Markovian dynamics for a quantum/classical system and quantum trajectories

Abstract

Quantum trajectory techniques have been used in the theory of open systems as a starting point for numerical computations and to describe the monitoring of a quantum system in continuous time. We extend this technique to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equations, we can describe a classical component and a quantum one which have their own intrinsic dynamics and which interact with each other. A mathematically rigorous construction is given, under the restriction of having a Markovian joint dynamics and of involving only bounded operators on the Hilbert space of the quantum component. An important feature is that, if the interaction allows for a flow of information from the quantum component to the classical one, necessarily the dynamics is dissipative. We show also how this theory is connected to a suitable hybrid dynamical semigroup, which reduces to a quantum dynamical semigroup in the purely quantum case and includes Liouville and Kolmogorov–Fokker–Planck equations in the purely classical case. Moreover, this semigroup allows to compare the proposed stochastic dynamics with various other proposals based on hybrid master equations. Some simple examples are constructed in order to show the variety of physical behaviors which can be described; in particular, a model presenting hidden entanglement is introduced.

Full text

Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 57 (2024) 315301 (31pp) https://doi.org/10.1088/1751-8121/ad5fd2
Markovian dynamics for a
quantum/classical system and quantum
trajectories
Alberto Barchielli1
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano, Milan, Italy
E-mail: alberto.barchielli@polimi.it
Received 25 March 2024; revised 29 May 2024
Accepted for publication 5 July 2024
Published 17 July 2024
Abstract
Quantum trajectory techniques have been used in the theory of open systems as
a starting point for numerical computations and to describe the monitoring of a
quantum system in continuous time. We extend this technique to develop a gen-
eral approach to the dynamics of quantum/classical hybrid systems. By using
two coupled stochastic differential equations, we can describe a classical com-
ponent and a quantum one which have their own intrinsic dynamics and which
interact with each other. A mathematically rigorous construction is given,
under the restriction of having a Markovian joint dynamics and of involving
only bounded operators on the Hilbert space of the quantum component. An
important feature is that, if the interaction allows for a ow of information
from the quantum component to the classical one, necessarily the dynamics
is dissipative. We show also how this theory is connected to a suitable hybrid
dynamical semigroup, which reduces to a quantum dynamical semigroup in the
purely quantum case and includes Liouville and Kolmogorov–Fokker–Planck
equations in the purely classical case. Moreover, this semigroup allows to com-
pare the proposed stochastic dynamics with various other proposals based on
hybrid master equations. Some simple examples are constructed in order to
show the variety of physical behaviors which can be described; in particular, a
model presenting hidden entanglement is introduced.
Keywords: hybrid dynamics, quantum trajectories,
positive operator valued measures, hybrid dynamical semigroup
1Also Istituto Nazionale di Alta Matematica (INDAM-GNAMPA).
Original Content from this work may be used under the terms of the Creative Commons Attribution
4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the
title of the work, journal citation and DOI.
© 2024 The Author(s). Published by IOP Publishing Ltd 1

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
1. Introduction
The interest in quantum/classical hybrid systems is old, see for instance [1–14] and references
therein. One of the main motivations of the study of hybrid systems is that the output of a
monitored system is classical; then, implicitly or explicitly, the dynamics of quantum/classical
systems is involved in the theory of quantum measurements in continuous time and quantum
ltering [1–4,15–28]. Moreover, hybrid systems could be used as an approximation to com-
plicate quantum systems, as an effective theory [1,4,9–12,29]. Another motivation is in the
connections between a theory of gravity and quantum theories; perhaps gravity is ‘classical’,
and a dynamical theory of gravity and matter needs a hybrid dynamics [1,4,5,28,30–33].
Classical environments could be involved even in revivals of quantum entanglement [34,35].
A common feature in many of these attempts is that a non-trivial hybrid dynamics is always
irreversible, due to the ow of information from the quantum system to the classical one [1,
3–7,13,14,31].
The main aim of this article is to construct a consistent formulation of the dynamics of quan-
tum/classical hybrid systems, by connecting the quantum trajectory theory, used to describe
open and monitored quantum systems [16,17,21,23,24,36,37], to the classical theory of
Markov processes and stochastic differential equations [38–44]. Some proposals of connect-
ing quantum trajectories to hybrid dynamics were given in [3,5,28]. The dynamics of the
hybrid systems is given in section 2in terms of coupled stochastic differential equations; only
the Markovian case is considered. We also show how this approach is connected to notions
appearing in the general formulation of quantum measurements, such as ‘positive operator
valued measures’ and ‘instruments’ [22–24].
In section 3we show how to associate a suitable semigroup to the constructed dynam-
ics. This hybrid semigroup reduces to a quantum dynamical semigroup [22–24] in the case
of a pure quantum system, while in the pure classical case it includes the Liouville and the
Kolmogorov–Fokker–Planck equations [44]. We give also the formal generator of this semig-
roup, which allows to compare the class of dynamics developed here with the ones obtained by
other approaches. This generator also shows analogies and links with the theory of innitely
divisible distributions [13,44–46].
To illustrate the possibilities of the constructed dynamics, various examples are presented
in section 4. In particular a two qubit system is considered and analogies with the phenomenon
of hidden entanglement [34,47] are discussed. Some nal conclusions are given in section 5.
We end this section by some preliminary notions and useful notations.
1.1. Some notations
The quantum component will be represented in a separable complex Hilbert space H. The
bounded linear operators are denoted by B(H)and the trace class by T(H); the unit element in
B(H)is denoted by 1and the adjoint of the operator a∈B(H)by a†. The complex conjugated
of α∈Cis denoted by α. We shall use also the notation
hρ,ai=Tr{ρa}, ρ ∈T(H),a∈B(H).
By S(H)⊂T(H)we denote the set of the statistical operators (the quantum states). Then,
general measurements are represented by positive operator valued measures and instruments
[22–24].
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Definition 1. Let Xbe a set and ΣXaσ-algebra of subsets of X. An instrument I(·)with
value space (X,ΣX)is a function from the σ-algebra ΣXto the linear maps on B(H)such
that: (i) (positivity) for every set E∈ΣX,I(E)is a completely positive and normal map
from B(H)into itself; (ii) (σ-additivity) we have ISiEi[a] = PiI(Ei)[a],∀a∈B(H)
and for every countable family of disjoint sets Ei∈ΣX; (iii) (normalization) I(X)[1] = 1.
’Normality’ is a suitable regularity requirement; for a positive map it is equivalent to require
the map to be the adjoint of a bounded map on T(H). Let us recall that an instrument gives
the probabilities for the observed values and the conditional state after the measurement. The
quantity I(·)[1]is a positive operator valued measure, and it gives only the probabilities.
Moreover, measurements on a quantum system can be interpreted as involving hybrid sys-
tems: a positive operator valued measure is a channel from a quantum system to a classical
one, an instrument is a channel from a quantum system to a hybrid system [2,7,9,48]. When
X=Rsit is usual to take for ΣXthe σ-algebra of the Borel sets, denoted by B(Rs).
We take a classical component with values in Rs, the classical phase space [7,13]. The
choice of observable and state spaces in the classical case is more involved [7, section 1.1].
To introduce the semigroups associated to the Markov processes on which our classical com-
ponent will be based, it is convenient to use the space Bb(Rs)of bounded, Borel measurable
complex functions on Rs[42–44] and its subspace Cb(Rs)of the continuous bounded func-
tions; both spaces are Banach spaces under the supremum norm [44, p 6]. We consider complex
functions instead of only real ones, because it is more convenient in constructing hybrid sys-
tems. Inside a C∗-algebraic approach, Cb(Rs)was one of the suggestions for hybrid systems
given in [7, p 7].
2. Hybrid dynamics and stochastic differential equations
In the quantum theory of measurements in continuous time, many conceptual and technical
results were obtained by the quantum trajectory approach, based on stochastic differential
equations (SDEs) [21–24,36,37]. By extending this technique, it is possible to develop a
quantum trajectory formulation of a very general class of hybrid dynamics.
The construction is based on two probability measures: a reference probability Qand a
physical probability P. The physical probability will be constructed in section 2.3.1, in terms
of Qand of the quantum/classical dynamics.
In the next Assumption we introduce the reference probability and the Wiener and Poisson
processes needed in the construction. They are dened in a ltered probability space satis-
fying the usual hypothesis, typically assumed in books on stochastic calculus [42, p 3 and
section I.5].
Assumption 1. Let Ω,F,Qbe a complete probability space with a ltration of σ-algebras
{Ft,t⩾0}, such that (a) F=Wt⩾0Ft, (b) Ft=Tr>tFr, (c) F∈F,Q(F) = 0⇒F∈F0; we
set also Ft−=W0⩽r<tFr[38, pp 3–4], [39, p 45].
Let Wk,k=1,...,d, be continuous versions of independent, adapted, standard Wiener pro-
cesses, with increments independent of the past [39, section I.7].
Let Ube a locally compact Hausdorff space with a topology with a countable basis and
B(U)be the Borel σ-algebra of U. Let νbe a Radon measure on (U,B(U)) with the property:
∃U0∈ B(U) : ν(U\U0)<+∞.(1)
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Finally, Π(du,dt)is an adapted Poisson point process on U×R+of intensity ν(du)dt;Πis
independent of the Wiener processes and with increments independent of the past [21, section
1], [39, sections I.8, I.9].
It is also useful to introduce the compensated Poisson measure:
e
Π(du,dt) := Π(du,dt)−ν(du)dt.
Stochastic integrals with respect to the Wiener and Poisson processes are dened, for instance,
in [38–42]. In particular, the stochastic integrals with respect to Π(du,dt)and with respect to
e
Π(du,dt)are dened in [39, section II.3], where also the classes of possible integrands are
introduced. In the following we shall make use of Itˆ
o’s formula and generalizations; see [39,
section II.5], [40, section I.4], [42, section II.7]. The simpler case of a nite collection of
Poisson processes is presented in section 4.2.
We shall denote by EQ[X]the mean value of the random variable Xwith respect to the
probability measure Q. As a real random variable is a measurable function from Ωto Rwe can
write X:ω∈Ω→X(ω)∈R; then, the mean value can be written as EQ[X] = ´ΩX(ω)Q(dω).
2.1. The classical component
The classical component of the hybrid system is the stochastic process X(t) taking values in
the phase space Rsand satisfying the stochastic differential equation
dXi(t) = ci(X(t−))dt+
d
X
k=1
bk
i(X(t−))dWk(t) + ˆU0
gi(X(t−),u)e
Π(du,dt)
+ˆU\U0
gi(X(t−),u)Π(du,dt),i=1,...,s.(2)
When considering stochastic processes with diffusion and jumps, it is useful to x some regu-
larity properties for their trajectories. We always understand to have regular right continuous
processes, which means processes adapted to the given ltration {Ft}t⩾0, with trajectories con-
tinuous from the right and with limits from the left [38, defenition 1.5]; indeed, X(t−)means
the limit from the left.
Assumption 2 (Growth and Lipschitz conditions) The bkand care Borel measurable func-
tions from Rsto Rs; the gare measurable functions from Rs×Uinto Rs. We have also, for
some K>0, and ∀x,y∈Rs,
kc(x)k2+X
k
bk(x)
2+ˆU0kg(x,u)k2ν(du)⩽K1+kxk2,
kc(x)−c(y)k2+X
k
bk(x)−bk(y)
2+ˆU0kg(x,u)−g(y,u)k2ν(du)⩽Kkx−yk2.
Proposition 1. Under assumptions 1and 2, given a random (F0-measurable) initial condition
X(0), there is a unique regular right continuous process solving (2); moreover, it turns out to
be a time-homogeneous Markov process. In particular, ∀x∈Rsthere is a unique solution Xx(t)
with deterministic initial condition Xx(0) = x.
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
The proof of this proposition is given in [39, section IV.9]; see, in particular, theorem 9.1
of [39]. Note that no condition is given on g(x,u)for u∈U\U0, apart from being jointly
measurable in (x,u). This is due to the fact that both Πand νare nite measures on U\U0.
It is usual to call transition probability (or transition function) [42–44] the function on
[0,+∞)×Rs×B(Rs), dened by
P(t,x,E) = Q[Xx(t)∈E],t∈[0,+∞),x∈Rs,E∈ B(Rs).(3)
Some properties of the solutions of (2) and of the transition probabilities are collected in
appendix A, where also the associated semigroup is given.
If for some iwe have ci=0, gi=0, and bj
i=δij, the corresponding Wiener process
Wiis among the observed classical processes. If for some iwe have ci=0, b·
i=0, and
gi(x,u) = 1U(u), with ν(U)<+∞, the Poisson process Π(U,(0,t]) is among the observed
classical processes.
2.2. Stochastic Schrödinger equations
We introduce now the stochastic equation which determines the dynamics of the quantum
component and the interaction of the two components. The coefcients involved in such an
equation are given in the following assumption; to avoid analytical complications, we consider
only bounded operators on the Hilbert space of the quantum component.
Assumption 3. Let H(x), Lk(x),J(x,u),k=1,...,d,x∈Rs,u∈U, be bounded operators on
the Hilbert space Hof the quantum component. All these functions are continuous in xin the
strong operator topology; (x,u)7→ J(x,u)is strongly measurable. We also have
H(x)†=H(x),sup
x∈RskH(x)k<+∞,
sup
x∈RskLk(x)k<+∞,sup
x∈Rs,u∈UkJ(x,u)k<+∞,
D(x) := ˆU
J(x,u)†J(x,u)ν(du)∈B(H),sup
x∈RskD(x)k<+∞;
(4)
the integral is dened in the weak topology of B(H).
These operators become operator-valued random variables when composed with X(t−).
Indeed, we dene, for x∈Rsand ω∈Ω,
K(x) = iH(x) + 1
2
d
X
k=1
Lk(x)†Lk(x) + 1
2D(x),(5)
Kt(ω) = K(X(t−;ω)),Ht(ω) = H(X(t−;ω)),
Lkt (ω) = Lk(X(t−;ω)),Jt(u;ω) = J(X(t−;ω),u),Dt(ω) = D(X(t−;ω)).(6)
We have inserted X(t−)in order to have predictable operator-valued processes, as was for the
coefcients in equation (2). Let us stress that these coefcients depend on the solution of (2)
and, so, on the initial condition X(0).
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Finally, we introduce the linear stochastic Schrödinger equation (SSE) in H[36, (2.3)]
dψt=−Ktψt−dt+
d
X
k=1
Lktψt−dWk(t) + ˆU
Jt(u)ψt−e
Π(du,dt),EQhkψ0k2i<+∞;(7)
the random initial condition ψ0∈His F0-measurable.
Let us note that the assumption of ‘no memory’ in the dynamics of the composed system
is implicitly contained in the fact that the coefcients in (2) and (7) depend on the past only
through X(t−)and that the involved noises are Wiener and Poisson processes, which have
increments independent of the past. Examples without these hypotheses are considered in [36].
However, with respect to what is done in that reference, the space Uinvolved in the point
process is slightly more general; similarly, the structure of the classical process X(t) has a
jump contribution more general than in [36, (4.32)]. In any case, the proofs of that article
apply to this case too and we can follow that construction.
Theorem 2. Up to Q-equivalence, equation (7) admits a unique solution ψt, t ∈R+, which is
an H-valued semimartingale [42, section II.1]. The process kψtk2is a non-negative martingale
[42, section I.2], such that
EQhkψtk2i=EQhkψ0k2i<+∞,dkψtk2=
ψt−
2dZt,
Zt=
d
X
k=1ˆ(0,t]
mks dWk(s) + ˆ(0,t]ˆU
(Is(u)−1)e
Π(du,ds),mkt =2Rehb
ψt−Lkt b
ψt−i,
It(u) = 

(Jt(u) + 1)b
ψt−


2,b
ψt(ω) = (ψt(ω)/kψt(ω)kif kψt(ω)k 6=0,
v (xed unit vector) if kψt(ω)k=0.
Proof. Assumptions 1,3, and the properties of X(t) easily imply properties 2.0.A, 2.0.B, 2.2,
2.3.A, 2.3.B in [36]; then, proposition 2.1 and theorem 2.4 of [36] hold, which is the thesis of
the theorem above.
Remark 1 (The non-linear SSE). The process b
ψt, 0 ⩽t⩽T, satises a non-linear stochastic
differential equation, the non-linear SSE, under the new probability kψT(ω)k2Q(dω), see [36,
section 2.2]. A rst example of SSE goes back to [16].
2.3. Stochastic master equations
We introduce now the stochastic differential equation in T(H)associated to the linear SSE (7).
Initial condition σ0.Let ψβ∈Hbe F0-measurable random vectors such that
EQhP+∞
β=1
ψβ
2i=1. Then, we dene the random trace-class operator σ0by
hσ0,ai=
+∞
X
β=1hψβ|aψβi,∀a∈B(H)⇒σ0⩾0,EQ[Tr{σ0}] = 1.(8)
Let ψβ
tbe the solution of (7) with initial condition ψβand dene σtby
hσt,ai=
+∞
X
β=1hψβ
t|aψβ
ti,∀a∈B(H).(9)
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
By the formal rules of stochastic calculus, we get the linear stochastic master equation (SME)
for σt:∀a∈B(H),
dhσt,ai=hσt−,Lt[a]idt+
d
X
k=1hσt−,Lkt†a+aLkt idWk(t)
+ˆUhσt−,Jt(u)†aJt(u) + Jt(u)†a+aJt(u)ie
Π(du,dt),(10)
Lt=L(X(t−)),(11)
L(x)[a] = i[H(x),a] +
d
X
k=1Lk(x)†aLk(x)−1
2nLk(x)†Lk(x),ao
+ˆUJ(x,u)†aJ(x,u)−1
2nJ(x,u)†J(x,u),aoν(du).(12)
Let us stress that the SME (10) is linear once the process X(t), solving (2), as been xed.
Instead, if we consider (2) and (10) as a system of SDEs, we have the non-linearity introduced
by (2).
Theorem 3. The T(H)-valued process σt, dened in (9), satises the weak-sense stochastic
differential equation (10). Moreover, it is the unique solution of (10) with the initial condi-
tion (8). We have also
σt⩾0,pt:= hσt,1i ≡ Tr{σt}⩾0,EQ[pt] = 1,(13)
dpt=pt− d
X
k=1
mk(t)dWk(t) + ˆU
(It(u)−1)e
Π(du,dt)!,(14)
mk(t) = 2Rehbσt−,Lkti,It(u) = Dbσt−,Jt(u)†+1(Jt(u) + 1)E,(15)
bσt(ω) = (σt(ω)/pt(ω)if pt(ω)6=0,
σv(xed statistical operator) if pt(ω) = 0.(16)
Proof. Again assumptions 1,3, and the properties of X(t) imply the validity of properties 3.1
and 3.3 of [36]. Then, propositions 3.2 and 3.4 of [36] give the thesis of the theorem.
By denition bσt(ω)∈S(H), i.e. bσtis a random quantum state. The random quantities mk(t)
and It(u)are dened by (6) and (15), so as to be Ft−-measurable (predictable processes [38,
section 3]). By construction, if the initial state bσ0is a pure state, then bσt(ω)too is a pure state.
2.3.1. The physical probability and the classical component. As in [36, section 3.1], we
use pt(13) as a probability density and we dene the new probability Pon the σ-algebra
F=Wt⩾0Ftby
∀t⩾0,∀F∈Ft,P(F) := EQ[pt1F]; (17)
for a similar construction of a probability measure see [38, section 30.2].
Note that pt(the trace of σt) is a probability density with respect to Qand that it gives the
probability Pon Ft, which contains the events related to the time interval (0,t]. By (14), pt
is a martingale and this implies the consistency of the probabilities constructed in different
time intervals: for 0 <s<t,E∈Fs, one has ´Ept(ω)Q(dω) = ´Eps(ω)Q(dω). This allows to
extend Pto the whole F.
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Remark 2. The probability P, dened by (17), represents the physical probability of the hybrid
system. By (6) and (10), the physical probability depends on the dynamics of both the classical
component and the quantum one. Moreover, the probability Pdepends on the initial conditions
X(0) and σ0used in solving equations (2) and (10).
Remark 3. By the so called Girsanov transformation and generalizations [42, section III.8],
we have that, under the probability P, the processes
b
Wk(t) := Wk(t)−ˆ(0,t]
mk(s)ds(18)
are independent standard Wiener processes adapted to the ltration {Ft,t⩾0}; moreover,
Π(du,dt)is no more a Poisson process, but it becomes a point process with compensator
It(u)ν(du)dt[36, proposition 2.5, remarks 2.6, 3.5]. We also introduce the notation
b
Π(du,dt) = Π(du,dt)−It(u)ν(du)dt.(19)
According to [38, section 31], b
Π(du,dt)is a white random measure.
By rewriting (2) in terms of the new processes (18) and (19), we obtain the dynamics of the
classical component under the physical probability P:
dXi(t) = ci(X(t−)) + C(1)
i(t) + C(2)
i(t)dt+
d
X
k=1
bk
i(X(t−))db
Wk(t)
+ˆU0
gi(X(t−),u)b
Π(du,dt) + ˆU\U0
gi(X(t−),u)Π(du,dt),(20)
C(1)
i(t) =
d
X
k=1
bk
i(X(t−))mk(t),C(2)
i(t) = ˆU0
gi(X(t−),u)(It(u)−1)ν(du).(21)
Here, Π(du,dt)is no more a Poisson measure and the processes mk(t)and It(u)depend on
the whole past through the quantum state; in particular, the dynamics (20) of the classical
component depends on the behavior of the quantum one.
From equations (20) and (21) we have that the classical component can extract some inform-
ation from the quantum one only when not all the terms C(1)(t),C(2)(t),It(u)vanish and they
effectively depend on the quantum state. So, rstly we need bk
i(x)and gi(x,u)not all identically
vanishing and this means that the diffusive and the jump terms are not both vanishing: some
dissipation must be present in the dynamics (20) of the classical component. Moreover, to
have dependence on the quantum state at least some of the operators Lk(x)and J(x,u)must be
not reducible to something proportional to the identity operator; this means that the Liouville
operator (12) must contain a dissipative contribution. As a consequence, also in this class of
models of hybrid dynamics, we have that, in order to extract information from the quantum
component, we need a dissipative dynamics for the quantum system. Furthermore, the clas-
sical system acting as a measuring apparatus, as it receives this information, has necessarily a
dissipative dynamics, because of the probabilistic nature of any quantum information.
Even without extraction of information from the quantum system, to know that there is a
jump in the counting process Πmeans to know that the quantum system has been subjected to a
certain kind of jump. But for this we have to know before the dynamical laws of the system; we
are not observing the quantum system, but a classical system which is acting on the quantum
component.
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
2.3.2. The non-linear SME and the conditional states. By the rules of stochastic calculus we
can obtain the non-linear SME satised by the normalized random states bσtdened in (16):
dhbσt,ai=hbσt−,Lt[a]idt+
d
X
k=1hbσt−,Lkt†a+aLkt −mk(t)aidb
Wk(t)
+ˆEtnIt(u)−1Dbσt−,Jt(u)†+1a(Jt(u) + 1)E−hbσt−,aiob
Π(du,dt); (22)
Etis the random subset of U, dened by Et(ω) = {u∈U:It(u;ω)6=0}. Equation (22) follows
from [36, remark 3.6].
The two coupled SDEs (20) and (22) have been obtained by construction. This proves that
these equations admit a solution and that it exists a probability Punder which the processes b
Wk
are independent Wiener processes and Π(du,dt)is a point measure with random compensator
It(u)ν(du)dt; the uniqueness remains open. In a simplied case, without jump contributions,
uniqueness has been proved in [23, theorem 5.2].
Remark 4 (Conditional state). Let Etbe the σ-algebra generated by X(r), 0 ⩽r⩽t; note that
it could depend on the choice of the initial condition, the random variable X(0). Then, we can
consider the quantum state conditioned on the observation of the process Xup to time t, the a
posteriori state (or conditional state)
ρt:= EP[bσt|Et]≡EP[bσt|E],E=_
t⩾0
Et.(23)
When the two ltrations coincide, Et=Ft, the conditional states satisfy the non-linear
SME (22). In general, the possibility of having a closed evolution equation for the conditional
states depends on the structure of the coefcients; examples are given in [36, section 4]. When
not all the components X1(t),...,Xs(t)of the classical system are observed, one has to take
as {Et}the ltration generated by the observed components of X(t); this is a case of partial
observation and it opens to the application of all the notions and results of classical ltering
theory [49], which indeed is at the origin of the idea of observations in continuous time in
quantum theories [16].
Remark 5 (Mean state). When the classical component is not of interest and not observed,
the quantum component is described by the a priori state, the mean state
ηt:= EQ[σt] = EP[bσt] = EP[ρt].(24)
We can say that {P(dω), ρt(ω)}is an ensemble of quantum states; then, ηtis the mean state of
this ensemble. Given a mixed state ηt, there are innitely many decompositions in ensembles
of states; the decomposition in terms of conditional states has a physical meaning, as it is xed
by the observation of the classical component of the hybrid system.
By taking the Q-mean of the linear SME (10) we get immediately
d
dtηt=EQLt∗σt− (25)
where the random Liouvillian Lt∗is the pre-adjoint of the operator dened by (11). When
the coefcients in the denition of L(x)do not depend on x, we get that Lt∗is non random
and the a priori state satises a closed quantum master equation. In general, equation (25)
is not closed and the reduced dynamics of the quantum component is not Markovian. The x-
dependence in the Liouville operator (12) gives the dependence of the quantum dynamics on
9

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
the classical component and it controls the transfer of information from the classical system
to the quantum one. This x-dependence represents a feedback of the classical system on the
quantum one; moreover, it could appear in the Hamiltonian term alone: the information transfer
from the classical to the quantum component is not related to dissipation.
The notions of conditional and mean states are typical of the approach to hybrid systems
based on measurement and ltering [16,20,22–24], but the idea of considering the full dynam-
ics and its reduced (or marginal) version appears also in other approaches [11].
2.4. Probabilities and instruments
A key point in the trajectory formulation is that the physical probabilities of section 2.3.1
can be expressed by means of instruments, which are completely positive linear maps (see
defenition 1); in this way the linearity of a quantum theory is respected. The SME (22) cannot
include arbitrary non-linearities: it is non-linear only due to normalization and the change of
probability; moreover, it is connected to the linear SME (10).
2.4.1. The stochastic map. By varying the initial condition in (10), we have that the linear
SME denes a linear map on the trace class. As in proposition 1,Xx(t)is the solution of the
stochastic equation (2) starting at x. Then, we consider the linear SME with the process Xx(t)
inside its coefcients (6); so, the random coefcients Lt,Lkt,Jt(u)in (10) are xed and the
equation is linear in the initial quantum state.
Let us denote by Λr,x
t∗[ρ],t⩾r, the solution of (10) with ρas initial condition at time r;
by construction, Λr,x
t∗is a random, completely positive, linear map on T(H). Let Λr,x
tbe the
adjoint map on B(H); its properties are collected in the following proposition.
Proposition 4. The random map Λr,x
tis linear, completely positive, normal, and it enjoys the
composition property
Λt1,x
t2◦Λt2,x
t3= Λt1,x
t3,0⩽t1⩽t2⩽t3.(26)
We have also
Λt′,x
t′+t= Λ0,Xx(t′)
t,Tr{ρΛr,x
t[1]}=1,∀ρ∈S(H).(27)
Proof. The rst properties and the composition rule (26) follow from the construction of σt
in section 2.3 and from (10). By the fact that Xis a Markov process under Qand that this
process appears in the right hand side of (10) only as X(t−), the rst property in (27) follows.
The second property in (27) follows from (13).
If σtis the solution of (10) with the initial condition σ0given in (8), we have
hσt,ai=hσ0,Λ0,X(0)
t[a]i,hσt′+t,ai=hσt′,Λt′,X(0)
t′+t[a]i=hσt′,Λ0,X(t′)
t[a]i.(28)
2.4.2. The transition instruments. By analogy with the transition probabilities, it is possible
to introduce instruments which depend on the initial value of the classical system, some kind
of transition instruments [36]. Indeed, we can dene the family of maps
hρ,It(E|x)[a]i=EQhhΛ0,x
t∗[ρ],ai1E(Xx(t))i,
∀ρ∈T(H),∀a∈B(H),∀E∈ B(Rs).
(29)
10

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
By 1Ewe denote the indicator function of a generic set E:
1E(x) = (1 if x∈E,
0 if x/∈E.
Moreover, we shall denote by δx(·)the trivial measure concentrated in the point x∈Rs(Dirac
measure).
Proposition 5. Equation (29) denes an instrument It(·|x)on the σ-algebra B(Rs)(see Def.
1). For t =0 we get the trivial instrument
I0(E|x)[a] = δx(E)a.(30)
Moreover, the family of instruments (29) enjoys the following composition property
It+t′(E|x) = ˆz∈RsIt′(dz|x)◦It(E|z).(31)
Equation (31) is a quantum analogue of the Chapman–Kolmogorov identity for transition
probabilities (A.2).
Proof. By using the properties of the maps Λ0,x
t, given in proposition 4, it is possible to check
that the maps dened in (29) satisfy all the properties needed in the denition of instrument
(defenition 1).
The composition property (31) is proved by direct computations. By using (26) and (27),
we have
⟨ρ,It+t′(E|x)[a]⟩=EQhEQh⟨Λ0,x
t′∗[ρ],Λt′
,x
t′+t[a]⟩1EXxt+t′

Et′ii
=ˆz∈Rs
EQhEQh⟨Λ0,x
t′∗[ρ],Λt′
,x
t′+t[a]⟩1E−zXxt+t′−Xxt′1dzXxt′

Et′ii
=ˆz∈Rs
EQh⟨Λ0,x
t′∗[ρ],It(E|z)[a]⟩1dzXxt′i=ˆz∈Rs
⟨ρ, It′(dz|x)◦ It(E|z)[a]⟩;
1dzXx(t′)is the measure-theoretic notation for δz−Xx(t′)dz; the quantity Xx(t′)is impli-
citly contained in Λ0,x
t′∗.
The transition instruments (29) are uniquely determined by their Fourier transform, the
characteristic operator: for k∈Rs,
hρ,Gt(k|x)[a]i=ˆRs
eik·yhρ,It(dy|x)[a]i=EQhhΛ0,x
t∗[ρ],aiexp{ik·Xx(t)}i.(32)
The ‘Markovian’ character of the hybrid dynamics is expressed by the fact that the instru-
ments contain only the length of the time interval (not initial and nal times) and that
equation (31) holds: the transition instruments are ‘time-homogeneous’, as the classical trans-
ition functions for a time-homogeneous Markov process, compare the properties of instru-
ments and equations (29)–(31) with equations (A.1) and (A.2).
Property (31) represents a compatibility condition among the various instruments at dif-
ferent times. Via Kolmogorov’s extension theorem [45, theorem 1.8], this property allows to
reconstruct the whole probability law of the process X(t) starting from the joint probabilities
at the times 0 ⩽t1<t2<···<tm, given by
11

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
P[X(t1)∈E1,X(t2)∈E2,...,X(tm)∈Em]
=EQ[hσtm,1i1E1(X(t1))1E2(X(t2))···1Em(X(tm))]
=ˆRs
Q0(dx)σ0(x),ˆE1It1(dx1|x)◦ˆE2It2−t1(dx2|x1)◦···
◦ˆEmItm−tm−1(dxm|xm−1)[1],(33)
σ0(x) = EQσ0X(0) = x,Q0(E) := Q[X(0)∈E];
here Pis the probability dened in section 2.3.1, which depends on the initial conditions σ0,
X(0). Note that we have hσtm,1i=Tr{σtm}=ptm, which is the probability density for events
up to the biggest time tm, in agreement with the discussion following the denition (17) of P.
In the construction of the hybrid dynamics, we have used a two-step procedure: rstly,
the SDEs (2) and (10) under the reference probability Qhave been introduced, and, then, the
SDEs (20) and (22) under the physical probability P. By following this path, already developed
in the eld of time-continuous measurements [23], we were able to give rise to the instru-
ments (29) with the right properties, given in proposition 5, and with the connection (33)
with the physical probability P. In this way, the connection with the general formulation of a
quantum theory is preserved.
3. Quantum–classical dynamical semigroup
The hybrid dynamics has been introduced through the formalism of quantum trajectories;
however, a Markovian hybrid dynamics is usually introduced by proposing suitable master
equations. The most standard choice is to take probability densities (with respect to Lebesgue
measure) as state space of the classical component [1–6,14], but this choice is too restrictive
in our case, as shown in appendix Afor the pure classical case.
A more general approach is to construct a dynamical semigroup in a suitable ‘space of
observables’ [7,13] and to take for the classical component the spaces of functions Cb(Rs)and
Bb(Rs), introduced in section 1.1. Then, for the hybrid system we introduce the C∗-algebras
[7]:
A1:= B(H)⊗CbRs=CbRs;B(H),A2:= B(H)⊗BbRs=BbRs;B(H).
The elements of A2are bounded functions from Rsto B(H), while A1⊂A2is made up by the
continuous bounded functions. The norm of F∈Aiis given by kFk=supx∈RskF(x)k, where
kF(x)kis the norm in B(H). The unit element Iin Aiis given by I=1⊗1; we can also write
I(x) = 1.
3.1. The hybrid semigroup
By using the stochastic process Xx(t), dened inside proposition 1, and the stochastic map
Λ0,x
t, dened in section 2.4.1, we can dene the linear map Tt,t⩾0, on A2by
hρ,Tt[F](x)i=EQhDρ, Λ0,x
t[F(Xx(t))]Ei,∀ρ∈T(H),∀F∈ A2,∀x∈Rs.(34)
12

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
As we shall see Ttis a contraction; so, it is enough to give Tton the product elements and
then to extend it by linearity and continuity. By using the instruments (29), we can also write
hρ,Tt[a⊗f](x)i=ˆy∈Rs
f(y)hρ,It(dy|x)[a]i,(35)
ρ∈T(H),a∈B(H),f∈Bb(Rs),x∈Rs.
On the other hand, given Tt, we can get the instruments (29) and their characteristic oper-
ator (32) by
hρ,It(E|x)[a]i=hρ,Tt[a⊗1E](x)i,hρ,Gt(k|x)[a]i=hρ,Tt[a⊗fk]i,fk(x) = eik·x.(36)
To get the characteristic operators it is enough to have Ttdened on A1, while to get directly
the instruments we need Ttdened on A2. As Tt,t⩾0, turns out to be a semigroup, we call it
the hybrid dynamical semigroup associated with the stochastic dynamics (2) and (10).
Proposition 6. The map Ttis linear and completely positive, which means: for any choice of
the integer N ⩾1, one has
N
X
i,j=1hψj|TtF∗
jFi(x)ψii⩾0,∀Fj∈ A2, ψj∈H,∀x∈Rs.(37)
We have also
T0=Id,Tt[I] = I,kTt[F]k⩽kFk,Tt:A27→ A2,(38)
Tt◦Tr=Tt+r.(39)
Proof. Linearity and complete positivity follow from the same properties of the integration
operation EQand of the stochastic map Λ0,x
t(proposition 4). Then, T0=Id follows from (30)
and (35); Tt[I] = Ifollows from (35) and the normalization of the instruments (defenition 1).
By using kFk21−F∗(x)F(x)⩾0, the positivity of Tt, and the preservation of the unit, we
get
Tt[F∗F](x)⩽kFk21.
By 2-positivity, linearity, and the preservation of the unit, one gets a Kadison-like inequality
[22, section 3.1.1], [50, example 6.7, p 107]:
Tt[F∗F](x)⩾Tt[F∗](x)Tt[F](x)≡ Tt[F](x)∗Tt[F](x).
By combining the two inequalities we get Tt[F]∗Tt[F]⩽kFk2I; this gives the contraction prop-
erty in (38) and the fact that Ttmaps A2into itself.
By using conditional expectations, we have
hρ,Tt+r[a⊗f](x)i=EQhhΛ0,x
t∗[ρ],Λt,x
t+r[a]if(Xx(t+r))i
=EQhDΛ0,x
t∗[ρ],EQΛt,x
t+r[a]f(Xx(t+r))Et,Xx(t)Ei.
By the Markov property of the process Xx, we get
EQΛt,x
t+r[a]f(Xx(t+r))Et,Xx(t)=Tr[a⊗f](Xx(t));
13

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
so, we obtain the composition property:
hρ,Tt+r[a⊗f](x)i=EQhDΛ0,x
t∗[ρ],Tr[a⊗f](Xx(t))Ei=hρ, Tt◦Tr[a⊗f](x)i.
This gives the composition property (39).
Remark 6 (Convolution semigroup of instruments). When all the coefcients introduced in
assumptions 2and 3do not depend on x∈Rs, from (2), (10), (29), (31) and (32) we obtain
Xx(t) = x+X0(t),It(E|x) = It(E−x|0),Gt+t′(k|0) = Gt′(k|0)◦Gt(k|0).
Under these conditions the associated dynamical semigroup (35) has been fully studied under
the name of convolution semigroup of instruments and generalized to cases with a classical
component not only in Rs[27,46,51–56]. In this construction a key point has been to exploit
the analogies with the classical innitely divisible distributions [44,45].
The dual C∗-algebra of A2is the space containing the hybrid states and the adjoint semig-
roup Tt∗gives the dynamics of the hybrid states; therefore, the hybrid dynamics is com-
pletely positive and linear. Another relevant point is equation (36), which denes the trans-
ition instruments It(·|x)once a semigroup Tt, satisfying the properties in proposition 6, is
given. Therefore, the dynamics Tt∗gives not only the hybrid state at the time t, but also all the
multi-time probabilities (33). Once again, to give a hybrid dynamical semigroup (linear and
completely positive) allows to respect the general structure of a quantum theory, as discussed
at the end of section 2.4.2 in the case of the stochastic formulation.
3.2. The generator
We have not studied the continuity properties in time of the dynamical semigroup (34); how-
ever, by using stochastic calculus we can see that it is differentiable when applied to the ele-
ments of A3:= C2
bRs;B(H)⊂A1, the space of the operator valued functions, which are
two-times differentiable and have bounded and continuous derivatives.
Proposition 7. The hybrid dynamical semigroup Ttcan be differentiated in weak sense
d
dthρ,Tt[F](x)i=hρ,(Tt◦K) [F] (x)i,∀ρ∈T(H),∀F∈A3,∀x∈Rs.(40)
The formal generator Kis given by
K[a⊗f](x) = f(x)L(x)[a] + a
s
X
i=1
∂f(x)
∂xi
ci(x) + a
s
X
i,j=1
1
2
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)
+
s
X
i=1
∂f(x)
∂xi
d
X
k=1
bk
i(x)aLk(x) + Lk(x)†a
+ˆUfx+g(x,u)−f(x)J(x,u)†+1aJ(x,u) + 1
−a1U0(u)
s
X
i=1
∂f(x)
∂xi
gi(x,u)ν(du); (41)
L(x)is dened in (12).
14

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Proof. By (10), (A.4), and stochastic calculus, we get
d(f(X(t))hσt,ai)X(t−)=x
=f(x) hσt−,L(x)[a]idt+
d
X
k=1hσt−,Lk(x)†a+aLk(x)idWk(t)
+ˆUhσt−,J(x,u)†aJ(x,u) + J(x,u)†a+aJ(x,u)ie
Π(du,dt)!
+hσt−,ai s
X
i=1
∂f(x)
∂xi ci(x)dt+
d
X
k=1
bk
i(x)dWk(t)!+1
2
s
X
i,j=1
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)dt
+dtˆU0 fx+g(x,u)−f(x)−
s
X
i=1
∂f(x)
∂xi
gi(x,u)!ν(du)
+ˆU0
(fx+g(x,u)−f(x))e
Π(du,dt) + ˆU\U0
(fx+g(x,u)−f(x))Π(du,dt)!
+
s
X
i=1
∂f(x)
∂xi
d
X
k=1
bk
i(x)hσt−,aLk(x) + Lk(x)†aidt
+ˆU
(fx+g(x,u)−f(x))hσt−,J(x,u)†aJ(x,u) + J(x,u)†a+aJ(x,u)iΠ(du,dt).
Then, by taking the mean we have
K[a⊗f](x) = f(x)L(x)[a] + a

s
X
i=1
∂f(x)
∂xi
ci(x) + 1
2
s
X
i,j=1
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)
+ˆU f(x+g(x,u)) −f(x)−1U0(u)
s
X
i=1
∂f(x)
∂xi
gi(x,u)!ν(du)!
+
s
X
i=1
∂f(x)
∂xi
d
X
k=1
bk
i(x)aLk(x) + Lk(x)†a
+ˆU
(f(x+g(x,u)) −f(x))J(x,u)†aJ(x,u) + J(x,u)†a+aJ(x,u)ν(du),
and this gives (41). The semigroup property gives the derivative at any time t(40).
Here we have not studied the problem of determining the dynamical semigroup Ttfrom the
generator (41). By construction Tt, dened by (34), is a solution of (40); what is not proved
is the uniqueness of this solution. The dynamics of the hybrid system is fully determined by
the stochastic differential equations (2) and (10) under assumptions 1–3(see proposition 1,
theorem 3).
15

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
3.3. A quasi-free dynamics
In [13] the structure of the most general quasi-free hybrid dynamical semigroup has been
obtained. The ‘quasi-free’ requirement means to map hybrid Weyl operators into a quantity
proportional to hybrid Weyl operators [57]. In this case, the formal generator involves unboun-
ded operators on the Hilbert space of the quantum component; so, the semigroup found in [13]
is not in the class studied here. However, the comparison of the two formal generators gives
hints to construct the stochastic formulation of the quasi-free dynamics.
Firstly, let us take
U= Ξ1⊕Ξ0,Ξ1=R2n,Ξ0=Rs,U0=S={u∈U:|u|⩽1},
ci(x) = α0
i+
s
X
j=1
xjZ00
ji , α0
i∈RZ00
ji ∈R,i=1,...,s,
bk
j(x) = bk
j∈R,g(x,u) = g(u) = P0u gi(u) = u2n+i,
where P0and P1are the orthogonal projections PiU= Ξi,i=1,2. With these choices assump-
tions 1and 2hold, and equation (2) becomes
dXi(t) = α0
idt+
s
X
j=1
Xj(t−)Z00
ji dt+
d
X
k=1
bk
idWk(t)
+ˆS
gi(u)e
Π(du,dt) + ˆU\S
gi(u)Π(du,dt),i=1,...,s.(42)
Then, we represent the quantum component in the Hilbert space H=L2(Rn); by ˆ
qjand
ˆ
pjwe denote the usual position and momentum operators. We collect these operators in the
vector RT= (ˆ
p1,...,ˆ
pn,−ˆ
q1,...,−ˆ
qn), where Tmeans transposition; moreover, we introduce
the usual Weyl operators
W1(P1u) = expiRTP1u=exp

i
n
X
j=1
(ujˆ
pj−un+jˆ
qj)

.
For the operators appearing in assumption 3we take
J(x,u) = J(u) = −W1(P1u)−1,(43a)
Lk(x) = Lk=−i
2n
X
l=1
Rldk
l=i
n
X
l=1dk
n+lˆ
ql−dk
lˆ
pl,dk
l∈C,(43b)
H(x) = Hq+Hx+ˆUi
2−W1(P1u) + W1(P1u)†−1S(P1u)TRν(du),(43c)
Hq=βTR+1
2RTP1σD11σTP1R, β ∈Ξ1,(43d)
Hx=xTP0ZP1R=
s
X
i=1
2n
X
j=1
xiZ01
ij Rj.(43e)
16

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
With these choices we have that the formal generator (41) becomes
K[a⊗f](x) = f(x)ni[Hq+Hx,a] +
d
X
k=1
2n
X
r,l=1
dk
ldk
r(σR)ra(σR)l−1
2(σR)r(σR)l,a
+a
s
X
i=1
α0
i+
s
X
j=1
xjZ00
ji 
∂f(x)
∂xi
+a
s
X
i,j=1
1
2
∂2f(x)
∂xi∂xj
d
X
k=1
bk
ibk
j
+
s
X
i=1
∂f(x)
∂xi
d
X
k=1
bk
i
2n
X
l=1i(σR)ldk
la−ia(σR)ldk
l
+ˆUfx+g(u)W1(σP1u)†aW1(σP1u)−af(x)
−1S(u)if(x)[(P1u)TσR,a]−1S(u)a
s
X
i=1
∂f(x)
∂xi
gi(u)ν(du); (44)
then, it is possible to check that it coincides with the formal generator given in
[13, proposition 5].
However, the operators Lkand H(x) are unbounded, so that the construction of sections 2.2
and 2.3 becomes purely formal. In any case (10) and (12) take the form
dhσt,ai=hσt−,Lt[a]idt+i
d
X
k=1
2n
X
l=1hσt−,Rldk
la−aRldk
lidWk(t)
+ˆUhσt−,W1(P1u)†aW1(P1u)−aie
Π(du,dt),(45)
L(x)[a] = i[Hq+Hx,a] +
d
X
k=1
2n
X
r,l=1
dk
ldk
rRraRl−1
2{RrRl,a}
+ˆUW1(P1u)†aW1(P1u)−a−i1S(u)h(P1u)TR,aiν(du).(46)
Due to the fact that assumption 3does not hold, the problem of existence and uniqueness of the
solution of (45) is open; however, the identication of the two generators has given an explicit
stochastic equation to study.
By (15), (43a) and (43b) we get
It(u) = 1,mk(t) = −2Im
2n
X
l=1
dk
lhbσt−,Rli.
The rst equality implies that, under the physical probability P,Π(du,dt)remains a Poisson
measure with the same compensator. In the second equality the unbounded operators Rlappear;
so, to have mk(t)well dened depends on the choice of the coefcients dk
land of the initial
state. When |mk(t)|<+∞, from equations (20), (21) and (22) we get the stochastic equation
for the classical component
17

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
dXi(t) = 
α0
idt+
s
X
j=1
Xj(t−)Z00
ji dt+
d
X
k=1
bk
imk(t)
dt+
d
X
k=1
bk
idb
Wk(t)
+ˆU0
gi(u)e
Π(du,dt) + ˆU\U0
gi(u)Π(du,dt),(47)
and the non-linear SME
dhbσt,ai=hbσt−,Lt[a]idt+i
d
X
k=1
2n
X
l=1hbσt−,Rldk
la−aRldk
lidb
Wk(t)
−
d
X
k=1
mk(t)hbσt−,aidb
Wk(t) + ˆUhbσt−,W1(P1u)†aW1(P1u)−aie
Π(du,dt).(48)
In [13] it has been shown also that the adjoint of a quasi-free dynamical semigroup leaves
invariant the sector of the state space based on classical densities with respect to Lebesgue
measure, a fact which could not hold in the general case as discussed in appendix A. So, the
‘quasi-free’ requirement allows for unbounded quantum operators, but at the end it is a strong
restriction on the possible structures.
3.4. The master equation for hybrid states
To have the explicit expression (41) of the formal generator can help in clarifying the various
physical interactions involved in the hybrid dynamics and it allows to compare the traject-
ory approach with the approaches based on master equations. Let us exclude the cases dis-
cussed at the end of appendix Aand assume that the pre-adjoint Tt∗is well dened and that
it leaves invariant T(H)⊗L1(Rs). Then, we can take as hybrid states the x-dependent trace
class operators ϱ(x)such that ϱ(x)⩾0 and Tr{ϱ(x)}is a probability density with respect to
the Lebesgue measure; then, we set ϱt(x) = Tt∗[ϱ0](x). To get the explicit master equation for
ϱt(x), we assume a smooth x-dependence and we simplify the jump contributions by taking
g(x,u) = g(u). By time-differentiation and using (41) and (12), we get the (heuristic) hybrid
master equation
d
dtϱt(x) = K∗[ϱt](x) = K0[ϱt](x) + Kdiff [ϱt](x) + Kjump [ϱt](x),(49)
K0[ϱt](x) = −i[H(x),ϱt(x)] −
s
X
i=1
∂
∂xi
(ci(x)ρt(x)),(50a)
Kdiff [ϱt](x) =
d
X
k=1 1
2
s
X
i,j=1
∂2
∂xi∂xjbk
i(x)bk
j(x)ρt(x)−1
2Lk(x)†Lk(x),ϱt(x)
+Lk(x)ϱt(x)Lk(x)†−
s
X
i=1
∂
∂xibk
i(x)Lk(x)ϱt(x) + ϱt(x)Lk(x)†!,(50b)
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Kjump [ϱt](x) = ˆU
ν(du) Jx−g(u),u+1ϱtx−g(u)Jx−g(u),u†+1
−1
2nJ(x,u)†+1(J(x,u) + 1),ϱt(x)o
+1
2J(x,u)†−J(x,u),ϱ(x)+1U0(u)
s
X
i=1
gi(u)∂ϱt(x)
∂xi!.(50c)
This structure allows to compare, at a heuristic level, the present approach to other
approaches based on master equations for hybrid states, so giving an idea of possible phys-
ical applications. [4,6,30,31] introduce master equations of jump type which are particular
cases of (49) with Kdiff =0. Then, the diffusive case is usually introduced by a suitable limit
in the jump case; [3–6,30,31] give master equations of diffusive type which are particular
cases of (49) with Kjump =0. Examples with two-level systems and harmonic oscillators are
constructed. The most relevant application is the attempt of connecting gravity and quantum
matter in the so called ‘Newtonian limit’ [30,31].
4. Examples
In this section we collect some simple examples of hybrid dynamics, just to illustrate the
possibilities of the general approach, developed in the previous sections.
4.1. A purely classical case
As discussed in section 2.3.1, the hybrid dynamics can produce a change of the underling
probability law. This can happen even in a purely classical case.
Let us take H=C, which means that no quantum component is present. Now the quantities
introduced in assumption 3become complex functions; conditions (4) continue to hold for the
functions Lk(x)and J(x,u). In this ‘classical’ case we get from (6) and (15)
mk(t) = 2ReLk(X(t−)),It(u) = |J(X(t−),u) + 1|2.
Then, there is no dynamics in the quantum component, as (22) gives dbσt=0, while
equation (10) reduces to (14). So, the probability density pthas the non trivial evolu-
tion (14) and it produces the new probability P(17), even without interaction with a quantum
component.
Under P, the classical process X(t) satises the stochastic differential equation (20), which
looks like very similar to the original stochastic equation (2) for a Markov process, apart from
a peculiar difference: under Pthe jump noise Π(du,dt)is no more a Poisson measure, but
a point process whose law depends on the solution X(t) of the stochastic equation itself, as
discussed before (19). The advantage of working under the reference probability Qis to avoid
the mathematical problems related to the involved stochastic equation (20); the construction of
the classical component X(t) is split into the two standard stochastic differential equations (2)
and (14).
19

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
The effect of a non trivial probability density is apparent also from the formal generator (41),
which now is purely of classical type:
K[f](x) = (s
X
i=1
∂f(x)
∂xi ci(x) + 2
d
X
k=1
bk
i(x)ReLk(x)!+
s
X
i,j=1
1
2
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)
+ˆU (f(x+g(x,u)) −f(x))|J(x,u) + 1|2−1U0(u)
s
X
i=1
∂f(x)
∂xi
gi(x,u)!ν(du)).
By comparing it with the formal generator (A.6), we see immediately the corrections due
to the change of probability: ci(x)→ci(x) + 2Pd
k=1bk
i(x)ReLk(x)in the rst term and 1 →
|J(x,u) + 1|2in the jump term. While the rst modication is only a redenition of ci(x), the
second modication is non trivial. Once again, the last modication is related to the fact that
Π(du,dt)is no more a Poisson process, but a counting process with a random compensator
JX(t−),u+12ν(du)dt.
To take a non trivial quantum component (H6=C), but with all the coefcients in assump-
tion 3proportional to the identity 1, would give the same result on the classical component;
indeed, this choice of the coefcients implies that the classical component does not receive any
back-action from the quantum component. Also intermediate situations can arise with suitable
choices of the coefcients. This example shows that a change of probability is not a purely
quantum effect, but it may be due to the reaction of the quantum component on the classical
one and to auto-interactions in the classical component.
4.2. Jumps of discrete type
In the classical case, described by (A.4), the integral over Uhas the role of giving rise to a
continuity of types of jumps; moreover, the integral over U0allows for ‘innitely many small
jumps’. These features are inherited by the hybrid case of section 2. A simpler case, with sums
instead of integrals, can help in the interpretation of the various terms.
In this section we consider the case of discrete types of jumps, when Uis a discrete and
nite set:
U=1,...,l,U0=∅, ν ({k}) = λk,Π({k},dt) = dNk(t).
In this case it is convenient a change of notation to simplify the expressions J(x,u) + 1appear-
ing in the generator (41) and in the SMEs (10) and (22). To get this, we make the replacements
J(x,k)→Jk(x)−1,H(x)→H(x)−i
2
l
X
k=1
λkJk(x)†−Jk(x).
The pre-adjoint of the Liouville operator (12), acting on the trace class, becomes
L(x)∗[ρ] = −i[H(x),ρ] +
d
X
k=1Lk(x)ρLk(x)†−1
2nLk(x)†Lk(x),ρo
+
l
X
k=1
λkJk(x)ρJk(x)†−1
2nJk(x)†Jk(x),ρo.(51)
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Moreover, as in (11) and (15) we introduce mk(t)and we set
Lt∗=L(X(t−))∗,Jkt =Jk(X(t−)),Ikt =hbσt−,J†
ktJkt i.(52)
Under the reference probability Q,Nk(t),k=1,...,l,are independent Poisson processes
of intensities λk; we set de
Nk(t) = dNk(t)−λkdt. Then, the hybrid dynamics is given by the
coupled SDEs (2) and (10), which now become
dXi(t) = ci(X(t−))dt+
d
X
k=1
bk
i(X(t−))dWk(t) +
l
X
k=1
gi(X(t−),k)dNk(t),(53)
dσt=Lt∗σt−dt+
d
X
k=1Lktσt−+σt−Lkt †dWk(t) +
l
X
k=1Jktσt−J†
kt −σt−de
Nk(t).(54)
Under the physical probability P, the Nk(t)become counting processes of intensities λkIkt
and we set db
Nk(t) = dNk(t)−λkIktdt. We assume to have Ikt 6=0. Now, the system of dynamical
SDEs (20) and (22) becomes
dXi(t) = ci(X(t−)) +
d
X
k=1
bk
i(X(t−))mk(t) +
l
X
k=1
gi(X(t−),k)λkIkt!dt
+
d
X
k=1
bk
i(X(t−))db
Wk(t) +
l
X
k=1
gi(X(t−),k)db
Nk(t),(55)
dbσt=Lt∗bσt−dt+
d
X
k=1bσt−Lkt†+Lkt bσt−−mk(t)bσt−db
Wk(t)
+
l
X
k=1 Jktbσt−J†
kt
Ikt −bσt−!db
Nk(t).(56)
We have written the linear SME (54) and the non-linear one (56) in strong form because only
nite sums are involved and not integrals as in (10) and (22). Recall that the conditional mean
of the driving increments, given the past frozen, vanishes; in symbols:
EPhdb
Wk(t)Ft−i=0,EPhdb
Nk(t)Ft−i=0.
This is because, under P,b
Wk(t)is a Wiener process and b
Nk(t)is a compensated counting
process; both processes are martingales.
Let us comment the dynamics (55) of the classical component. By the vanishing of the
means of the increments of the driving processes, the terms in the second line of (55) can
be interpreted as uctuating forces; the rst term is a continuous contribution and the second
one gives also jumps. The quantum component contributes to these two terms only through
the probability law of the processes b
Nk(t). Instead, in the rst line, a drift term appears. The
rst term in the drift does not depend on the quantum component and it represents some auto-
interaction of the classical component; it could be generated by a classical Hamiltonian func-
tion. The two other terms in the drift can again depend on the classical component, as all
the coefcients can contain X(t−), but they depend also on the quantum state through the
‘quantum expectations’ mk(t)and Itk. They represent a back-reaction of the quantum compon-
ent on the classical one. Let us stress that the presence of these back-reaction drifts needs the
non-vanishing of the uctuating forces, because they are connected by the coefcients bk(x)
and g(x,k): the classical evolution must be random if there is transfer of information from the
21

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
quantum component. In the case without jumps, particular cases of (20), including the back-
reaction, have been obtained also in other approaches, see for instance [3, (49)], [5, (7)]. Again
similar non-linear back-reaction terms are obtained in completely different schemes in which
the hybrid dynamics is obtained from microscopic models via Ehrenfest-like approximations
[11].
A way to compare the trajectory approach to hybrid systems to approaches based on
moments [11,12] would be to compute the multi-time correlations of the classical component
by using (55) and stochastic calculus, or by using the characteristic functions of the multi-time
probabilities (33) and the characteristic operator given in (36). The computations of the multi-
time moments in the case of continuous measurements and purely diffusive case are developed
in [23, section 4.3]. For the rst moment it is enough to take the mean of (55); we get directly
d
dtEP[X(t)] = EP[c(X(t))] +
d
X
k=1
EPbk(X(t))mk(t)+
l
X
k=1
EP[g(X(t),k)λkIkt].
Here we recognize immediately the terms discussed above: the mean of the auto-interaction of
the classical component and of the two terms giving the back-action of the quantum component
on the classical system.
About the dynamics (56) of the quantum component, we note the dependence on X(t−)
in the Liouville operator (51) and (52). This gives the non-Markovian behavior of the mean
quantum dynamics, already commented after equation (25). When present, this x-dependence
in (51) represents the action of the classical component on the quantum one, say a classical
feedback or an external control [24,28,37,58]. It can be shown that (56) preserves the pure
states, as it is equivalent to the non-linear SSE recalled in remark 1. Often, this equation is
introduced as stochastic unravelling of the dissipative dynamics of the mean state [3,5,24].
Finally, the formal hybrid generator (41) becomes
K[a⊗f](x) = f(x)e
L(x)[a] + a
s
X
i=1
∂f(x)
∂xi
ci(x) + a
s
X
i,j=1
1
2
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)
+
s
X
i=1
∂f(x)
∂xi
d
X
k=1
bk
i(x)aLk(x) + Lk(x)†a+
l
X
k=1
λkf(x+g(x,k))Jk(x)†aJk(x),(57)
e
L(x)[a] = i[H(x),a] +
d
X
k=1Lk(x)†aLk(x)−1
2nLk(x)†Lk(x),ao
−1
2
l
X
k=1
λknJk(x)†Jk(x),ao.
As an example, by using the classical component given by (A.8), and by taking Lk(x) = 0, it
is possible to construct discrete hybrid master equations, based on the classical Pauli equation,
as those introduced in [2,3].
22

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
4.2.1. Unitary jumps. A particularly simple case is when the jump operators Jk(x)are pro-
portional to unitary operators (as in section 3.3) and there is no diffusion contribution; so we
take Lk(x) = 0 and
Jk(x) = βk(x)Uk(x), βk(x)>0,Uk(x)†Uk(x) = 1,
⇒Jk(x)†Jk(x) = βk(x)21,Ik(x) = βk(x)2.
Now, from equations (54), (51) and (56) we get
L(x)∗[ρ] = −i[H(x),ρ] +
l
X
k=1
λkβk(x)2Uk(x)ρUk(x)†−ρ,(58)
dσt=−iHt,σt−dt+
l
X
k=1
β2
kt Uktσt−U†
kt −σt−dNk(t), βkt =βk(X(t−)),(59)
dbσt=−iHt,bσt−dt+
l
X
k=1Uktbσt−U†
kt −bσt−dNk(t),Ukt =Uk(X(t−)).(60)
Note that Ikt =βkX(t−)2does not depend on the quantum state and this gives that, under P,
the full law of the processes Nk(t)remains independent of the quantum state.
4.3. Entanglement creation and preservation
An interesting application of the trajectory formalism is to the entanglement dynamics of two
quantum systems without direct interaction; we can have an a priori dynamics (25) which
destroys any initial entanglement, while it is totally or partially protected in a single quantum
trajectory [35,59–61]. To give an example, we consider the simple dynamics discussed in
section 4.2.1; now, H=C2⊗C2. As classical component we take the Poisson processes them-
selves:
l=s,Xi(0) = 0,Xi(t) = Ni(t).
With this choice, (60) gives the dynamics of the conditional state (23); as in the general case,
it sends pure states into pure states. So, we take a pure state at time t=0 and we can write
bσt=|b
ψtihb
ψt|.
To quantify the entanglement we consider the concurrence [62], whose denition is recalled
in appendix B. Now we have the a priori state and the conditional state, which is random. Due
to the denition (B.2) of the concurrence of a mixed state, it is easy to see that the concurrence
of the a priori state ηtis a lower bound for the mean concurrence of the conditional state b
ψt:
EPhCb
ψti=EP[|χ(t)|]⩾C(ηt), χ (t) = χb
ψt=hTb
ψt|σ1
yσ2
yb
ψti.
By (60), we get the evolution equation for the random chi-quantity:
dχ(t) = −ihTHtb
ψt|σ1
yσ2
yb
ψti− ihTb
ψt|σ1
yσ2
yHtb
ψtidt
+
s
X
k=1hTUkt b
ψt|σ1
yσ2
yUkt b
ψti− χ(t)dNk(t).(61)
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J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
Local operators. Let us consider the case of only local operators in (60), say
H=ω
2σ1
z+σ2
z,Uk(x) = Uk=(eiϕkσ·ϵk⊗1k=1,...,l1,
1⊗eiϕkσ·ϵkk=l1+1,...,l,
with ϕk∈R,ϵk∈R3,ϵk=1. By (61) and (B.3) we get dχ(t) = 0: the conditional concurrence
is constant in time. If the initial state is maximally entangled, we have EPC(b
ψt)=1, ∀t⩾0.
On the contrary, the concurrence of the a priori state can decay. For instance let us take
ω > 0,l=2,l1=1, λk=λ > 0, βk=β6=0, ϕk=π2, ϵk= (1,0,0)
⇒Uk=iσk
x.
Then, by taking the P-mean of equation (60), we get the master equation for the a priori states
d
dtηt=
2
X
k=1−iω
2σk
z,ηt+λβ2σk
xηtσk
x−ηt.(62)
For this equation there is a unique equilibrium state η∞=1/4, which is separable; so, the a
priori concurrence decays to zero. In spite of this, if we know the values of N1(t)and N2(t)at all
times, we know the state of the quantum component, which maintains the initial entanglement.
Non-local operators. By taking the same Hamiltonian and the same constants λand β, but
U1=1
√2σ1
x+iσ2
xand U2=1
√2σ1
x−iσ2
x, we get again the master equation (62) for the
a priori states. However, in this case, the entanglement of the conditional states can vary; in
particular entanglement can be created [35]. For instance, if the initial state is |11i, at a jump
we have
|11i → Uk|11i=1
√2(|01i± i|10i) :
we get a maximally entangled state from a separable one.
We have constructed two simple models with the same quantum master equation (62) for
the mean state, a master equation which destroys any initial entanglement. On the contrary,
the two decompositions in ensembles of conditional states are different; the rst one preserves
any initial entanglement, while the second one allows for increasing and decreasing of entan-
glement. Let us stress that the two decompositions are physically different, as they depend on
different interactions with the classical component of the hybrid system.
Hidden entanglement. For the two models above we can speak of the phenomenon of hidden
entanglement, a term which is sometimes used for situations in which strong entanglement is
present in a statistical ensemble, while it is less relevant in the associated mean state, see for
instance [34,47].
The systems considered in [34,47] are different; typically, they consider a unitary local
transformation which hits an entangled system with a probability 1/2, so generating a mixed
state. Eventually, revivals of the hidden entanglement can be obtained by local pulses. We can
take this as a suggestion: some kinds of ‘one shot’ transformations can be introduced also in
our ‘Markovian’ framework
24

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
For instance, we can have a single shot which hits the second qubit only once at a random
time and leaves free the rst qubit. Let us take
H=ω
2σ1
z+σ2
z,l=1,X(t) = N(t),J(x) = g(x)β1⊗U, β > 0,
where Uis a unitary operator on C2and g(x) is a continuous real function such that g(0) = 1
and g(x) = 0 for x⩾1. As X(t) = N(t)takes only integer values, we have that gN(t)behaves
as Kronecker delta in 0, and equation (56) gives
dbσt=−iH,bσt−dt+(Ubσt−U†−bσt−dN(t),if N(t−) = 0,
0if N(t−)⩾1;
N(t) turns out to be a Poisson process of intensity λβ2. The function g(x) has been taken to be
continuous to have that J(x) satises assumption 3. This trick allows to introduce single shots
as in [34,47], with the difference that the time of the jump is the random waiting time of a
Poisson process.
4.4. Control
Firstly, let us note that we could use a component of the classical process just to introduce some
noise in the Hamiltonian H(x), or in some of the operators Lk(x),J(x,u). If this component is
deterministic, simply we get a time dependent Hamiltonian. By this, we have that at least open
loop control can be included in the Markovian hybrid dynamics.
Moreover, we have that the classical subsystem satises equation (20) under the physical
probability P; so, its dynamics depends on the state of the quantum subsystem through the
coefcients C(i)(t). On the other side, the classical component can affect the quantum dynam-
ics through the x-dependence of the operators H(x), Lk(x),J(x,u). In this way we have realized
a closed loop feedback; applications in quantum optics can be found in [23,24,58]. This opens
the question of the connections with a full quantum control theory, where already the quantum
trajectory formalism (quantum ltering) started to be explored, see [28,63,64] and references
therein.
5. Conclusions
In the present approach, the hybrid dynamics has been introduced by a couple of stochastic
differential equations, one for the classical component and one for the quantum component,
and this has been done in two different ways. In the rst case a reference probability Qis used,
and the two stochastic equations are the evolution equation (2) for the classical process and
the linear SME (10) for the (non-normalized) quantum state; the quantum/classical interaction
is contained only in the SME (10). The physical probability Pis determined by the dynamics
itself, and it is dened in (13) and (17). By suitable hypothesis (assumptions 1–3) it is possible
to prove existence and uniqueness of the solution of these coupled equations.
An equivalent description can be given by working under the physical probability P; now
we have the stochastic equation (20) for the classical process and the non-linear SME (22) for
the quantum states. However, the mathematical setting is more involved as even the probability
law depends on the solution of the two coupled equations. On the other side, in this formulation
the quantum/classical interaction is more transparent, as it appears in both equations.
25

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
We have also shown that the general axiomatic and the linear structure of a quantum theory
is respected. Indeed, we have shown how to connect the whole construction to the general
notion of quantum observable (the positive operator valued measures) and to the notion of
‘instrument’ (section 2.4.2). We have shown that, as expected, the extraction of information
from the quantum subsystem is necessarily connected to the presence of suitable dissipation in
the dynamics (see the comments in section 2.3.1, after equation (21)). The absence of memory
is underlined by the fact that probabilities and dynamics can be summarized in a notion of
‘transition instruments’ (29), a quantum analogue of the transition probabilities in the theory
of classical Markov processes.
Moreover, the Markovian character of the dynamics appears also in the fact that it can be
expressed as a hybrid dynamical semigroup. However, in the pure classical case it is shown
that the classical states cannot always be reduced to densities with respect to the Lebesgue
measure (appendix A); then, the hybrid semigroup needs a C∗-algebraic formulation. In any
case, we have given its formal generator, which allows to compare our results with less general
approaches based on master equations.
Finally, some simple examples on entanglement and control show that we can have a
considerable freedom in the construction of physical models, even under the restriction of
a Markovian hybrid dynamics.
Appendix A. The classical Markov process under the reference probability Q
In this appendix we collect known results on the Markov process dened by (2), when there is
no interaction with the quantum component; in particular, we discuss the associated semigroup.
Equation (2) is studied in [44, section 6.2].
By following [44, sections 6.4.1, 6.4.2], we can introduce the stochastic ow Φassociated
to the stochastic differential equation (2). For 0 ⩽s⩽t, let Φs
t(x)be the solution of (2) with
initial condition xat time s; in particular, with the notation introduced at the end of proposition
1, we have Φ0
t(x) = Xx(t). The solution of (2) is an homogeneous Markov process and we have
P(t,x,E)≡QΦ0
t(x)∈E=QΦs
s+t(x)∈E.(A.1)
Then, the transition probabilities (3) satisfy the properties of a time-homogeneous transition
function [43, chapter 4, section 1, p 156]: (a) P(t,x,·)is a probability measure on B(Rs),
∀t∈[0,+∞),∀x∈Rs; (b) P(0,x,·) = δx(·),∀x∈Rs; (c) P(·,·,E)is a bounded Borel function
on [0,+∞)×Rs,∀E∈ B(Rs); (d) the Chapman–Kolmogorov identity holds, which is
P(t+r,x,E) = ˆRs
P(r,y,E)P(t,x,dy)r,t⩾0,x∈Rs,E∈ B(Rs).(A.2)
We also dene the map
Tt[f](x) = EQfΦ0
t(x)=ˆRs
f(y)P(t,x,dy),f∈Bb(Rs),x∈Rs;(A.3)
{Tt,t⩾0}is a semigroup of positive, bounded linear maps from Bb(Rs)into itself, with
Tt(1) = 1, T0=Id. Moreover, Ttmaps Cb(Rs)into itself [44, Note 3, p 402]. The expres-
sion (A.3) is continuous in t, for xed fand x.
Let fbe in the space C2
b(Rs)of the two times differentiable functions; the functions and
their rst two derivatives are continuous and bounded. By (2) and Itˆ
o’s formula, we get
26

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
df(X(t))X(t−)=x=
s
X
i=1
∂f(x)
∂xi ci(x)dt+
d
X
k=1
bk
i(x)dWk(t)!
+1
2
s
X
i,j=1
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)dt
+dtˆU0 f(x+g(x,u)) −f(x)−
s
X
i=1
∂f(x)
∂xi
gi(x,u)!ν(du)
+ˆU0
(f(x+g(x,u)) −f(x)) e
Π(du,dt)
+ˆU\U0
(f(x+g(x,u)) −f(x))Π(du,dt).(A.4)
Then, we have
d
dtTt[f](x) = Tt◦Kcl [f](x),∀f∈C2
b(Rs),∀x∈Rs,(A.5)
Kcl [f](x) =
s
X
i=1
∂f(x)
∂xi
ci(x) + 1
2
s
X
i,j=1
∂2f(x)
∂xi∂xj
d
X
k=1
bk
i(x)bk
j(x)
+ˆU f(x+g(x,u)) −f(x)−1U0(u)
s
X
i=1
∂f(x)
∂xi
gi(x,u)!ν(du).(A.6)
Under some additional sufcient assumptions, it is possible to prove that Ttis strongly con-
tinuous and that it is generated by Kcl, see [44, (6.36) p 402]. We do not need these results,
because existence and uniqueness of the solution of (2) is enough to give the intrinsic dynam-
ics of the classical component of the hybrid system. In [42, pp 56, 349–50], (A.5) is used as
denition of ‘generator of the time-homogeneous Markov process’; the law of the process is
determined by the Dynkin’s expectation formula
EQf(X(t)) −f(X(0)) −ˆt
0Kcl [f](X(r))dr=0.(A.7)
In some cases the adjoint of Tt, dened in the dual Banach space, leaves invariant the space
of the probability densities with respect to the Lebesgue measure and some known classical
master equations are obtained. In the deterministic case, i.e. bk
i(x) = 0 and g(x,u) = 0, we can
take seven and the coefcients ci(x)givingan Hamiltonian evolution; then, the master equation
is the Liouville equation. Instead, with only g(x,u) = 0, we can get the Kolmogorov–Fokker–
Planck equation [44, section 3.5.3], [41, sections 9.3, 14.2.2].
In general, it can happen that X(t) has not a density with respect to Lebesgue meas-
ure, even if we start with X(0) having a density. As an example, take the equation dX(t) =
x0−X(t−)dN(t)with x0∈Rs;X(0) is taken with a density and independent of the Poisson
process Nof intensity λ > 0. Then, we obtain P[X(t) = x0] = 1−e−λtand a density with
respect to the Lebesgue measure does not exist for t>0.
27

J. Phys. A: Math. Theor. 57 (2024) 315301 A Barchielli
By generalizing the example above, we can construct models with jumps among xed points
at random times. Let us take ndistinct points xj∈Rsand lindependent Poisson processes Nk
of intensities λk>0,
dX(t) =
l
X
k=1
g(X(t−),k)dNk(t),g(x,k) = (yk−xifx∈Fk,
0 ifx/∈Fk,
Fk∈ B(Rs),
l
[
k=1
Fk=Rs,yk∈x1,...,xn.
(A.8)
After the rst jump, the process reduces to a random walk on the points {x1,...,xn}. When the
initial condition has a probability concentrated in the set of points {x1,...,xn}, the probability
at time tsatises a Pauli rate equation [2, (13)], [3, (14)].
Appendix B. Concurrence
To quantify the entanglement of two qubits we use the concurrence [62]. Let φ∈H≡C2⊗
C2be a generic normalized vector and expand it on the computational basis: φ=φ11|11i+
φ10|10i+φ01 |01i+φ00|00i. Let Tbe the complex conjugation of the coefcients in this basis:
Tφ=φ11 |11i+φ10 |10i+φ01 |01i+φ00 |00i. Then, the concurrence C(φ)of the pure state
φis dened by
C(φ) := |χ(φ)|, χ(φ) := hTφ|σy⊗σyφi=2(φ10φ01 −φ11 φ00).(B.1)
The concurrence goes from 0 (for a separable state) to 1 (for a Bell state).
If ρis a generic statistical operator, the concurrence is dened by
C(ρ) := inf X
i
piC(ψi),(B.2)
where the inmum is taken over all decompositions of ρinto pure states, ρ=Pipi|ψiihψi|.
Finally, let Aand Bbe linear operators on C2. In studying the dynamics of the concurrence,
the following formulae are very useful:
χ((A⊗B)φ)=(detC2A)(detC2B)χ(φ),
hTφ|(σyA)⊗σyφi=hTA⊗1φ|σy⊗σyφi=1
2(TrC2A)χ(φ).(B.3)
ORCID iD
Alberto Barchielli https://orcid.org/0000-0002-6988-3855
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