Quantum Markov semigroups and quantum stochastic flows ...belton/www/notes/25xi13.pdf · Quantum Markov semigroups and quantum stochastic ﬂows – construction and perturbation

Quantum Markov semigroups

and quantum stochastic flows

– construction and perturbation

Alexander Belton

(Joint work with Martin Lindsay and Adam Skalski, and Stephen Wills)

Department of Mathematics and StatisticsLancaster UniversityUnited Kingdom

[email protected]

Noncommutative Geometry Seminar

Mathematical Institute of the Polish Academy of Sciences

25th November 2013

Structure of the talk

Construction

The classical theory

Non-commutative probability

Solving a quantum stochastic differential equation [BW]

Examples [BW]

Perturbation

The classical theory

Quantum Feynman–Kac on von Neumann algebras [BLS]

Quantum Feynman–Kac on C∗ algebras [BW]

Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 2 / 67

Construction

Classical Markov semigroups

Markov processes

Markov semigroups Infinitesimal generators


Markov processes

Definition

A Markov process with state space S is a collection of S-valued randomvariables (Xt)t>0 on a common probability space such that

E[f (Xs+t)|σ(Xr : 0 6 r 6 s)

]= E

[f (Xs+t)|Xs

](s, t > 0)

for all f ∈ L∞(S).

Such a Markov process is time homogeneous if

E[f (Xs+t)|Xs = x

]= E[f (Xt)|X0 = x

](s, t > 0, x ∈ S)

for all f ∈ L∞(S).


Markov semigroups

Given a time-homogeneous Markov process, setting

(Tt f )(x) = E[f (Xt)|X0 = x

](t > 0, x ∈ S)

defines a Markov semigroup on L∞(S).

Definition

A Markov semigroup on L∞(S) is a family (Tt)t>0 such that

1 Tt : L∞(S) → L∞(S) is a linear operator for all t > 0,

2 Ts Tt = Ts+t for all s, t > 0 and T0 = I (semigroup),

3 ‖Tt‖ 6 1 for all t > 0 (contraction),

4 Tt f > 0 whenever f > 0, for all t > 0 (positive).

If Tt1 = 1 for all t > 0 then T is conservative.


Feller semigroups

Definition

Suppose the state space S is a locally compact Hausdorff space. TheMarkov semigroup T is Feller if

Tt

(C0(S)

)⊆ C0(S) (t > 0)

and‖Tt f − f ‖∞ → 0 as t → 0

(f ∈ C0(S)

).

Every sufficiently well-behaved time-homogeneous Markov process isFeller: Brownian motion, Poisson process, Levy processes, . . .

Theorem 1

If the state space S is separable then every Feller semigroup gives rise to a

time-homogeneous Markov process.


Infinitesimal generators

Definition

Let T be a C0 semigroup on a Banach space E . Its infinitesimal generator

is the linear operator τ in E with domain

dom τ =f ∈ E : lim

t→0t−1(Tt f − f ) exists

and actionτ f = lim

t→0t−1(Tt f − f ).

The operator τ is closed and densely defined.

If T comes from a Markov process X then

E[f (Xt+h)− f (Xt)|Xt

]= (Thf − f )(Xt) = h(τ f )(Xt) + o(h),

so τ describes the change in X over an infinitesimal time interval.


The Lumer–Phillips theorem

Theorem 2 (Lumer–Phillips)

A closed, densely defined operator τ in the Banach space E generates a

strongly continuous contraction semigroup on E if and only if

im(λI − τ) = E for some λ > 0

and (dissipativity)

‖(λI − τ)x‖ > λ‖x‖ for all λ > 0 and x ∈ dom τ.


The Hille–Yosida–Ray theorem

Theorem 3 (Hille–Yosida–Ray)

A closed, densely defined operator τ in C0(S) is the generator of a Feller

semigroup on C0(S) if and only if

λI − τ : dom τ → X has bounded inverse for all λ > 0 and

τ satisfies the positive maximum principle.

Definition

Let S be a locally compact Hausdorff space. A linear operator τ in C0(S)satisfies the positive maximum principle if whenever f ∈ dom τ and s0 ∈ S

are such thatsups∈S

f (s) = f (s0) > 0

then (τ f )(s0) 6 0.


Quantum Markov semigroups

Quantum Markov processes

Quantum Feller semigroups Infinitesimal generators


Quantum Feller semigroups

Theorem 4

Every commutative C ∗ algebra is isometrically isomorphic to C0(S), whereS is a locally compact Hausdorff space.

Definition

A quantum Feller semigroup on the C ∗ algebra A is a family (Tt)t>0 suchthat

1 Tt : A → A is a linear operator for all t > 0,

2 Ts Tt = Ts+t for all s, t > 0 and T0 = I ,

3 ‖Ttx − x‖ → 0 as t → 0 for all x ∈ A,

4 ‖Tt‖ 6 1 for all t > 0,

5 (Ttaij) ∈ Mn(A)+ whenever (aij) ∈ Mn(A)+, for all n > 1 and t > 0.

If A is unital and Tt1 = 1 for all t > 0 then T is conservative.


From generators to semigroups

Problem

Given a densely defined operator τ on the C ∗ algebra A, show that τ isclosable and its closure that generates a quantum Feller semigroup T .

It can be very difficult to verify the conditions of the Lumer–Phillipstheorem.

Is there a non-commutative version of the positive maximumprinciple?

Strategy

Rather than construct the semigroup T directly, we shall instead constructa quantum Markov process which is a dilation of T .



Random variables

If X : Ω → S is a classical S-valued random variable then

A → M; f 7→ f X

is a unital ∗-homomorphism, where A = C0(S) and M = L∞(Ω,F,P).A non-commutative random variable is a unital ∗-homomorphism

j : A → M

from a unital C ∗ algebra A ⊆ B(h) to a von Neumann algebra M.Classical stochastic calculus has a universal sample space: cadlagfunctions.In quantum stochastic calculus, B(F) plays this role, where F is BosonFock space over L2(R+; k), and

A⊗m B(F) ⊆ M := B(h⊗F).



Adaptedness

L2(R+) ∼= L2[0, t)⊕ L2[t,∞)

=⇒ F ∼= Ft) ⊗F[t ; ε(f ) ∼= ε(f |[0,t))⊗ ε(f |[t,∞)),

where E := linε(f ) : f ∈ L2(R+; k), the linear span of the exponential

vectors (such that 〈ε(f ), ε(g)〉 = exp〈f , g〉) is dense in F .

A quantum flow is a family of unital ∗-homomorphisms

(jt : A → A⊗m B(F)

)t>0

which is adapted:

jt(a) = jt)(a)⊗ I[t ∈ A⊗m B(Ft))⊗ CI[t (t > 0, a ∈ A).



Dilation

A quantum flow (jt : A → M)t>0 is a dilation of the quantum Fellersemigroup T on A if there exists a conditional expectation E : M → Asuch that Tt = E jt for all t > 0.

Markovianity

In this framework, Markovianity is given by a cocycle property:

js+t = s σs jt (s, t > 0),

where

σs : A⊗m B(F)∼=−→ A⊗m B(F[s)

is the shift (CCR flow) and

s = js) ⊗m id[s : A⊗m B(F[s) → A⊗m B(F).



Cocycles produce semigroups on ALet EΩ : M → A be such that

〈u,EΩ(X )v〉 = 〈u ⊗ ε(0),Xv ⊗ ε(0)〉(u, v ∈ h, X ∈ M).

Then EΩ j is a semigroup on A if the quantum flow j is a Markoviancocycle.

If, further, t 7→ EΩ jt is norm continuous then the quantum flow j is aFeller cocycle and EΩ j is a quantum Feller semigroup.



Let A0 ⊆ A ⊆ B(h) be a norm-dense ∗-subalgebra of A whichcontains 1 = Ih.

Definition

A family of linear operators (Xt)t>0 in h⊗F with domains including h⊙Eis an adapted operator process if

〈uε(f ),Xtvε(g)〉 = 〈uε(1[0,t)f ),Xtvε(1[0,t)g)〉〈ε(1[t,∞)f ), ε(1[t,∞)g)〉

for all u, v ∈ h, f , g ∈ L2(R+; k) and t > 0.

An adapted mapping process on A0 is a family of linear maps

(jt : A0 → L(h⊙E ; h⊗F)

)t>0

such that(jt(x)

)t>0

is an adapted operator process for all x ∈ A0.


A quantum stochastic differential equation

The generator

Let φ : A0 → A0⊙B be a linear map, where B = B(k) and k := C⊕ k.Distinguish the unit vector ω := (1, 0) ∈ k and let x := (1, x) for all x ∈ k.

For all z , w ∈ k, let φzw : A0 → A0 be the linear map such that

〈u, φzw (x)v〉 = 〈u ⊗ z , φ(x)v ⊗ w〉 (u, v ∈ h, x ∈ A0).

Definition 5

An adapted mapping process j on A0 satisfies the QSDE

j0(x) = x ⊗ IF , djt(x) = (t φ)(x) dΛt (x ∈ A0) (1)

if and only if

〈uε(f ), (jt(x) − x ⊗ IF )vε(g)〉 =∫ t

0〈uε(f ), js

(φf (s)

g(s)(x)

)vε(g)〉 ds

for all x ∈ A0 and all t > 0, u, v ∈ h and f , g ∈ L2(R+; k).


Feller cocycles from QSDEs

Theorem 6

If the quantum flow j satisfies the QSDE (1) then j is a Feller cocycle and

the generator of the Feller semigroup EΩ j is an extension of φωω.

Theorem 7

Let the quantum flow j satisfy (1). Then φ : A0 → A0⊙B is such that

φ is ∗-linear,φ(1) = 0 and

φ(xy) = φ(x)(y ⊗ Ik) + (x ⊗ Ik)φ(y) + φ(x)∆φ(y) for all x, y ∈ A0,

where ∆ := Ih ⊗ Pk =[ 0 00 Ih⊗k

]∈ A0⊙B(k) and Pk := |ω〉〈ω|⊥ ∈ B(k) is

the orthogonal projection onto k ⊂ k.

Question

When are these necessary conditions sufficient for the solution of (1) to bea quantum flow?


Flow generators

Lemma 8

The map φ : A0 → A0⊙B is a flow generator if and only if

φ(x) =

[τ(x) δ†(x)

δ(x) π(x)− x ⊗ Ik

]for all x ∈ A0, (2)

where

π : A0 → A0⊙B(k) is a unital ∗-homomorphism,

δ : A0 → A0⊙B(C; k) is a π-derivation, i.e., a linear map such that

δ(xy) = δ(x)y + π(x)δ(y) (x , y ∈ A0)

δ† : A0 → A0⊙B(k;C) is such that δ†(x) = δ(x∗)∗ for all x ∈ A0,

τ : A0 → A0 is ∗-linear and such that

τ(xy)− τ(x)y − xτ(y) = δ†(x)δ(y) (x , y ∈ A0).


Quantum Wiener integrals

Theorem 9

For all n ∈ N and T ∈ B(h⊗ k⊗n) there exists a family(Λnt (T )

)t>0

of

linear operators in h⊗F , with domains including h⊙E , that is adaptedand such that

〈uε(f ),Λnt (T )vε(g)〉 =

∫

Dn(t)〈u ⊗ f ⊗n(t),Tv ⊗ g⊗n(t)〉 dt 〈ε(f ), ε(g)〉

for all u, v ∈ h, f , g ∈ L2(R+; k) and t > 0, where the simplex

Dn(t) := t := (t1, . . . , tn) ∈ [0, t]n : t1 < · · · < tn

and

f ⊗n(t) := f (t1)⊗ · · · ⊗ f (tn) et cetera.

We include n = 0 by setting Λ0t (T ) := T ⊗ IF for all t > 0.


An estimate for quantum Wiener integrals

Proposition 10

If n ∈ Z+, t > 0, T ∈ B(h⊗ k⊗n) and f ∈ L2(R+; k) then

‖Λnt (T )uε(f )‖ 6

Knf ,t√n!

‖T‖ ‖uε(f )‖,

where Kf ,t :=√

(2 + 4‖f ‖2)(t + ‖f ‖2).


A quantum random walk

Definition

Given a flow generator φ, the family of linear maps

φn : A0 → A0 ⊙B⊙ n

is defined by setting

φ0 := idA0 and φn+1 :=(φn ⊙ idB

) φ for all n ∈ Z+,

where idA0 is the identity map on A0 and similarly for idB.

Let

Aφ := x ∈ A0 : ∃Cx ,Mx > 0 with ‖φn(x)‖ 6 CxMnx ∀ n ∈ Z+

be those elements of A0 for which(φn(x)

)n>0

has polynomial growth.


Solutions to the QSDE

Theorem 11

If x ∈ Aφ then the series

jt(x) :=∞∑

n=0

Λnt

(φn(x)

)

is strongly absolutely convergent on h⊙E for all t > 0. The family of

operators(jt(x)

)t>0

is an adapted mapping process on Aφ which satisfies

the QSDE (1) on Aφ.

Questions1 When does Aφ = A0?

2 When does the adapted mapping process j given by Theorem 11extend to a quantum flow?


Answers

If Aφ = A0 then the adapted mapping process j given by Theorem 11extends to a quantum flow as long as A is sufficiently well behaved.

In practice it will be difficult to find directly constants Cx and Mx

such that ‖φn(x)‖ 6 CxMnx for all x ∈ A0.

Key to both: the higher-order Ito formula.

Two easy cases when Aφ = A0

If k is finite dimensional and φ is bounded then so is each φn, with

‖φn‖ 6

(dim k

)n−1‖φ‖n (n ∈ N).

If φ is completely bounded then so is each φn, with

‖φn‖ 6 ‖φ‖ncb (n ∈ Z+).


The higher-order Ito formula

Notation

Let α ⊆ 1, . . . , n, with elements arranged in increasing order andcardinality |α|. The unital ∗-homomorphism

A0⊙B⊙|α| → A0⊙B⊙ n; T 7→ T (n, α)

is defined by linear extension of the map

A⊗ B1 ⊗ · · · ⊗ B|α| 7→ A⊗ C1 ⊗ · · · ⊗ Cn,

where

Ci :=

Bj if i is the jth element of α,Ik

if i is not an element of α.

Example

(A⊗ B1 ⊗ B2 ⊗ B3)(5, 1, 3, 4

)= A⊗ B1 ⊗ I

k⊗ B2 ⊗ B3 ⊗ I

k.


More notation

For all n ∈ Z+ and α ⊆ 1, . . . , n, let

φ|α|(x ; n, α) :=(φ|α|(x)

)(n, α) for all x ∈ A0.

Also let∆(n, α) := (Ih ⊗ P

⊗|α|k )(n, α),

so that Pk acts on the components of k⊗n which have indices in α and Ik

acts on the others.

Theorem 12

Let φ be a flow generator. For all n ∈ Z+ and x, y ∈ A0,

φn(xy) =∑

α∪β=1,...,n

φ|α|(x ; n, α)∆(n, α ∩ β)φ|β|(y ; n, β),

where the summation is taken over all α and β with union 1, . . . n.


Two corollaries

Corollary 13

If φ : A0 → A0⊙B is a flow generator then Aφ is a unital ∗-subalgebraof A0, which is equal to A0 if Aφ contains a ∗-generating set for A0.

Corollary 14

Let φ : A0 → A0⊙B be a flow generator and let j be the adapted

mapping process on Aφ given by Theorem 11. If x, y ∈ Aφ then

x∗y ∈ Aφ, with

〈jt(x)uε(f ), jt(y)vε(g)〉 = 〈uε(f ), jt(x∗y)vε(g)〉

for all u, v ∈ h and f , g ∈ L2(R+; k).


The first dilation theorem

Theorem 15

Let φ : A0 → A0⊙B be a flow generator and suppose A0 contains its

square roots: for all non-negative x ∈ A0, the square root x1/2 lies in A0.

If Aφ = A0 then there exists a quantum flow such that

t(x) = jt(x) on h⊙E (x ∈ A0),

where j is the adapted mapping process given by Theorem 11.

Remark

If A is an AF algebra, i.e., the norm closure of an increasing sequence offinite-dimensional ∗-subalgebras, then its local algebra A0, the union ofthese subalgebras, contains its square roots.


The second dilation theorem

Theorem 16

Let A be the universal C ∗ algebra generated by isometries si : i ∈ I, andlet A0 be the ∗-algebra generated by si : i ∈ I.If φ : A0 → A0⊙B is a flow generator such that Aφ = A0 then there

exists a quantum flow such that

t(x) = jt(x) on h⊙E (x ∈ A0),

where j is the adapted mapping process given by Theorem 11.


Random walks on discrete groups

The group algebra

Let G is a discrete group and set A = C0(G )⊕C1 ⊆ B(ℓ2(G )

), where

x ∈ C0(G ) acts on ℓ2(G ) by multiplication.

Let A0 = lin1, eg : g ∈ G, where eg (h) := 1g=h for all h ∈ G .

Permitted moves

Let H be a non-empty finite subset of G \ e and let the Hilbert space khave orthonormal basis fh : h ∈ H; the maps

λh : G → G ; g 7→ hg (h ∈ H)

correspond to the permitted moves in the random walk to be constructedon G .



Lemma 17

Given a transition function

t : H × G → C; (h, g) 7→ th(g),

the map φ : A0 → A0⊙B such that

x 7→[ ∑

h∈H |th|2(x λh − x)∑

h∈H th(x λh − x)⊗ 〈fh|∑

h∈H th(x λh − x)⊗ |fh〉∑

h∈H(x λh − x)⊗ |fh〉〈fh|

]

is a flow generator with φn(eg ) equal to

∑

h1∈H∪e

· · ·∑

hn∈H∪e

eh−1n ···h−1

1 g⊗mhn(h

−1n · · · h−1

1 g)⊗ · · · ⊗mh1(h−11 g)

for all n ∈ N and g ∈ G.



One-step matrices

For all g ∈ G and h ∈ H, let

me(g) :=

[−∑

h∈H |th(g)|2 −∑h∈H th(g)〈fh|

−∑

h∈H th(g)|fh〉 −Ik

]

and

mh(g) :=

[|th(g)|2 th(g)〈fh|th(g)|fh〉 |fh〉〈fh|

].

Then‖me(g)‖ = 1 +

∑

h∈H

|th(g)|2

and‖mh(g)‖ = 1 + |th(g)|2.



A sufficient condition for Aφ = A0

If

Mg := limn→∞

sup|th(h−1

n · · · h−11 g)| : h1, . . . , hn ∈ H ∪ e, h ∈ H

<∞

(3)then

‖φn(eg )‖ 6(1 + |H|+ 2|H|M2

g

)n(n ∈ Z+),

where |H| denotes the cardinality of H.

Hence Aφ = A0 if (3) holds for all g ∈ G .



Examples

1 If t is bounded then (3) holds for all g ∈ G .

2 If G = (Z,+), H = ±1 and the transition function t is bounded,with t+1(g) = 0 for all g < 0 and t−1(g) = 0 for all g 6 0, then theFeller semigroup T which arises corresponds to the classicalbirth-death process with birth and dates rates |t+1|2 and |t−1|2,respectively.

3 If G = (Z,+), H = +1 and t+1 : g 7→ 2g then Mg = 2g and thecondition (3) holds for all g ∈ G . Thus the construction applies toexamples where the transition function t is unbounded.


The symmetric quantum exclusion process

The CAR algebra

For a non-empty set I, the CAR algebra is the unital C ∗ algebra A withgenerators bi : i ∈ I, subject to the anti-commutation relations

bi , bj = 0 and bi , b∗j = 1i=j (i , j ∈ I).

Let A0 be the unital algebra generated by bi , b∗i : i ∈ I.

Lemma 18

For each x ∈ A0 there exists a finite subset I0 ⊆ I such that x lies in the

finite-dimensional ∗-subalgebra

AI0 := linb∗j1 · · · b

∗jqbi1 · · · bip : distinct i1, . . . , ip ∈ I0, j1, . . . , jq ∈ I0.

Consequently, A is an AF algebra and A0 contains its square roots.



Amplitudes

Let αi ,j : i , j ∈ I ⊆ C be a fixed collection of amplitudes, so that(I, αi ,j) is a complex digraph. For all i ∈ I, let

supp(i) := j ∈ I : αi ,j 6= 0 and supp+(i) := supp(i) ∪ i.

Thus supp(i) is the set of sites with which site i interacts and | supp(i)| isthe valency of the vertex i . Suppose

| supp(i)| <∞ (i ∈ I).

The transport of a particle from site i to site j with amplitude αi ,j isdescribed by the operator

ti ,j := αi ,j b∗j bi .



Energies

Let ηi : i ∈ I ⊆ R be fixed. The total energy in the system is given by

h :=∑

i∈I

ηi b∗i bi ,

where ηi gives the energy of a particle at site i .

Lemma 19

Setting

τ(x) := i∑

i∈I

ηi [b∗i bi , x ]− 1

2

∑

i ,j∈I

τi ,j(x)

defines a ∗-linear map τ : A0 → A0, where

τi ,j(x) := t∗i ,j [ti ,j , x ] + [x , t∗i ,j ]ti ,j .


Lemma 20

Let k be a Hilbert space with orthonormal basis fi ,j : i , j ∈ I. Setting

δ(x) :=∑

i ,j∈I

[ti ,j , x ]⊗ |fi ,j〉 (x ∈ A0),

where

|fi ,j〉 : C 7→ k; λ 7→ λfi ,j ,

defines a linear map δ : A0 → A0⊙B(C; k) such that

δ(xy) = δ(x)y + (x ⊗ Ik)δ(y)

and δ†(x)δ(y) = τ(xy)− τ(x)y − xτ(y) (x , y ∈ A0),

with τ defined as in Lemma 19. Hence

φ : A0 → A0 ⊙B; x 7→[τ(x) δ†(x)δ(x) 0

]

is a flow generator.



Lemma 21

If the amplitudes satisfy the symmetry condition

|αi ,j | = |αj ,i | for all i , j ∈ I (4)

then

φn(bi0) =∑

i1∈supp+(i0)

· · ·∑

in∈supp+(in−1)

bin ⊗ Bin−1,in ⊗ · · · ⊗ Bi0,i1

for all n ∈ N and i0 ∈ I, where

Bi ,j := 1j=iλi |ω〉〈ω| + |ω〉〈αi ,j fi ,j | − |αj ,i fj ,i〉〈ω| (i , j ∈ I)

and

λi := −iηi − 12

∑

j∈supp(i)

|αj ,i |2 (i ∈ I).



Example

Suppose that the amplitudes satisfy the symmetry condition (4), andfurther that there are uniform bounds on the amplitudes, valencies andenergies, so that

M := supi ,j∈I

|αi ,j |, V := supi∈I

| supp(i)| and H := supi∈I

|ηi |

are all finite. Then

|λi | 6 |ηi |+ 12VM

2 and ‖Bi ,j‖ 6 |λi |+ 2M 6 H + 12VM

2 + 2M

for all i , j ∈ I. Hence

‖φn(bi )‖ 6 (V + 1)n(H + 1

2VM2 + 2M

)n(n ∈ Z+, i ∈ I)

and Aφ = A0.


The universal rotation algebra

Definition

Let A be the universal rotation algebra: this is the universal C ∗ algebrawith unitary generators U, V and Z satisfying the relations

UV = ZVU, UZ = ZU and VZ = ZV .

It is the group C ∗ algebra corresponding to the discrete Heisenberg groupΓ := 〈u, v , z | uv = zvu, uz = zu, vz = zv〉.

A pair of derivations

Letting A0 denote the ∗-subalgebra generated by U, V and Z , there areskew-adjoint derivations

δ1 : A0 → A0; UmV nZ p 7→ mUmV nZ p

and δ2 : A0 → A0; UmV nZ p 7→ nUmV nZ p.


The universal rotation algebra

Theorem 22

Fix c1, c2 ∈ C, let δ = c1δ1 + c2δ2 and define the Bellissard map

τ : A0 → A0;

UmV nZ p 7→

−(12 |c1|

2m2 + 12 |c2|

2n2 + c1c2mn + (c1c2 − c1c2)p)UmV nZ p.

Then

φ : A0 → A0 ⊙B(C2); x 7→[τ(x) δ†(x)

δ(x) 0

]

is a flow generator. Furthermore, U, V , Z ∈ Aφ and Aφ = A0.

Remark

If c1c2 = c1c2 then this construction specialises to the non-commutativetorus.


The non-commutative torus

Definition

Let A be the non-commutative torus with parameter λ ∈ T, so that A isthe universal C ∗ algebra with unitary generators U and V subject to therelation

UV = λVU,

and letA0 := 〈U,V 〉 = linUmV n : m, n ∈ Z.

An automorphism

For each (µ, ν) ∈ T2, let πµ,ν be the automorphism of A such that

πµ,ν(UmV n) = µmνnUmV n for all m, n ∈ Z.


The non-commutative torus

Theorem 23

Fix (µ, ν) ∈ T2 with µ 6= 1. There exists a flow generator

φ : A0 → A0⊙B(C2); x 7→[τ(x) −µδ(x)δ(x) πµ,ν(x)− x

],

where the πµ,ν-derivation

δ : A0 → A0; UmV n 7→ 1− µmνn

1− µUmV n

is such that δ† = −µδ and the map

τ :=µ

1− µδ : A0 → A0; UmV n 7→ µ(1− µmνn)

(1− µ)2UmV n.

Furthermore, U, V ∈ Aφ and so Aφ = A0.


Perturbation

Classical Feynman–Kac perturbation

Ito’s formula for Brownian motion

If f : R → R is twice continuously differentiable then

f (Bt) = f (B0) +

∫ t

0f ′(Bs) dBs +

1

2

∫ t

0f ′′(Bs) ds (t > 0).

Corollary

Define a Feller semigroup (Tt)t>0 on C0(R) by setting

(Tt f )(x) := E[f (Bt)|B0 = x

](t > 0, f ∈ C0(R), x ∈ R).

Then1

t

[f (Bt)− f (B0)|B0 = x

]→ 1

2f ′′(x) as t → 0,

so the generator of this semigroup extends f 7→ 12 f

′′.



Proposition

If v : R → R is well behaved and Yt := exp(∫ t

0 v(Bs) ds)then

Yt f (Bt) = f (B0) +

∫ t

0Ys f

′(Bs) dBs

+

∫ t

0

(v(Bs)Ys f (Bs) +

1

2Ys f

′′(Bs))ds.

Proof

This follows from the classical Ito product formula:

d(Yt f (Bt)

)= (dYt)f (Bt) + Yt d

(f (Bt)

)+ dYt d

(f (Bt)

).



Theorem (F–K)

Let

(St f )(x) := E

[exp

(∫ t

0v(Bs) ds

)f (Bt)|B0 = x

].

Then (St)t>0 is a Feller semigroup on C0(R) with generator which extendsf 7→ 1

2 f′′ + vf .


Introducing non-commutativity

Idea

There are two ways in which non-commutativity can appear:

1 quantum stochastic noises replace Brownian motion;

2 C0(R) is replaced by a general C ∗ algebra.


Feynman–Kac perturbation (von Neumann)

The free flow

Until further notice, A is a von Neumann algebra, so ⊗m = ⊗, and j is anultraweakly continuous, normal quantum flow and a Feller cocycle.

Multipliers

A multiplier for j is an adapted operator process Y ⊆ A⊗ B(F) such that

Y0 = Ih⊗F and Ys+t = Js(Yt)Ys (s, t > 0).

Theorem 24

If Y and Z are multipliers for j and

kt : A → A⊗ B(F); a 7→ Y ∗t jt(a)Zt (t > 0)

then k is a Feller cocycle and EΩ k is a semigroup on A.


Some questions

Generators1 How is the generator of EΩ k related to that of EΩ j?

2 How is the generator of k related to that of j?

Existence1 How do we produce multipliers?


The multiplier QSDE

Theorem 25

Let F ∈ A⊗ B(k) be such that

q(F ) := F + F ∗ + F∆F ∗6 0,

where ∆ :=[0 0

0 Ih⊗k

].

There exists a unique operator process Y ⊆ A⊗m B(F) such that

Yt = Ih⊗F +

∫ t

0s(F )Ys dΛs (t > 0),

where s := js ⊗m idB(k)

and Ys := Ys ⊗ Ik.

Each Yt is a contraction, and is an isometry if and only if q(F ) = 0.

The adapted operator process Y is a multiplier for j with generator F .


Classical multipliers

Feynman–Kac

Yt = exp(∫ t

0v(Bs) ds) ⇒ Yt = 1 +

∫ t

0v(Bs)Ys ds = 1 +

∫ t

0js(v)Ys ds.

Cameron–Martin

Yt = exp(Bt −

1

2t)

⇒ Yt = 1 +

∫ t

0Ys dBs = 1 +

∫ t

0s

([0 1

1 0

])Ys dΛs .

Then EΩ k has generator f 7→ 12 f

′′ + f ′.

C–M: Setting (Tt f )(x) := E[f (Bt + t)|B0 = x

]gives the same generator.

See Pinsky (1972), Stroock (1970).


A QSDE for the free flow

Assumption

Suppose the free flow j satisfies the QSDE (1) on A0 ⊆ A, i.e.,

djt(x) = (t φ)(x) dΛt (x ∈ A0),

where φ : A0 → A⊗m B is the generator of j .

Example: Brownian motion

If jt : f 7→ f (Bt) then A0 ⊆ C 2(R) and

jt(g) = g(Bt) = g(B0) +

∫ t

0g ′(Bs) dBs +

1

2

∫ t

0g ′′(Bs) ds

= j0(g) +

∫ t

0s

([ 12δ2(g) δ(g)

δ(g) 0

])dΛs (g ∈ A0),

where δ : h 7→ h′.


The flow generator

Structure of φ

As in Lemma 8, the generator φ has the form

φ =

[L δ†

δ π − ι

],

where

π : A0 → A⊗m B(k) is a unital ∗-homomorphism,

ι : A0 → A0⊙B(k); x 7→ x ⊗ Ik,

δ : A0 → A⊗m B(C; k) is a π-derivation: δ(xy) = δ(x)y + π(x)δ(y)

δ† : x 7→ δ(x∗)∗

L “generates” EΩ j , L(xy)− L(x)y − xL(y) = δ†(x)δ(y).

The Brownian case

(fg)′′ − f ′′g − fg ′′ = 2f ′g ′.


Quantum Feynman–Kac perturbation

Theorem 26

If the multipliers Y and Z have generators F and G respectively then

dkt(x) = (kt ψ)(x) dΛt (x ∈ A0),

where

ψ(x) = φ(x)+F ∗(∆φ(x)+ι(x)

)+F ∗∆

(φ(x)+ι(x)

)∆G+

(φ(x)∆+ι(x)

)G .

Corollary

The semigroup EΩ k has generator which extends

x 7→ L(x) + ℓ∗F δ(x) + ℓ∗F π(x) ℓG + δ†(x) ℓG +m∗F x + x mG ,

where

F =

[mF ∗ℓF ∗

]and G =

[mG ∗ℓG ∗

].


Quantum Feynman–Kac perturbation – proof

Quantum Ito product formula

If X =∫F dΛ and Y =

∫G dΛ then

XtYt =

∫ t

0Xs dYs + (dXs)Ys + dXs dYs

=

∫ t

0

(XsGs + FsYs + Fs∆Gs

)dΛs (t > 0).

Key step in proving of Theorem 26

d(jt(x)Zt) = ( ˜jt(x)t(G )Zt + t(φ(x)

)Zt + t

(φ(x)

)∆t(G )Zt

)dΛt

= t(ι(x)G + φ(x) + φ(x)∆G )Zt dΛt .


Quantum Feynman–Kac for C ∗-algebras

The matrix-space product

A⊗m B(F) := T ∈ B(h⊗F) : T xy ∈ A for all x , y ∈ F,

where T xy ∈ B(h); 〈u,T x

y v〉 = 〈u ⊗ x ,Tv ⊗ y〉.

If k : A1 → A2 is completely bounded then

k ⊗m idB(H) : A1 ⊗m B(H) → A2 ⊗m B(H); (k ⊗m idB(H))(T )xy = k(T xy ).

Problems

Henceforth A is only a C ∗ algebra. Lack of ultraweak continuity/closuremeans

1 A⊗m B(F) is not an algebra ⇒ kt(a) = Y ∗t jt(a)Zt 6∈ A ⊗m B(F),

2 t := jt ⊗m idB(k)

is not multiplicative ⇒ key step for Theorem 26

fails.


Semigroup decomposition for a Markovian cocycle

Theorem 27

A mapping process(kt : A → B(h⊗F)

)t>0

of completely bounded maps

is a Feller cocycle if

1 im kt ⊆ A⊗m B(F) ( t > 0 );

2 ks+t = ks σs kt ( s, t > 0 );

3 EΩ k0 = idA.

A mapping process k is a Feller cocycle if and only if there exists a total

subset T ⊆ k such that 0 ∈ T,

t 7→ Pz ,wt : A → B(h); a 7→ kt(a)

(1[0,t)z)

(1[0,t)w)

is a semigroup on A for all z, w ∈ T,whenever f and g are step functions subordinate to some partition

0 = t0 < t1 < · · · ,

kt(a)(1[0,t)f )

(1[0,t)g)= P f (t0),g(t0)

t1−t0 · · · P f (tn),g(tn)

t−tn

(t ∈ [tn, tn+1)

).


C∗ Feller cocycles from QSDEs

Theorem 28

Let k be a mapping process with locally bounded norm and such that

k0(x) = x ⊗ IF , dkt(x) =(kt ψ)(x) dΛt on E(T) (x ∈ A0)

for a norm-densely defined linear map ψ : A0 → A⊗m B(k) and a total

set 0 ∈ T ⊆ k, where

E(T) := linε(f ) : f is a T-valued, right-continuous step function.

If, for all z, w ∈ T, there exists a C0-semigroup generator ηz ,w with a core

Az ,w ⊆ A0 such that

ψ(x)zw = ηz ,w (x) (x ∈ Az ,w )

then k is a Feller cocycle.


Existence of multipliers

Obstruction

Let F ∈ A⊗m B(k). The construction of Theorem 25 applies, but theproof of contractivity fails as t is not multiplicative.

Solution

Choose the generator F such that q(F ) 6 0 and

t(F )∗∆t(F ) = t(F

∗∆F ) (t > 0). (5)


A QSDE for the perturbed process

Theorem 29

If Y and Z are contractive multipliers with generators F and G

respectively such that

∆F ,∆G ∈T ∈ B

(h⊗ (k)

): Ty ∈ A⊗ B(C; k) for all y ∈ k

then the mapping process

kt : A → B(h⊗F); a 7→ Y ∗t jt(a)Zt (t > 0)

is such that

k0(x) = x ⊗ IF , dkt(x) =(kt ψ)(x) dΛt on E(T) (x ∈ A0)

where ψ is as in Theorem 26.


Simplifying the generator

Bounded parts

Note that

ψ(x) = φ(x) + F ∗(∆φ(x) + ι(x)

)+ F ∗∆

(φ(x) + ι(x)

)∆G

+(φ(x)∆ + ι(x)

)G

= (Ih⊗k

+∆F )∗φ(x)(Ih⊗k

+∆G )

+F ∗ι(x) + F ∗∆ι(x)∆G + ι(x)G .

Let ∆F =[

0 0

ℓF wF−Ih⊗k

]and ∆G =

[0 0

ℓG wG−Ih⊗k

]. Then

(Ih⊗k

+∆F )∗φ(x)(Ih⊗k

+∆G )

=

[L(x) + ℓ∗F δ(x) + δ†(x) ℓG δ†(x)wG

w∗F δ(x) 0

]+

[ℓ∗F

w∗F

](π−ι)(x)

[ℓG wG

].


Applying Theorem 28

Conclusion

If, for all z , w ∈ k, there exist a subspace Az ,w ⊆ A0 such that

Az ,w → A; x 7→ L(x) + ℓ∗F δ(x) + δ†(x) ℓG +(w∗F δ(x)

)z+

(δ†(x)wG )w

is closable with closure which generates a C0 semigroup then k is a Fellercocycle and EΩ k is a semigroup on A with generator which extends

x 7→ L(x) + ℓ∗F δ(x) + ℓ∗F π(x) ℓG + δ†(x) ℓG +m∗F x + x mG .


References

Construction

A. C. R. Belton & S. J. Wills, An algebraic construction of quantum flows

with unbounded generators, Annales de l’Institut Henri Poincare (B)Probabilites et Statistiques, to appear. arXiv:math/1209.3639

Perturbation

A. C. R. Belton, A. G. Skalski & J. M. Lindsay, Quantum Feynman–Kac

perturbations, Journal of the London Mathematical Society, to appear.arXiv:math/1202.6489

A. C. R. Belton & S. J. Wills, Feynman–Kac perturbation of C ∗ quantum

flows, in preparation.


http://arxiv.org/abs/1209.3639

http://arxiv.org/abs/1202.6489

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Quantum Markov semigroups and quantum stochastic flows ...belton/www/notes/25xi13.pdf · Quantum Markov semigroups and quantum stochastic ﬂows – construction and perturbation