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 Statistics Lancaster University United Kingdom [email protected]Noncommutative Geometry Seminar Mathematical Institute of the Polish Academy of Sciences 25th November 2013
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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
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 11 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 12 / 67
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 .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 13 / 67
Non-commutative probability
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).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 14 / 67
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 15 / 67
Non-commutative probability
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).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 16 / 67
Non-commutative probability
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 17 / 67
Non-commutative probability
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
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 29 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 30 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 31 / 67
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 .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 32 / 67
Random walks on discrete groups
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 33 / 67
Random walks on discrete groups
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 34 / 67
Random walks on discrete groups
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 .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 35 / 67
Random walks on discrete groups
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 36 / 67
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
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 39 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 40 / 67
The symmetric quantum exclusion process
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).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 41 / 67
The symmetric quantum exclusion process
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 42 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 43 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 44 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 45 / 67
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.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 46 / 67
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
′′.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 48 / 67
Classical Feynman–Kac perturbation
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)
).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 49 / 67
Classical Feynman–Kac perturbation
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 .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 50 / 67
Introducing non-commutativity
Idea
There are two ways in which non-commutativity can appear: