An introduction to quantum stochastic calculus. Robin L Hudson. Loughborough University. July 21, 2014. (Institute). July 21, 2014. 1 / 31 ...
An introduction to quantum stochastic calculus Robin L Hudson Loughborough University
July 21, 2014
(Institute)
July 21, 2014
1 / 31
What is Quantum Probability? Quantum probability is the generalisation of the classical theory of probability made necessary by the noncommutative multiplication of quantum observables, which are usually represented by self-adjoint operators in a Hilbert space.
(Institute)
July 21, 2014
2 / 31
What is Quantum Probability? Quantum probability is the generalisation of the classical theory of probability made necessary by the noncommutative multiplication of quantum observables, which are usually represented by self-adjoint operators in a Hilbert space. For example the momentum and position observables p and q for a one-dimensional particle satisfy the Heisenberg commutation relation
[p, q ] =
i
h 2π
where h is Planck’s constant, h = 6.626
10
34
m2 kg /s,
which is quite small in everyday units. It is more convenient to take h = 4π so that [p, q ] = 2i but to remeber that these are not everyday units. (Institute)
July 21, 2014
2 / 31
Some things don’t work in quantum probability. For example if you try to de…ne a joint probability distribution ρp,q for such a canonical pair (p, q ) by Z
R2
D E D E =: ψ, e i (xp +yq ) ψ , e i (xu +yv ) ρp,q (u, v ) dudv = e i (xp +yq ) ψ
you …nd a nice joint Gaussian distribution ρp,q (u, v ) = (2π )
1
e
1 2
when ψ is the ground state of the oscillator it’s the …rst excited state you …nd ρp,q (u, v ) = (2π )
1
u2 + v 2
( u 2 +v 2 ) 1 2
p 2 + q 2 , but when
1 e
1 2
(u 2 +v 2 )
which may look OK till you try to calculate the probability that (p, q ) lies inside the unit circle.
(Institute)
July 21, 2014
3 / 31
Some things don’t work in quantum probability. For example if you try to de…ne a joint probability distribution ρp,q for such a canonical pair (p, q ) by Z
R2
D E D E =: ψ, e i (xp +yq ) ψ , e i (xu +yv ) ρp,q (u, v ) dudv = e i (xp +yq ) ψ
you …nd a nice joint Gaussian distribution ρp,q (u, v ) = (2π )
1
e
1 2
when ψ is the ground state of the oscillator it’s the …rst excited state you …nd ρp,q (u, v ) = (2π )
1
u2 + v 2
( u 2 +v 2 ) 1 2
p 2 + q 2 , but when
1 e
1 2
(u 2 +v 2 )
which may look OK till you try to calculate the probability that (p, q ) lies inside the unit circle. Another thing that doesn’t always work is the notion of conditional expectation and the associated probabilistic concept of martingale. (Institute)
July 21, 2014
3 / 31
Notice that if (p1 , q1 ) , (p2 , q2 ) , ... is a sequence of canonical pairs satisfying the canonical commutation relations
[pj , qk ] =
2i δj ,k , [pj , pk ] = [qj , qk ] = 0,
then. for N = 1.2, ... p1 + p2 + p N
+ pN q1 + q2 + p , N
+ qN
=
2i.
Assuming that the input canonical pairs are "independent, identically distributed and of zero means and …nite variance", this new canonical pair converges in distribution as N ! ∞ to a Gaussian limit, in the manner of the central limit theorem.
(Institute)
July 21, 2014
4 / 31
Notice that if (p1 , q1 ) , (p2 , q2 ) , ... is a sequence of canonical pairs satisfying the canonical commutation relations
[pj , qk ] =
2i δj ,k , [pj , pk ] = [qj , qk ] = 0,
then. for N = 1.2, ... p1 + p2 + p N
+ pN q1 + q2 + p , N
+ qN
=
2i.
Assuming that the input canonical pairs are "independent, identically distributed and of zero means and …nite variance", this new canonical pair converges in distribution as N ! ∞ to a Gaussian limit, in the manner of the central limit theorem. But what can these words mean and what is the meaning of convergence in distribution when there is no joint probability distribution? (Institute)
July 21, 2014
4 / 31
The answer lies in the Stone-von-Neumann uniqueness theorem, according to which there is, up to unitary equivalence, exactly one irreducible representation of the canonical commutation relations for …nitely many degrees of freedom. In the one degree of freedom case, denoting this representation by (p0 , q0 ) and the carrier Hilbert space by H0 , a consequence is that, for an arbitrary canonical pair (p, q ) in a Hilbert space H and an arbitrary state in H , there exists a unique unit vector Ψ(p,q ) 2 H0 H¯ 0 which is invariant under the ‡ip, the conjugate-unitary map from H0 H¯ 0 to itself for which ¯ such that each ψ φ¯ 7! φ ψ, D E D E Ψ(p,q ) , e i (xp0 +yq9 ) Ψ(p,q ) = e i (xp +yq ) .
Ψ(p,q ) is called the distribution vector of (p, q ) (in this state). Two canonical pairs are identically distributed if they have the same distribution vector. A sequence of canonical pairs converges in distribution if the sequence of distribution vectors converges in the usual Hilbert space sense. (Institute)
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To de…ne independence we …rst de…ne the joint distribution vector Ψ(p,q ),(p 0 ,q 0 ) 2 H0 H¯ 0 H0 H¯ 0 of two mutually commuting canonical pairs using the two-dimensional Stone-von-Neumann theorem analogously. Then (p, q ) and (p 0 , q 0 ) are independent if Ψ(p,q ),(p 0 ,q 0 ) = Ψ(p,q ) Ψ(p 0 ,q 0 ) . A canonical pair (p, q ) is of zero mean and variance σ2 1 if hp i = hq i = 0 and the covariance matrix is of form p 2 hpq i σ2 i = 2 i σ2 hqp i q Finally we say that such a canonical pair (p, q ) is Gaussian : If σ2 = 1, if Ψ(p,q ) = ψ0 ψ¯ 0 where ψ0 is the harmonic oscillator ground state. ! If
σ2
> 1, if Ψ(p,q ) =
1/2
∞
∑
n =0
e
2β(2n +1 )
∞
∑e
β(2n +1 ) ψ
n
ψ¯ n where
n =0
ψn is the n-th excited oscillator state and the reciprocal temperature is β given by coth 4 = σ2 .
(Institute)
July 21, 2014
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Notes (1) Provided all second moments are …nite, an arbitrarily distributed canonical pair (p, q ) can be reduced to one of zero mean and variance σ2 by an inhomogeneous linear canonical transformation (p, q ) 7! (αp + βq + a, γq + δq + b ) , where αδ βγ = 1. (2) For sequence of iid canonical pairs (p1 , q1 ) , (p2 , q2 ) , ...of zero means and …nite variance σ2 the sequence p1 , p2 , ... of mutually commuting observables is iid of zero means and …nite variance σ2 in the classical sense. In particular, by the classical central limit theorem, as N ! ∞, 1 + pN ) converges in distribution to the standard (N!) 2 (p1 + p2 + Gaussian limit distribution N 0, σ2 . Likewise for q1 , q2 , ....The limit state for the sequence of pairs is consistent with this. (3) By Donsker’s invariance principle, the process PN (t ) de…ned by PN (t ) = N
1 2
p1 + p2 +
+ p[Nt ] + (Nt
[Nt ]) p[Nt ]+1
must converge to a Brownian motion P of variance σ2 . Similarly QN (t ) = N
1 2
q1 + q2 +
+ q[Nt ] + (Nt
[Nt ]) q[Nt ]+1
converges to a Brownian motion Q, also of variance σ2 . (Institute)
July 21, 2014
7 / 31
Quantum planar Brownian motion. Since
[PN (s ) , QN (t )] h i = N 1 p1 + p2 + + p[Ns ] , q1 + q2 + + q[Nt ] h i +N 1 (Ns [Ns ]) p[Ns ]+1 , (Nt [Nt ]) q[Nt ]+1
=
[N (s ^ t )] N ! 2is ^ t 2i
2i
(Ns
[Ns ]) (Nt N
[Nt ])
δ[Ns ]+1,[Nt ]+1
N !∞
we expect the limit Brownian motions P and Q to satisfy the commutation relation
[P (s ) , Q (t )] =
(Institute)
2is ^ t.
July 21, 2014
8 / 31
Quantum planar Brownian motion. Since
[PN (s ) , QN (t )] h i = N 1 p1 + p2 + + p[Ns ] , q1 + q2 + + q[Nt ] h i +N 1 (Ns [Ns ]) p[Ns ]+1 , (Nt [Nt ]) q[Nt ]+1
=
[N (s ^ t )] N ! 2is ^ t 2i
2i
(Ns
[Ns ]) (Nt N
[Nt ])
δ[Ns ]+1,[Nt ]+1
N !∞
we expect the limit Brownian motions P and Q to satisfy the commutation relation
[P (s ) , Q (t )] =
2is ^ t.
How can we construct two such Brownian motions? We again distinguish the cases σ = 1 and σ > 1. (Institute)
July 21, 2014
8 / 31
The Fock space F (H) over a Hilbert space H is usually de…ned by physicists as the Hilbert space in…nite direct sum
F (H) = C H (H
H)sym
(H
H
H)sym
of symmetric parts of the n-fold tensor product of H with itself. A more useful de…nition for our purposes is that F (H) is a Hilbert space generated by a family (e (f ))f 2H of socalled exponential vectors, satisfying
he (f ) , e (g )iF (H) = exp hf , g iH .
The connection between the two de…nitions is made by realising the exponential vectors in the …rst de…nition as f f f f f p . e (f ) = 1 f p 2! 3! One reason why this second de…nition is useful is that it makes clear the exponential property of Fock spaces, that there is a natural isomorphism allowing us to identify the Fock space over a direct sum with the tensor product of the Fock spaces over the summands;
F (H1
H2 ) = F (H1 )
(Institute)
F (H2 ) , with e (f1
f2 ) = e ( f1 )
e ( f2 ) .
July 21, 2014
9 / 31
To construct two Brownian motions P and Q satisfying
[P (s ) , Q (t )] =
2i
(1)
of unit variance = 1 in the Fock space F (R+ ) over (R+ ) , …rst de…ne the mutually adjoint creation and annihilation processes A† = A† (t ) t 0 and A = (A (t ))t 0 by their actions on exponential vectors D E d e f + zχ[0,t [ , A (t ) e (f ) = χ[0,t [ , f e (f ) . A† (t ) e (f ) = dz These satisfy the commutation relations h i h i A† (s ) , A† (t ) = [A (s ) , A (t )] = 0, A (s ) , A† (t ) = s ^ t, (2) σ2
L2
L2
in the sense that for example, for arbitrary exponential vectors, D E A† (s ) e (f ) , A† (t ) e (g ) hA (s ) e (f ) , A (t ) e (g )i = s ^ t he (f ) , e (g )i De…ne P and Q by
P = i A†
A , Q = A + A†
Then (2) implies (1) and commutativity of the process P (likewise Q). (Institute)
July 21, 2014
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Moreover in the vacuum state e (0) they are both Brownian motions of unit variance in the sense of the following Theorem: Denote by (Ω, F, W) the standard Wiener probability space and by X = (X (t ))t 0 the standard realisation on it of unit variance Brownian motion, so that for ω 2 Ω, X (t ) (ω ) = ω (t ) and F is the σ-…eld generated by the X (t ) . Then there exists a unique Hilbert space isomorphism DP (resp. DQ ) from the Fock space F L2 (R+ ) onto L2 (Ω, F, W) with the properties DP e (0) (ω ) (resp. DQ e (0) (ω ) ) 1
DP P (t ) DP (resp. DQ Q
(t ) DQ=1 )
= 1 for all ω 2 Ω, = mult X (t ) for all t 2 R+
where mult X (t ) denotes the operator of multiplication by X (t ). But because P and Q don’t commute with eachother they cannot be simultaneously diagonalized; DP 6= DQ . Note: Although they don’t commute, they have a property amounting to stochastic D independence, namely Efactorization of Ejoint D D characteristic E i Σ x P s Σ y Q t i Σ x P s ( ( )+ ( )) ( ) j j j j j j k k k functions, e = e e i Σk yk Q (tk ) , so we regard them as jointly a quantum planar Brownian motion. (Institute)
July 21, 2014
11 / 31
Now assume σ2 > 1. Then we can write σ2 = α2 + β2 where α2 β2 = 1. In the tensor product F F¯ of the Fock space F = F L2 (R+ ) with its dual Hilbert space, equipped with the unit vector e (0) e (0), de…ne Pσ (t ) = αP (t ) I¯ + βI P¯ (t ) , Qσ (t ) = αQ (t ) I¯ + βI Q¯ (t ) , where for example P¯ (t ) is the dual operator to P (t ) , P¯ (t ) ψ¯ = P (t ) ψ. Then, in the product state e (0) e (0), αP I¯ is a Brownian motion of variance α2 and βI P¯ is an independent Brownian motion of variance β2 which commutes with it. Hence the sum αP I¯ + βI P¯ is itself a Brownian motion of variance α2 + β2 = σ2 . Similarly αQ I¯ + βI Q¯ is a Brownian motion of variance σ2 . On the other hand [Pσ (s ) , Qσ (t )] = α2 [P (s ) , Q (t )] I¯ + β2 I [P¯ (s ) , Q¯ (t )]
=
α2
β2 ( 2is ^ t ) =
2is ^ t
since [P¯ (s ) , Q¯ (t )] = [P (s ) , Q (t )] = 2is ^ t = 2is ^ t. The (Pσ , Qσ ) with σ > 1 is technically easier to work with than (P, Q ) because e (0) e (0) is cyclic and separating for fPσ , Qσ g00 . (Institute)
July 21, 2014
12 / 31
Quantum stochastic calculus. Brownian motion X is non-di¤erentiable. Intuitively this is because (for s > t) X (s ) X (t ) is N (0, s t ) , so it is of order of magnitude p X (s ) X (t ) s t. So when you form the di¤erence quotient there can be s t no sensible limit as s ! t. However the two components P and Q of quantum planar Brownian motion show some hints of di¤erentiability. For example Q = A† + A and so, for exponential vectors e (f ) and e (g ) ,
he (f ) , Q (t ) e (g )i = hA (t ) e (f ) , e (g )i + he (f ) , A (t ) e (g )i D E D E χ[0,t [ , f + χ[0,t [ , g = he (f ) , e (g )i =
Zt 0
(f¯ (s ) + g (s )) he (f ) , e (g )i
which suggests that, at least for well-behaved f and g , e (f ) , (Institute)
dQ (t ) e (g ) dt
= (f¯ (t ) + g (t )) he (f ) , e (g )i .
July 21, 2014
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dQ (t )
Unfortunately this does not de…ne an operator dt . But it does suggest a theory of operator-valued quantum stochastic integrals, of the form M (t ) =
Zt
F (s ) dA† (s ) + G (s ) dA (s ) + H (s ) dT (s )
0
for which the "…rst fundamental formula"
he (f ) , M (t ) e (g )i =
Zt 0
he (f ) , (F (s ) f¯ (s ) + G (s ) g (s ) + H (s )) e (g )i ds
holds, provided that dA† (s ) may be commuted through F (s ) to reach the left of the inner product. Assuming that di¤erentials like dA† "point into the future" this requires that the integrand processes F = (F (t ))t 0 are adapted, meaning that each operator F (t ) is of form F ( t ) = Ft
It,
in so far as the Fock space
F L2 (R+ ) = F L2 ([0, t [) (Institute)
F L2 ([t, ∞[) July 21, 2014
14 / 31
Everyone agrees how to de…ne the stochastic integral L 2 A† , A, T of an elementary adapted process F (t ) = χ[a,b [ (t ) F (a) = χ[a,b [ (t ) Fa
Rt 0
F (s ) dL (s ) ,
I a;
Rt it is 0 F (s ) dL (s ) = F (a) (L (t ^ b ) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only …nitely many di¤erent values.
(Institute)
July 21, 2014
15 / 31
Everyone agrees how to de…ne the stochastic integral L 2 A† , A, T of an elementary adapted process F (t ) = χ[a,b [ (t ) F (a) = χ[a,b [ (t ) Fa
Rt 0
F (s ) dL (s ) ,
I a;
Rt it is 0 F (s ) dL (s ) = F (a) (L (t ^ b ) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only …nitely many di¤erent values. Note: The products of unbounded operators occurring here can be de…ned rigorously on the exponential domain as tensor product operators.
(Institute)
July 21, 2014
15 / 31
Everyone agrees how to de…ne the stochastic integral L 2 A† , A, T of an elementary adapted process F (t ) = χ[a,b [ (t ) F (a) = χ[a,b [ (t ) Fa
Rt 0
F (s ) dL (s ) ,
I a;
Rt it is 0 F (s ) dL (s ) = F (a) (L (t ^ b ) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only …nitely many di¤erent values. Note: The products of unbounded operators occurring here can be de…ned rigorously on the exponential domain as tensor product operators. The …rst fundamental formula is easily proved for such integrands:
(Institute)
July 21, 2014
15 / 31
Everyone agrees how to de…ne the stochastic integral L 2 A† , A, T of an elementary adapted process F (t ) = χ[a,b [ (t ) F (a) = χ[a,b [ (t ) Fa
Rt 0
F (s ) dL (s ) ,
I a;
Rt it is 0 F (s ) dL (s ) = F (a) (L (t ^ b ) L (t ^ a)) and hence by additivity also of a simple adapted process, ie one taking only …nitely many di¤erent values. Note: The products of unbounded operators occurring here can be de…ned rigorously on the exponential domain as tensor product operators. The …rst fundamental formula is easily proved for such integrands: Theorem: For simple adapted processes F , G and H * + Zt F (s ) dA† (s ) + G (s ) dA (s ) + H (s ) dT (s ) e (g )
e (f ) ,
0
=
Zt 0
he (f ) , (F (s ) f¯ (s ) + G (s ) g (s ) + H (s )) e (g )i ds.
(Institute)
July 21, 2014
15 / 31
The second fundamental formula is the heart of quantum stochastic calculus. It is a rule for expressing the product of two stochastic integrals as a sum of iterated stochastic integrals. The prototype is the rule for Brownian motion X :
(X (t ))2 = =
Z
Z
2
dX (s ) 0 s
