Quantum Probability and Applications III: Proceedings of a Conference held in Oberwolfach, FRG, January 25–31, 1987

378 Pages·1988·7.363 MB·English

by  Luigi Accardi (auth.), Luigi Accardi, Wilhelm von Waldenfels (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
1303
.L Accardi .W von Waldenfels ).sdE(
Quantum Probability
and Applications III
Proceedings of a Conference held ni
Oberwolfach, FRG, January 25-31, 1987
Springer-Verlag
Heidelberg Berlin NewYork London Paris oykoT
Editors
Luigi Accardi
Dipartimento di Matematica, Universit& di Roma II
Via Orazio Raimondo, 00173, Roma, Italy
Wilhelm von Waldenfeis
Institut fLir Angewandte Mathematik, Universit&t Heidelberg
Im Neuenheimer Feld 294, 6900 Heidelberg, Federal Republic of Germany
Mathematics Subject Classification (1980): 43A35, 46LXX, 46M20, 47DXX,
60FXX, 60GXX, 60HXX, 60JXX, 80A05, 81C20, 81K05, 81L05, 81M05,
82A15, 83C40, 83C75
ISBN 3-540-18919-X Springer-Verlag Berlin Heidelberg New York
ISBN 0o387-18919-X Springer-Verlag New York Berlin Heidelberg
This work subject is to copyright. whether All the rights are reserved, whole or part of the material
concerned, is specifically the rights of translation, reprinting, re-use of illustrations, recitation,
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of this publication or permitted thereof the provisions parts only under the German of is Copyright
Law of September 9, 1965, ni 24, version of June its 1985, copyright and a fee always must be
paid, fall the Violations under prosecution act of i the German Copyriph~ ....
© Springer-Verlag Berlin Heidelberg 1988
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and Printing binding: Druckhaus Beltz, Hemsbach/Bergstr,
2146/3140-543210
Introduction
This volume contains the proceedings of the first Quantum Probability meeting held in
Oberwolfach which is the fourth of a series begun with the 1982 meeting of Mondragone and
continued in Heidelberg ('84) and in Leuven ('85). The main topics discussed during the meeting
were: quantum stochastic calculus, mathematical models of quantum noise and their appfications to
quantum optics, the quantum Feynman-Kac formula, quantum probability and models of quantum
statistical mechanics, the notion of conditioning in quantum probability and related problems
(dilations, quantum Markov processes), quantum central limit theorems.
We are grateful to the Mathematisches Forschungsinstitut Oberwolfach and to its director
Prof. M. Barner for giving us the unique opportunity of scientific collaboration and mutual
exchange.
We would like to thank also the speakers and all the participants for their contributions to the
vivid and sometimes heated discussions.
L. Accardi
W. v. Waldenfels
C O N T E N T S
L. Accardi, A Note on Meyer's Note .............................. 1
L. Accardi and F. Fagnola, Stochastic Integration ............... 6
D. Applebaum, Quantum Stochastic Parallel Transport on Non-Commu-
tative Vector Bundles ........................................... 20
A. Barchielli, Input and Output Channels in Quantum Systems and
Quantum Stochastic Differential Equations ....................... 37
C. Cecchini, Some Noncommutative Radon-Nikodym Theorems on von
Neumann Algebras ................................................ 52
M. Evans and R.L. Hudson, Multidimensional Quantum Diffusions ... 69
M. Fannes, An Application of de Finetti's Theorem ............... 89
G.W. Ford, The Quantum Langevin Equation from the Independent-
Oscillator Model ................................................ 103
A. Frigerio, Quantum Poisson Processes: Physical Motivations and
Applications .................................................... 107
A.S. Holevo, A Noncommutative Generalization of Conditionally
Positive Definite Functions ..................................... 128
2
R. Jajte, Contraction Semigroups in L over avon Neumann Algebra 149
B. Kftmmerer, Survey on a Theory of Non-Commutative Stationary
Markov Processes ................................................ 154
G. Lindblad, Dynamical Entropy for Quantum Systems .............. 183
M. Lindsay and H. Maassen, An Integral Kernel Approach to Noise . 192
P.A. Meyer, A Note on Shifts and Cocycles ....................... 209
Vi
K.R. Parthasarathy, Local Measures in Fock Space Stochastic Cal-
culus and a Generalized Ito-Tanaka Formula ...................... 213
K.R. Parthasarathy and K.B. Sinha, Representation of a Class of
Quantum Martingales II ......................................... 232
D. Petz, Conditional Expectation in Quantum Probability ......... 251
J. Quaegebeur, Mutual Quadratic Variation and Ito's Table in
Quantum Stochastic Calculus ..................................... 261
U. Quasthoff, On Mixing Properties of Automorphisms of yon Neu-
mann Algebras Related to Measure Space Transformations ......... 275
J.-L. Sauvageot, First Exit Time: A Theory of Stopping Times in
Quantum Processes ............................................... 285
M. Sch~rmann and W. von Waldenfels, A Central Limit Theorem on
the Free Lie Group .............................................. 300
G.L. Sewell, Entropy, Observability and The Generalised Second
Law of Thermodynamics ........................................... 319
A.G. Shuhov and Yu. M. Suhov, Remarks on Asymptotic Properties of
Groups of Bogoliubov Transformations of CAR C*-Algebras ........ 329
R.F. Streater, Linear and Non-Linear Stochastic Processes ....... 343
A. Verbeure, Detailed Balance and Critical Slowing Down ......... 354
I.F. Wilde, Quantum Martingales and Stochastic Integrals ........ 363
A NOTE ON MEYER' S NOTE
Luigi Accardi
Dipartimento di Matematica
Universita' di Roma II
(1.) NOTATIONS AND STATEMENT OF THE PROBLEM
Let us denote
)+R(2L(r
- the Boson Fock space over the one-particle space L2(R+)
.)+R(2L(P
- "~ = {¢(f) : f E L2(R+)} the set of exponential vectors in
-<p = ¢(0) the vacuum state in P(L2(R+) .
- P(xlo.,]) the orthogonal projector defined by
l*,oIX(C )¢(f) = ¢(XI,~.tl f)
W(f) f E L2(R+) the meyl operator characterized by the property
w(:)¢M = :(¢>"-':<-:"q%-e + h)
A - + A , N , the annihilation, creation and number (or gauge or conservation fields ) defined, on ° c by the
relations :
A(f)¢(g) =< f,g > ¢(g)
d
A+(f)¢(g) = ~ It=o ¢(g + t f)
d
Nt¢(g) = G o=~,[ ¢(e"×t""lg)
we write N(s,t) for Nt - ~,N ,
The W(f) are unitary operators on ~ satisfying the CCR
W(f)W(g) = e- I~l~-</"J>¢(f +h)
ft] X[o.e}f ; fit Xlt.~}f
- = =
Ho a complex Hilbert space , called the initial space.
-
¢) = Ho®F(L~(R+)
- - ~,I = Ho ® r(L2([0, tI) ® ¢I~
8 - (u,,
= = ®
- }tB = B(H,, ® r(L:([o, ))}t ® ,1 I
- ,131 = B(IH. ® lq® F(L~(It,~))
,~ eht shift no L'~(R+) .
- at the shift ou F(L2(R+)
= ['(~t)
- '~u = oz ® at the free time shift on ~. where oz is the identity map on 23(H.).
Tile objects described above provide a simple and , for a certain class of models, canonical example of a
quantum Markov process (in fact also of a quantum independent increment process in the sense of [2]) and the
Feynman-Kac formula allows to perturb such structures by means of unitary cocycles (for the free time shift )
giving rise to new quantum processes [1]. In particular, the generator L of the quantum Markovian semigroup
canonically associated to the new process is the sum of the generator L,~ of the semigroup associated to the
original process and of an additional perturbative piece, denoted Lz . The problem with the above class of
quantum Markov processes is that the free tinLe shift u' t' acts trivially on the initial algebra and therefore
the corresponding semigroup is zero so that , as remarked in }1[ , in this case one is ill fact deMing with
FK perturbations of the identity semigroup . As a consequence of this one looses one of the most attractive
analytical advantages of the classical FK formula, uamely the possibility of dealing with perturbations L1
so singular that the operator Lo + L~ is not well defined ( a typical example is the possibility of giving a
meaning ,via tile FK formula, to the formal generator -& + V where ~Z is the Laplacian on R ~' and V is a
highly singular potential). By analogy with the classical case we would like to have a time shift v; 'wich shifts
also the initial random variables (observables) and not only the increments. Moreover , ill order to be able to
apply the quantum FK perturbation technique , the free time shift should be such that the structure of the
associated unitary cocycles should be deterlnined quite explicitly , which is rarely the case if this time shift
is itself a Feynman-Kac perturbation of u~. This problem was posed by A. Meyer during the Obervolfach
meeting and in the following I want to outline a possible general scheme for a solution and illustrate it with
all example.
(2.) A POSSIBLE SCHEME FOR A SOLUTION
Let us look for a time shift t v o of tile fornl
where
*3 : 3(Ito) ~ 3(H<,) ® 1 ~ 3t) (2.2)
is a *-homomorphism. For all ao E ~(g,,) , b E B(F(L2(R+)) one has
oa(;'~2;~ ® b) : 3; ® ~,(3i(~o) ® ~,(b)) = (3; ® ~,)(3i(~,,)). (3; ® #,)(L ® ~,,(b)) (2.a)
-- ~:7.( @cr,)(y;(a,,)) ' (lt] ®a,+t(b))
and since we want v~' to be a 1-paranieter senii-group of *-endomorphisms, .ti follows that, for ally a, @
B(H+,) , b E B(F(L2(R+)) the right hand side of (2.3) must be equal to
v~+, (a,, ® b) - j,+t(a,,) ® o',~+t(b) (2.4)
Titus v" will be a 1-parameter semi-group if and only if
,q
(3, ® ~,)(j,(a,,)) - J.,+,(~,4 wo e S(Ho) )~.2(
Here we give all example of a jt satisfying condition (2.5) above. First shall we give tile expression of 3;
ill unbounded form and then shall write the corresponding bounded form .
Ill the notations of Section (1) choose ,<H = L2(R) with a,, + a denoting the usual annihilation and creation
operators. Define
3,(a,~) ' ~ = ,~a + Xlo., I ; a;, = ~,a or ,~a + (2.6)
with (s, t) H Xlo.t} a e-homogeneous normal additive process, i.e.
xlo.'} =x+lo,t} g l"., ® 8 (P(L= (I0, el)) ", {Xlo.~l, Xl;.q t = 0 )7~2(
(2.s)
Xi~,,q + Xl,,q = X[~,tt ; r < s < t
ar(X[.~,tl) = Xl.,+r.t+r ] (2.9)
,q~.~IXI Xb,.q] =0 if (u,t) C~(r,s) =0 (2,10)
where [ . , . ] denotes , as usual , the commutator. In bounded form and under the additional assumption
that X[o,t] is self-adjoint, t3 can be defined on the Weyl operators on H, by :
3t(Wo(z)) = )~(expi(za + + z+ a,,)) = expi(zjt(a +) + z+ J~(a.)) -- (2,11)
= eilZ,,2"+~+,~.,l+I2n,:~xt,,.,ll = W,,(z)eq~S~':z)x[,,.,I
An example of Xlo.t ] sastisfying the required conditions is the momentum operator P(X[0.t]) - Another
example is X[o.q = W~ - W, , which gives the usual free shift in Wiener space (not only in the increment
space el. Meyer ' s contribution to these proceedings ) . Other examples could be obtained using the position
or number processes or mixtures of them i.e., Weyl shifts of the form :
Ji (W,,(z) ) = W,,(z)W (zxi,,.,l ; e'zx~"' ) L ( 12)
(cf. the remark at the end of this note ).
(8.) THE SEMI-GROUP ASSOCIATED TO THE CHOICE X[ot] = P(X[--I)
The senti-group P~ , associated to the "free" evolution '~v is
~(Ho) )1.3(
P~ = E,,I(v';(a,,)) ; a,, ~
E,,] =~,® < ¢,( . )(P > : B(H,,)® B ~ B(It,)) ~ B(H,)® I (3.2)
Ill our case, choosing b = W,,(z) (z ~ C) and X[0.t I - P(XN.t]] , one finds
P~(W,(z)) = E,,,(,';(w,,(z))) = E,,j (w, (z)e. ~"~'~ ,~,,,,) = W,,(z)e- 2(m~) h (3.3)
tlence the Weyl operators are ill the domain of the generator of P~ and one has:
pt = exptL (3.4)
with
L(W,~(z)) = -2(Rez)~Ve,~(z) ; z e C) (3.5)
The explicit form of the generator can be obtained with the following semiheuristic, considerations:
w,~(z) = exp~(z,~ + ~+~,~) (3,6)
therefore
0 Wo(z) = (iz)W,,(z) ' _O W (z) = (iz+)W,,(z)
Oa + ' Oao "
hence
and therefore
(3.7)
Now~ iworf
I~,, ~+,] = 1 (3.8)
we deduce
a 3 (3.9)
[ao, • 1-- +aa [ a+ 8ao
In conclusion
)]2 1 + ]2 (3,16)
] = ibo- % , "
and since 7~[a(, - a ....
L=-(p, ]2:_(p, IP, ] (3.11)
So the free semigroup is a quasifree semigroup of the type considered by Lindbtad in I4}. Notice moreover
that, if f is a smooth function and Mf is multiplication by f in L2(R) , then with the identification p =7,Tz I a
one has
M 1
[P'MI}= 7 (~/~ (3.12)
llence
-[p, [p, M/ ~} = M(~:) (3.13)
which gives the right answer when we restrict our attention to the classical Wiener process.
(4.) v~'- Markovian cocycles : an example
Consider tile stochastic differential equation (SDE)
dU = ((Lo + X[o.q)dA + - (L + + X[o.fl)dA + Zdt) U (4,1)
the unitarity condition (using the Fock Ito table for dA , dA +) is
Z = iH - ~ 1 o L + X[o.fl + 21 (4.2)
with H self-adjoint. By shifting the equation (4.1) with H = 0 with the free shift v,, o we obtain the
equation for v~(U~) namely
1
dv','(Ut) = ((L(, + Xio,t+,l)dA+(t) - (L + + Xlo,t+~l)dA~(t ) - + L l + Xiv,~+,, 21 dt)v'~'(U~) (4.3)
where we have used the notation
dA~(t) = A(s + t + dr) - A(s + t) (4.4)
which means that, by definition
f(T Y~+~dA,(t) := f,f+T YtdA(t) (4.5)
Now, written in integral form, the equations (4.1} and (4.3) look respectively like
U, = + 1 ~' ((Lo + XIo.r])dA+(r ) - (L + + X[o.r])dA(r) - ~ t 1 + LX[(',d : 21 dr) rU (4.6 I
v:'(U,) = 1+ (Lo+X[o.~+rl)dA+(s+r)-(L2+Xio.,,+d)dA(s+r)-~ I i2+Xio.,,+d
So that v~(Ut) • U, and U,,+t satisfy the same SDE (in tile t-variable) with the same initial condition i.e. ,,U
at t = 0. Therefore , if the Xi0.tl-process is regular enough to assure the existence and uniqueness of the
solutions of the above SDE , it will follow that
(4.8)
,,;(ud . u,, = u,+,
which means that Ut is a '~v - Markovian cocycle . The formal unitarity of follows from (4.2) and in many
interesting cases the mfitarity can be effectively proved . Having the unitary cocycle, we can apply the FK
perturbation scheme to the free semigroup associated to '~v . Denoting ~o the generator of this semigroup,
a simple calculation shows that the formal generator of the perturbed semigroup
:: E,,j (u;. (4.9)
will be
[',, + L+b + + bL~, L+bL,, (4.10)
If the operators ~o , Lo are unbounded, the expression (4.10) will not be in general a well defined operator
• However, for Xlo.t I as in Section (2) , the operators L, + Xi0.t I ; + L + Xi0 N are always well defined and
therefore equation (1) makes sense and in some cases the conditions for the existence of a solution of this
equation are much weaker than those which allow to realize ~o + L +b + bLo + L+bL,, as a well defined
operator.
Applying the considerations above to the additive functional X[0.t] = P(x{o.t]) , for which the regularity
problems lnentioned above.can be solved with standard techniques, one can produce singular perturbations
of the noncommutative Laplacian in full analogy with the classical case.
The physical meaning of tile operator Xtt,.t } in (4.1) call be understood ill terms of Barchielli' s analysis
13[ : Xio.q is the input field ( the number process in Barchielli's paper ) which interacts with all apparatus
described by the operators Lo, Z ill (4.1) . The free evolution of the system is given by the time shift '~v and
equation (4.1) describes the interaction cocycle giving the evolution of the observables of the coupled system
(input field + apparatus) according to the scheme proposed in ]1[ . The advantage of the present approach
with respect to ]3[ is that, due to the v~'-cocycle property of the solution of (4.1), the interacting evolution
x ~ v~'(U+xUt) will now be a 1-parameter automorphism group, in agreement with tile basic principles of
quantum physics.
BIBLIOGRAPHY
1) ACCARDI L. Quantum stochastic processes ill : Statistical Physics and Dynamical Systems (Rigorous
results) J.Fritz, A.Jaffe,D.Szasz (eds.) Birkhauser 1985
2) ACCARDI L., SCHURMANN M., yon WALDENFELS W.: Quantum independent increment processes
on super algebras. Heidelberg preprint 1987
3) BARCHIELLI A. Input and output channels in quantum systems and quantum stochastic differential
equations. These Proceedings.
4) LINDBLAD G. Brownian motion of a quantum harmonic oscillator. Reports on Math. Phys. 10, (1976),
393-406.
4) MEYER P.A. A note on shifts and cocycles. These Proceedings.