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arXiv:2307.02611v2 [quant-ph] 30 May 2024
Hybrid quantum-classical systems: Quasi-free Markovian dynamics
Alberto Barchielli*
, Reinhard F. Werner†
31st May 2024
Abstract
In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of
characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free. This is a
generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl
operators in the Heisenberg description. The result is a quantum generalization of the L´evy-Khintchine formula;
Gaussian and jump contributions are included. As a byproduct, the most general quasi-free quantum-dynamical
semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov-Fokker-Planck equation are
included. As a classical subsystem can be observed, in principle, without perturbing it, information can be extracted
from the quantum system, even in continuous time; indeed, the whole construction is related to the theory of quantum
measurements in continuous time. While the dynamics is formulated to give the hybrid state at a generic time t, we
show how to extract multi-time probabilities and how to connect them to the quantum notions of positive operator
valued measure and instrument. The structure of the generator of the dynamical semigroup is analyzed, in order to
understand how to go on to non quasi-free cases and to understand the possible classical-quantum interactions; in
particular, all the interaction terms which allow to extract information from the quantum system necessarily vanish if
no dissipation is present in the dynamics of the quantum component. A concrete example is given, showing how a
classical component can input noise into a quantum one and how the classical system can extract information on the
behaviour of the quantum one.
Contents
1 Introduction 2
2 Setting and main result 3
2.1 Some notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Quantum-classical Weyl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Other properties of the Weyl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Characteristic function of a state and Wigner function . . . . . . . . . . . . . . . . . . . . . . 5
2.3 A quasi-free dynamical semigroup for a hybrid system . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.1 Step 1: Composition properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.2 Step 2: Twisted positive definiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.3 Step 3: L ´evy-Khintchine analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.4 Step 4: Complete positivity and sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Wigner function and equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1 Approach to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The generator 15
3.1 The reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The classical-quantum interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 The role of the dissipative terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
*Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano,Italy; also Istituto Nazionale di Alta Matematica (INDAM-GNAMPA)
†Institut f¨ur Theoretische Physik, Leibniz Universit¨at, Hannover, Germany
1
4 Quasi-free quantum dynamical semigroups 18
4.1 The state dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.1 Quasi-free quantum linear Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.2 An optomechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 The pure classical case 22
5.1 The probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1.1 State evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1.2 Louville equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1.3 Transition probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 The process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2.1 The mean values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 An example: a dissipative harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 A generic hybrid system 26
6.1 Instruments and probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.1.1 Multi-time probabilities and instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1.2 Conditional probabilities and conditional states . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2.1 The input classical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2.2 The reduced quantum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2.3 The observed output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Conclusions 33
1 Introduction
Quantum-classical hybrid systems have been studied for various reasons, ranging from computational advantages,
description of m esoscopic systems. . . , to fou ndational problem s, such as the for mulation of quantum measurements,
see [1–7] and references there in. Quantum measurements can be interpreted as involving hybrid systems: a positive
operator valued measure is a channel from a quantum system to a classical one, an instrument [8, Sec. 4.1.1] is a
channel from a quantum system to a hybrid system [2, 5, 7, 9]. Also the quantum measurements in continuous time
can be interpreted in terms of hybrid systems; here the classical component is the monitored signal extracted from
the quantum system [8, 10–13]. A presentation of the theory of measurements in continuous time is developed in
Maassen’s contribution [14].
Our aim is to study a class of possible hybrid dynamics in the “Markov” case (no memory); so, we need to
generalize quantum dynamical semigroups on the quantum side [8, 15], and semigroups of transition probabilities on
the classical side [16,17]. With respect to the theory of quantum measurements in continuous time, we are generalizing
the notions of “convolution semigroups of instruments” and “semigroups of probability operators” [10–13, 18–25].
As well known, hard mathematical problems arise in the study of dynamical semigroups, when unbounded gen-
erators are involved; see for instance [8, 26, 27] for the quantum case. To have a significant class of semigroups, but
avoiding these problems, we consider only quasi-free transformations, introduced and developed in [5]. Such trans-
formations are defined by their action on Weyl operators [5,28–30]; they generalize the Gaussian case [8]. Quasi-free
processes can be characterized by a translation covariance property involving an additional linear map between the
phase spaces of input and output systems. The translation covariance would specialize in the classical case to a process
whose increments have a specified distribution. These increments will then be independent by virtue of the Markov
property. It is therefore expected that the generators are described by a version of the L´evy-Khintchine formula. This
is indeed a description of our main result. The appearance of quantum versions of the L ´evy-Khintchine formula and of
infinitely divisible distributions goes back to the first attempts of constructing a general framework for measurements
in continuous time [18, 20, 35].
Our paper is organized as follows.
2
The hybrid Weyl operators are introduced in Sec. 2.2. The notion of hybrid dynamical semigroup is introduced in
Sec. 2.3. The main result is the explicit structure of the most general quasi-free hybrid semigroup. This structure is
given in terms of the action on the Weyl operators, and it turns out to to be a quantum generalization of the classical
L´evy-Khintchine formula. The structure of the generator is discussed in Sec. 3, because it is linked to non quasi-free
cases and it helps in understanding the physical interactions.
In Sec. 4 we give the most general quasi-free quantum dynamical semigroup, in the pure quantum case. With
respect to the known Gaussian dynamics [31], also “jump” terms are introduced; similar contributions appeared in
symmetry based approaches [32, 33]. A couple of simple examples are given at the end of the section: a particle in a
noisy environment and a quantum harmonic oscillator.
Section 5 is dedicated to the pure classical case; as a comparison with the quantum case, an example based on the
classical harmonic oscillator is given at the end of the section. We also show that the deterministic classical Liouville
equation is included in the treatment. However, the essential point of this section is to show that not only probability
densities at a single time can be constructed, but that the semigroup gives rise also to transition probabilities and
multi-time probabilities; then, a whole stochastic process in time is constructed.
The dynamics of a generic hybrid system is studied in Sec. 6. Now we have an interaction between the quantum and
the classical components and a flow of information from the classical sub-system to the quantum one and viceversa.
By generalizing the concept of transition probabilities of the classical case, we are able to show that the hybrid
dynamics implies the existence of “transition instruments” and of multi-time probabilities for the classical component,
but which contain information on the quantum system. An example showing the possibility of flow of information in
both directions is presented at the end of the section. Conclusions and possible developments are given in Sec. 7.
2 Setting and main result
To introduce the notion of quasi-free dynamics we need position and momentum operators for the quantum component
and we take the Hilbert space H=L2(Rn); the Lebesgue measure is understood. Then, we take a classical system
with sdegrees of freedom, living in Rs.
The choice of the spaces of states and of observables can depend on the aims of the construction one wants to
realize; a detailed discussion on this point is given in [5, Secs. 2, 3]. We shall try to define the dynamical semigroup
of interest by asking a set of minimal properties and, then, to search for the natural setting. Essentially, there is the
possibility of a W∗-algebraic approach, which is often well suited to develop a theory of dynamical semigroups for
finitely many degrees of freedom, [5, Secs. 2.8, 4.4, 5.3, 5.7], [12]. Alternatively, there are C∗-algebraic approaches,
needed for instance when classical pure states are important [5, Sec. 2.5].
2.1 Some notations
Let us start by introducing the various spaces we shall need in the following.
Firstly, we introduce the spaces of operators on H=L2(Rn):
•B(H): bounded operators,
•T(H): trace class,
•K(H): compact operators.
Then, we introduce the spaces of complex functions on Rs:L∞(Rs),L1(Rs), and
•Cb(Rs): bounded continuous functions,
•C0(Rs): continuous functions vanishing at infinity.
A first choice as observable space is the W∗-algebra N=B(H)⊗L∞(Rs). We can identify a generic element
F∈Nwith a function F(x)from Rsinto B(H). Then, the state space is the predual N∗=T(H)⊗L1(Rs). A
state ˆπis a trace-class valued function ˆπ(x)∈T(H),x∈Rs, such that ˆπ(x)≥0and RRsdxTr{ˆπ(x)}= 1.
Measurability and integrability properties are always understood. This choice of state space allows to include generic
quantum master equations from one side (Sec. 4), and, on the classical side, both the Liouville equation and the
Kolmogorov-Fokker-Planck equation (Sec. 5).
3
Another possible choice, given in [5, Prop. 6], is to take K(H)⊗C0(Rs)as observable space and its dual as state
space.
We denote by Id the identity operator on Nand by 1the unit element in B(H). The adjoint of the operator
a∈B(H)is a†and αis the complex conjugated of α∈C. For a matrix M, not necessarily square, MTdenote its
transpose.
The translation operator on the classical component is denoted by Rz:∀f∈L∞(Rs),Rz[f](x) = f(x+z).
Rzand Id ⊗ Rzare identified.
We denote by Qj,Pithe position and momentum operators for the quantum component and by Xlthe multiplica-
tion operators in the classical component. As in [5], we denote by Rthe vector of the fundamental operators:
R=
Q
P
X
, Rj=
Qjfor j= 1,...,n,
Pj−nj=n+ 1,...,2n,
Xj−2nj= 2n+ 1,...,2n+s.
Then, formally, the commutation rules take the form:
[Ri, Rj] = iσij 1, σij =
1 1 ≤i≤n j =i+n,
−1n+ 1 ≤i≤2n j =i−n,
0otherwise.
(1)
Commutator and anti-commutator are denoted by [•,•]and {•,•}, respectively.
It is useful to combine the spaces R2n, related to the Hilbert space H, and Rs, appearing in the classical component,
in a unique space; so, we define the phase space
Ξ = Ξ1⊕Ξ0,Ξ1=R2n,Ξ0=Rs, d = 2n+s⇒Ξ = Rd.
It is also useful to see Ξand its components as commutative groups under addition (translation groups). Moreover, let
Pjbe the orthogonal projection on Ξj,j= 1,0. By using the column notation for vectors, for ξ, η ∈Ξ, their scalar
product is ξTξ=Pd
i=1 ξiηi. Finally, let σbe the matrix defined by the elements σij (1).
2.2 Quantum-classical Weyl operators
We introduce now the Weyl operators for the hybrid system [5, (6)]. The parameters of the Weyl operators will live
in the space Ξ, which is always identified with its dual. We denote by ξ,ζ,k, . . . column vectors a nd by ξT, . . . their
transposed versions (row vectors). Often we use
ζ=u
v∈R2n≡Ξ1, k ∈Rs≡Ξ0, ξ =ζ
k∈Ξ.
The Weyl operators [5, 8] are defined by
W(ξ) = exp iRTξ=W1(ζ)W0(k),
W0(k) = exp iXTk= exp iRTP0ξ,
W1(u, v) = exp iuTQ+vTP= exp iRTP1ξ.
(2)
Remark 1.W0(k)can be seen as a function from Rsinto C:W0(k)(x) = exp {ixTk}. Moreover, W1(ζ)∈B(H)
and W0(k)∈Cb(Rs); therefore, W(ξ)∈N.
The Weyl operators satisfy the following composition property:
W(ξ+η) = W(ξ)W(η) exp i
2ξTση=W(η)W(ξ) exp −i
2ξTση.(3)
More rigorously, the Weyl operators W1are defined as projective unitary representations of the translation group
Ξ1[8], or as displacement operators acting on coherent vectors [15, 34]. Then, (3) represents the rigorous version of
the canonical commutation rules [8].
4
2.2.1 Other properties of the Weyl operators
By using W1and the classical translation operator Rz, defined in Sec. 2.1, we have the translation properties [5, (5),
(6), (9), Sec. 2.6]:
Rz[W0(k)](x) = W0(k)(x+z) = exp ikT(x+z)=W0(k)(x)eikTz,
Rz[W(ξ)] = W(ξ)eizTP0ξ,(4a)
W1(u, v)†QiW1(u, v) = Qi−vi, W1(u, v )†PiW1(u, v ) = Pi+ui, i = 1,...,n. (4b)
In a shorter form we can write
W(ξ)†RiW(ξ) = Ri+σTξi, i = 1,...,2n,
⇒[(σR)i, W (ξ)] = ξiW(ξ), i = 1,...,2n. (5)
From (3), we get the properties:
W(η)†W(ξ)W(η) = W(−η)W(ξ+η)e−i
2ξTση = eiηTσξ W(ξ),(6a)
W(η)†W(ξ)W(η)−1
2W(η)†W(η), W (ξ)=W(η)†W(ξ)W(η)−W(ξ) = eiηTσξ −1W(ξ).(6b)
Also explicit forms of the derivatives of the Weyl operators are very useful in computations:
∂W (ξ)
∂ξj
=i
2{Rj, W (ξ)}, j = 1,...,d,
ikiW0(k)(x) = ∂
∂xi
W0(k)(x), i = 1,...,s.
(7)
2.2.2 Characteristic function of a state and Wigner function
As in the pure quantum case, the states ˆπ∈N∗are uniquely determined by their characteristic function χˆπ(ξ)[5, Sec.
2.4] or by their Wigner function Wˆπ(z)[15]:
χˆπ(ξ) = ZRs
dxTr {ˆπ(x)W(ξ)(x)},Wˆπ(z) = 1
(2π)dZΞ
dξe−izTξχˆπ(ξ).(8)
By Bochner’s Theorem, the Wigner function is non-negative, Wˆπ(z)≥0,∀z∈Rd, if and only if χˆπ(ξ)is positive
definite: for any choice of the integer N, of ξk∈Ξ,k= 1,...,N,ck∈C,
0≤
N
X
l,k=1
clχˆπ(ξk−ξl)ck≡
N
X
l,k=1 ZRs
dx clTr W(ξl)(x)†ˆπ(x)W(ξk)(x)ckexp −i
2ξT
kσξl.(9)
Due to the last factor, the characteristic function is not positive definite for any state; so, for some states, the Wigner
function can become negative for some z.
Remark 2.By [5, Theor. 4], a function χ: Ξ →Cis the characteristic function of a state if and only if
(1) χis continuous, (2) χ(0) = 1, (3) for every integer Nand every choice of ξ1,...,ξN,ξj∈Ξ, the
N×N-matrix with elements χ(ξk−ξl) exp i
2ξT
kσξlis positive semi-definite.
To guarantee that the state belongs to N∗the following property has to be added: (4) χ∈L1(Ξ).
2.3 A quasi-free dynamical semigroup for a hybrid system
Here we consider a Markovian dynamics, in the Heisenberg description. With the further condition of translation in-
variance in the classical component, similar semigroups were introduced inside the theory of quantum measurements
in continuous time under the name of convolution semigroups of instruments. In that case their generator was char-
acterized under a strong continuity in time, implying that only bounded operators on the quantum component were
5
involved [8, 10–12, 14, 18–24]. Now we want to treat a generic hybrid system, without the hypothesis of translation
invariance and of bounded operators. However, as a first step, we consider only quasi-free dynamical semigroups,
according to the definition of quasi-free channels given in [5, Sec. 4.1].
As motivated in [5] and recalled at the beginning of Sec. 2, it is better to leave open the choice of the spaces of
observables and states. To this end, we give a minimal definition of the semigroup {Tt, t ≥0}, involving only its
action on the Weyl operators.
Definition 1. Aquasi-free hybrid dynamical semigroup is a family {Tt, t ≥0}of linear operators defined at least on
the linear span of the Weyl operators, satisfying the following properties: ∀t, s ≥0,
a. (normalization) Tt[1] = 1;
b. (initial condition) T0= Id;
c. (semigroup property) Tt◦ Ts=Tt+s;
d. (complete positivity) for any integer Nand any choice of the vectors φk∈Hand ξk∈Ξ,k= 1,...,N,
N
X
k,l=1
hφk|Tt[W(ξk)†W(ξl)]|φli ≥ 0; (10)
e. (quasi-free property) for all ξ∈Ξ
Tt[W(ξ)] = ft(ξ)W(Stξ),(11)
where Stis a linear operator from Ξto Ξ, and ftis a continuous function from Ξto C;
f. (continuity in time) the functions t→ft(ξ)and t→Stare continuous.
By the property (3), the product of Weyl operators is proportional to a Weyl operator, so that the positivity property
(10) involves the action of Ttonly on Weyl operators. In [5] ftis called noise function. By using the composition
property (3) of the Weyl operators and the quasi-free structure (11), we get
Tt[W(ξk)†W(ξl)] = ft(ξl−ξk) exp i
2ξkTσ−St
TσStξlW(Stξk)†W(Stξl).(12)
Finding the structure of ft(ξ)and Stand constructing explicitly the semigroup requires various steps and involves
the theory of classical stochastic processes, such as processes with independent increments, additive processes, L´evy
processes, and processes of Ornstein-Uhlenbeck type [16, 17, 36, 37]. So, here we give only the final results and we
leave to Sec. 2.4 the proof of the theorems and the discussion of the construction.
Quasi-free semigroups (on general CCR algebras) were considered also in [30], and preliminary results on their
structure were obtained. Here we arrive to a complete characterization in the case of a finite-dimensional phase space.
The expression of the noise function ft(ξ)and the proof of Theor. 1 involve infinitely divisible distributions and
L´evy-Khintchine formula. The function ψ(ξ)appearing in the formulation of the theorem is sometimes called L´
evy
symbol [17, p. 31], while A, ν, α, appearing in (16), is said to be its generating triplet [16, Def. 8.2]; νis the L´
evy
measure. As it is customary for L´evy measures, the integral (16) describing the non-Gaussian jump part is separated
into small jumps and large jumps using the indicator function 1Sof the unit ball S=ξ∈Rd:|ξ|<1. This cutoff
function can be changed to any function which is = 1 for small arguments and vanishes for large ones; this replacement
can be compensated by a change of the vector α(see [16]).
Theorem 1. The objects ft(ξ)and Stdescribe a quasi-free hybrid dynamical semigroup {Tt, t ≥0}in the sense of
Definition 1, if and only if they have the following structure:
1. The matrices Stform a semigroup: there is a real d×d-matrix Zsuch that
St= eZt ,∀t≥0.(13)
6
2. The noise function ft(ξ)is parameterized by a vector α∈Ξ, a real symmetric d×d-matrix A, and a L´
evy
measure νon Ξ, i.e., a σ-finite measure such that
ν({0}) = 0,ZS
|η|2ν(dη)<+∞, ν({Ξ\S})<+∞.(14)
Then, ftis determined by
ft(ξ) = exp Zt
0
dτ ψ(Sτξ),(15)
where
ψ(ξ) = iαTξ−1
2ξTAξ +ZΞ
ν(dη)eiηTξ−1−i1S(η)ηTξ,∀ξ∈Ξ = Rd.(16)
3. The matrix Asatisfies the further positivity condition
A±iB≥0, B := 1
2σZ −ZTσT.(17)
For every t≥0, the operator Ttextends from the linear span of the Weyl operators to N, and this extension
is bounded. Moreover, this extension, always denoted by Tt, is the adjoint of a completely positive, normalization
preserving map Tt∗on the predual N∗such that {Tt∗, t ≥0}forms a strongly continuous semigroup.
Perhaps the most remarkable feature of this result is that the positivity condition (17), which captures all quantum
uncertainty constraints of such evolutions, involves only the Gaussian part, i.e., the matrix A, and not the jump part.
Since this is also the only place where the symplectic form σenters, the conditions as stated imply those for σ= 0,
i.e., for a purely classical system. Understanding this process essentially carries over to the quantum and hybrid cases:
whenever the initial state has positive Wigner function, this will be true through the entire evolution, and the Wigner
functions follow just the classical process with the same ftand St. By linearity this will even be true for non-positive
Wigner function, although in this case the probabilistic interpretation is lost. But it is worth noting that the evolution
drives towards states with positive Wigner function.
Apart from the above Theorem, the general results given below and in Sec. 3 include the form of the generator of
the semigroup, giving the dynamical equations in a way which connects a Lindblad-like form with a Fokker-Planck
structure, and a criterion for the existence of an equilibrium state, balancing the effects of a semigroup Stcontracting
to the origin and of the noise driving the process away.
2.4 Proof of the main results
The proof of Theorem 1 will be in the following main steps.
Step 1. We evaluate the composition laws for the ftand St. If we could assume ft(ξ)to be differentiable in
t, this would already imply the form (15), so this is heuristically clear. However, we are not assuming this;
differentiability in time will be a final result.
Step 2. We show that the associated classical process (σ= 0) is well-defined. This is in contrast to the positivity
conditions for states, where the twisted definiteness does not imply the positivity of a classical object with the
same characteristic function (the Wigner function). The difference is made by the divisibility of the process.
Step 3. We use the theory of classical additive processes [16] to get the L´evy-Khintchine structure (15), (16). This
would be just the standard result for independent increment processes, if we had St=1for all t. The variant
we need for general Stis sometimes called a generalized Ornstein-Uhlenbeck process [37].
Step 4. We go back to quantum processes to evaluate the complete positivity condition. Heuristically this is the
question of how much noise, as described by the Gaussian part Aand the jump measure ν, is needed to meet
the quantum uncertainty requirement, a conditional complete positivity property for the functional ψ. For suf-
ficiency of (17) we consider the Gaussian case, and observe that the jumps add a classically positive definite
part. This leaves open the possibility of positivity partly ensured by jump noise, with maybe even a vanishing
Gaussian part. However, this is ruled out by a scaling argument.
7
2.4.1 Step 1: Composition properties
Let us translate the basic semigroup properties to ftand St, using that the map Ttuniquely determines these objects
and is determined by them.
Proposition 2. Given a semigroup Ttaccording to Definition 1, its defining objects Stand ft(ξ), enjoy the following
properties.
1. The family of matrices Stforms a semigroup: St= exp(Zt)for some real matrix Z.
2. The function ft(ξ)satisfies the normalization condition ft(0) = 1 for all t≥0, and the initial condition
f0(ξ) = 1 for all ξ∈Ξ.
3. It satisfies the product formula:
ft+s(ξ) = fs(Stξ)ft(ξ),∀t, s ≥0,∀ξ∈Ξ.(18)
Proof. By using Eq. (11), we have that Property a. of Def. 1 is equivalent to ft(0) = 1, while Property b. is equivalent
to f0(ξ) = 1, S0ξ=ξ. By Properties c. and e., we get
ft+s(ξ)W(St+sξ) = ft(Ssξ)fs(ξ)W(StSsξ) = fs(Stξ)ft(ξ)W(SsStξ).
If for a certain ξwe have ft+s(ξ)6= 0 we get (18) and St+sξ=SsStξ. If ft+s(ξ) = 0 also the product fs(Stξ)ft(ξ)
must vanish and (18) holds again. By continuity in Ξand ft(0) = 1, for fixed ξand t+sit exists α > 0such
that ft+s(αξ)6= 0; then, αSt+sξ=αSsStξ. As a consequence, Stis a continuous semigroup of real matrices with
S0=1, so that St= eZt .
Remark 3.By iteration, the product formula (18) gives
ft(ξ) =
m−1
Y
ℓ=0
ft/m(Sℓt/m ξ),(19)
for any integer m≥1. Let us also recall that ft(ξ)is continuous in tand ξ.
2.4.2 Step 2: Twisted positive definiteness
The condition on the noise function ftin (11), which ensures the complete positivity (10) of Tt, is a generalization
of Bochner’s criterion for the positivity of Fourier transforms. Following [5], we will say that fis twisted positive
definite with respect to the antisymmetric form σ, if the matrix
Mjk =f(ξj−ξk) exp −i
2ξT
jσξk
is positive semidefinite for any ξ1, ..., ξN∈Ξ. Since M≥0implies M†=M, twisted definiteness implies that f
is hermitian in the sense that f(−ξ) = f(ξ). Moreover, by taking ξk7→ −ξkwe see that twisted definiteness with
respect to σimplies the same for −σ. We cannot conclude that fis also positive definite in the usual sense, i.e.,
twisted definite with respect to σ= 0. Since for a state the function fis the Fourier transform of the Wigner function,
this is the same as saying that there are quantum states with non-positive Wigner function.
For a quasi-free channel T(W(ξ)) = f(ξ)W(Sξ)to be completely positive, it is necessary and sufficient [5] that
fis definite for the difference of input and output symplectic form, i.e., for the antisymmetric matrix ∆σ=σ−STσS.
Again this holds for both signs, and there is no conclusion about zero definiteness. However, we are now looking at
the product (19) and this implies that ftis positive definite.
We collect these results in the following Proposition (and we give the explicit proofs).
Proposition 3. The noise function ftof Definition 1 enjoys the following properties.
8
1. The function ftis twisted positive definite: for every integer Nand any choice of ξk∈Ξand ck∈C,
k= 1,...,N, we have
N
X
k,l=1
ckft(ξl−ξk) exp ±i
2ξkTσ−St
TσStξlcl≥0.(20)
2. The function ftis also positive definite: for every choice of N,ξ,ckas above, we have
N
X
k,l=1
ckft(ξl−ξk)cl≥0.(21)
3. For every fixed time t≥0, the complex function ξ→ft(ξ)is the characteristic function of a probability
measure on Rd. Also ξ→fs(Stξ)is the characteristic function of a probability measure on Rd.
Proof. By using (12) and the vectors φl=W(Stξl)†clψin the positivity condition d. of Definition 1, we get (20) with
the plus sign in the exponent. By using N= 2,ξ1=ξ,ξ2=−ξ,c1= 1,c2= 1 or = i, we get ft(ξ) = ft(−ξ). In
(20) with the plus sign we make the replacements cl→cl,ξl→ −ξland we take the complex conjugated of the full
expression; by using ft(ξ) = ft(−ξ), we get (20) with the minus sign in the exponent. So, property 1 is proved.
By adding together the two positive expressions appearing in (20), we get
N
X
k,l=1
ckft(ξl−ξk) cos 1
2ξkTσ−St
TσStξlcl≥0.
This gives
N
X
k,j=1
ckft/mS(l−1)t/m (ξj−ξk)cos 1
2S(l−1)t/mξkTσ−St/m
TσSt/mS(l−1)t/mξjcj≥0.
As the element-wise (Schur-Hadamard) product of non-negative matrices is non-negative, we have
N
X
k,j=1
ckcj
m
Y
l=1ft/mS(l−1)t/m (ξj−ξk)cos 1
2S(l−1)t/mξkTσ−St/m
TσSt/m S(l−1)t/mξj≥0;
by (19), this gives
N
X
k,j=1
ckft(ξj−ξk)cj
m
Y
l=1 cos 1
2S(l−1)t/mξkTσ−St/m
TσSt/m S(l−1)t/mξj≥0.
Being Sta matrix semigroup with S0=1, clearly, for large m, the argument of each cosine is O(t/m), so each cosine
is 1 + O(t/m)2. Hence the whole product goes to 1as m→ ∞:
lim
m→+∞
m
Y
l=1 cos 1
2S(l−1)t/mξkTσ−St/m
TσSt/m S(l−1)t/mξj= 1.
Then, we obtain (21).
By the normalization ft(0) = 1, the positive definite condition (21), and the continuity of ξ→ft(ξ), we have that
ft(·)is the characteristic function of a probability measure on Ξ(Bochner’s theorem, see [16, Prop. 2.5, point (i)]).
The same arguments give that fs(St·)is the characteristic function of some probability measure. So, also property 3
is proved.
Remark 4.Being fta characteristic function, we have also ft(ξ) = ft(−ξ); then, equation (11) implies Tt[W(ξ)]†=
Tt[W(ξ)†].
9
2.4.3 Step 3: L´
evy-Khintchine analysis
In this section we completely ignore the quantum aspects of the problem; we prove the special case of the theorem for
purely classical systems (σ= 0). We take into account the necessary positivity condition (21), but we postpone to the
next step the introduction of the more restrictive condition (20). Now, {ft}t≥0is a collection of classical characteristic
functions satisfying the product formula (18), and the problem of finding its structure fits exactly into the framework
of additive processes in the sense of Sato [16, Sec. 9].
An additive process ˜
Xtis defined in [16, p.3] to be a stochastic process with independent increments, starting
from the origin ˜
X0= 0, and enjoying stochastic continuity. In [36, Chapt. 3] these processes are simply called
“stochastically continuous processes with independent increments”. In our context the “independence of the incre-
ments” essentially follows from the product properties (18), (19), as we shall see in the following.
Let us set ˆ
fs,t(ξ) := ft−s(Ssξ),0≤s≤t. (22)
By property 3 of Proposition 3, ˆ
fs,t(·)is the characteristic function of a probability measure on Rdwhich we denote
by µs,t. By the product formula (18) written with ξreplaced by St1ξ,t1≥0, and by setting t2=t1+t,t3=t2+s,
we get ˆ
ft1,t2(ξ)ˆ
ft2,t3(ξ) = ˆ
ft1,t3(ξ),0≤t1≤t2≤t3.
This means that for the associated probability measures we have
µt1,t2∗µt2,t3=µt1,t3;(23)
the symbol ∗denotes the convolution. By applying the results of [16, Sec. 9], we can find the main properties of the
process associated to these probability measures and the explicit structure of our noise function ft(ξ).
Proposition 4. The following statements hold:
1. There exists an additive process in law ˜
Xton Rdsuch that µs,t is the probability measure of the increment
˜
Xt−˜
Xs,t≥s≥0.
2. The probability measure µs,t is infinitely divisible.
3. There is a unique continuous function Ψt(·)from Rdinto C, such that
ft(ξ) = eΨt(ξ),∀ξ∈Rd,∀t≥0; (24)
Ψt(ξ)is also continuous in time and
Ψ0(ξ) = 0,Ψt(0) = 0.
4. For all t≥0, the function Ψt(ξ)admits the representation: ∀ξ∈Ξ = Rd,
Ψt(ξ) = iα(t)Tξ−1
2ξTAtξ+ZRd
νt(dη)eiηTξ−1−i1S(η)ηTξ,(25)
where α(t)∈Rd,Atis a symmetric nonnegative definite d×dmatrix, and νtis a measure on Rdsatisfying
νt({0}) = 0 and ZRd|ξ|2∧1νt(dξ)<+∞.
The representation of Ψt(ξ)by the triplet (At, νt, α(t)) is unique.
5. The triplet (At, νt, α(t)) enjoys the following properties:
(a) A0= 0, ν0= 0, α(0) = 0;
(b) if 0≤s≤t < +∞, then As≤Atand νs(E)≤νt(E)for every Borel subset Eof Rd;
(c) as s→tin (0,+∞), we have As→At,α(s)→α(t), and νs(E)→νt(E)for every Borel subset Eof
Rdsuch that E⊂ {x:|x|> ǫ},ǫ > 0.
10
Proof. Let us consider Properties (9.13)-(9.16) in Theorem 9.7 of [16]. Condition (9.13) is exactly (23). By the
definition (22), we have ˆ
fs,s(ξ) = f0(ξ) = 1, so that µs,s =δ0, which is condition (9.14). By the continuity in
time of Stand ft(ξ)and the definition (22), we have the continuity in sand tof ˆ
fs,t(ξ). The weak convergence of
probability measures is defined in [16, Def. 2.2] and it is equivalent to the convergence of the characteristic functions,
see Properties (vi) and (vii) of Proposition 2.5. These facts easily imply that also conditions (9.15) and (9.16) hold.
Then, by point (ii) of [16, Theor. 9.7], point 1 is proved.
By Theor. 9.1 in Ref. [16], the probability measure µs,t is infinitely divisible, which is point 2.
ftis the characteristic function of the infinitely divisible distribution µ0,t , which gives ft(ξ)6= 0,∀ξ∈Rd. Let us
define f−t(ξ) = 1 for t > 0. By the continuity in time and on the phase space, the complex function (t, ξ)→ft(ξ)on
Rd+1 is continuous; moreover, ft(0) = 1,ft(ξ)6= 0,∀ξ∈Rd. Then, the statement 3 follows from [16, Lemma 7.6].
By points 2 and 3, Ψt(ξ)is the L´evy symbol associated to the infinitely divisible distribution µ0,t. By [16, Theor.
8.1 and Def. 8.2], Ψt(ξ)has the structure given in point 4 and this representation is unique.
By the existence of the additive process of point 1, we have that the hypotheses of point (i) in [16, Theor. 9.8] hold.
Then, we get Properties (1)-(3) of that theorem, which correspond to Properties (a)-(c) in point 5.
Remark 5.By Proposition 4, ft(ξ) = eΨt(ξ)is the characteristic function of an infinitely divisible distribution. The
structure (25) is due to the L ´evy-Khintchine formula; At, νt, α(t)is the generating triplet of Ψt(ξ). This represent-
ation is unique once the cutoff function 1Shas been fixed. Other choices of the cutoff function are possible; a change
of cutoff implies a consequent change of the vector α(t)[16, Remark 8.4].
Remark 6.By the result (24), the product formula (18) is equivalent to
Ψt+s(ξ) = Ψt(ξ) + Ψs(Stξ),∀t, s > 0,∀ξ∈Ξ.(26)
We have also ˆ
fs,t(ξ) = exp {Ψs,t (ξ)},Ψs,t(ξ) = Ψt−s(Ssξ),0≤s≤t.
Proposition 5. The triplet (At, νt, α(t)), introduced in point 4 of Proposition 4, has the structure
At=Zt
0
dτ ST
τASτ, νt(E) = ZRd
ν(dη)Zt
0
dτ1EST
τη,(27)
Eis any Borel subset of Rd,
α(t) = Zt
0
dτ ST
τα+ZRd
ν(dη)Zt
0
dτ ST
τη1SST
τξ′−1S(η).(28)
Here, α∈Rd,Ais a real symmetric d×d-matrix with A≥0, and νis a L´
evy measure on Rd, that is, it satisfies (14).
Proof. By iterating the decomposition (26) we obtain
Ψt(ξ) =
m−1
X
l=0
Ψt/m Slt/mξ,∀m≥1, t ≥0, ξ ∈Rd.
By (25) and the uniqueness of the Gaussian contribution in the generating triplet, we have
ξTAtξ=ξT
m−1
X
l=0 Slt/mξTAt/m Slt/mξ, (29)
iα(t)Tξ+ZRd
νt(dη)eiηTξ−1−i1S(η)ηTξ
=
m−1
X
l=0 iα(t/m)TSlt/mξ+ZRd
νt/m(dη)eiηTSlt/m ξ−1−i1S(η)ηTSlt/mξ.(30)
11
Moreover, by point 5 in Proposition 4, we have
lim
m→+∞At/m = 0,lim
m→+∞αt/m = 0,lim
m→+∞νt/m(E) = 0 (31)
for every Borel subset Eof Rdsuch that E⊂ {x:|x|> ǫ},ǫ > 0.
From (29) we get
ξTAtξ−Zt
0
dτ ξTST
τ
m
tAt/mSτξ=ξT
m−1
X
l=0 ST
lt/mAt/m Slt/m −Z(l+1)t/m
lt/m
dτST
τ
m
tAt/mSτ!ξ
=ξT
m−1
X
l=0
ST
lt/m At/m −Zt/m
0
dτST
τ
m
tAt/mSτ!Slt/m ξ,
lim
m→+∞ξTAtξ−Zt
0
dτ ξTST
τ
m
tAt/mSτξ= lim
m→+∞ξT
m−1
X
l=0
ST
lt/m At/m −Zt/m
0
dτST
τ
m
tAt/mSτ!Slt/m ξ
=−lim
m→+∞ξT
m−1
X
l=0
t
2mST
lt/m At/mZ+ZTAt/m Slt/mξ
=−1
2lim
m→+∞ξTZt
0
dτST
τAt/mZ Sτ+ST
τZTAt/mSτξ= 0;
the last equality is due to (31). The map K→Rt
0dτ ST
τKSτ, at least for small times, is invertible, because it is near
to the identity map and positivity preserving; then, we have the existence of the limit
lim
m→+∞
m
tAt/m =A≥0;
then, the first equality in (27) holds.
By (30) we have
iα(t)Tξ−im
tα(t/m)TZt
0
dτ Sτξ+ZRd
νt(dη)eiηTξ−1−i1S(η)ηTξ
−m
tZRd
νt/m(dη)Zt
0
dτeiηTSτξ−1−i1S(η)ηTSτξ
=ZRd
νt/m(dη)
m−1
X
l=0 eiηTSlt/mξ−1−i1S(η)ηTSlt/m ξ−m
tZt/m
0
dτeiηTSlt/mSτξ−1−i1S(η)ηTSlt/m Sτξ
+ iα(t/m)T m−1
X
l=0 Slt/m −m
tZtl+t/m
tl
dτ Sτ!!ξ
=ZRd
νt/m(dη)
m−1
X
l=0 eiηTSlt/mξ1−m
tZt/m
0
dτeiηTSlt/m(Sτ−1)ξ
−i1S(η)ηTSlt/m 1−m
tZt/m
0
dτSτ!ξ+ iα(t/m)T m−1
X
l=0
Slt/m 1−m
tZt/m
0
dτ Sτ!!ξ.
12
Then, for m→+∞, we get
lim
m→+∞iα(t)Tξ−im
tα(t/m)TZt
0
dτ Sτξ+ZRd
νt(dη)eiηTξ−1−i1S(η)ηTξ
−m
tZRd
νt/m(dη)Zt
0
dτeiηTSτξ−1−i1S(η)ηTSτξ
=−lim
m→+∞
m−1
X
l=0 ZRd
νt/m(dη)it
2meiηTSlt/mξ−1S(η)ηTSlt/m Zξ +it
2mα(t/m)TSlt/mZ ξ
=−i
2lim
m→+∞Zt
0
dτZRd
νt/m(dη)eiηTSτξ−1S(η)ηT˙
Sτξ+α(t/m)T˙
Sτξ
=1
2lim
m→+∞ZRd
νt/m(dη)eiηTξ−eiηTStξ+ i1S(η)ηT(St−1)ξ−iα(t/m)T(St−1)ξ= 0;
the last equality is due to (31). Then, we must have
νt(E) = lim
m→+∞
m
tZt
0
dτZRd
1E(ST
τη)νt/m(dη), α(t)Tξ= lim
m→+∞
m
tα(t/m)TZt
0
dτ Sτξ;
these equations give
lim
ǫ↓0
νǫ(E)
ǫ=ν(E),lim
ǫ↓0
α(ǫ)
ǫ=α∈Ξ,
the measure νsatisfies (14).
Then, one easily checks that (27) and (28) hold.
By Eqs. (24), (25) and Proposition 5, we get the structure of Eqs. (14), (15), (16) in Theor. 1:
ft(ξ) = eΨt(ξ),Ψt(ξ) = Zt
0
dτ ψ(Sτξ),(32)
where the function ψis given in (16).
2.4.4 Step 4: Complete positivity and sufficiency
Up to now we have proved that conditions 1 and 2 (with A≥0) in Theorem 1 are necessary. Now we have to prove
condition 3 (due to complete positivity) and the sufficiency of the three conditions. We have also to prove the final
statements in Theorem 1 about the extension of Tt. Here we give this final part of the proof of the main theorem.
Proof of Theorem 1. Let us consider that ft(ξ)is twisted positive definite (point 1 in Proposition 3). Under the condi-
tion PN
k=1 ck= 0, the time derivative of (20) in t= 0 is not negative. By using (13) and (32), we get
N
X
k,l=1
ckψ(ξk−ξl)±i
2ξkTσZ +ZTσξlcl≥0.
By inserting (16) into this equation, we obtain
N
X
k,l=1
ckξkAξl±i
2ξkTσZ +ZTσξl+ZΞ
ν(dη)eiηT(ξk−ξl)cl≥0,
N
X
k=1
ck= 0.
By dilating ξkinto λξk,λ∈R, and dividing by λ2, we get
0≤
N
X
k,l=1
ckξkAξl±i
2ξkTσZ +ZTσξl+1
λ2ZΞ
ν(dη)eiληT(ξk−ξl)cl,
13
which goes to PN
k,l=1 ckξkAξl+i
2ξkT(σZ +ZTσ)ξlclfor λ→ ±∞; then, (17) is a necessary condition for
positivity. This ends the proof of the necessity of conditions 1–3.
By inserting (15) and (16) into (11) and by using (12), it is easy to check that the conditions 1, 2, 3 in Theor. 1 are
also sufficient to get all the points in Definition 1.
By linearity, the operator Ttis extended to the linear span in Nof the Weyl operators (2). By (10), Ttsends positive
elements into positive ones; if F∈Nis in the linear span of the Weyl operators and F≥0, we have F≤ kFk1, and,
by the normalization of Tt, we get
Tt[F]≤ kFk Tt[1] = kFk1.(33)
Let ˆπ0be a generic state in N∗=T(H)⊗L1(Rs)with characteristic function χˆπ0(ξ), defined in (8). Then, we define
ZRs
dxTr {Tt∗[ˆπ0](x)W(ξ)(x)}:= ZRs
dxTr {ˆπ0(x)Tt[W(ξ)](x)}.
By (11), we get
ZRs
dxTr {Tt∗[ˆπ0](x)W(ξ)(x)}=ft(ξ)ZRs
dxeixTP0StξTr {ˆπ0(x)W1(P1Stξ)}=: χˆπt(ξ).
By the properties of ft(ξ)and St, one can check that all the four properties in Remark 2 hold; then, χˆπt(ξ)is the
characteristic function of a state ˆπtin N∗. Then, we can define Tt∗on N∗by Tt∗[ˆπ0] = ˆπtand linear extension.
One can check that {Tt∗, t ≥0}is a strongly continuous semigroup of completely positive, normalization preserving
maps.
By defining Tt=Tt∗∗on the whole N, by (33), we get that Ttis bounded with norm 1. This is the unique linear
extension to Nof the operator of Definition 1.
This ends the proof of the theorem.
Remark 7.Being Zthe generator of St, from (17) we get the following positivity condition on At, given by the first
equality in (27):
0≤Zt
0
dτ ST
τA±i
2σZ +ZTσSτ=At±i
2ST
tσSt−σ.(34)
2.5 Wigner function and equilibrium
We can introduce the characteristic function of the hybrid state ˆπt(see Sec. 2.2.2) [5, Theor. 4]; by using the explicit
form (11), (15) of the action of Tton the Weyl operators, we get
χˆπt(ξ) = ZRs
dyTr {ˆπt(y)W(ξ)(y)}= eΨt(ξ)χˆπ0(Stξ).(35)
As one can check, condition (34) guarantees the positivity condition (9) for χˆπt(ξ).
The quantity ft(ξ) = eΨt(ξ)is the characteristic function of an infinitely divisible distribution. Its Fourier trans-
form is non-negative because it is a probability density; possibly, the probability has discrete components and the
“density” has singular components. Formally, we can write
WΨt(z) := 1
(2π)2n+sZΞ
dξexp −izTξ+ Ψt(ξ)≥0.
Then, by (8), we get the Wigner function of the hybrid state ˆπt:
Wˆπt(z) = 1
(2π)2n+sZΞ
dξe−izT˜σξ eΨt(ξ)χˆπ0(Stξ) = 1
(2π)2n+sZΞ
dxZΞ
dξWΨt(x)χˆπ0(Stξ)e−i(z−x)T˜σξ .(36)
This Wigner function can be also expressed as the convolution
Wˆπt(z) = 1
|det St|ZΞ
dxWΨt(x)Wˆπ0ST
t(z−x).
The presence of possible negative zones in Wˆπ0depends on the choice of the state ˆπ0; the convolution with the positive
function WΨt(x)tends to diminish the possible negative contributions. We can say that the quasi-free evolution tends
to diminish the quantum signature of the Wigner function.
14
2.5.1 Approach to equilibrium
By using the Wigner function (36) of the hybrid state, it is easy to give a formula for the final equilibrium state of a
quasi-free semigroup. As ft(ξ)is the characteristic function of a probability distribution, the study of its behaviour
for large times is a purely classical problem; results on this problem are given in [37].
Let us assume that Stis a contracting semigroup, that is Stξ→0as t→ ∞. Since it is a matrix semigroup, this
condition implies also that |Stξ| ≤ e−ct |ξ|for some constant c > 0. Then, in (35) the second factor converges to
χˆπ0(0) = 1, i.e. the initial state becomes irrelevant for the limit. For what concerns the first factor, we assume that the
following integral converges:
Z|ξ|>1
ln |ξ|ν(dξ)<+∞.
By [37, Theor. 4.1], under these hypotheses we have the existence of the limit Ψ∞(ξ) = R+∞
0dτ ψ(Sτξ); moreover,
eΨ∞(ξ)is the characteristic function of an infinitely divisible distribution.
Under the same hypotheses, from (36), we have also Wˆπt(z)→ WΨ∞(z)which is non-negative, but not necessar-
ily a L1-function: one can have a limit state, but, perhaps, not in N∗. A C∗- algebraic approach is needed to include
the limit state; see the discussion in Sec. 2.1 and in [5].
3 The generator
In Theorem 1 we constructed the explicit form of the quantities involved in the action of Tton the Weyl operators;
then, by linearity and weak∗-continuity, we obtained the action on the whole W∗-algebra N. So, the generator of the
semigroup was not needed to determine the semigroup, but it is useful to better understand the physical interactions
and to connect the quasi-free case to other situations.
By differentiating (11) and using (7), we get the action of the generator Kof Tton the Weyl operators:
K[W(ξ)] = ψ(ξ)W(ξ) + i
2RTZξ , W (ξ).
Then, by dividing a “Gaussian” part from a “compensated jump” part, we can write this generator as
K=K1+K2,K1[W(ξ)] = iαTξ−1
2ξTAξW(ξ) + i
2RTZ ξ, W (ξ),(37a)
K2[W(ξ)] = ZΞ
ν(dη)eiηTξ−1−i1S(η)ηTξW(ξ).(37b)
We recall that (A, ν, α)is the generating triplet of ψgiven in Theor. 1, and Zis the real (2n+s)×(2n+s)-matrix
generating the semigroup St(13). Let us stress that the above decomposition is not unique, as there are equivalent ways
of writing the compensator in the L´evy-Khintchine formula [16, Remark 8.4]; a change in the compensator amounts
to a change in α. This generator can be extended at least to the linear span of the Weyl operators and expressed by
introducing suitable operators on the Hilbert space H.
Remark 8.In the following we use the block notation with respect to the decomposition Ξ = Ξ1⊕Ξ0:
α=β
α0, A =A11 A10
A01 A00, Z =Z11 Z10
Z01 Z00;(38)
recall that Ais a real symmetric matrix. With this notation, the matrix Bdefined in (17) becomes
B=1
2σZ −ZTσT=1
2 σZ11 −Z11TσTσZ10
−Z10TσT0!=B11 B10
B01 0.(39)
We define also
D11 =1
2P1Zσ +σTZTP1=1
2Z11σ+σTZ11 T.(40)
By inverting the previous equations with respect to Z, we get
Z11 =D11σT−σB11 , Z10 = 2σTB10 .(41)
15
Proposition 6. When ais in the linear span of the Weyl operators and fis bounded and twice differentiable, the
generator (37) can be written in the following form: K=K1+K2,
K1[a⊗f](x) = 1
2a
s
X
i,j=1
A00
ij
∂2f(x)
∂xi∂xj
+ i
2n
X
i=1
s
X
j=1
RiaσTA10 −iB10ij
∂f (x)
∂xj
−i
s
X
i=1
2n
X
j=1
∂f (x)
∂xiA01 −iB01 σij aRj+
2n
X
i,j=1 σTA11 −iB11σij RiaRj−1
2{RiRj, a}f(x)
+a
s
X
j=1
α0
j
∂f (x)
∂xj
+a
s
X
i,j=1
xiZ00
ij
∂f (x)
∂xj
+ i [Hq+Hx, a]f(x),(42a)
Hq=βTσP1R+1
2RTP1D11P1R, (42b)
Hx=xTP0ZP1σR =
s
X
i=1
2n
X
j=1
xiZ01
ij (σR)j,(42c)
K2[a⊗f](x) = ZΞ
ν(dη)f(x+y)W1(σζ′)†aW1(σζ′)−f(x)a
−1S(η)if(x)hζ′TσR, ai+a
s
X
j=1
yj
∂f (x)
∂xj, η =ζ′
y.(42d)
Recall that B,Dand Zare connected by (40) and (41).
Proof. By Eqs. (5) and (7), we get the correspondence rules: (P1ξ)i=ζi→[(σR)i,•],(P0ξ)i=ki→ −i∂
∂xi. By
applying these rules to K1(37a), we obtain (42a)-(42c) by direct computations.
By using (4a), (6a), (7), and ηT= (ζ′T, yT)we get
eiηTξW(ξ) = eiyTkW0(k)eiζ′TζW1(ζ) = W1(σζ′)†W1(ζ)W1(σζ′)⊗ Ry[W0(k)],
iηTξW (ξ)(x) = i hζ′TσR, W1(ζ)iW0(k)(x) + W1(ζ)
s
X
j=1
yj
∂
∂xj
W0(k)(x);
the classical translation operator Ryis defined in Sec. 2.1. Then, we have (42d).
Eventually, the domain of the generator can be extended by weak∗-closure. It is possible to check that the structure
of this generator is analogous to the structure of [43, (4.39)], while in that reference only bounded operators on the
Hilbert space were allowed.
The first four addends in (42a) form a hybrid dissipative contribution and they are connected by the positivity
condition (17); indeed we have
0≤σT+P0(A−iB) (σ+P0) = σTA11 −iB11σ σTA10 −iB10
A01 −iB01σ A00
= σTA11σ+i
2σTZ11T−Z11σσTA10 −i
2Z10
A01σ+i
2Z10TA00 !.(43)
16
3.1 The reduced dynamics
Let us start from the reduced dynamics of the quantum component. By setting f(x) = 1 in the expression (42) of the
generator we get
K[a⊗1](x) = Lq[a] + i [Hx, a],(44)
Lq[a] = i [Hq, a] +
2n
X
i,j=1 σTA11 −iB11σij RiaRj−1
2{RiRj, a}
+ZΞ
ν(dη)nW1(σζ′)†aW1(σζ′)−a−1S(η)i hζ′TσR, aio, ζ ′=P1η. (45)
The last term in (44) can be seen as a random Hamiltonian evolution, because the classical variables xj,j= 1,...,s,
appear in Hx, which is defined in (42c). The dynamics of the quantum component alone is not autonomous, unless
Z01 vanishes; we can say that Z01 controls the information flow from the classical system to the quantum one. As
we shall see in Sec. 4, Lqcan be written in the usual Lindblad form; the jump term and the term involving A11 are of
dissipative type.
We consider now the reduced dynamics of the classical component. By setting a=1in the expression (42) of the
generator and by taking into account (43) and that Ais a symmetric matrix and Ban antisymmetric one, we get
K[1⊗f](x) = Kcl[f](x) +
s
X
i=1
2n
X
j=1
RjZ10
ji
∂f (x)
∂xi
,(46)
Kcl[f](x) =
s
X
j=1
α0
j
∂f (x)
∂xj
+
s
X
i,j=1
xiZ00
ij
∂f (x)
∂xj
+1
2
s
X
i,j=1
A00
ij
∂2f(x)
∂xi∂xj
+ZΞ
ν(dη)f(x+y)−f(x)−1S(η)
s
X
j=1
yj
∂f (x)
∂xj, y =P0η. (47)
The last term in (46) contains the quantum operators Rj; so, also the dynamics of the classical component alone is not
autonomous, unless Z10 vanishes. We can say that Z10 controls the information flow from the quantum system to the
classical one. Again the jump term and the term with the second derivatives are of dissipative type; we shall discuss
their role in Sec. 5. The classical component Kcl of the generator is exactly the generator given in [37, (1.1)], for the
so called processes of Ornstein-Uhlenbeck type.
3.2 The classical-quantum interaction
To clarify the meaning of the various terms in the generator (42), it is useful to rewrite it in a way which puts in
evidence the quantum-classical interaction terms:
K[a⊗f](x) = Lq[a]f(x) + aKcl[f](x) +
4
X
l=1
Kl
int[a⊗f](x),(48a)
K1
int[a⊗f](x) = i [Hx, a]f(x),(48b)
K2
int[a⊗f](x) = 1
2
s
X
i=1
2n
X
j=1
{a, Rj}Z10
ji
∂f (x)
∂xi
,(48c)
K3
int[a⊗f](x) =
s
X
i=1
2n
X
j=1
i[(σR)j, a]A10
ji
∂f (x)
∂xi
,(48d)
K4
int[a⊗f](x) = ZΞ
ν(dη) (f(x+P0η)−f(x)) W1(σP1η)†aW1(σP1η)−a.(48e)
17
The interaction K1
int (48b) involves the random Hamiltonian Hx, defined in (42c); its role has been discussed in
Sec. 3.1: it represents a kind of force exerted on the quantum system by the classical one. On the other side, the
interaction K2
int (48c) represents some action of the quantum system on the classical one. Note that Z10 ≡2σTB10
is involved in the positivity condition (17), (43); in some sense this interaction term injects some quantum uncertainty
into the classical output.
The interaction term K3
int (48d) has a peculiar structure, as it vanishes either when the reduced classical dynamics
is considered (a=1), either when the reduced quantum dynamics is considered (f(x) = 1). Also this term is involved
in the positivity condition (17), (43).
Finally, K4
int (48e) represents the interaction contained in the jump part; also this term vanishes either from the
reduced classical dynamics, either from the reduced quantum dynamics. This is a property of the quasi-free character
of the dynamics, because the Lindblad operators involved in the jump part are proportional to unitary operators; it could
be seen that this fact makes the probability law of the inter-jump time independent from the quantum state [15, 51].
3.2.1 The role of the dissipative terms
It is interesting to see what happens in the case of no-dissipation in the purely quantum component of the dynamics,
i.e. in Lqgiven by (45). Firstly, we have to take A11 = 0; by the positivity condition (17) and the structure (38), (39),
(41), this gives also the vanishing of other terms:
A11 = 0 ⇒A01 =A10T= 0, B01 =−B10 T= 0,
B11 = 0, Z10 = 0, Z11 =D11σT.
To have no dissipation in Lq, also the compensated jump term in (45) has to vanish. These conditions imply the vanish-
ing of the interaction terms K2
int,K3
int,K4
int; the only classical-quantum interaction which survives is the Hamiltonian
contribution K1
int, linked to the flux of information from the classical system to the quantum one. So, we have the
vanishing of all the terms which can transfer information from the quantum to the classical system, which in principle
could be observed without disturbance; again, also for the introduced hybrid Markovian quasi-free dynamics we have
that no information can be extracted from the quantum system without dissipation.
Also the specular statement holds for the hybrid quasi-free dynamics: if the classical component extracts inform-
ation from the quantum one, it acquires necessarily a stochastic behaviour. Indeed, if we require the vanishing of the
diffusive term in Kcl (47) we have:
A00 = 0 ⇒A01 =A10T= 0, B01 =−B10 T= 0,
Z10 = 0,K2
int = 0,K3
int = 0.
To have no dissipation in Kcl, also the compensated jump term in (47) has to vanish; then, we have also K4
int = 0.
Again, only the interaction term K1
int, which contains Hx, survives: a deterministic classical system can act only as a
kind of control on the quantum system.
Before discussing the meaning and the properties of the general hybrid dynamics, it is useful to see the case of
a pure quantum dynamical semigroup (Sec. 4) and the pure classical case (Sec. 5). From the pure quantum case we
can have some hints on the possible interactions among quantum components and with the external environment. In
the pure classical case we can see that the dynamical semigroup does not give only the state at time t, the probability
distribution at a single time, but a whole stochastic process can be constructed with all multi-time probabilities. This
idea can be extended to the hybrid case and we can interpret the observation of the classical component as a monitoring
in continuous time of the quantum system.
4 Quasi-free quantum dynamical semigroups
By taking s= 0 in Sec. 2.3 we obtain the most general quasi-free quantum dynamical semigroup. In this case we have
St= eZ11t, where Z11 =Zis a generic 2n×2nreal matrix; moreover, A11 =Aand the positivity condition (17)
reduces to
A11 ±iB11 ≥0, B11 =1
2σZ11 −Z11TσT.(49)
18
Then, Eq. (11) becomes
Tt[W1(ζ)] = exp Zt
0
dτ ψ1(Sτζ)W1(Stζ),∀ζ∈R2n,∀t≥0; (50)
ψ1(ζ)is obtained by particularizing (16), (14):
ψ1(ζ) = iβTζ−1
2ζTA11ζ+ZR2n
ν1(dζ′)eiζ′Tζ−1−i1S1(ζ′)ζ′Tζ,
S1=ζ∈R2n:|ζ|<1,ZS1\0
|ζ′|2ν1(dζ′)<+∞, ν1({1≤ |ζ|′<+∞})<+∞.
(51)
In the general hybrid case, we can obtain the dynamics of the quantum subsystem alone by taking ξT= (ζT,0) in
(11). As seen in Sec. 3.1, the reduced quantum dynamics is not a semigroup and it depends on the dynamics of the
classical system. The reduced quantum dynamics is a semigroup if and only if we have
Z01 = 0.
Under this assumption, we get ψ(P1ζ) = ψ1(ζ)by identifying the generating triplet:
A11 =P1AP1, ν1(E) = ZΞ
ν(dη)1E(P1η), β =P1α+Z{η∈Ξ:|P1η|2<1,|η|≥1}
ν(dη)P1η.
A similar reduction problem is considered in [16, Prop. 11.10], in the pure classical case.
The generator Lqof the most general quasi-free quantum dynamical semigroup on B(H)takes the form (45),
(42b). We can say that Tt, given by (50) and (51), solves the Markovian quantum master equation (in the Heisenberg
description) ˙
Tt[a] = TtLq[a]. By separating the Gaussian part from the compensated jump part in Lq, we can write
Lq=L1+L2,(52a)
L1[a] = i [Hq, a] +
2n
X
i,j=1 σA11 −iB11σTij RiaRj−1
2{RiRj, a},(52b)
Hq=1
2
2n
X
i,j=1
RiD11
ij Rj+
2n
X
i,j=1
βiσij Rj, D11 =1
2Z11σ+σTZ11T, βj∈R,(52c)
L2[a] = ZR2n
ν1(dζ′)W1(σζ′)†aW1(σζ′)−a−i1S1(ζ′)hζ′TσR, ai.(52d)
Let us note that Lqis an unbounded generator. It is known that many analytical problems are involved in the
construction of the semigroup from a formal unbounded generator [26, 27, 32]; however, in the quasi-free case the
semigroup is defined directly by its action on the Weyl operators (50), (51), and these problems do not arise.
Remark 9.By the positivity condition in (49) we can write the matrix σA11 −iB11σTin terms of its eigenvalues
and eigenvectors:
σA11 −iB11σTji =
2n
X
l=1
λlaj
lai
l, λl≥0,
2n
X
i=1
ai
lai
l′=δll′.
Then, the Gaussian contribution can be written in a Lindblad-like form as
L1[a] = i [Hq, a] +
2n
X
l=1 L†
laLl−1
2nL†
lLl, ao, Ll=pλl
2n
X
i=1
ai
lRi.
Similarly, by using −a=−1
2W1(σζ′)†W1(σζ′), ainside the integral in (45), we see that the compensated jump
contribution can be written as an integral of Lindblad-like expressions.
Remark 10.If we start from the operator expression (52), with the matrices B11 and D11 given, we can obtain the
action of the generator on the Weyl operators by taking Z11 from (41), which gives it in terms of B11 and D11 .
Note that βand D11 come out from the Hamiltonian contribution, while A11 and B11 come out from the dissipative
part of the Gaussian contribution. Moreover, the compensated jump part does not contribute to the “deterministic”
dynamics St= eZ11 t.
19
4.1 The state dynamics
As already recalled in Sec. 2.2.2 for the hybrid case, a state ρis completely determined by its characteristic function
χρ(ζ) = Tr {W1(ζ)ρ}. The anti-Fourier transform of χρ(ζ)is the Wigner function of ρ; when the Wigner function is
non-negative, it is a probability density and we say that the corresponding state is “classical”.
Let Tt∗be the preadjoint of Tt; then, ρt=Tt∗[ρ0]is the state at time t. The characteristic function of the state at
time tis
χρt(ζ) = Tr {Tt[W1(ζ)]ρ0}= exp Zt
0
dτ ψ1(eZ11τζ)χρ0(eZ11 tζ).(53)
Note that the exponential pre-factor does not depend on the initial quantum state; by Remark 5 this factor is the
characteristic function of a probability law (which is non-negative). Therefore, when ρ0is a classical state, ρttoo
turns out to be a classical state. When ρ0is non-classical, the product structure in the characteristic function implies a
convolution structure in the Wigner function, which tends to eliminate non-classicality.
By using (5) and (6a), from the generator structure (52) we get easily
W1(z)†L1[W1(z)W1(ζ)W1(z)†]W1(z) = L1[W1(ζ)] −izTZ11σζW1(ζ),
W1(z)†L2[W1(z)W1(ζ)W1(z)†]W1(z) = L2[W1(ζ)].
So, we have that the jump component L2is Weyl covariant [32, 33]. The Gaussian component L1enjoys the same
covariance property only if Z11 = 0, which means D11 = 0 and B11 = 0; then, L1takes the form
L1[a] = i βTσ R, a−1
2
2n
X
i,j=1 σA11 σTji [Rj,[Ri, a]] .(54)
By Proposition 1 in [33] the most general Weyl covariant quantum dynamical semigroup turns out to be quasi-free
and, indeed, the adjoint of the generator (2.12) in Proposition 2 of [33] coincides with the generator defined by the
sum of (52d) and (54). The same generator was already found in [32, Theor. 1].
Remark 11 (Existence of a unique equilibrium state).If we have eZ11 tζ≤e−κt|ζ|,∀ζ∈R2n,∀t≥0, with κ>0,
and R|ζ|>1ln |ζ|ν1(dζ)<+∞, by the results of Sec. 2.5.1, we have
χρeq (ζ) = lim
t→+∞χρt(ζ) = exp Z+∞
0
dτ ψ1(eZ11τζ).
The equilibrium state has a non-negative Wigner function, given by an infinitely divisible probability distribution.
Note that, by (34), we have A∞±i
2σ≥0, and the equilibrium state is not singular.
By the property (7) of the Weyl operators, we get that the quantum means are given by
hRit=St
ThRi0+Zt
0
dτ Sτ
T˜
β, d
dthRit=Z11ThRit+˜
β,
hRjit:= Tr {Rjρt},˜
β:= β+Z|ζ|≥1
ζν1(dζ).
(55)
The quantum means exist when hRji0is finite and the integral in the definition of ˜
βexists.
4.2 Examples
Quasi-free quantum dynamical semigroups, with a non-vanishing jump part, already appeared in the literature, mainly
under symmetry requirements; for instance, the Galilean covariant evolutions considered in [33, Sec. III] turn out to be
quasi-free semigroups. When less symmetry is considered, the resulting semigroup is not necessarily quasi-free, but
jump structures are included. For instance, in [32, Theor. 2] the most general generator of a space-translation covariant
semigroup is obtained; in this case to ask the semigroup to be quasi-free is a restriction.
Under the name of quantum linear Boltzmann equation, translation covariant quantum dynamical semigroups have
been used to study the motion of a test particle in a gas and the so called dynamical decoherence [38,39].
20
4.2.1 Quasi-free quantum linear Boltzmann equation
As a first example, we give here the quasi-free version of the quantum linear Boltzmann equation studied in [39]. We
take n= 3 and consider the case of covariance under spatial translations and rotations.
We use a notation with 3×3-blocks (position variables and momentum variables) and we take
Z11T=01
m
0−γ1, A11 =a11a31
a31a21, β = 0,
a1≥0, a2≥0, a1a2≥a2
3+γ2
4, m > 0, γ > 0;
this gives also
D11 =0γ
21
γ
211
m, B11 =γ
20−1
10, A11 ±iB≥0,
St= 10
1−e−γt
mγ 1e−γt 1!.
Then, we take the measure ν1to be rotationally invariant; as pointed out in [33] after Eq. (3.14), the compensator
formally vanishes and we can write
ψ1(ζ) = −1
2ζTA11ζ+ lim
ǫ↓0Z|ζ′|>ǫ
ν1(dζ′)ei(ζTζ′)−1.
Finally, we ask that the quantum expectations hRjitexist.
In this example there is not a stationary state at large times and, moreover, the memory of the initial state is not lost.
However, the quantum distribution of the momentum reaches a stationary expression and looses any contribution of
the initial state. Indeed, by using the improper eigenfunctions |piof the momentum operators, the momentum density
at time tis given by
hp|ρt|pi=1
(2π)3ZR3
dke−ikTpχρt(0, k).
Then, we get
lim
t→+∞χρt(0, k) = exp (−a2|k|2
4γ+ZΞ
ν1(dζ′)Z+∞
0
dτexp i(0, kT)ζ′e−γ τ −1),
which is the characteristic function of an infinitely divisible distribution. Let us note that γ > 0implies a2>0and
the Gaussian contribution cannot vanish.
4.2.2 An optomechanical system
We consider now a quantum harmonic oscillator with dissipation. The choice of the dynamics depends on the physical
system we want to describe, an optical mode in a cavity or a micro-mechanical system [40–42]. Here we take as
example an oscillating micro-mirror, hit by free photons (not in a cavity) and damped by a phonon bath [41, 42].
The interaction between the mechanical mirror and the photons is by the radiation pressure. The micro-mirror is a
mechanical system; at least at the level of the quantum means, the forces must appear only in the equation for the
momentum.
Let us take n= 1; we assume the existence of the quantum expectations of Qand P. By the previous consider-
ations we want to have d
dthQit= ΩhPit; by (55) we must have Z11T
11 = 0. Moreover, we choose the underdamped
case; the dynamical matrix Z11 Tand its eigenvalues are given by the expressions
Z11T=0 Ω
−Ω−γ, γ > 0,Ω>0,Ω2>γ2
4,
λ±=−γ
2±iω, ω =rΩ2−γ2
4>0.
(56)
21
Then, we have
D11 =1
2Z11σ+σTZ11 T=Ωγ
2
γ
2Ω, B11 =1
2σZ11 −Z11TσT=−γ
2σ.
The matrix A11 cannot have vanishing eigenvalues because of the positivity condition (49), which gives
A11 =a1a3
a3a2, a1>0, a2>0, a1a2≥a2
3+γ2
4.
The jump contribution has to give rise only to the radiation pressure force (momentum kicks); so, we take the measure
ν1to be concentrated on the second component:
ZR2
ν1(dζ′)f(ζ′
1, ζ′
2) = ZR
m1(dv)f(0, v).
Apart from damping and radiation pressure, no other force must be present; so, we take also β1= 0 and β2=
R{|v|<1}m1(dv), which we assume to exists. Then we have
˜
β1= 0,˜
β2=ZR
vm1(dv), ψ1(ζ) = −1
2ζTA11ζ+ZR
m1(dv)eivζ2−1.
The generator (52) reduces to
Lq[a] = i [Hq, a] +
2
X
i,j=1 σA11 σT+iγ
2σij RiaRj−1
2{RiRj, a}+ZR
m1(dv)e−ivQaeivQ −a,
R1=Q, R2=P, Hq=1
2
2
X
i,j=1
RiD11
ij Rj=Ω
2Q2+P2+γ
4{Q, P }.
An interesting point is that the damping γcomes out from a combined action of the Hamiltonian term (through D11)
and of the dissipative Gaussian term (through B11).
The dynamical evolution ST
tis the same as for a classical damped oscillator:
ST
t=eλ+t
2iω−λ−Ω
−Ωλ++eλ−t
2iωλ+−Ω
Ω−λ−.(57)
Then, the characteristic function of the state at time tis given by (53). As Stdecays exponentially to 0and the first
moments are assumed to exist, the equilibrium state exists and does not depend on the initial state:
χρeq (ζ) = lim
t→+∞χρt(ζ) = exp Z+∞
0
dτ−1
2(Sτζ)TσA11σTSτζ+ZR
m1(dv)eiv(Sτζ)2−1,(58)
By taking the derivatives of (58) with respect to ζ1,ζ2, the quantum moments of the equilibrium state can be computed.
We assumed the existence of the first moments; the existence of higher moments depends on the properties of the
measure m1.
5 The pure classical case
Let us take n= 0; then St= eZ00 t, where Z00 =Zis a generic s×sreal matrix. Moreover, A00 =Aand condition
(17) reduces to the fact that A00 is a real positive-semidefinite s×s-matrix. Now, Eq. (11) becomes
Tt[W0(k)] = exp Zt
0
dτ ψ0(Sτk)W0(Stk),∀k∈Rs,∀t≥0,(59)
22
where the structure of ψ0(16) reduces to the usual LK-formula
ψ0(k) = ikTα0−1
2kTA00k+ZRs
ν0(dy)eiyTk−1−i1S0(y)yTk,
∀k∈Ξ0=Rs, α0∈Rs,S0={y∈Rs:|y|<1}.
(60)
Conditions (14) hold for ν0.
Similarly to what is done at the beginning of Sec. 4, the same result can be obtained in the general hybrid case,
with the assumption
Z10 = 0.
This gives StP0=P0StP0= eZ00tP0. Then, equations (11), (16) give (59), (60) for ξ= (0, k). Indeed, we get
ψ(P0k) = ψ0(k)by identifying the generating triplet:
A00 =P0AP0, ν0(E) = ZΞ
ν(dη)1E(P0η),
α0=P0α+Z{η∈Ξ:|P0η|<1,|η|≥1}
ν(dη)P0η;
Eis any Borel subset of Ξ0.
The generator of the reduced classical dynamics takes the form (47). Semigroups like Tt(59) are well known in the
theory of classical stochastic processes; they are semigroups of transition probabilities of time-homogeneous Markov
processes [16,17]; ˙
Tt[f] = TtKcl[f]is a version of the Kolmogorov-Fokker-Planck equation [17, Secs. 3.5 .2, 3.5.3].
5.1 The probabilities
In the classical case the initial condition is a probability distribution, having a density: P0(E) = REdx p0(x),
p0∈L1(Rs),p0≥0,RRsdx p0(x) = 1. Moreover, the duality form reduces to an expectation: E0[f] =
RRsdx p0(x)f(x),∀f∈L∞(Rs).
5.1.1 State evolution
The semigroup Tt∗gives the probability distribution at time t:Et[f] = RRsdx pt(x)f(x) = E0Tt[f],∀f∈
L∞(Rs). For f=W0(k)we get Et[W0] = RRsdx pt(x)eikTx=: ˆpt(k), the Fourier transform of pt:
ˆpt(k) = E0Tt[W0(k)]=ZRs
dy p0(y) exp Zt
0
dτ ψ0(Sτk)eiyTStk= exp Zt
0
dτ ψ0(Sτk)ˆp0(Stk).(61)
As the initial probability distribution has a density, we have that ˆp0(Stk)is the characteristic function of a distri-
bution admitting a density. By Remark 5 and Proposition 5, exp nRt
0dτ ψ0(Sτk)ois the characteristic function of an
infinitely divisible distribution. Then, the product ˆpt(k)is the characteristic function of a distribution with density,
given by
pt(x) = 1
(2π)sZRs
dke−ixTkˆpt(k) = ZRs
dy p0(y)1
(2π)sZRs
dkei(ST
ty−x)Tkexp Zt
0
dτ ψ0(Sτk).(62)
5.1.2 Louville equation
Here we show that, in the case of a pure Hamiltonian system, the evolution equation for the probability density (62)
reduces to the Liouville equation.
We consider a classical system with s= 2m, and canonical coordinates xi=qi,xm+i=pi,i= 1,...m. We
take the case of no noise: in (47) we have
A00 = 0, ν(dη) = 0.(63)
23
We write pt(x) = pcl (q, p, t)and we use the duality form to find the preadjoint generator acting on pcl . By integration
by parts, we shift the derivatives appearing in the generator (47) from fto pcl and we get
∂pcl (q, p, t)
∂t =−
2m
X
i=1
Z00
ii −
m
X
j=1 α0
j
∂pcl (q, p, t)
∂qj
+α0
j+m
∂pcl (q, p, t)
∂pj
−
m
X
i,j=1 qiZ00
ij +piZ00
i+m,j ∂pcl (q, p, t)
∂qj
+qiZ00
i,j+m+piZ00
i+m,j+m∂pcl(q , p, t)
∂pj.
To reduce this equation to an Hamiltonian evolution, we have to take
Z00
i+m,j =Z00
j+m,i, Z 00
i,j+m=Z00
j,i+m, Z00
ji =−Z00
i+m,j+m, i, j = 1,...,m. (64)
Now, we introduce the classical Hamiltonian
Hcl(q, p) =
m
X
i=1 α0
ipi−α0
i+lqi+
m
X
i,j=1 1
2piZi+m,j pj+1
2qiZi,j+mqj+qiZi,j pj;(65)
the Hamiltonian is of second order because we started with a quasi-free dynamics. Then, the evolution equation above
reduces to ∂pcl (q, p, t)
∂t +{pcl (q, p, t), Hcl (q, p),}cl = 0,(66)
where {·,·}cl denotes the Poisson brackets. Eq. (66) is the Liouville equation with Hamiltonian (65); note that we
asked to have a conservative dynamics (64) and no noise (63).
5.1.3 Transition probabilities
Let us go back to the general case (59), (60). The important point of the classical case is that the dynamical semigroup
does not give only the probability distribution (62) at a single time, but also all the multi-time probabilities.
By using [17, Eq. (3.3)] we can identify the transition probabilities by
Tt[f](x) = ZRs
f(y)Pt(dy|x).(67)
By (62) the characteristic function of this transition probability turns out to be
ZRs
eikTyPt(dy|x) = eikTST
txexp Zt
0
dτ ψ0(Sτk);(68)
again, these transition probabilities are infinitely divisible. By construction, the transition probabilities satisfy the
Chapman-Kolmogorov identity (for the time-homogeneous case) [16, (10.1)], [17, Theor. 3.1.5 and (3.6)]:
ZRs
Pt(C|y)Pr(dy|x) = Pt+r(C|x).(69)
Having the transition probabilities, also joint probabilities at different times can be introduced (for every choice
of a finite number of times); then, by Kolmogorov’s extension theorem [16, Theor. 1.8], it is possible to construct the
probability measure for a stochastic process X(t)in continuous time.
5.2 The process
Let B(t)be the L´evy process with characteristic function for the increments
Eexp ikT(B(t+ ∆t)−B(t))= exp {ψ0(k)∆t}.(70)
A L ´evy process has independent and stationary increments, B(0) = 0, and it is stochastically continuous [17, Sec.
1.3]. Let us recall that L´evy processes can be used to define a class stochastic integrals [17, Sec. 4.3].
24
The process X(t)associated with the semigroup Ttis a Markov process, which satisfies the stochastic differential
equation
dX(t) = Z00TX(t) dt+ dB(t),(71a)
whose solution is
X(t) = St
TX(0) + Zt
0
St−τ
TdB(τ); (71b)
the initial condition X(0) is independent of the process Band it has distribution with density p0. These processes
are sometime called of Ornstein-Uhlenbeck type [37], and they are a sub-class of the processes with independent
increments [17, p. 43], [16, Property (1) in Def. 1.6].
To check this result, we can show that the transition probabilities associated with the process (71) coincide with
the transition probabilities (67), (68). From (71) we get
X(t) = St−t0
TX(t0) + Zt
t0
St−τ
TdB(τ).
By working with the characteristic functions and using (68), we have
EheikTX(t)X(t0) = zi= exp izTSt−t0kEexp iZt
t0
(St−τk)TdB(τ)
= exp izTSt−t0k+Zt
t0
dτ ψ0(St−τk)= exp izTSt−t0k+Zt−t0
0
dτ ψ0(Sτk)=ZRs
eikTyPt−t0(dy|z);
this result gives the identification.
Let us recall that the L´evy process B(t)can be represented in terms of Wiener processes and random Poisson
measures (the L ´evy-Itˆo decomposition [17, Theor. 2.4.16]).
5.2.1 The mean values
By assuming the convergence of the integral
ZRs
|y|ν0(dy)<+∞,
we get the existence of the mean values, which are given by
E[Xj(t)] = ZRs
dx xjpt(x) = −i∂ˆpt(k)
∂kjk=0 =ST
tE[X(0)]j−iZt
0
dτ∂ψ0(Sτk)
∂kjk=0,
−i∂ψ0(Sτk)
∂kjk=0 =Sτ
Tα0j+Z|y|≥1
ν0(dy)Sτ
Tyj.
Then, we can write (60) as
ψ0(k) = i˜α0Tk+˜
ψ0(k),˜η=η0−ZS0
yν0(dy),
˜
ψ0(k) = −1
2kTA00k+ZRs
ν0(dy)eiyTk−1.
Now Eqs. (71) become
dX(t) = Z00TX(t)dt+ ˜ηdt+ d ˜
B(t),
X(t) = St
TX(0) + Zt
0
St−τ
T˜ηdτ+Zt
0
St−τ
Td˜
B(τ);
the increments of ˜
B(t)have characteristic function
Ehexp nikT˜
B(t+ ∆t)−˜
B(t)oi= exp n˜
ψ0(k)∆to.
Note that the jump contribution in ˜
ψ0is not compensated.
25
5.3 An example: a dissipative harmonic oscillator
Let us take s= 2 and set x(t) = X1(t),p(t) = X2(t). We take the deterministic part of the dynamics to be equal
to the quantum case of Sec. 4.2.2: the generator of ST
tis given by Eq. (56) with Z11T→Z00T. Then, ST
tis given by
(57). Moreover, in the classical case there is no restriction on the matrix A00 , apart from being non-negative definite.
So, we are allowed to take A00
11 = 0 and we can restrict also the noise and the forces to the second component (as it
must be in classical equations of motion):
˜η1= 0,˜
ψ0(k) = −1
2A00
22k22+ZR
m0(dv)eivk2−1.(72)
Note that this gives ˜
B1(t) = 0 and
dx(t) = Ωp(t)dt,
dp(t) = −Ωx(t)dt−γp(t)dt+ ˜η2dt+ d ˜
B2(t).(73)
All the forces appear only in the second equation, while the first equation gives the meaning and the units of measure
of the momentum: p(t) = ˙x(t)/Ω. We assume also the existence of the mean values; so, we have
dhx(t)i
dt= Ωhp(t)i,dhp(t)i
dt=−Ωhx(t)i − γhp(t)i+ ˜η2+d
dth˜
B2(t)i,
d
dth˜
B2(t)i=ZR
y˜ν0(dy).
The last term is the mean pressure force due to the jump noise; if the noise is symmetric in both sides, the mean
pressure force can vanish.
Similarly to (56) and (57), Eqs. (73) can be explicitly solved and the characteristic function (68) of the transition
probabilities can be computed. In particular, we get that it exists the limit for large times of the probability distribution
at time t: from (61) we get
lim
t→+∞ˆpt(k) = exp Z+∞
0
dτikTST
τ˜η+˜
ψ0(Sτk).
This is the characteristic function of an infinitely divisible distribution. By the choice (72) the matrix in the Gaussian
component of the generating triplet (27) turns out to be
A00
∞=A00
22 Z+∞
0
dτ ST
τ0 0
0 1Sτ=A00
22
2γ1 0
0 1.
For A00
22 >0, we have det A00
∞>0and the Gaussian component is not degenerate; then, the probability distribution
with characteristic function limt→+∞ˆpt(k)admits a density with respect to the Lebesgue measure.
6 A generic hybrid system
The semigroup Tt, constructed in Theor. 1, acts on the W∗-algebra Nand it represents the dynamics in the Heisenberg
description. Then, if ˆπ0is the state at time t= 0, its evolution is given by the preadjoint semigroup: ˆπt=Tt∗[ˆπ0].
Let us recall that a generic element F∈Nis a function F(x)from Rsinto B(H); then, the state ˆπtis a trace-class
valued function ˆπt(x)∈T(H),x∈Rs, such that ˆπt(x)≥0and RRsdxTr{ˆπt(x)}= 1.
The characteristic function of a hybrid state has been introduced in (8); similarly, the characteristic function of the
trace-class operator ˆπt(x)can be defined:
χˆπt(ξ) = ZRs
dyTr {ˆπt(y)W(ξ)(y)},
χˆπt(x)(ζ) = Tr {ˆπt(x)W1(ζ)}=1
(2π)sZRs
dke−ixTkχˆπt(ζ, k).
26
By using the explicit form (15) giving the action of Tton the Weyl operators, we get
χˆπt(ξ) = ZRs
dxTr {ˆπ0(x)W(Stξ)(x)}eΨt(ξ)=ZRs
dx χˆπ0(x)(P1Stξ) exp Ψt(ξ) + ixTP0Stξ.
Let X(t)be the random variable representing the classical component at time t, as in Sec. 5.2. Then, we have
P[X(t)∈B] = ZB
dx pt(x), pt(x) = Tr {ˆπt(x)}=χˆπt(x)(0),(74)
while the reduced quantum state ρtis given by
ρt=ZRs
dxˆπt(x),
χρt(ζ) = χˆπt(ζ, 0) = ZRs
dx χˆπ0(x)(P1StP1ζ) exp Ψt(ζ , 0) + ixTP0StP1ζ.
(75)
The expressions (74) and (75) can be seen as the marginals of the classical/quantum state ˆπt.
In the pure classical case of Sec. 5.1, we have seen that the dynamical semigroup gives rise not only to the
probabilities at time t(the state at time t), but also to the joint probabilities at different times and to a whole stochastic
process in continuous time. In a quantum theory probabilities and state changes are unified in the notion of instrument
[8, Sec. 4.1], [44, Sec. B.4], [15, Sec. 1.4]. As in the pure classical case transition probabilities are needed (see Sec.
5.1.3) to construct the related process, in the hybrid system we need to introduce a kind of “transition” instruments [43].
6.1 Instruments and probabilities
In the theory of measurements in continuous time, the classical component is the output of the measurement (usually
without a proper dynamics and backaction on the quantum system) [10–12]; in this situation, the connection between
the semigroup Ttand the instruments is given by the equation [11, (2.5)]. In the classical case we have the transition
probabilities (67); by analogy, we need to introduce instruments depending on the initial value of the classical system,
some kind of transition instruments as done in [43, Sec. 4.4]. So, we define the family of instruments
It(E|x)[a] = Tt[a⊗1E](x),∀a∈B(H),∀x∈Rs;(76)
E⊂Rsis a generic Borel set.
Proposition 7. Equation (76) actually defines an instrument It(·|x)on the σ-algebra of the Borel sets in Rs, which
means that the following properties hold:
1. for every Borel set E⊂Rsand x∈Rs(almost everywhere with respect to the Lebesgue measure), It(E|x)is
a completely positive and normal map from B(H)into itself;
2. (normalization) It(Rs|x)[1] = 1;
3. (σ-additivity) for every countable family of Borel disjoint sets Ei,ItSiEix[a] = PiIt(Ei|x)[a],∀a∈
B(H).
Moreover, the family of instruments (76) enjoys the following composition property:
It+t′(E|x) = Zz∈Rs
It′(dz|x)◦ It(E|z).(77)
Finally, the action of the transition instrument on the Weyl operators is given by
It(E|x)[W1(ζ)] = 1
(2π)sZRs
dkZE
dzexp Ψt(ξ)−izTkW(Stξ)(x), ξ =ζ
k.(78)
Properties 1.–3. in the proposition above come from the definition of instrument. Equation (77) is the quantum
analogue of the Chapman-Kolmogorov identity (69).
27
Proof. By using the defining equation (76) and the properties of Ttin Definition 1, we have that
• point a. implies property 1.,
• point b. implies property 2.,
• again normality (point a.), linearity, and the structure with the indicator function imply property 3.
For any F(·)∈Nwe have RRsIt(dy|x)[F(y)] = Tt[F](x). Then, we have
ZRs
It+t′(dy|x)[F(y)] = Tt+t′[F](x) = Tt′◦ Tt[F](x)
=Zy∈Rs
Tt′[It(dy|·)[F(y)]] (x) = Zy∈RsZz∈Rs
It′(dz|x)◦ It(dy|z)[F(y)];
this gives (77).
By using 1E(x) = 1
(2π)sRRsdkREdzei(x−z)Tk, we get by easy computations
It(E|x)[W1(ζ)] = 1
(2π)sZRs
dkZE
dze−ikTzTt[W(ξ)](x);
by (11) and (15), Eq. (78) follows.
From (78) we obtain also the explicit form of the Fourier transform of the transition instrument, the characteristic
operator:
Γt(k|x)[W1(ζ)] := ZRs
It(dz|x)[W1(ζ)]eikTz=Tt[W(ξ)](x) = exp {Ψt(ξ)}W(Stξ)(x), ξ =ζ
k.(79)
6.1.1 Multi-time probabilities and instruments
By repeated application of the single-time instrument, also multi-time instruments can be obtained, again determined
by their Fourier transform. Let us take the arbitrary times 0< t1< t2<··· < tm; the multi-time analogue of (79) is
the quantity
Γ(k1, t1;...;km, tm|ζ, x) = Z(Rs)m
It1(dx1|x)◦ It2−t1(dx2|x1)◦ · · ·
◦ Itm−tm−1(dxm|xm−1)[W1(ζ)] exp (m
X
l=1
ixT
lkl).
By recursion, from (11), (15), and (76), we get
Γ(k1, t1;...;km, tm|ζ, x) = exp (m
X
l=1
Ψtl−tl−1(ξl) + ixTP0St1ξ1)W1(P1St1ξ1),
t0= 0, ξl=Stl+1−tlξl+1 +P0kl, l = 1,...,m−1, ξm=ζ
km.
Condition (77) implies the consistency of the multi-time probabilities and the existence of a stochastic process in
continuous time as in the purely classical case (Sec. 5.1.3). The joint probabilities for the process X(t)at the times
0< t1< t2<···< tm(for an initial state ˆπ0) are given by
P[X(t1)∈E1, X(t2)∈E2,...,X(tm)∈Em|ˆπ0]
=ZRs
dxTrˆπ0(x)ZE1
It1(dx1|x)◦ZE2
It2−t1(dx2|x1)◦ · · · ◦ ZEm
Itm−tm−1(dxm|xm−1)[1].(80)
28
Then, the quantity RRsdxTr {ˆπ0(x)Γ(k1, t1;k2, t2;...;km, tm|0, x)}turns out to be the characteristic function of
these multi-time probabilities. To better understand this result let us consider the two times case: 0< t1< t2. By
simple computations we get the characteristic function
Eexp ikT
1X(t1) + ikT
2X(t2)=ZRs
dxTr {ˆπ0(x)Γ(k1, t1;k2, t2|0, x)}
=ZRs
dxTr {ˆπ0(x)W1(P1St1ξ1)}exp ixTP0St1ξ1+ Ψt1(ξ1) + Ψt2−t1(ξ2),(81)
where
ξ2=P0k2, ξ1=St2−t1P0k2+P0k1, St1ξ1=St1P0k1+St2P0k2,
Ψt2−t1(ξ2) = Zt2−t1
0
dτψ(SτP0k2),Ψt1(ξ1) = Zt1
0
dτψ(SτP0k1+St2−t1+τP0k2).
6.1.2 Conditional probabilities and conditional states
Having constructed multi-time probabilities (80) and instruments (76), it is possible to introduce both probabilities and
quantum states conditioned on the previous observations. Let us exemplify the introduction of conditional probabilities
and states in a simple case: a single application of the transition instrument. The role of the notion of instrument in
a quantum theory is to give the probabilities for the associated measurement and the state after the measurement
conditioned on the observed result. In a time interval (0, t), the probability of observing the increment X(t)−X(0)
of the classical subsystem in a Borel set E⊂Ξ0, conditional on the initial value X(0) = xand on the state ρ0of the
quantum subsystem, is given by
Pt(E|x) := PX(t)−X(0) ∈E|X(0) = x, ρ0= Tr {ρ0It(E|x)[1]}.(82)
Then, the quantum state at time t, conditional on the observation of the classical component in E, is
ρt(E;x) = It(E|x)∗[ρ0]
Pt(E|x);
also the limit case of the set Ereducing to a point zcan be considered. By using the characteristic function for the
quantum state we get
χρt(E;x)(ζ) = Tr {ρt(E;x)W1(ζ)}=Tr {ρ0It(E|x)[W1(ζ)]}
Pt(E|x).
In the quasi-free case, using ξT= (ζT, kT), we have
Tr {ρ0It(E|x)[W1(ζ)]}=1
(2π)sZRs
dkZE
dzexp Ψt(ξ)−izTk+ ixTP0Stξχρ0(P1Stξ).
In particular, the probabilities (82) become
Pt(E|x) = 1
(2π)sZRs
dkZE
dzexp Ψt(P0k)−izTk+ ixTP0StP0kχρ0(P1StP0k);
the quantum state ρ0affects these probabilities only through P1StP0. For a small time interval ∆t, we have P1S∆tP0
≃Z10∆tand χρ0(P1S∆tP0k)≃exp i∆thRTi0P1Z10k. By comparing with (48c) we see once more that the
classical probabilities acquire information from the quantum system via the interaction term K2
int. Moreover, in the
approximation for small times, we have that P∆t(•|x)is an infinitely divisible distribution, because we have
P∆t(E|x)≃1
(2π)sZRs
dkZE
dzexp n∆tψ(P0k) + i (x−z)Tk+ i∆txTZ00 k+ i∆thRTi0P1Z10ko.
29
6.2 An example
To illustrate the properties of hybrid systems and the role of the interaction terms discussed in Sec. 3.2, we present a
simple example with a classical system injecting noise in the quantum component and an observed output signal; as
quantum system we take the quantum mechanical oscillator introduced in Sec. 4.2.2. So, we take n= 1,s= 2; as
generator of the deterministic evolution St= eZt we take
Z=
0−Ω 0 g
Ω−γ0 0
0b−c0
0 0 0 0
,
Ω>0, γ > 0, c > 0, b, g ∈R,
λ±=−γ
2±iω, ω =qΩ2−γ2
4>0.
By setting ξ(t) = Stξ(0) we can easily solve the first order equations ˙
ξ(t) = Zξ(t); under the initial condition
ξ(0) = (ζ1, ζ2, k1, k2)T, we get
ξ1(t) = x+eλ+t+x−eλ−t+γgk2
Ω2,
ξ2(t) = y+eλ+t+y−eλ−t+gk2
Ω,
ξ3(t) = e−ctk1+bℓ(t),
ξ4(t) = k2,
ℓ(t) = y+eλ+t−e−ct
λ++c+y−eλ−t−e−ct
λ−+c+gk21−e−ct
Ωc,
2iωx+=−λ−ζ1−Ωζ2−gk2,
2iωy+= Ωζ1+λ+ζ2+gk2
Ω,
x−=x+, y−=y+.
By (39), the form of Zimplies
B=σZ −ZTσT
2=1
2
0−γ0 0
γ0 0 −g
0 0 0 0
0g0 0
, A −iB≥0.
We take a very simple choice for A, compatible with the positivity condition above:
A=
A11 A12 0 0
A12 A22 0A24
0 0 A33 0
0A24 0A44
,
Aii ≥0, A22 =A1
22 +A2
22,
A1
22 ≥0, A11A1
22 ≥A2
12+γ2
4,
A2
22 ≥0, A2
22A44 ≥A24 2+g2
4.
The action of the dynamical semigroup on the Weyl operators is given by (11) and (15), where ψ(ξ)has the expression
(16). We choose a simpler expression also for this quantity, with the jump part only in the first component of the
classical system:
ψ(ξ) = −1
2ξTAξ + iα0ξ3+J3(ξ3), J3(ξ3) = ZR\{0}
ν(dv)eivξ3−1−i1{|v|<1}(v)vξ3.(83)
To have the limit for large times, according to the discussion in Sec. 2.5.1, we add the assumption
Z|v|>1
ln |v|ν(dv)<+∞.
With these choices, the quantum/classical interaction terms in (48) become
K1
int[a⊗f](x) = i [Hx, a]f(x) = −ibx1[Q, a]f(x),(84a)
K2
int[a⊗f](x) = g
2{a, Q}∂f (x)
∂x2
,(84b)
K3
int[a⊗f](x) = −i[Q, a]A24
∂f (x)
∂x2
,K4
int = 0.(84c)
30
6.2.1 The input classical noise
Let us start by considering only the input classical noise X1(t), which means to take ζ1= 0,ζ2= 0,k2= 0. This
gives x±=y±= 0,ξ1(t) = ξ2(t) = ξ4(t) = 0,ξ3(t) = e−ctk1,
Tt[W(0,0, k1,0)](x) = exp Zt
0
dτiα0k1e−cτ −A33
2k2
1e−2cτ +J3k1e−cτ 1.
By this, we see that the quantum system is not involved in the evolution of the classical process X1(t). So, we can
directly apply the results of Sec. 5.2; by (71) and (70), X1(t)is the process
X1(t) = e−ctX(0) + Zt
0
e−c(t−τ)dB1(τ);
B1(t)is a L´evy process, whose increments have characteristic function
Eexp ik1B1(t+ ∆) −B1(t)= exp ∆iα0k1−A33
2k2
1+J3k1.
6.2.2 The reduced quantum state
The reduced quantum state does not satisfy a Markovian master equation, but, being the evolution quasi-free, we have
implicitly the expression of the reduced state at any time. To see the noise acquired from the classical component, we
study the quantum state at large times. We take k1= 0,k2= 0, which gives
ξ1(t) = x+eλ+t+x−eλ−t,
ξ2(t) = y+eλ+t+y−eλ−t,
ξ3(t) = by+eλ+t−e−ct
λ++c+y−eλ−t−e−ct
λ−+c,
ξ4(t) = 0,
(85a)
x+=x−=1
2ζ1−i
2γ
2ωζ1−Ω
ωζ2=−λ−y+
Ω,
y+=y−=1
2ζ2+i
2γ
2ωζ2−Ω
ωζ1=−λ+x+
Ω;
(85b)
by the choice of the quantum system, ξ1(t)and ξ2(t)have the expression (57) of the example of Sec. 4.2.2. The
reduced quantum state is given by (75). The expressions for ξj(t)in (85) decay exponentially; then, the quantum
characteristic function at large times turns out to be
χρeq(ζ)= lim
t→+∞χρt(ζ) = exp Z+∞
0
dτ ψξ(τ).
By (83) and (85), we get
Z+∞
0
dτ ψξ(τ)= i bα0ζ2
cΩ−1
2ζTAqζ+Z+∞
0
dτ J3ξ3(τ),
Aq
11 =Aq
22 +A12
Ω, Aq
12 =Aq
21 = 0, Aq
22 =A11 +A22
2γ+A33b2γ
2+c
2cγ (Ω2+γc +c2).
Without the interaction with the first component X1(t)of the classical system (b= 0, which in this example means
K1
int = 0), the asymptotic state of the quantum system becomes purely Gaussian, as the jump integral J3vanishes to-
gether with the terms proportional to A33 . The interaction of the quantum oscillator with X1(t)modifies the Gaussian
component of the state, by adding the terms proportional to A33. Moreover, while this interaction is linear (the first
expression in (84)), the asymptotic quantum state is not Gaussian, but it contains also noise of “jump type” received
from the classical noise X1(t).
31
6.2.3 The observed output
We consider now the observed output, the classical component X2(t)alone. Without interaction, i.e. g= 0,A14 = 0,
only the term with A44 survives and X2(t)is a Wiener process with variance tA44 . For the case with interaction, we
discuss only some of the involved probabilities, not the full process. By taking ζ1= 0,ζ2= 0,k1= 0,k2=κ, we
get
ξ1(t;κ) = κgeλ−t−eλ+t
2iω+γ
Ω2,
ξ2(t;κ) = κg
Ωλ+eλ+t−λ−eλ−t
2iω+ 1,
ξ3(t;κ) = κbg
Ω2 Re λ+(eλ+t−e−ct )
2iω(λ++c)+1−e−ct
c,
ξ4(t;κ) = κ.
By using (79) we obtain the characteristic function of the one-time probability law
EheiκX2(t)i=ZR2
dxTr ˆπ0(x)W1P1ξ(t;κ)exp ixTP0ξ(t;κ) + Zt
0
dτ ψξ(τ;κ).
For large times we have
ξ(t;κ)≃ξ(∞;κ) = κgγ
Ω2,g
Ω,bg
cΩ,1T
,
Zt
0
dτ ψξ(τ;κ)≃Z+∞
0
dτψξ(τ;κ)−ψξ(∞;κ)+tψξ(∞;κ),
ψξ(∞;κ)=J3bg
cΩ+ i bg
cΩα0κ−κ2
2g2γ2A11
Ω4+g2A22
Ω2+2g2γA12
Ω3+ 2 gA24
Ω+A44 +b2g2A33
c2Ω2.
Obviously, we have to assume the existence of the integral in the central expression. These expressions show that the
process X2(t)cumulates contributions coming from the quantum system (the terms with the coupling constants gand
A14), but also from the classical component X1(t); however, these last contributions have to pass through the quantum
system as shown by the terms with the product bg of coupling constants.
To see the contribution of the interaction terms in a short time, we consider the distribution of the increment for
unit of time: ∆−1X2(t+ ∆)−X2(t), with ∆small. It is not possible to take the limit of vanishing ∆because X2(t)
contains a Wiener component, whose derivative is a white noise (a generalized stochastic process). In the two-time
probability (81) we take t1=t,t2=t+ ∆,k2=κ
∆(0,1)T,k1=−κ
∆(0,1)T, and we get, for small ∆,
ξ1=κ
∆(S∆−1) P00
1≃κZP00
1=κgP11
0,
ξ2=κ
∆P00
1, Stξ1≃gκY(t),Ψt(ξ1) = Zt
0
dτ ψ (gκY(τ)) ,
Y(t) = 1
2iω
λ+eλ−t−λ−eλ+t
Ωeλ+t−eλ−t
bΩeλ+t−e−ct
λ++c−eλ−t−e−ct
λ−+c
0
,
Ψ∆(ξ2)≃∆Z1
0
dx ψ κ
∆Sx∆P00
1≃∆Z1
0
dx ψ κ1
∆+xZP00
1
= ∆ Z1
0
dx ψ κ
∆P00
1+κgxP11
0=−A44
2∆ κ2−A11
6∆gκ2.
32
We also change the initial time from 0 to t0, which means ˆπ0→ˆπt0. Then (81) becomes
Eexp iκX2(t0+t+ ∆) −X2(t0+t)
∆
≃ZR2
dxTr {ˆπt0(x)W1(gκP1Y(t))}exp ix1gY3(t)κ−A44
2∆ κ2+Zt
0
dτ ψgκY(τ).(86)
Note that there is no dependence on x2inside the exponential which means that these probabilities do not depend
on the distribution of X2(t0). For large t,Y(t)vanishes and any memory of the state ˆπt0is lost. For t= 0 we get
Yi(0) = δi1and (86) becomes
Eexp iκX2(t0+ ∆) −X2(t0)
∆≃Tr ρt0eiQgκexp −A44
2∆ κ2, ρt0=ZR2
dxˆπt0(x).
This means that the distribution of the increment per unit of time is the quantum distribution of the position in the state
ρt0smoothed by a large Gaussian. The state ρt0depends on the whole previous history, and it could be the conditional
state with respect to previous observations (an argument not developed here).
7 Conclusions
In this article we have afforded the problem of giving a unified treatment of interacting classical and quantum systems.
The formalism we adopted allows to unify quantum master equations, Liouville equation, Kolmogorov-Fokker-Planck
equation. The treatment has been restricted to the case of a Markovian and quasi-free dynamics. The notion of quasi-
free dynamics, Sec. 2.3, generalizes the well known Gaussian case by including contributions of “jump type”. The
problem of finding the most general Markovian quasi-free dynamics has been solved, see Theor. 1; in the construction,
connections with the classical L´evy-Khintchine formula have been used, Sec. 2.4.3.
The restriction to the quasi-free case allows to give explicitly the dynamical map, without the need of solving the
evolution equations, involving unbounded generators. However, also the formal structure of the generator is given
(Sec. 3), because this gives hints to construct hybrid dynamics out of the quasi-free case. As a byproduct, we have
obtained the most general quasi-free quantum dynamical semigroup, together with the structure of its generator, and
we have presented examples which underline the role of the “jump” contributions (Sec. 4).
The results in the pure classical case (Sec. 5) are all essentially known. However, this case allows to introduce
the idea that not only probabilities at one time are involved in the theory, but also transition probabilities, multi-
time probabilities, probabilities for stochastic processes in continuous time. In the general case (Sec. 6), these ideas
are translated in the introduction of transition instruments (Sec. 6.1), which allow for the construction of multi-time
probabilities for the classical component. By analyzing the generator of the hybrid dynamics (Sec. 3.2), it is possible
to identify the interactions responsible of the action of the classical system on the quantum one and the interactions
responsible of the flux of information from the quantum system to the classical one. The classical component can
be observed, in principle without disturbance, and information about the quantum system can be collected. This last
feature connects dynamical semigroups for hybrid systems to the theory of quantum measurements in continuous
time. Let us stress that the whole construction respects the general idea that no information can be extracted from a
quantum system without dissipation; indeed, by asking the vanishing of the dissipative terms in the quantum part of
the generator, we get that also the terms responsible of the flux of information towards the classical component have to
vanish. Finally, in Sec. 6.2, an example is elaborated, which shows how a classical system can inject noise of Gaussian
and jump type into the quantum one, and how it can extract information from the quantum component.
Many points are left open for future developments. In the theory of open systems, the notion of stochastic dila-
tion or stochastic unraveling has been introduced, to mean the case of quantum master equations written in form of
(classical) stochastic differential equations [15, 33, 44]. This can be done also for the hybrid case, as sh own in [4, 43],
but not all possibilities have been explored. Another old idea for hybrid systems is to treat the classical component
as quantum, but with only commuting operators appearing as observables. The connections of L ´evy processes with
Bose quantum fields and quantum stochastic calculus have been worked out in detail [34, 45]. By using these means,
in the case of the monitoring in time of a quantum system, the observed signal has been dilated to a quantum system
and represented by quantum observables, commuting among themselves also at differen times in the Heisenberg pic-
ture [46,47]. The same construction for the dynamics of general hybrid systems is to be done, as it is open the problem
of not quasi-free hybrid systems, possibly with unbounded generators [27, 48].
33
Inside the theory of quantum measurements in continuous time, also the notions of quantum filtering,quantum
trajectories,conditional states have been developed [15, 44, 47, 49–52]; the introduction of these notions for hybrid
systems is clearly possible and we expect this to be fruitful. The classical component of the hybrid can be observed, or
partially observed, and the whole filtering theory for stochastic processes enters into play. As we wrote, in the hybrid
case there is transfer of information also from the classical to the quantum component and it could be interesting to
connect this to the idea of feedback on quantum systems [15]. Obviously, the final open problem is to find physical
application of the new ideas on the dynamics of hybrid systems in quantum optics and quantum information, beyond
the quasi-free case.
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