covariant quantum markovian evolutions

Covariant quantum Markovian evolutionsA. S. Holevo Citation: Journal of Mathematical Physics 37, 1812 (1996); doi: 10.1063/1.531481 View online: http://dx.doi.org/10.1063/1.531481 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/37/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Translation-covariant Markovian master equation for a test particle in a quantum fluid J. Math. Phys. 42, 4291 (2001); 10.1063/1.1386409 Heat engines in finite time governed by master equations Am. J. Phys. 64, 485 (1996); 10.1119/1.18197 Practical measurement of diffusion constants in sintered zirconias by using a lightscattering method J. Appl. Phys. 69, 3016 (1991); 10.1063/1.348588 The use of Einstein’s coefficients to predict the theory of operation of a semiconductor laser J. Appl. Phys. 68, 3122 (1990); 10.1063/1.346407 Theory of lasers with intracavity optical mixing J. Appl. Phys. 68, 3114 (1990); 10.1063/1.346406

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Covariant quantum Markovian evolutionsA. S. Holevoa)Fachbereich Physik, Phillips-Universita¨t Marburg, Renthof 7, D-3550 Marburg, Germany

~Received 3 April 1995; accepted for publication 30 November 1995!

Quantum Markovian master equations with generally unbounded generators, hav-ing physically relevant symmetries, such as Weyl, Galilean or boost covariance, arecharacterized. It is proven in particular that a fully Galilean covariant zero spinMarkovian evolution reduces to the free motion perturbed by a covariant stochasticprocess with independent stationary increments in the classical phase space. Ageneral form of the boost covariant Markovian master equation is discussed and aformal dilation to the Langevin equation driven by quantum Boson noises isdescribed. ©1996 American Institute of Physics.@S0022-2488~96!01503-X#

I. INTRODUCTION

The classification of Galilean or Poincare covariant elementary systems is a cornerstone in themathematical foundations of quantum mechanics.1,2 The question of covariant irreversible evolu-tions ~such as unstable particles! was also raised some time ago. However, in that case the answerremained far from being complete; essentially quasi-free dynamical semigroups, corresponding toGaussian reservoirs were studied in detail~see Refs. 3 and 4 and the references therein!.

In this paper we fill this gap by giving a complete characterization of Galilean covariantquantum Markovian evolutions~with zero spin!. We prove in particular that in the case of the fullGalilean covariance~Sec. III! these reduce to the free motion perturbed by a covariant stochasticprocess with stationary independent increments in the classical phase space. The reservoir of aGalilean covariant zero spin system is thus a classical noise, but not necessarily Gaussian: thevariety of possible noises is described by the Levy–Khinchin formula. This result is based on thecharacterization of Weyl covariant dynamical semigroups given in Sec. II.

In the second part of this paper~Secs. IV and V! we consider much a broader class ofevolutions, covariant only with respect to the Galilean boosts. The boost covariance is of funda-mental importance as this is essentially the symmetry of particle motion in arbitrary potential field.In Sec. IV the general boost covariant Markovian master equation is discussed, based on a non-commutative generalization of the Levy–Khinchin formula obtained in Ref. 5, which is presentedhere in a more accessible form. Then in Sec. V the quantum Langevin equations are given, dilatingthis master equation with Boson quantum noise.

Being covariant with respect to noncompact symmetry groups, the generators of evolutionsunder consideration are, as a rule, unbounded operators. Existence and uniqueness of a dynamicalsemigroup with the generator given by a densely defined operator expression become nontrivialproblems in this case. A recently developed framework for these problems, briefly outlined below~see also Appendix A!, shifts the accent from the semigroup to the Markovian master equation itsatisfies.

Let B~H! be the algebra of all bounded operators in a Hilbert spaceH. By a dynamicalsemigroupin B~H! we shall call a semigroupFt , t>0, of normal completely positive maps inB~H!, weak*-continuous, satisfyingF05Id @the identity map ofB~H!#, andF t[ I ]<I ~the unitoperator inH!. HereFt is calledunital if F t[ I ]5I . Let T ~H! be the Banach space of trace-classoperators inH, so thatT ~H!*5B~H!. There is a unique strongly continuous preadjoint semi-

a!Permanent address: Steklov Mathematical Institute, Vavilova 42, 117966 Moscow.Electronic mail: [email protected]

0022-2488/96/37(4)/1812/21/$10.001812 J. Math. Phys. 37 (4), April 1996 © 1996 American Institute of Physics


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groupC t5(F t)*in T ~H!, such thatC t*5F t . Denoting its generatorL

*one has the differ-

ential equation

d

dtTr rF t@X#5Tr L* @r#F t@X#, XPB~H!. ~1.1!

If r runs through domL*

,T ~H!, then this equation determinesFt uniquely. However, inapplicationsL

*is usually given only on a dense subspaceD. Typically a dense domainD,H is

given and

D5 lin$r:r5uf&^cu, f,cPD%, ~1.2!

while L*is defined by an expression of thestandard form:

L* @r#5(i

uL jf&^L jcu2uKf&^cu2uf&^Kcu, r5uf&^cu, ~1.3!

whereL j ,K are operators defined onD ~or by similar expression with the sum replaced by anintegral!. Thedissipativity condition

(j

iL jci2<2 Re cuKc&, cPD , ~1.4!

is assumed~implying in particular thatK is accretive: RecuKc&>0, cPD!. Equation~1.1! thenreduces to the equation for the matrix elements ofF t[X]:

d

dt^cuF t@X#f&5L~c;F t@X#,f!; f,cPD , ~1.5!

where

L~c;X;f!5Tr L* @ uf&^cu#X5(j

^L jcuXLjf&2^KcuXf&2^cuXKf& ~1.6!

is the form-generator.6,7 We call ~1.5! the (backward) Markovian master equation.If D is a core forL

*, then this equation determinesFt uniquely, otherwise it may have a

nonunique solution. Of special interest is the case of aunital generator, satisfying TrL*@c&^fu#[0

or

(j

iL jci252 Re cuKc&, cPD . ~1.7!

Then under the condition thatK is maximal accretive andD is an invariant domain fore2Kt, t>0,there exists a dynamical semigroupFt

` giving theminimal solutionof Eq. ~1.5! in the sense thatfor any other solutionFt the differenceF t2F t

` is completely positive~see Appendix A!. IngeneralFt

` may not be unital; however, if it is, thenFt` is the unique solution of~1.5!.8–10

Under the additional assumption that operatorsL j* , K* are defined on a dense domainD* ,and

(j

iL j*ci2,`, cPD* , ~1.8!

one can write also theforwardMarkovian master equation for the preadjoint semigroupCt :

1813A. S. Holevo: Covariant quantum Markovian evolutions

J. Math. Phys., Vol. 37, No. 4, April 1996


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d

dt^fuC t@r#c&5L* ~f;C t@r#;c!, f,cPD* , ~1.9!

where

L* ~f;r;c!5Tr rL@ uc&^fu#5(j

^L j*furL j*c&2^K*furc&2^furK*c&. ~1.10!

HereL denotes the generator of the semigroupFt . AssumingK* to be maximal accretive andD*to be an invariant domain fore2K* t, t>0, we can prove thatC t

`5(F t`)*is the minimal solution

of the forward equation~see Appendix A!. Thus the situation is similar to that for theKolmogorov–Feller differential equations in the theory of Markov processes.11 If Ft

` is not unital,then there is a positive probability of ‘‘explosion,’’ i.e., exit from the phase space for the corre-sponding Markov process. Additional ‘‘boundary conditions’’ are required to specify the solution,which amounts to a certain maximal extension ofL

*from D.

From the results in Secs. II and III it follows in particular that an explosion can never occurfor the fully Galilean covariant Markovian evolutions. However, this is possible for the boostcovariant evolutions potentially interesting for applications. The mathematical study of these is farfrom being complete; the presentation in Secs. IV and V is thus on a formal level, and we onlyoutline a few rigorous results and a number of problems waiting for their mathematical solutions.

II. WEYL COVARIANT DYNAMICAL SEMIGROUPS

Let Z be a finite-dimensional symplectic space with a nondegenerate symplectic formD(z,z8)and letz→W(z) be an irreducible representation of the Weyl–Segal CCR:

W~z!W~z8!5exp iD~z,z8!W~z8!W~z! ~2.1!

in a Hilbert spaceH.Proposition 1:A dynamical semigroupFt in B~H! is Weyl covariant,

F t@W~z!*XW~z!#5W~z!*F t@X#W~z!, zPZ, ~2.2!

if and only if

F t@W~z!#5W~z!etl ~z!, ~2.3!

wherel (z) is a continuous conditionally positive definite function satisfyingl ~0!<0.Proof: By weak*-continuity the semigroupFt is uniquely defined by the values ofF t[W(z)],

zPZ. From ~2.2! and ~2.1! F t[W(z8)] satisfies the same relation~2.1! as W(z8), henceF t[W(z8)]W(z8)* commutes withW(z), zPZ. Since the representationz→W(z) is irreducible,

F t@W~z!#5W~z!f t~z!,

wheref t(z) is a complex function. From the definition of dynamical semigroup, it is continuousin t, z and satisfiesf t1s(z)5f t(z)fs(z), f0(z)51, ft~0!<1. Hencef t(z)5expt l (z), with l (z)continuous andl ~0!<0. From complete positivity ofFt it follows

12,13 that f t(z) is positivedefinite inz for all t>0. Indeed for any finite collections$zj%,Z, $cj%,C,

(j ,k

c jckf t~zj2zk!5(j ,k

c jck^W~zj !cuF t@W~zj !W~zk!* #W~zk!c&>0,

1814 A. S. Holevo: Covariant quantum Markovian evolutions



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wherecPH is arbitrary unit vector. Then by a theorem of Schoenbergl (z) is conditionallypositive definite.

Conversely, let the operatorF t[W(z)] be defined by~2.3!. There exists a stochastic processzt , t>0, with stationary independent increments inZ, having the characteristic function

MeiD~z,z t!5et„l ~z!2 l ~0!… ~2.4!

~see, e.g., Refs. 11 and 14!. Then for allXPB~H!

F t@X#5MW~z t!*XW~z t!etl ~0!, ~2.5!

since for X5W(z) this follows from ~2.1! and ~2.4!, and then can be extended byweak*-continuity. One easily sees that the right-hand side defines Weyl-covariant dynamical semi-group; in particular the semigroup property follows from the fact thatzt has stationary independentincrements. h

ObviouslyFt is unital if and only if l ~0!50. In this case the relation~2.5! gives a dilation ofthe Markovian evolutionFt to the unitary stochastic evolution

X→X~ t !5W~z t!*XW~z t!. ~2.6!

Defining the canonical observablesR(z) as self-adjoint operators, linearly depending onz, andsatisfyingW(z)5exp iR(z), one has from~2.6! R(z;t)5R(z)1D(z,z t)I . This is equivalent to theLangevin–Heisenberg equation

dR~z;t !5D~z,dz t!I , R~z;0!5R~z!, ~2.7!

which due to~2.1! can be viewed as an infinitesimal canonical transformation implemented by theinteraction HamiltoniandHint52R(dz t).

Let us describe the general master and Langevin equations for the evolution~2.5!. By themultidimensional Levy–Khinchin formula11,14

l ~z!2 l ~0!5 ib~z!21

2a~z!1E

z8Þ0@eiD~z,z8!212 iD~z,z8!1e~z8!#n~dz8!, ~2.8!

whereb(z) is real linear,a(z) is non-negative quadratic function onZ, andn(dz) is positive Levymeasure onZ\$0% satisfying

EzÞ0

@1e~z!uzu21„121e~z!…#n~dz!,`. ~2.9!

Here 1e(z) is the indicator of the setuzu,e for some fixed normu•u in Z, ande.0. One has

b~z!5D~z,z0!, a~z!5(j51

r

D~z,zj !2, ~2.10!

wherez0PZ, $zj% is a linearly independent system inZ, r<dim Z. The processzt has the Itorepresentation14

dz t5dz tc1E

zÞ0z1e~z!P ~dzdt!1E z@121e~z!#P~dzdt!, ~2.11!

whereztc is the Wiener process inZ with the moments




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MD~z,z tc!5b~z!dt, DD~z,z t

c!5a~z!dt,

P(dzdt) is the Poisson random measure such that

MP~dzdt!5n~dz!dt,

andP (dzdt)5P(dzdt)2n(dz)dt is the compensated Poisson random measure. Formula~2.11!gives the decomposition ofzt into the continuous, ‘‘small-jumps’’ and ‘‘big-jumps’’ components,the stochastic integral for small jumps~of magnitudesuzu,e! converging in the mean-squaresense, while for big jumps converges pathwise, via the condition~2.9!.

Consider the domain

D5 ùzPZ

domR~z!2.

ThenD is a dense domain inH, invariant underW(z). LetD be defined by~1.2! with this domainD .

Proposition 2:The dynamical semigroup~2.5! is the unique solution of the Markovian masterequation~1.5! with

L* @r#5 l ~0!r1 i @R~z0!,r#21

2 (j51

r

†R~zj !,@R~zj !,r#‡

1Ez8Þ0

~W~z8!rW~z8!*2r2 i @R~z8!,r#1e~z8!!n~dz8! ~2.12!

for rPD. The domainD is a core forL*, andFt is unital if and only ifL

*is such.

The random density operator

r~ t !5W~z t!rW~z t!* etl ~0!, rPD, ~2.13!

satisfies the Langevin–Schroedinger equation

dr~ t !5 i @R~dz tc!,r~ t !#1E

z8Þ0„W~z8!r~ t !W~z8!*2r~ t !…P ~dz8dt!1L* @r~ t !#dt,

~2.14!

understood as stochastic equation in the weak operator topology.Proof: From ~2.3!

d

dtTr rF t@W~z!#5 l ~z!Tr rF t@W~z!#, rPdomL* ,

where

Tr L* @r#W~z!5 l ~z!Tr rW~z!.

For rPD

D~z,z8!Tr rW~z!5Tr@R~z8!,r#W~z! ~2.15!

~see Ref. 15!, hence also

D~z,z8!2 Tr rW~z!5Tr†R~z8!,@R~z8!,r#‡W~z8!. ~2.16!




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By using ~2.8!, ~2.9!, and~2.1! we obtain~2.12!.To show thatD is a core forL

*, we remark thatrPD impliesr(t)PD, sinceD is invariant

under the Weyl automorphismsr→W(z)rW(z)* . Assuming thatD is not a core forL*, we can

find l0.0 such that~l0 Id2L*!~D! is not dense inT ~H! ~see Ref. 16, theorem X.49!. Then there

existsX0PB~H!, X0Þ0, such that Tr„l0r(t)2L*@r(t)#…X050. From~2.5! and ~1.1!

d

dtTr rF t@X#5Tr rF t†L@X#‡5M Tr rW~z t!*L@X#W~z t!e

tl ~0!

5M Tr r~ t !L@X#5M Tr L* @r~ t !#X

for XPdomL andrPD. Integrating,

Tr rF t@X#2Tr rX5E0

t

M Tr L* @r~s!#Xds

for XPD and hence for allXPB~H! by weak*-continuity. TakingX5X0, we obtain

Tr rF t@X0#5el0t Tr rX0 , rPD,

which contradicts the contraction property ofFt . ThusD is a core forL*.

It is sufficient to establish the relation~2.14! for r5uc&^cu, cPD . Thenr t5uc t&^c tu, wherect5exp 1

2[ t l (0)1 i*0t D(zs ,dzs)]W(z t)c. The vectorct satisfies the following exponential sto-

chastic differential equation

dc t5H iR~dz tc!1E

zÞ0„W~z!2I …P ~dzdt!

1F12 S l ~0!I2(j51

r

R~zj !2D 1E

zÞ0„W~z!2I2 iR~z!1e~z!…n~dz!GdtJ c t . ~2.17!

To prove this formula we can use a concrete form of the CCR, e.g., the so-called regular repre-sentation in the spaceL2(Z):

W~z8!c~z!5c~z1z8!expi

2D~z,z8!,

R~z8!c~z!5@2 i“~z8!1 12D~z,z8!#c~z!.

Then

c t~z!5c~z1z t!exp1

2 F t l ~0!1 i E0

t

D~z1zs ,dzs!G .By using the Ito formula14 one arrives at~2.17!. Then the stochastic equation~2.14! for the matrixelements cur(t)f&5^cuc t&^c tuf& follows from ~2.17! by application of the Ito product rule.

Apparently unitality of the generator~2.12! implies unitality ofFt , since both are equivalentto l ~0!50. h

The Langevin equation~2.14! gives the dilation of the dynamical semigroupFt with theclassical Wiener process and Poisson random measure as the driving noises. The structure of thegenerator~2.12! is in formal agreement with a result of Ref. 17, in which finite-dimensional




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dynamical semigroups admitting dilation with a classical noise were described. It was shown therethat the generator of such semigroup is a sum of ‘‘Gaussian’’ and ‘‘Poissonian’’ terms

†L, @L, X#‡, V*XV2X,

whereL is Hermitean andV is unitary operator. In our case we have unbounded self-adjointoperators in the ‘‘Gaussian’’ part and the sum of ‘‘Poissonian’’ terms is replaced by integral overall possible magnitudes of jumps. Also the ‘‘small-jumps’’ correction@the commutator term underthe integral in~2.12!# is peculiar to the infinite-dimensional case.

The generator~2.12! has the standard form since it can be represented as

L* @r#5 l ~0!r1(j51

r

R~zj !rR~zj !1EzÞ0

@W~z!21e~z!I #r@W~z!21e~z!I #* n~dz!2Kr2rK* ,

where

K51

2 (j51

r

R~zj !21E

zÞ0H 1e~z!@ I2W~z!1 iR~z!#1@121e~z!#

I

2 J n~dz!.

From these formulas one sees that the analog of condition~1.8! holds with D*5D . ThusC t5(F t)*

is the unique solution of the forward Markovian master equation~1.9! with

L@X#5 l ~0!X1 i @X,R~z0!#21

2 (j51

r

†@X, R~zj !#,R~zj !‡

1EzÞ0

$W~z!*XW~z!2X2 i @X, R~z!#1e~z!%n~dz!.

III. GALILEAN COVARIANT EVOLUTIONS

Let ~j,t!, jPR3, tPR, be a point in the four-dimensional nonrelativistic space–time, and let(x,v,R,t):~j,t!→~j8,t8! be the Galilei transformation

z85Rj1x1vt, t85t1t, ~3.1!

wherexPR3 is the space shift of the reference system,vPR3 is the Galilean boost,R is the matrixof rotation inR3, andtPR is the time shift. The Galilean covariant elementary quantum system isgiven by an irreducible projective unitary representation (x,v,R,t)→U(x,v,R,t) of the group oftransformations~3.1!, satisfying

U~x1 ,v1 ,R1 ,t1!U~x2 ,v2 ,R2 ,t2!5expim

2~v1•R1x22x1•R1v22t2v1•R1v2!

3U~R1x21x11t2v1 ,R1v21v1 ,R1R2 ,t11t2!, ~3.2!

wherem is the mass constant.1,2 In particular, the kinematics is described by the unitary repre-sentations

~x,v !→Wx,v5U~x,v,E,0!, R→UR5U~0,0,R,0!,

whereE is the unit matrix,Wx,v is projective representation ofR33R3, satisfying the Weylcommutation relation




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Wx1 ,v1Wx2 ,v2

5exp im~v1•x22x1•v2!Wx2 ,v2Wx1 ,v1

, ~3.3!

andUR is a unitary representation of the rotation group, satisfying

URWx,v5WRx,RvUR . ~3.4!

The reversible dynamics is given by the one-parameter unitary groupt→Ut5U(0,0,E,2t), sat-isfying

Wx,vUt5UtWx2vt,v , ~3.5!

URUt5UtUR . ~3.6!

Restricting to the case of zero-spin elementary system, we assume that the representationWx,v isirreducible. The well-known solution of~3.3!–~3.6! is

Wx,v5exp i ~mv•Q2x•P!, ~3.7!

Ut5expS 2 i tuPu2

2m D , ~3.8!

whereQ5(Q1 ,Q2 ,Q3), P5(P1 ,P2 ,P3) are canonical position and momentum observables.Turning to the Markovian dynamics, we assume, following Ref. 3, that it is described by a

dynamical semigroupFt , t>0 in B~H!. Then the covariance conditions~3.5! and ~3.6! arereplaced by

F t@Wx,v* XWx,v#5Wx2vt,v* F t@X#Wx2vt,v , ~3.9!

F t@UR*XUR#5UR*F t@X#UR . ~3.10!

LetL*be the generator of the preadjoint semigroup~Ft!*

. LetD,H be the dense domain

D5 ùx,vPR3

dom~mv•Q2x•P!2, ~3.11!

and letD,B~H! be the domain defined by the relation~1.2!.Theorem: Let Ft be a dynamical semigroup satisfying the covariance conditions~3.9! and

~3.10!, and assumeD,domL*. Then

F t@Wx,v#5Wx2vt,v exp E0

t

l ~x2vs,v !ds, ~3.12!

where

l ~x,v !5 l 021

2~aPPuxu212aPQmx•v1aQQm

2uvu2!

1E Euxu21uyu2.0

@eim~x•v82v•x8!21#n~dx8dv8!%. ~3.13!

Here l 0<0, the real 232 matrix @aPQ

aPPaQQ

aPQ# is positive definite andn is a positive measure on

R33R3\$0%, satisfying the Levy condition




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E E0,uxu21uvu2,e

~ uxu21uvu2!n~dxdv !1E Ee<uxu21uvu2

n~dxdv !,`, ~3.14!

and invariant in the sensen„(Rdx)(Rdy)…5n(dxdy) for all rotationsR. @The integration in~3.13!should be taken first over the spheresuxu5r 1, uvu5r 2, and then with respect tor 1 ,r 2 .#

This semigroup is the unique solution of the Markovian master equation~1.5! with

L* @r#5 l 0r2 i F uPu2

2m,rG2

1

2 (j51

3

$aPP†Pj ,@Pj ,r#‡12aPQ†Pj ,@Qj , r#‡1aQQ†Qj , @Qj , r#‡%

1E Euxu21uvu2.0

$Wx,vrWx,v* 2r%n~dxdv !, ~3.15!

MoreoverD is a core forL*, andFt is unital if and only ifL

*is such.

Proof: From the covariance relation~3.9! and irreducibility ofWx,v it follows as in the proofof Proposition 1

F t@Wx,v#5Wx2vt,vf t~x,v !,

wheref t(x,v) is a continuous function. The semigroup property implies~cf. Ref. 13!

f t1s~x,v !5fs~x2vt,v !f t~x,v !. ~3.16!

From the assumptionD,domL*it follows that

f t~x,v !5^fuF t@Wx,v#c&^fuWx2vt,vc&21, f,cPD ,

is differentiable with respect tot; the relation~3.16! then implies

d

dtf t~x,v !5 l ~x2vt,v !f t~x,v !,

wherel (x,v)5(d/dt)f t(x,v)u t50, whence~3.12! follows.On the other hand, differentiating~3.9! at t50, we have forrPD

Wx,v* L* @Wx,vrWx,v* #Wx,v5L* @r#1 i @v•P,r#. ~3.17!

This is an inhomogeneous linear equation with respect toL*, thus we can write

L* @r#5L*0 @r#2 i F uPu2

2m, rG , ~3.18!

where the second term is the generator of the free particle motion, giving a particular solution of~3.17!, andL

*0 is the general solution of the homogeneous equation

Wx,v* L*0 @Wx,vrWx,v* #Wx,v5L

*0 @r#, ~3.19!

expressing Weyl covariance ofL*0 . Applying proposition 2@relation ~2.12! for a Weyl covariant

generator# and using rotational covariance~3.10! results in the expression~3.13!, wherel 05 l (0,0)~single commutator terms disappear due to rotational covariance, provided the order of integrationis as stated in the theorem!. From ~3.12! and ~3.17!,

Tr L*0 @r#Wx,v5 l ~x,v !Tr r,




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where~3.15! follows, taking into account~3.3! and ~2.16!.We now describe a stochastic dilation of the semigroupFt . Let (h t

j t ) be a classical stochastic

process with stationary independent increments inR33R3, defined by the characteristic function

M exp i ~mv•j t2x•h t!5exp t„l ~x,v !2 l 0….

Consider the stochastic differential equations

dQt5Pt

mdt1dj t , dPt5dh t , ~3.20!

with initial conditionsQ05Q andP05P. The solution of these equations

Qt5Q1Pt

m1j t1

1

m E0

t

hs ds, Pt5P1h t . ~3.21!

We define stochastic unitary operators

Ut~j,h!5W„j t1~1/m!*0

t hsds…,ht/mUt5US j t1

1

m E0

t

~hs2h t!ds,h t

m,E,2t D ,

so thatUt(j,h)* (PQ)Ut(j t ,h) 5 (Pt

Qt), and prove that

F t@X#5MUt~j,h!*XUt~j,h!etl0. ~3.22!

It is sufficient to establish this forX5Wx,v . Then

Ut~j,h!*Wx,vUt~j,h!5exp i ~mv•Qt2x•Pt!5Wx2vt,v exp i Fmv•S j t11

m E0

t

hs dsD 2x•h tG .However,

M exp i Fmv•S j t11

m E0

t

hs dsD 2x•h tGetl05M exp i E0

t

@mv•djs2~x2„v~ t2s!…•dhs!#etl0

5exp E0

t

l „x2v~ t2s!,v…ds5f t~x,v !,

and the relation~3.22! follows from ~3.12!.To prove thatD is a core forL

*, we observe thatD is invariant underU(x,v,R,t) and hence

underUt~j,h!. Then the rest can be proved as in proposition 2. h

The equations~3.20! are the Langevin–Heisenberg equations for the canonical observablesP,Q. They can be interpreted as infinitesimal canonical transformation implemented by theHamiltonian

dHt5uPu2

2mdt1P•dj t2Q•dh t .

The random density operator

r~ t !5Ut~j,h!rUt~j,h!* etl0




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satisfies the Langevin–Schroedinger equation similar to~2.14!.

IV. BOOST COVARIANT MARKOVIAN MASTER EQUATIONS

If we abandon covariance with respect to space shifts and rotations, we are left with theboostcovariance, the condition obtained from~3.9! by puttingx50:

F t@Vv*XVv#5Vv*U2vt* F t@X#U2vtVv , ~4.1!

where

v→Vv5exp~ imv•Q!, x→Ux5exp~2 ix•P! ~4.2!

are the unitary representations of the shifts in velocity and position, respectively. The generatorL

of the semigroupFt satisfies the equation@the following formal calculation can be made preciseby using the form-generator~1.6! with the appropriate domainD#

VvL@Vv*XVv#Vv*5L@X#1 i @v•P,X#. ~4.3!

The general solution of this equation

L@X#5L0@X#1 i @H,X#, ~4.4!

whereH5uPu2/2m1b•P1U (Q), U (Q) being a real function, andL0 is a solution of thehomogeneous equation

VvL0@Vv*XVv#Vv*5L0@X#. ~4.5!

By making a gauge transformation we replaceP1mb with P, so that

H5uPu2

2m1U~Q!, ~4.6!

whereU~•! is again a real function.The mathematical study of Eq.~4.5!, under the regularity assumption that an analog of the

conditions~1.7! and ~1.8! holds withD*5D5C02~R3! in the Schroedinger representation, is pre-

sented in Ref. 5. The main tool of this study is harmonic analysis of operator-valued cocycles ofthe symmetry group. The general solution has the Levy–Khinchin structure similar to~2.12!:

L0@X#5K0~Q!+X1L1@X#1L2@X#1L3@X#, ~4.7a!

whereK0~•! is a nonpositive function~vanishing for a unital generator!, + means the Jordanproduct, andL1, L2, L3 are ‘‘continuous,’’ ‘‘big-jumps,’’ and ‘‘small-jumps’’ components, re-spectively. The continuous component is

L1@X#5 (k51

r

„P k1Lk~Q!…*X„P k1Lk~Q!…2K1*X2XK1 , ~4.7b!

wherer<3, P k5( j513 bk jPj ~bk jPR!, Lk~•! are complex functions, and

K151

2 (k51

r

„P k212P kLk~Q!1uLk~Q!u2…. ~4.8a!




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The essential ingredient of the discontinuous components are a positive Levy measurem(dx) onR3, and a family of complex functionsL jx(•); j51,2,...;xPR3, satisfying

ER3(j

uL jx~• !u2m~dx!,`. ~4.9!

The simpler big-jumps component has the form

L2@X#5Ee<uxu

(jL jx~Q!*Ux*XUxL jx~Q!2K2+X, ~4.7c!

where

K251

2 Ee<uxu

(j

uL jx~Q!u2m~dx!, ~4.8b!

while the small-jumps component is

L3@X#5Euxu,e

(j„UxL jx~Q!1~Ux2I !v j~x!…*X„UxL jx~Q!

1~Ux2I !v j~x!…m~dx!2K3*X2XK3 , ~4.7d!

where

K35Euxu,e

(j

H ~ I2Ux2 ix•P!uv j~x!u21~ I2Ux!L jx~Q!v j~x!11

2uL jx~Q!u2J m~dx!,

~4.8c!

andv j (x) are complex functions satisfying

Euxu,e

uxu2(j

uv j~x!u2m~dx!,`. ~4.10!

Several remarks are in order.~1! The shift covariance~4.5! of the expressions~4.7b!–~4.7d! can be checked by using the

Weyl CCR

UxVv5e2 imx•vVvUx ~4.11!

and its consequence

Vv*PVv5P1mv. ~4.12!

The regularity assumption onL implies certain local properties of the functionsK0~•!,U~•!,L j (•),L jx(•) ~see Ref. 5!, which are omitted here.

~2! In concrete problemsL usually reduces to one of its components. Particular generators ofthe formL1 or L2 appeared in Refs. 18 and 19 as a result of taking special~weak-coupling orlow-density! Markovian limit for an open quantum system. To our knowledge the general form ofL1 andL3 was not observed before. Some insight into the physical meaning of the functionsL j ~•! andL jx~•! can be obtained from the Langevin equations to be described in Sec. V.

The termi @b•P1U(Q), X#, incorporated into the Hamiltonian component in~4.4!, may bealso included in any ofL j . The pointx50 is not excluded from the integral in the small-jumpscomponent. Therefore, ifm(dx) has positive massm0 at x50, it contributes the ‘‘zero-jump’’ term




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m0(j

@L j0~Q!*XLj0~Q!2uL j0~Q!u2+X#

to L3. Inclusion of this term intoL3 is a convention and it may appear inL1 or L2 as well.~3! By definingv j (x)50 for uxu>e, one can write the discontinuous partL2,35L21L3 in

the form ~4.7d! with the integral extended to the wholeR3. On the other hand, if the functionsv j (x) are such that

Euxu,e

(j

uv j~x!u2m~dx!,`, ~4.13!

then by replacingL jx(•)1v j (x) with L jx~•! and by redefining the Hamiltonian term it is possibleto write the discontinuous part in the form~4.7c! with the integral extended toR3.

~4! Previous remarks show that the decomposition~4.7a! is not unique, the nonuniquenessbeing essentially related to the fact that the operatorsL j (Q) andL jx(Q) are defined up to scalaradditive terms. The transformationL j (Q)→L j (Q)1cj , L jx(Q)→L jx(Q)1v j (x) @with v j (x)satisfying~4.13!# does not change the form of the components, provided the proper compensationin the Hamiltonian term is made.8

~5! The shift-covariance~4.5! implies that the maximal Abelian algebra of operatorsX5 f (Q)is invariant under the corresponding evolution. LetfPC2~R3!. Using formulas

i @Pk , f ~Q!#5] f ~Q!

]Qk, Ux* f ~Q!Ux5 f ~Q1x!, ~4.14!

one obtains

L0@ f ~Q!#5K0~Q! f ~Q!1 (k51

3

bk~Q!] f ~Q!

]Qk11

2 (l ,k51

3

s lk

]2f ~Q!

]Ql]Qk

1E F f ~Q1x!2 f ~Q!21e~x!(k51

3

xk] f ~Q!

]QkGm~dxuQ!, ~4.15!

where

bk~Q!52(j51

r

bk j Im L j~Q!1Euxu,e

xk(j

@ uL jx~Q!1v j~x!u22uv j~x!u2#m~dx!,

~4.16!

s lk5(j51

r

b j lb jk , m~dxuQ!5(j

uL jx~Q!1v j~x!u2m~dx!.

This is the generator of a classical~sub-!Markov process in the position spaceR3 ~called processwith locally independent increments20!. It comprises diffusion with the driftb j (Q) and the con-stant diffusion matrixslk , and jump process with intensitiesm(dxuQ) for jumps of magnitudesxfrom a pointQ.

~6! We shall finally briefly comment on the existence and uniquness of the solution of theboost covariant Markovian master equation~1.5!. According to theorem A1~see Appendix A!, toestablish existence of the minimal solution one has to prove that the operator of the form

K5 iH1K01K11K21K3 ,




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whereH is given by~4.6!, andK1 ,K2 ,K3 by ~4.8a!–~4.8c!, is maximal accretive with a coreD .Although this problem resembles the well-studied problem of essential self-adjointness for quan-tum mechanical Hamiltonians~see, e.g., Ref. 16!, there are only few specific results on it~see Ref.8 for the caseK5K1 in one dimension!.

AssumingK0(Q)50, uniqueness reduces to the unitality of the minimal solution. There is ageneral criterion for covariant dynamical semigroups,7 which implies that forL5L0 unitality isequivalent to nonexplosion of the classical Markov process with the generator~4.15!. Applicationof this result to the boost covariant Markovian master equation will be considered elsewhere.

V. THE QUANTUM LANGEVIN EQUATIONS

We start with a formal description of the quantum noise, which will give driving terms in theLangevin equation, corresponding to the boost covariant generator~4.4! and ~4.7a!. For the con-tinuous componentL1 it arises from the the representation of CCR

@aj~ t !,al†~s!#5d j ld~ t2s!, j ,l51,...,r , t,sPR1 , ~5.1!

in the Fock spaceH15G„H1^L2~R1!…, whereH15Cr ~see Appendix B!. The Ito stochasticdifferentials

dAj~ t !5Et

t1dt

aj~s!ds, dAj†~ t !5E

t

t1dt

aj†~s!ds, ~5.2!

with dt.0, obey the quantum Ito rule@cf. ~B2!#, to be described below.The driving noises for the discontinuous partL2,3 live in the Fock space

H2,35G~H2,3^L2~R1!…, whereH2,35L2~R3,m!^l 2, and arise from the CCR:

@ajx~ t !, aly† ~s!#5d j ldm~x2y!d~ t2s!, j ,l51,2,..., x,yPR3;t,sPR1 , ~5.3!

wheredm is the delta-function inL2~R3,m!. The stochastic differentials

dAj ,dx~ t !5Et

t1dt

ajx~s!dsm~dx!, dAj ,dx† ~ t !5E

t

t1dt

ajx† ~s!dsm~dx!,

~5.4!

dL j ,dx~ t !5Et

t1dt

ajx† ~s!ajx~s!dsm~dx!

obey the corresponding Ito rule; to express it we shall use more spectacular formal notation for allstochastic differentials:

dAj~ t !5aj~ t1 !dt,...,dAj ,dx5ajx~ t1 !m~dx!dt,...,

dL jx~ t !5ajx† ~ t1 !ajx~ t1 !dtm~dx!, ~5.5!

having in mind that the symbolsaj (t1),..., have meaning only in combination withdt. Then theIto rule takes the form

aj~ t1 !aj†~ t1 !~dt!25dt, ~5.6a!

ajx~ t1 !ajx† ~ t1 !~dt!2m~dx!5dt, ~5.6b!




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with all other products containing (dt)2 vanishing. Due to the CCR~5.1!, ~5.3! and the definitionof the Ito stochastic differentials~5.2! and ~5.4! the symbolsaj (t1),... must be considered ascommuting with any operator depending on system observables in the Hilbert spaceH and on thevalues of all noisesaj (s),..., at timess<t.

To describe interaction of the system with the noise, we introduce the family of operatorsUt ;tPR1 , in the Hilbert spaceH^G, whereG5G„~H1%H2,3!^L2~R1!… is the full Fock space,satisfying the left quantum stochastic differential equation of the type~B4!:

dUt5dZtUt , ~5.7!

where

dZt5Hdt1dZt11dZt

2,3, ~5.8a!

with

dZt15 (

k51

r

@ak†~ t1 !„P k1Lk~Q!…2h.c.#dt2K1dt, ~5.8b!

dZt2,35E (

j$~Ux2I !ajx

† ~ t1 !ajx~ t1 !1ajx† ~ t1 !@UxL jx~Q!1~Ux2I !v j~x!#

2@L jx~Q!1v j~x!~ I2Ux* !#* ajx~ t1 !%m~dx!dt2~K21K3!dt, ~5.8c!

where from now on we systematically use the convention thatv j (x)50 for uxu>e ~see Remark 3in Sec. IV!.

These equations are written by analogy with Eq.~B5!, where the coefficientsW,L,K are takenfrom the generatorsL1,L2,L3 @see~4.7b!–~4.7d!#. These coefficients satisfy the formal condi-tions for the unitarity ofUt . In fact, by applying the exponential formula~B6!, we get therepresentation

Ut5expQS 2 i E0

t

dHsD , ~5.9!

where

dHt5Hdt1dHt11dHt

2,3 ~5.10a!

is formally self-adjoint with

dHt15 (

k51

r

$ i @„ak~ t1 !1Lk~Q!…†„P k1Lk~Q!…2h.c.#%dt, ~5.10b!

dHt2,35E (

jH @h.c.#~x•P!@ajx~ t1 !1„~ I2eix•P!211~ ix•P!21

…@L jx~Q!1v j~x!##

1L jx~Q!* S x•P21

2ctg

x•P

2 DL jx~Q!2~x•P!uv j~x!u2J m~dx!dt, ~5.10c!

where one should keep in mind thate2 ix•P5Ux and @h.c.# means Hermitean conjugate of theexpression in squared brackets.




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AssumingUt unitary, the time evolution of a system observableX is

X~ t !5Ut* ~X^ I G!Ut ,

whereI G is the unit operator in the noise Fock space. From~5.7a! and~B.9!, we have the quantumstochastic differential equation

dX~ t !5dZ~ t !*X~ t !Z~ t !1Z~ t !*X~ t !dZ~ t !1dZ~ t !*X~ t !dZ~ t !, ~5.11!

wheredZ(t) 5 Ut* dZtUt . Since

Ut* aj~ t1 !Ut5aj~ t1 !, Ut* ajx† ~ t1 !Ut5ajx

† ~ t1 !,

dZ(t) is given by the same expression asdZt with P,Q replaced withP(t), Q(t). Due to the factthat

dZj~ t !*X~ t !dZk~ t !50 for jÞk

@the driving noises indZj (t) 5 Ut* dZtjUt involve different modes#, we obtain the Langevin–

Heisenberg equation in the form

dX~ t !5 i @H~ t !, X~ t !#dt1dX1~ t !1dX2,3~ t !, ~5.12a!

whereH(t)5uP(t)u2/2m1U„Q(t)…, anddXj (t) are calculated from~5.11! with dZ(t) replacedwith dZj (t). By analogy with~B8! we obtain

dX15 (k51

r

$@ P 1ReLk~Q!, X#„ak~1 !2ak†~1 !…

2 i @ Im Lk~Q!, X#„ak~1 !1ak†~1 !…1L1@X#%dt, ~5.12b!

dX2,35E (j

$~Ux*XUx2X!ajx† ~1 !ajx~1 !1@„L jx~Q!1v j~x!…*Ux* , X#Uxajx~1 !

2Ux* @Ux„L jx~Q!1v j~x!…,X#ajx† ~1 !%m~dx!dt1~L2@X#1L3@X# !dt, ~5.12c!

whereL j@X# 5 Ut*Lj@X#Ut , and the argumentt is omitted from all observables to simplify

notations. The Langevin–Heisenberg equation~5.12a! is a dilation of the forward Markovianmaster equation~1.9!, corresponding to the unital generatorL, given by ~4.4! and ~4.7a! withK0(Q)50, in the sense that averaging Eq.~5.12a! with respect to the vacuum state of the noisesgives this Markovian master equation.

TakingX5 f (Q) in ~5.12a! gives

d f~Q!5 i F uPu2

2m, f ~Q!Gdt1d f1~Q!1d f2,3~Q!, ~5.13a!

where

d f1~Q!5 (k51

r

(j51

3

bk j

] f ~Q!

]Qj$ i „ak

†~1 !2ak~1 !…12 Im Lk~Q!%dt11

2 (j ,l51

3

s j l

]2f ~Q!

]Qj]Qldt,

~5.13b!




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d f2,3~Q!5E (j

H @ f ~Q1x!2 f ~Q!#•„ajx~1 !1L jx~Q!1v j~x!…†„ajx~1 !

1L jx~Q!1v j~x!…2uv j~x!u2(k51

3] f ~Q!

]QkxkJ m~dx!dt. ~5.13c!

Thusd f1(Q) describes the diffusion component driven by the classical Wiener process@cf. ~B12!#

dWk~ t !5 i „ak†~ t1 !2ak~ t1 !…dt, 1<k<r ,

while d f2,3(Q) describes the jump component driven by the classical martingale measure

P~dxdt!5(j

~ajx~ t1 !1L jx~Q!1v j~x!!†~ajx~ t1 !1L jx~Q!1v j~x!!m~dx!dt

with the compensator14 m(dxuQ) @see~4.16!#.On the other hand, by takingX5Pl , we obtain

dPl5]U~Q!

]Qldt1dPl

11dPl2,3, ~5.14a!

where

dPl152 i(

k51

r H S ak~1 !11

2Lk~Q! D † ]Lk~Q!

]Ql2h.c.J dt, ~5.14b!

dPl2,352 i E (

jH S ajx~1 !1

1

2L jx~Q! D † ]L jx~Q!

]Ql2h.c.J m~dx!dt. ~5.14c!

Thus the momentumP is subject to a force represented by a quantum diffusion with coefficientsdepending onQ.

The problems of existence, uniqueness, and unitarity of solution of the quantum stochasticdifferential equation~5.7! and ~5.8a! with sufficiently general functionsL j ~•! andL jx~•! will betreated elsewhere. Some results on abstract quantum stochastic differential equations with un-bounded operator coefficients were obtained in Refs. 21–23, where mainly right equations wereconsidered. In particular, it was proved in Ref. 22 that the solution of the right equationUt isisometric if and only if the related dynamical semigroupFt

` is unital, and a similar condition forthe co-isometry ofUt holds. Contrary to the case of bounded operators~see Appendix B! theapproaches via right and left equations become inequivalent in general, as they lead, correspond-ingly, to the backward and forward Markovian master equations. Other questions to be investi-gated are the rigorous treatment of quantum stochastic integrals and equations with ‘‘big jumps’’~that are treated pathwise in the classical case! and extension of the time-ordered exponentialrepresentation~B6! to the case of unbounded operator coefficients.

ACKNOWLEDGMENTS

The question of characterization of Galilean covariant dynamical semi-groups arose from theauthor’s conversations with V. Gorini and L. Lanz during the author’s visit to the University ofMilan. The work was completed during the author’s stay at the University of Marburg. The authoris grateful to H. D. Doebner, G. C. Hegerfeldt, O. Melsheimer, H. Neumann, and R. F. Werner forstimulating discussions.

The grants from DFG and INFN are acknowledged.




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APPENDIX A: THE BACKWARD AND FORWARD QUANTUM MARKOVIAN MASTEREQUATIONS

We assume the following regularity properties for a solution of the backward equation~1.5!:this should be a familyFt ,t>0, of normal completely positive maps inB~H!, uniformlybounded in norm, satisfyingF05Id, and such that all functionst→Tr rFt[X], rPT ~H!,XPB~H! are continuous.

Theorem A1: Let K be maximal accretive andD be an invariant domain of the semigroupexp(2Kt)t,>0. Then there exists the minimal solutionFt

` of Eq. ~1.5!, which is a dynamicalsemigroup.

Proof (Sketch):Introducing the dynamical semigroupC t[r]5e2Ktre2K* t, we see thatD,defined by~1.2!, is an invariant domain forC t . DefiningL[r]5( j uL jf&^L jcu for r5uf&^cuPD,one can show7 that ~1.5! is equivalent to the integral equation

Tr rF t@X#5Tr C t@r#X1E0

t

Tr L†Cs@r#‡F t2s@X#ds, rPD,XPB~H!. ~A1!

Indeed, under the assumptions of the theorem both equations are equivalent to

d

dsTr Cs@r#F t2s@X#52Tr L†Cs@r#‡F t2s@X#, 0<s<t.

Note that in~A1! B~H! may be replaced by any weak*-dense subspace.The existence of the minimal solution of~A1! is proved by considering iterations10,22

Tr rF tn11@X#5Tr Ct@r#X1E

0

t

Tr L†Cs@r#‡F t2sn @X#ds ~A2!

with F t1[X]5F t[X]5e2K* tXe2Kt. Complete positivity ofL implies thatF t

n112F tn is com-

pletely positive, and~1.4! implies F tn[ I ]<I , by induction. By bounded monotone convergence

there exists limn→` F tn5F t

`, satisfying ~A1!. Since for any other solutionFt the differenceF t2F t

n is completely positive by induction,Ft` is the minimal solution. For detailed proof of

properties ofFt` see Refs. 10, 8, and 22. h

Assuming~1.8!, let us consider the forward equation~1.9!. For a solutionC t ;t>0, of ~1.9!we demand thatC t* should satisfy the regularity properties of solution of the backward equation.Defining

D*5 lin$X5uc&^fu;f,cPD* %

andL* @X# 5 ( j uL j*c&^L j*fu for X5uc&^fuPD* , we have

Tr L@r#X5Tr rL* @X#, rPD,XPD* . ~A3!

Theorem A2: Let K ~henceK* ! be maximal accretive andD* an invariant domain of thesemigroup exp(2K* t);t>0. ThenC t

`5(F t`)*is the minimal solution of the forward equation

~1.9!.Proof (Sketch):As in the previous theorem, one can prove that~1.9! is equivalent to

Tr C t@r#X5Tr rFt@X#1E0

t

Tr C t2s@r#L* †Fs@X#‡ds, rPT ~H!,XPD* ,

whereT ~H! can be replaced by any norm-dense subspace. Consider the iterations




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Tr C tn11@r#X5Tr rFt@X#1E

0

t

Tr C t2sn @r#L* †Fs@X#‡ds, ~A4!

with C t1[r]5C t[r]. If these iterations converge, then the limit is the minimal solution, by the

same argument as in previous theorem. We shall prove by induction thatC tn5(F t

n)*; then the

convergence will follow from the proof of theorem A1.We can write~A2! and ~A4! shortly as

F~• !n115F~• !1A@F~• !

n #,

C~• !n115C~• !1B@C~• !

n #,

whereA andB are the corresponding integral operators. Following argument of Refs. 11 and 24,one can see that

A@F~• !#5B@C~• !#*

and

A†B@F~• !* #* ‡5B†A@F~• !#* ‡* .

It follows thatC (•)n 5(F (•)

n )*. h

APPENDIX B: QUANTUM STOCHASTIC CALCULUS

There is a profound connection between the symmetric Fock space and the Levy–Khinchinformula for infinitely divisible representations of Lie groups,25 which underlies quantum processeswith independent increments. We give here a brief informal account of quantum stochastic calcu-lus with one-mode Boson noise in the Fock space. See Ref. 26 for the mathematical presentationof the general case and Refs. 27 and 28 for physical motivation and applications.

The role of the one-particle space is played by the Hilbert spaceL2~R1! of complex square-integrable functions oftPR1 @in the case of many modes one takesK^L2~R1!, whereK is theHilbert space representing modes#. LetG„L2~R1!… be the Boson Fock space overL2~R1! ~see, e.g.,Ref. 16! with the irreducible representation of the CCR

@a~ t !, a†~s!#5d~ t2s!, t,s>0,

satisfyinga(t) u0&50, whereu0& is the vacuum state vector. The basic operator processes

A~ t !5E0

t

a~s!ds, A†~ t !5E0

t

a†~s!ds, L~ t !5E0

t

a†~s!a~s!ds ~B1!

obey the quantum Ito rule

dA~ t !dA†~ t !5dt, dL~ t !25dL~ t !,

dA~ t !dL~ t !5dA~ t !, dL~ t !dA†~ t !5dA†~ t !, ~B2!

with all other products, including those comprisingdt, vanishing. Moreover, the quantum stochas-tic differentials should be regarded as commuting with anyadaptedprocess, i.e., a family ofbounded operators depending for eacht on a(s), a†(s) with s<t, and possibly on operators insome fixedinitial spaceH. Quantum stochastic integral




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Y~ t !5E0

t

@X1~s!dL~s!1X2~s!dA†~s!1X3~s!dA~s!1X4~s!ds#

can be defined for sufficiently regular adapted processesXj (t), and the Ito product formula

d~Y1•Y2!5dY1•Y21Y1•dY21dY1•dY2 ~B3!

can be established, where the products are to be calculated by using the quantum Ito table.LetW be unitary andL andK bounded operators in the initial Hilbert spaceH, satisfying

L* L5K1K* ,

so thatK5 12L* L1 iH , whereH is bounded Hermitean operator. The left and right quantum

stochastic differential equations

dUt5dZt Ut , dU t5U t dZt , ~B4!

where

dZt5~W2I !dL~ t !1LdA†~ t !2L*WdA~ t !2Kdt, ~B5!

have unique unitary solutionsUt ,U t ,t>0, satisfyingU05U 05I ~see Ref. 26!. These solutionscan be represented in the form of time-ordered exponentials

Ut

U tJ 5H exp←

exp→ J i E

0

tH FdL~s!2F

eiF2ILdA†~s!2h.c.2FH1L*

sin F2F

~2 sinF/2!2LGdsJ , ~B6!

whereF is a bounded Hermitean operator such thatW5eiF andF/(eiF2I ),..., are the corre-sponding meromorphic functions of this operator.28

Let

X~ t !5Ut* ~X^ I G!Ut , X ~ t !5U t* ~X^ I G!U t , ~B7!

whereX is a bounded operator inH. From ~B2! and~B3! one derives the following equation forthe familyX(t)

dX5~W*XW2X!dL2W* @L,X#dA†1@L* ,X#WdA1L@X#dt, ~B8!

where

L@X#5L@X#~ t !5Ut*L@X#Ut , L@X#5L*XL2K*X2XK, ~B9!

and the argument (t) is omitted from all operators in~B8! to simplify notations. The familyX (t)satisfies similar equation, in which, however,W, L, andL are time independent and the argument(t) is omitted only fromX and the noises.

Consider the vacuum expectations

F t@X#5^0uX~ t !u0&, F t@X#5^0uX ~ t !u0&, t>0. ~B10!

Averaging ~B8! and the similar equation forX (t), and taking into account that the vacuumexpectation of all noises vanish, one obtains the forward and the backward equations




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dF t@X#5F t†L@X#‡dt, dFt@X#5L†Ft@X#‡dt, ~B11!

whereF t5F t5exptL is a unital quantum dynamical semigroup with the generatorL.The relations to the classical stochastic processes are as follows. Each of the operator families

Q~ t !5A~ t !1A†~ t !, P~ t !5 i „A†~ t !2A~ t !…,~B12!

P~ t !5L~ t !1mA†~ t !1mA~ t !1umu2t5E0

t

„a~s!1m…†„a~s!1m…ds

is a family of commuting self-adjoint operators in the Fock spaceG„L2~R1!…, thus unitary equiva-lent to a classical stochastic process in the correspondingL2 space with 1 as the vacuum vector.The processesQ(t) andP(t) are unitary equivalent to the standard Wiener process~via Segal’s‘‘duality maps’’!, whileP(t) is unitary equivalent to the Poisson process of intensityumu2 ~see Ref.26!. HoweverQ(t), P(t), andP(t) do not commute and hence cannot be diagonalized simulta-neously.

1V. Bargmann, Ann. Math.59, 1 ~1954!.2V. S. Varadarajan,Geometry of Quantum Theory, II. Quantum Theory of Covariant Systems~Van Nostrand, New York,1970!.

3A. Barchielli and L. Lanz, Nuovo Cimento B44, 241 ~1978!.4R. Alicki, M. Fannes, and A. Verbeure, J. Phys. A19, 919 ~1986!.5A. S. Holevo, Izvestija RAN, Ser. Math.59, 205 ~1995!.6A. S. Holevo, Russ. Acad. Sci. Dokl. Math.47, 161 ~1993!.7A. S. Holevo, Rep. Math. Phys.33, 95 ~1993!, Theorem 2.8A. S. Holevo, J. Funct. Anal.131, 255 ~1995!.9E. B. Davies, Rep. Math. Phys.11, 169 ~1977!.10A. M. Chebotarev, J. Sov. Math.56, 2697~1991!.11W. Feller,An Introduction to Probability Theory, II~Wiley, New York, 1966!.12B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math. Phys.2, 161 ~1977!, Theorem 2.3.13B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Rep. Math. Phys.15, 27 ~1979!.14R. Sh. Liptser and A. N. Shiryaev,Theory of Martingales~Kluwer, Dodrecht, 1986!.15A. S. Holevo,Probabilistic and Statistical Aspects of Quantum Theory~North–Holland, Amsterdam, 1982!, LemmaV.4.2.

16M. Reed, and B. Simon,Methods of Modern Mathematical Physics, II~Academic, New York, 1975!.17B. Kummerer and H. Maassen, Commun. Math. Phys.109, 1 ~1987!.18A. Frigerio, Lect. Notes. Math.1303, 107 ~1988!.19D. D. Botvich, V. A. Malyshev, and A. D. Manita, Helv. Phys. Acta64, 1072~1991!.20D. W. Stroock and S. R. S. Varadhan,Multidimensional Diffusion Processes~Springer-Verlag, Berlin, 1979!.21A. Mohari and K. B. Sinha, Sankhya ser. A52, 43 ~1990!.22F. Fagnola, Quant. Prob. Rel. Topics8, 143 ~1993!.23G. F. Vincent-Smith, Proc. Lond. Math. Soc.63, 401 ~1991!.24A. S. Holevo, to be published in Prob. Theory Rel. Fields~1996!.25H. Araki, Publ. RIMS Kyoto Univ.5, 361 ~1970!.26R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys.93, 301 ~1984!.27C. W. Gardiner and M. J. Collett, Phys. Rev. A31, 3761~1985!.28A. S. Holevo, Proc. Steklov Inst. Math.191, 145 ~1992!.




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covariant quantum markovian evolutions

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