Classical and quantum Markov semigroupsbelton/www/notes/23iv14.pdf · 2014-04-23 · Classical...
Transcript of Classical and quantum Markov semigroupsbelton/www/notes/23iv14.pdf · 2014-04-23 · Classical...
Classical and quantum Markov semigroups
Alexander Belton
Department of Mathematics and StatisticsLancaster UniversityUnited Kingdom
http://www.maths.lancs.ac.uk/~belton/
Young Functional Analysts’ Workshop
Lancaster University
23rd April 2014
Classical Markov semigroups
Markov processes
Markov semigroups Infinitesimal generators
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 2 / 27
Markov processes
Definition 1
Let S be a topological space. A Markov process with state space S is acollection of S-valued random variables X = (Xt)t>0 on a commonprobability space such that
E[
f (Xs+t) | Xr : 0 6 r 6 s]
= E[
f (Xs+t) | Xs
]
(s, t > 0)
for all
f ∈ Bb(S) := g : S → R | g is bounded and Borel measurable.
A Markov process X is time homogeneous if
E[
f (Xs+t) | Xs = x]
= E[f (Xt) | X0 = x]
(f ∈ Bb(S), s, t > 0, x ∈ S).
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 3 / 27
Markov semigroups
Definition 2
A Markov semigroup on Bb(S) is a family T = (Tt)t>0 such that
1 Tt : Bb(S) → Bb(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) and
4 Tt f > 0 whenever f > 0, for all t > 0 (positive).
If Tt1 = 1 for all t > 0 then T is conservative.
Proposition 3
Given a time-homogeneous Markov process X , setting
(Tt f )(x) = E[
f (Xt) | X0 = x]
(t > 0, x ∈ S)
defines a conservative Markov semigroup T on Bb(S).
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 4 / 27
Proof of Proposition 3
Easy part
Properties 1, 3 and 4 follow immediately from basic properties ofconditional expectation, as does the fact that T is conservative.
The semigroup property
Note that
(Ts+t f )(x) = E[f (Xs+t) | X0 = x ] (definition)
= E[
E[f (Xs+t) | Xr : 0 6 r 6 s]∣
∣ X0 = x]
(tower property)
= E[
E[f (Xs+t) | Xs ]∣
∣ X0 = x]
(Markov property)
= E[
(Tt f )(Xs) | X0 = x]
(homogeneity)
= Ts(Tt f )(x).
The identity T0f = f is immediate.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 5 / 27
Feller semigroups
Definition 4
Suppose the state space S is a locally compact Hausdorff space. TheMarkov semigroup T is Feller if
1 Tt
(
C0(S))
⊆ C0(S) for all t > 0 and
2 ‖Tt f − f ‖∞ → 0 as t → 0 for all f ∈ C0(S).
Remark
Every sufficiently well-behaved time-homogeneous Markov process isFeller: Brownian motion, Poisson processes, Levy processes, . . . .
Theorem 5
If the state space S is metrisable then a conservative Feller semigroupgives rise to a time-homogeneous Markov process.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 6 / 27
Proof of Theorem 5
Probabilistic version
Let
pt(x ,A) := (Tt1A)(x) (t > 0, x ∈ S , ABorel⊆ S)
be the probability of moving from x to A in time t. Then
(Tt f )(x) =
∫
S
f (y)pt(x ,dy) (t > 0, f ∈ Bb(S), x ∈ S) (⋆)
and ps+t(x ,A) =(
Ts(Tt1A))
(x) (semigroup property) (1)
=
∫
S
(Tt1A)(y)ps(x ,dy) (by (⋆))
(2)
=
∫
S
pt(y ,A)ps(x ,dy) (definition). (3)
The second identity is the Chapman–Kolmogorov equation.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 7 / 27
Proof of Theorem 5
Probabilistic version (ctd.)
Let µ be a probability measure on S . If tn > · · · > t1 > 0 and A1, . . .An
are Borel subsets of S then
pt1,...,tn(A1 × · · · × An)
=
∫
S
µ(dx0)
∫
A1
pt1(x0,dx1) · · ·
∫
An
ptn−tn−1(xn−1,dxn).
These finite-dimensional distributions are consistent, by C–K.The Daniell–Kolmogorov extension theorem yields a probability measureon the product space
Ω := SR+ = ω = (ωt)t>0 : ωt ∈ S for all t > 0
such the coordinate projections Xt : Ω → S ; ω 7→ ωt form atime-homogeneous Markov process X with associated semigroup T .
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 8 / 27
Proof of Theorem 5
Functional-analytic version
Without loss of generality, suppose that S is compact. Then Ω is acompact Hausdorff space and the algebraic tensor product
⊙
t>0
C (S) = linf1 Xt1 · · · fn Xtn : f1, . . . , fn ∈ C (S), t1, . . . , tn > 0
is dense in C (Ω) by the Stone–Weierstrass theorem.If µ is a state on C (S) then
f1 Xt1 · · · fn Xtn 7→ µ(Tt1(f1 · · · (Ttn−tn−1 fn) · · · ))
extends to a state φ on C (Ω).By the Riesz–Markov theorem, there exists a probability measure on Ωcorresponding to φ, and X is a time-homogeneous Markov process withrespect to this measure, with associated semigroup T .
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 9 / 27
Infinitesimal generators
Definition 6
Let T be a C0 semigroup on a Banach space E . Its infinitesimal generatoris the linear operator L in E with domain
domL =
x ∈ E : limt→0+
t−1(Ttx − x) exists
and actionLx = lim
t→0+t−1(Ttx − x).
The operator L is closed and densely defined.
Remark
If T comes from a time-homogeneous Markov process X then
E[
f (Xt+h)− f (Xt) | Xt
]
= (Thf − f )(Xt) = h(Lf )(Xt) + o(h),
so L describes the change in X over an infinitesimal time interval.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 10 / 27
Examples
Uniform motion
If S = R and Xt = X0 + t for all t > 0 then
E[f (Xs+t)|Xs = x ] = f (x + t) = E[f (Xt)|X0 = x ](
f ∈ C0(R))
.
It follows that X is a time-homogeneous Feller process with semigroupgenerator L such that Lf = f ′.
Brownian motion
If S = R and X is a standard Brownian motion then Ito’s formula givesthat
f (Xt) = f (X0) +
∫ t
0f ′(Xs)dXs +
1
2
∫ t
0f ′′(Xs)ds
(
f ∈ C 2(R))
.
It follows that X is a time-homogeneous Feller process with semigroupgenerator L such that Lf = 1
2 f′′.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 11 / 27
Examples (ctd.)
A Poisson process
If S = R and X is a Poisson process with unit intensity and unit jumpsthen
E[f (Xt)|Xs = x ] = e−(t−s)∞∑
n=0
(t − s)n
n!f (x + n) (t > s > 0).
It follows that X is a time-homogeneous Feller process with semigroupgenerator L such that (Lf )(x) = f (x + 1)− f (x) for all x ∈ R.(Note that
(Tt f − f )(x)
t=
e−t − 1
tf (x) + e−t f (x + 1) + O(t) (t → 0+).)
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 12 / 27
The Lumer–Phillips theorem
Theorem 7 (Lumer–Phillips)
A closed, densely defined operator L in the Banach space E generates astrongly continuous contraction semigroup on E if and only if
ran(λI − L) = E for some λ > 0 and
the operator L is dissipative:
‖(λI − L)x‖ > λ‖x‖ for all λ > 0 and x ∈ domL.
Remark
If the operator L is dissipative then ran(λI − L) = E for some λ > 0 ifand only if ran(λI −L) = E for all λ > 0.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 13 / 27
The Hille–Yosida–Ray theorem
Definition
Let S be a locally compact Hausdorff space. A linear operator L in C0(S)satisfies the positive maximum principle if whenever f ∈ domL and x0 ∈ Sare such that supx∈S f (x) = f (x0) > 0 then (Lf )(x0) 6 0.
Theorem 8 (Hille–Yosida–Ray)
A closed, densely defined operator L in C0(S) is the generator of a Fellersemigroup on C0(S) if and only if
ran(λI − L) = C0(S) for some λ > 0 and
L satisfies the positive maximum principle.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 14 / 27
Proof of Theorem 8
Sufficiency
Suppose L satisfies the positive maximum principle.Let f ∈ domL and λ > 0. To see that
‖(λI − L)f ‖∞ > λ‖f ‖∞,
note first that there exists x0 ∈ S such that |f (x0)| = ‖f ‖∞; without lossof generality, suppose f (x0) > 0. Then
‖(λI − L)f ‖∞ > |λf (x0)− (Lf )(x0)|
and (Lf )(x0) 6 0, by the positive maximum principle. Consequently,
‖(λI − L)f ‖∞ > λf (x0)− Lf (x0) > λf (x0) = λ‖f ‖∞.
Hence T is a strongly continuous contraction semigroup, by L–F. Positivityis left as an exercise: show that (λI − L)−1 is positive for all λ > 0.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 15 / 27
Proof of Theorem 8 (ctd.)
Necessity
Suppose that L generates a Feller semigroup on C0(S). By L–P, it sufficesto show that L satisfies the positive maximum principle.Given f ∈ domL, let x0 ∈ S be such that f (x0) = supx∈S f (x).Let f + := x 7→ maxf (x), 0 and note that
(Tt f )(x0) 6 (Tt f+)(x0) 6 ‖Tt f
+‖∞ 6 ‖f +‖∞ = f (x0).
Then
(Lf )(x0) = limt→0+
(Tt f − f )(x0)
t6 0,
as required.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 16 / 27
Quantum Markov semigroups
Quantum Markov processes
Quantum Markov semigroups Infinitesimal generators
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 17 / 27
Quantum Feller semigroups
Theorem 9
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 ‖Tt‖ 6 1 for all t > 0,
4 (Ttaij) ∈ Mn(A)+ whenever (aij) ∈ Mn(A)+, for all n > 1 and t > 0(complete positivity) and
5 ‖Ttx − x‖ → 0 as t → 0 for all x ∈ A.
If A is unital and Tt1 = 1 for all t > 0 then T is conservative.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 18 / 27
Complete positivity
Exercise
A linear map Φ : A → B between C ∗ algebras is completely positive if andonly if
n∑
i ,j=1
b∗i Φ(a∗
i aj)bj > 0
for all n > 1, a1, . . . , an ∈ A and b1, . . . , bn ∈ B.
Theorem 10
A positive linear map φ : A → B between C ∗ algebras is completelypositive if A is commutative (Stinespring) or B is commutative (Arveson).
Theorem 11 (Kadison)
A CP unital linear map Φ : A → B between unital C ∗ algebras is such that
Φ(a∗a) > Φ(a)∗Φ(a) (a ∈ A) (CP-Schwarz)
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 19 / 27
Stinespring’s theorem
Theorem 12 (Stinespring)
Let Φ : A → B be a linear map, where A is a unital C ∗ algebra andB ⊆ B(H). Then Φ is completely positive if and only if there exists arepresentation π : A → B(K) and a bounded operator V : H → K suchthat
Φ(a) = V ∗π(a)V (a ∈ A).
Corollary 13
If Φ : A → B is as above, with Φ(1) = I , then
n∑
i ,j=1
〈vi ,(
Φ(a∗i aj)− Φ(ai)∗Φ(aj)
)
vj〉 > 0
for all n > 1, a1, . . . , an ∈ A and v1, . . . , vn ∈ H.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 20 / 27
Proof of Corollary 13
Note first that I = Φ(1) = V ∗π(1)V = V ∗V , so V ∗ has norm 1. Hence
n∑
i ,j=1
〈vi ,Φ(a∗
i aj)vj〉 =n
∑
i ,j=1
〈Vvi , π(a∗
i aj)Vvj〉
=∥
∥
∥
n∑
i=1
π(ai )Vvi
∥
∥
∥
2
>
∥
∥
∥V ∗
n∑
i=1
π(ai )Vvi
∥
∥
∥
2
=∥
∥
∥
n∑
i=1
Φ(ai)vi
∥
∥
∥
2
=n
∑
i ,j=1
〈vi ,Φ(ai )∗Φ(aj)vj〉.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 21 / 27
Infinitesimal generators
Theorem 14
Let T be a quantum Feller semigroup on B(H) which is uniformlycontinuous:
limt→0+
‖Tt − I‖ = 0.
The generator L is bounded, ∗-preserving and conditionally completelypositive: if n > 1, a1, . . . , an and v1, . . . , vn ∈ H then
n∑
i ,j=1
〈vi ,L(a∗
i aj)vj〉 > 0
whenevern
∑
i=1
aivi = 0.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 22 / 27
Proof of Theorem 14
Easy parts
The boundedness of L is standard semigroup theory. The fact thatL(a∗) = L(a)∗ for all a ∈ A follows by continuity of the involution.
Proof of conditional complete positivity
Let a1, . . . , an ∈ A and v1, . . . , vn ∈ H. By Corollary 13,
n∑
i ,j=1
〈vi ,(
Tt(a∗
i aj)− Tt(ai )∗Tt(aj)
)
vj〉 > 0.
Differentiating with respect to t gives that
n∑
i ,j=1
〈vi ,(
L(a∗i aj)− L(ai)∗aj − a∗i L(aj)
)
vj〉 > 0;
if∑n
i=1 aivi = 0, the second and third terms vanish.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 23 / 27
Characterisation of bounded generators
Theorem 15 (Lindblad, Evans)
Let L be a ∗-preserving bounded linear map on the unital C ∗ algebra A.Then Tt = exp(tL) is completely positive for all t > 0 if and only if L isconditionally completely positive (in the appropriate sense).
Since CP unital linear maps between unital C ∗ algebras are automaticallycontractive, this characterises the generators of uniformly continuousconservative quantum Feller semigroups on unital C∗ algebras.
Theorem 16 (Gorini–Kossakowski–Sudarshan, Lindblad)
A bounded map L on B(H) is the generator of a uniformly continuousconservative quantum Feller semigroup composed of normal maps if andonly if
L(X ) = i [H,X ] − 12
(
L∗LX − 2L∗(X ⊗ I )L + XL∗L) (
X ∈ B(H))
,
where H = H∗ ∈ B(H) and L ∈ B(H;H⊗ K) for some Hilbert space K.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 24 / 27
Quantum Markov processes
Random variables
Let S be a compact Hausdorff space. If X : Ω → S is a random variablethen
jX : A → B; f 7→ f X
is a unital ∗-homomorphism, where A = C (S) and B = L∞(Ω,F ,P).
Definition 17
A non-commutative random variable is a unital ∗-homomorphism jbetween unital C ∗ algebras.
A family (jt : A → B)t>0 of non-commutative random variables is adilation of the quantum Feller semigroup T on A if there exists aconditional expectation E : B ։ A such that Tt = E jt for all t > 0.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 25 / 27
Construction of dilations
Many authors have tackled this problem: Evans and Lewis; Davies;Accardi, Frigerio and Lewis; Vincent-Smith; Kummerer; Sauvageot; Bhatand Parthasarathy; . . . . Essentially, one attempts to mimic thefunctional-analytic proof of Theorem 5. The state
f1 Xt1 · · · fn Xtn 7→ µ(Tt1(f1 · · · (Ttn−tn−1 fn) · · · ))
becomes a sesquilinear form
(f1 ⊗ · · · ⊗ fn, g1 ⊗ · · · ⊗ gn) 7→ µ(Tt1(f∗
1 · · · (Ttn−tn−1(f∗
n gn)) · · · g1)).
The key to proving positivity of this form is the complete positivity of thesemigroup maps.There are many technical details which must be addressed. This wouldrequire another talk.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 26 / 27
References
Classical
D. Applebaum, Levy processes and stochastic calculus, secondedition, Cambridge University Press, 2009.
T.M. Liggett, Continuous time Markov processes, AmericanMathematical Society, 2010.
L.C.G. Rogers and D. Williams, Diffusions, Markov processes andmartingales, volumes I and II, second edition, Cambridge UniversityPress, 2000.
Quantum
D.E. Evans and J.T. Lewis, Dilations of irreversible evolutions inalgebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. ANo.24 (1977), v+104 pp.
F. Fagnola, Quantum Markov semigroups and quantum flows,Proyecciones 18 no.3 (1999), 144 pp.
Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 27 / 27