)) denote the universal enveloping algebra of sl(2;R) with generators B+, M , B− satisfyingthe commutation relations

[B−, B+] = M , [M,B+] = 2B+ , [M,B−] = −2B−

with involution

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(B−)∗ = B+ , M∗ = M

Fixing an orthonormal basis {en, n = 0, 1, 2, · · · } of l2(N), and defining the mapping

ρ+ : U(sl(2;R))→ L(l2(N)) (=linear, densely defined operators on l2(N))

by

ρ+(B+nMkB−l) em = θn,k,l,m en+m−l(2.1)

where

θn,k,l,m = H(n+m− l)√

m−l+n+1m+1 2k(m− l + 1)n(m+ 1)(l)(m− l + 1)k

H(x) =

1 if x ≥ 0is the Heaviside function

0 if x < 0

00 = 1, (B+)n = (B−)n = Nn = 0, for n < 0

and “factorial powers” are defined by

x(n) = x(x− 1) · · · (x− n+ 1)

(x)n = x(x+ 1) · · · (x+ n− 1)

(x)0 = x(0) = 1

we obtain a representation of sl(2;R), hence of U(sl(2;R)), on l2(N). We define the basic stochasticdifferentials in the following:

Definition 1. For n, k, l,m ∈ {0, 1, ...},

dΛn,k,l(t) := dΛt(ρ+(B+nMkB−

l))

dAm(t) := dAt(em)

dA†m(t) := dA†t(em)

where Λt, At, and A†t are the conservation, annihilation, and creation operator processes of [HuPa84].

Λt, At, and A†t are associated with the four dimensional oscillator algebra obtained from the three dimen-sional Heisenberg-Weyl Lie algebra with basis {A†, A,E}, commutations

[A,A†] = E , [E,A†] = [E,A] = 0

and involution

(A)∗ = A† , E∗ = E

by adding (see [AcFrSk00] for details) a hermitian element Λ satisfying

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[Λ, A†] = A† , [Λ, A] = −A , [E,Λ] = 0

The usual Hudson-Partasarathy quantum stochastic differentials of [HuPa84], corresponding to the firstpower of the white noise functionals, are dA0(t), dA+

0 (t), and dΛ0,0,0(t).

Definition 2. Let λ, k ∈ R and z ∈ C. We call “ first order white noise Weyl operator” any operator of theform

E(t) = λt+ zA0(t) + zA+0 (t) + kΛ0,0,0(t)(2.2)

Remark 1. E(t) is also called a “Poisson-Weyl ” operator. This terminology is justified by the fact thatthe process {E(t) : t ≥ 0} is a classical Poisson process expressed in terms of Weyl operators.

The following proposition was proved in [AcBou01d] and is a standard result in the Hudson-Parthasarathytheory.

Proposition 1. Let U(t) = eiE(t) where E(t) is as in Definition 2. Then {U(t) : t ≥ 0} is a unitary processsuch that

(a) if k 6= 0 then

dU(t) = U(t)[(iλ+ |z|2k2 M)dt+ (iz + z

k M)dA0(t) + (iz + zk M)dA+

0 (t) + (ik +M)dΛ0(t)](2.3)

where

M = eik − 1− ik.(b) if k = 0 then

dU(t) = U(t)[(iλ− |z|2

2 )dt+ izdA0(t) + izdA+0 (t)](2.4)

The above two equations are of the form (1.1) with

W = eikI

L = zk (eik − 1)I

H = (λ− |z|2

k −i2|z|2k2 [eik − e−ik − 1])I

and

H = λI, L = izI, W = −Irespectively. Here, and in what follows, I denotes the identity operator in the appropriate context.

Definition 3. We call “square of white noise Weyl operator” any operator of the form

E(t) = λ t+ z B−t + z B+t + kMt(2.5)

= (λ+ k) t+ z A0(t) + z A†0(t) + z Λ0,0,1(t) + z Λ1,0,0(t) + kΛ0,1,0(t)

where B−t , B+t , Mt are the square of white noise processes of [AccLuVol99] expressed in terms of the basic

processes A0(t), A†0(t), Λ0,0,1(t), Λ1,0,0(t), Λ0,1,0(t) of [AcBou03b] (see also Definition 1).

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The following proposition was also proved in [AcBou01d].

Proposition 2. Let U = {U(t) = ei E(t), t ≥ 0} where E(t) is as in Definition 3. Then U is a unitaryprocess satisfying

dU(t) = U(t)[τ(λ, z, k) dt+∑+∞m=0[am(z, k) dAm(t) + am(z, k) dA†m(t)] +

∑0<i+j+r<+∞ li,j,r(z, k) dΛi,j,r(t)]

where the coefficients τ(λ, z, k), am(z, k), am(z, k), and li,j,r(z, k) are analytic functions of z and k.

It was shown in [AcBou01d] that τ(λ, z, k), am(z, k), am(z, k), and li,j,r(z, k) have the form

τ(λ, z, k) =∑+∞n=1 τn(λ, z, k)in/n!

am(z, k) =∑+∞n=1 am,n(z, k)in/n!

am(z, k) = am(z, k)

li,j,r(z, k) =∑+∞n=1 li,j,r,n(z, k)in/n!

where the coefficient processes am,n(z, k), li,j,r,n(z, k), and τn(λ, z, k) can be computed with the use ofthe recursions

am,n(z, k) = z θ0,0,1,m+1am+1,n−1(z, k) + k θ0,1,0,mam,n−1(z, k) + z θ1,0,0,m−1am−1,n−1(z, k)

with

a0,1(z, k) = z,

τn(λ, z, k) = a0,n−1(z, k) z

with

τ1(λ, z, k) = λ

and

li,j,r,n = z∑

cr,ρ,j−ω,ω,0β,γ,1,0 li+γ−r−1,β,γ,n−1 + k∑

cr,ρ,j−ω−ε,ω,εβ,γ,0,1 li+γ−r,β,γ,n−1

+z∑

cr−1,ρ,j−ω,ω,0β,γ,0,0 li+γ−r+1,β,γ,n−1

with

l1,0,0,1 = z, l0,1,0,1 = k, l0,0,1,1 = z

Here, as in [AcBou01d],

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cλ,ρ,σ,ω,εβ,γ,a,b =

cλ,ρ,σ,ω,εβ,γ,a,b if 0 ≤ λ ≤ γ, 0 ≤ ρ ≤ γ − λ,0 ≤ σ ≤ γ − λ− ρ, 0 ≤ ω ≤ β, 0 ≤ ε ≤ b

0 otherwise

cλ,ρ,σ,ω,εβ,γ,a,b =

(γλ

)(γ−λρ

)(βω

)(bε

)2β+b−ω−εSγ−λ−ρ,σa

(γ−λ)(a+ λ− 1)(ρ)(a− γ + λ)β−ωλb−ε

Sγ−λ−ρ,σ are the ”Stirling numbers of the first kind”, and

∑=∑γλ=0

∑γ−λρ=0

∑γ−λ−ρσ=0

∑βω=0

∑bε=0

3. Module form of Weyl evolutions

Quantum stochastic differential equations driven by the square of white noise were elegantly described in[AccBou03b] as Hudson-Partasarathy type equations on the module B(HS) ⊗ Γ(K), where HS is a systemHilbert space, K = l2(N) and Γ(K) denotes the Fock space over K.

Let

E(t) = (λ+ k) t+ z A0(t) + z A†0(t) + z Λ0,0,1(t) + z Λ1,0,0(t) + kΛ0,1,0(t)

as in Definition 3.Letting

T = z ⊗ e0

and

N = z ρ+(B+0M0B−

1) + z ρ+(B+1

M0B−0) + k ρ+(B+0

M1B−0)

we can write

E(t) = (λ+ k) t+At(T ) +A†t(T ) + Lt(N)(3.1)

where the module differentials dAt(·), dA†t(·), and dLt(·) were defined in [AcBou03b] and can be multipliedwith the use of the Ito table of [AcBou03b], namely

dAt(D−) dA†t(D+) = (D−|D+) dt

dLt(D1) dLt(E1) = dLt(D1 ◦ E1)

dLt(D1) dA†t(D+) = dA†t(l(D1)D+)

dAt(D−) dLt(E1) = dAt(r(E1)D−)

where

D+ =∑nD+,n ⊗ en

D− =∑mD−,m ⊗ em

D1 =∑α,β,γ D1,α,β,γ ⊗ ρ+(B+αMβB−

γ)

E1 =∑a,b,cE1,a,b,c ⊗ ρ+(B+aM bB−

c)

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with n,m,α, β, γ, a, b, c ∈ {0, 1, 2, ...}, D+,n, D−,m, D1,α,β,γ , E1,a,b,c ∈ B(HS), r(·) and l(·) the right andleft module actions respectively , defined by

l(D1)D+ =∑n,α,β,γ D1,α,β,γθα,β,γ,n−α+γD+,n−α+γ ⊗ en

r(E1)D− =∑n,α,β,γ E

∗1,α,β,γθγ,β,α,n+α−γD−,n+α−γ ⊗ en

(·|·) the module inner product, defined linearly on elementary tensors by

(a⊗ ξ|b⊗ η) = a∗b〈ξ, η〉

and D1 ◦ E1 defined in [AcBou03b] by

D1 ◦ E1 =∑α,β,γ

∑a,b,c

∑cλ,ρ,σ,ω,εβ,γ,a,b D1,α,β,γE1,a,b,c ⊗ ρ+(B+a+α−γ+λ

Mω+σ+εB−λ+c

)

All other products of stochastic differentials (including dt) are equal to zero.A simple form of the equation satisfied by the operator process U = {U(t) = ei E(t), t ≥ 0} of Proposition

2 can be derived as follows.

Proposition 3. Let U = {U(t) = ei E(t), t ≥ 0} where E(t) is as in (3.1). Then

dU(t) = U(t)[(i(λ+ k) + (T |f(l(N))T )) dt+ dAt((h(r(N))− i)T )(3.2)

+dA†t((g(l(N)) + i)T ) + dLt(e◦iN − 1)]

where the analytic functions f, g, h are defined by

eix = 1 + ix+ x2 f(x)

eix = 1 + ix+ x g(x)

e−ix = 1− ix+ xh(x)

and

e◦iN =∑+∞n=0

in

n!N◦n

where

N◦n = N ◦N ◦ · · · ◦N (n-times)

.

Remark 2. The QSDE satisfied by U = {U(t) = ei E(t), t ≥ 0} is of the form (1.2), with

W = e◦iN

H = λ+ k

and

D− = (h(r(N))− i)T

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Proof. Computing the differential of U(t) we find

dU(t) = d(ei E(t)) = ei E(t+dt) − ei E(t) = ei (E(dt)+E(t)) − ei E(t)

= ei E(dt) ei E(t) − ei E(t) (by the commutativity of E(dt) and E(t))

= ei E(t) [ei dE(t) − I] = U(t)∑∞n=1

(i dE(t))n

n!

By the module form of the square of white noise Ito table

dE(t)2 = ((λ+ k) dt+ dAt(T ) + dA†t(T ) + dLt(N))

·((λ+ k) dt+ dAt(T ) + dA†t(T ) + dLt(N))

= (T |T ) dt+ dAt(r(N)T ) + dA†t(l(N)T ) + dLt(N ◦N)

dE(t)3 = ((λ+ k) dt+ dAt(T ) + dA†t(T ) + dLt(N))

·((T |T ) dt+ dAt(r(N)T ) + dA†t(l(N)T ) + dLt(N ◦N))

= (T |l(N)T ) dt+ dAt(r(N)2T ) + dA†t(l(N)2T ) + dLt(N ◦N ◦N)

and, in general, for n ≥ 2

dE(t)n = (T |l(N)n−2T ) dt+ dAt(r(N)n−1T ) + dA†t(l(N)n−1T ) + dLt(N◦n)

Thus

dU(t) = U(t)∑∞n=1

(i dE(t))n

n! = U(t)[i dE(t) +∑∞n=2

(i dE(t))n

n! ]

= U(t)[i dE(t) + (T |∑∞n=2

in

n! l(N)n−2T ) dt+ dAt(∑∞n=2

(−1)nin

n! r(N)n−1T )

+dA†t(∑∞n=2

in

n! l(N)n−1T ) + dLt(∑∞n=1

in

n!N◦n)]

= U(t)[(i(λ+ k) + (T |f(l(N))T )) dt+ dAt((h(r(N))− i)T ) + dA†t((g(l(N)) + i)T ) + dLt(e◦iN − 1)]

�

4. Matrix elements

Proposition 4. Let E(t) be as in Definition 2 and let ψ(f), ψ(g) be two “exponential vectors” in the BosonFock space Γ(L2([0,+∞),C)), in the sense of [HuPa84]. Then, for each t ≥ 0 and f and g in L2([0,+∞),C))

< ψ(f), E(t)ψ(g) >= [(λ+ k)t+ z∫ t

0g(s)ds+ z

∫ t0f(s)ds+ k

∫ t0f(s)g(s)ds]e

∫ +∞0

f(s)g(s)ds(4.1)

Proof. The proof follows directly from the formulas for the matrix elements of the basic Hudson-Parthasarathynoise processes, provided in [HuPa84]. �

Proposition 5. Let E(t) be as in Definition 3 and let ψ(f), ψ(g) be two “exponential vectors” in theBoson Fock space Γ(L2([0,+∞), l2(N))). Then, for each t ≥ 0 and functions f , g in L2([0,+∞), l2(N)) withf(s) = {fn(s)}+∞n=0 , g(s) = {gn(s)}+∞n=0 and

< f, g >=∑+∞n=0[

∫ +∞0

fn(s)gn(s)ds]

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< ψ(f), E(t)ψ(g) >= [(λ+ k)t+ z∫ t

0g0(s)ds+ z

∫ t0f0(s)ds(4.2)

+z∑+∞n=0

√(n+ 2)(n+ 1)

∫ t0fn(s)gn+1(s)ds+ z

∑+∞n=0

√n(n+ 1)

∫ t0fn(s)gn−1(s)ds

+k∑+∞n=0(2n+ 2)

∫ t0fn(s)gn(s)ds]e<f,g>

Proof. By (2.5), Definition 1, and (2.1), with en = (0, 0, ..., 0, 1, 0, ...) where 1 is in the n+ 1-st position,

< ψ(f), At(e0)ψ(g) >=< ψ(f), < χ[0,t]e0g > ψ(g) >=∫ t

0g0(s)ds < ψ(f), ψ(g) >

and by the duality of A and A†

< ψ(f), A†t(e0)ψ(g) >=∫ t

0f0(s)ds < ψ(f), ψ(g) >

Moreover

< ψ(f),Λ0,0,1(t)ψ(g) >=< ψ(f),Λ(χ[0,t]ρ+(B−))ψ(g) >

=< f, χ[0,t]ρ+(B−)g >< ψ(f), ψ(g) >=

∫ t0< f(s), ρ+(B−)g(s) > ds < ψ(f), ψ(g) >

=∑+∞n=0[

∫ +∞0

fn(s)(ρ+(B−)g)n(s)ds] < ψ(f), ψ(g) >

and since

g(s) =∑+∞n=0 gn(s)en ⇒ ρ+(B−)g(s) =

∑+∞n=0 gn(s)

√n(n+ 1)en−1

⇒ (ρ+(B−)g)n(s) = gn+1(s)√

(n+ 1)(n+ 2)

we obtain

< ψ(f),Λ0,0,1(t)ψ(g) >=∑+∞n=0

√(n+ 1)(n+ 2)[

∫ t0fn(s)gn+1(s)ds] < ψ(f), ψ(g) >

Similarly

< ψ(f),Λ1,0,0(t)ψ(g) >=∑+∞n=0

√n(n+ 1)[

∫ t0fn(s)gn−1(s)ds] < ψ(f), ψ(g) >

and

< ψ(f),Λ0,1,0(t)ψ(g) >=∑+∞n=0(2n+ 2)[

∫ t0fn(s)gn(s)ds] < ψ(f), ψ(g) >

�

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