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Open Systems & Information Dynamics
Vol. 28, No. 1 (2021) 2150001 (39 pages)
DOI:S1230161221500013
©World Scientific Publishing Company
Gaussian Quantum Markov Semigroups
on a One-Mode Fock Space:
Irreducibility and Normal Invariant States
J. Agredo1, F. Fagnola2, and D. Poletti2
1Department of Mathematics
Escuela Colombiana de Ingenier´ıa Julio Garavito
Autopista Norte AK 45 No. 205–59
111166 Bogot´a, Colombia
e-mail: julian.agredo@escuelaing.edu.co
2Dipartimento di Matematica, Politecnico di Milano
Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
e-mail: franco.fagnola@polimi.it (F. Fagnola)
e-mail: damiano.poletti@polimi.it (D. Poletti)
(Received: November 4, 2020; Accepted: March 1, 2021; Published: August 30, 2021)
Abstract. We consider the most general Gaussian quantum Markov semigroup on a
one-mode Fock space, discuss its construction from the generalized GKSL representation
of the generator. We prove the known explicit formula on Weyl operators, characterize
irreducibility and its equivalence to a H¨ormander type condition on commutators and es-
tablish necessary and sufficient conditions for existence and uniqueness of normal invariant
states. We illustrate these results by applications to the open quantum oscillator and the
quantum Fokker-Planck model.
Keywords: Quantum Markov semigroup, quasi-free semigroup, GKSL generator, Gaus-
sian, irreducibility.
1. Introduction
Gaussian semigroups are of utmost importance in many fields because they
arise in several relevant models, they form a class with a rich structure al-
lowing one to establish and take advantage of a number of explicit formulas.
This happens also in quantum theory of open systems with quantum Markov
semigroups (QMS), namely weakly∗-continuous semigroups (Tt)t≥0of com-
pletely positive, identity preserving, normal maps on a von Neumann algebra.
When this is the algebra B(Γ(Cd)) of all bounded operators on the Fock space
Γ(Cd) a QMS is called Gaussian if the predual semigroup (T∗t)t≥0acting on
trace class operators on Γ(Cd) preserves Gaussian states (see Sect. 5 for the
definition).
2150001-1
J. Agredo, F. Fagnola, and D. Poletti
There are two typical approaches to this class of semigroups. Physicists
usually consider generators represented in a Gorini-Kossakowski-Sudharshan-
Lindblad [19, 22] (GKSL) form (see (1), (3), (2) below), which is only formal
because it involves unbounded operators, and compute moments of Gaussian
observables without concern about the existence and well-definiteness of the
dynamics (see, for instance [1, 3, 4, 20, 23] and the references therein). Math-
ematicians introduce Gaussian QMSs by their action on Weyl operators (see,
e.g., [15, 30]) of a regular representation of canonical commutation relations
(CCR) but they just show ([30] Proposition 4.8, Theorem 4.9) that the ac-
tion of generator, on a certain restricted domain, admits a generalized GKSL
representation with unbounded Hamiltonian and noise operators.
The joining link for handling both techniques and exploiting the advan-
tages of each one of them is the characterization of the unbounded generator
with a generalized GKSL form involving unbounded operators that are either
linear or quadratic in creation and annihilation operators. In this way one
can go beyond explicit computations on Gaussian states and observables and
study, for instance, the evolution of any initial state applying general results
from the theory of QMS.
In this paper we consider the most general Gaussian quantum Markov
semigroup on the one mode Fock space Γ(C) of the regular representation
of one-dimensional CCR. First we discuss its construction starting from the
unbounded generator in its generalized GKSL form and give a proof of the
known explicit formula for the action on Weyl operators. Second, we fully
characterize irreducibility in terms of parameters of the model. This is an
important property of the dynamics because it implies that the system has
to be regarded as a whole and reduction to subsystems is not possible. In
particular, the support of any initial state cannot remain confined in a proper
subspace (see, e.g., [18]). Third, still in terms of these parameters, establish
necessary and sufficient conditions for existence and uniqueness of normal
invariant states. As a corollary, for any initial state, we also deduce conver-
gence towards the unique invariant state.
In this way, we provide a unified treatment of both approaches and a
thorough study of the one-dimensional case. This is quite complex because
it depends on many parameters and a detailed (somewhat lengthy) analysis
of several special subcases is necessary.
Gaussian QMSs on the von Neumann algebra of all bounded operators
on the one-mode Fock space Γ(C) are uniquely defined by pre-generators in
a generalized (GKSL) form
L(x) = i [H, x]−1
2
2
X
`=1
(L∗
`L`x−2L∗
`xL`+x L∗
`L`),(1)
where unbounded operators L1, L2, called noise operators or Kraus operators,
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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
depend linearly on Bosonic creation and annihilation operators on Γ(C)
L`=v`a+u`a†v`, u`∈C, ` = 1,2,(2)
and His the operator
H= Ω a†a+κ
2a†2+κ
2a2+ζ
2a†+ζ
2a(3)
with Ω ∈R, κ, ζ ∈C. The operators L1, L2will be assumed to be linearly
independent. However, we shall also consider the special case when L2does
not appear in (1), namely the multiplicity of the completely positive part
of (1) is one. We will not consider when there are no noise operators L`,
corresponding to a closed system.
In this paper, we find three main new results. The first one is the com-
plete characterization of irreducibility and its equivalence to a H¨ormander
type condition on certain commutators. More precisely, we show that the
QMS with pre-generator (1) is irreducible if the completely positive part
has multiplicity two, namely the operators L1, L2are linearly independent
(Theorems 5). While, in the case where the completely positive part has
multiplicity one (i.e., formally, L2does not appear in (1)), we show (Theo-
rem 7) that the QMS is irreducible if and only if the operators L1and [H, L1]
are linearly independent. This is clearly a H¨ormader type multiple commu-
tator condition in which one needs only the first order commutator because
of one-dimensionality of the CCR. Here, however, H¨ormander condition is
established for a differential operator with second order anti-selfadjoint part,
not first order as in the classical case.
The second main result is the characterization of Gaussian QMSs with
normal invariant states by two simple inequalities on the parameters of the
model. We show (Theorems 8, 9) that one can find a normal invariant state,
which is explicit and is a quantum Gaussian state, if and only if
γ=1
2X
`=1,2
(|v`|2− |u`|2)>0 and γ2+ Ω2− |κ|2>0.
Note that normal invariant states may exist also when the Hamiltonian has
no eigenvalues, however transitions to lower-level states induced by the dis-
sipative part must be stronger to compensate the effect of the Hamiltonian
Hwithout eigenstates (see Remark after Theorem 8). The third main result
is uniqueness of Gaussian invariant states in the set of all normal invariant
states and convergence towards invariant states (Theorem 8) which follows
from irreducibility in most cases.
The paper is organized as follows. In Sect. 2 we begin by describing the
generator in the generalized GKSL form and construct Gaussian QMSs by the
2150001-3
J. Agredo, F. Fagnola, and D. Poletti
minimal semigroup method. After proving Markovianity (i.e., preservation of
the identity operator) in Theorem 1 we have a characterization of its domain
at our disposal that we exploit for proving the known explicit formula for the
action on Weyl operators (Theorem 2).
Then we turn our attention to irreducibility. In Sect. 3 we prove that
it always holds when the completely positive part of the generator has two
noise operators (Theorem 5). The case in which there is only one noise oper-
ator where the H¨ormander type commutator condition appears is studied in
Sect. 4 (see the decision tree at the end of the section). In Sect. 6 we study
normal invariant states showing that, when they exist, they are also unique.
Moreover, it turns out that they are either faithful or pure (Proposition 5).
Finally, we illustrate these results by applications to the open quantum os-
cillator and the quantum Fokker-Planck model.
2. Gaussian QMSs
In this section we introduce the class of QMS that we will analyze in this
paper.
Let h= Γ(C) be the Fock space on Cwith canonical orthonormal basis
(en)n≥0. Each vector enis called n-particle vector. For each z∈Cthe vector
e(z) = X
n≥0
zn
√n!en
is called exponential or coherent vector with parameter z. The vector space
of finite linear combinations of vectors of the canonical orthonormal basis,
denoted by D, is a natural common domain for all unbounded operators
that we will consider. One could consider as domain Dthe linear span of
exponential vectors, or the linear span of (en)n≥0together with exponential
vectors, without further complications.
The number operator is the selfadjoint operator on hdefined by
Dom(N) = nξ=X
n≥0
ξnen∈h:X
n≥0
n2|ξn|2<∞oNξ =X
n≥0
n ξnen.
Annihilation and creation operators on hare defined on the domain Dom(N1/2)
by
a ξ =X
n≥1
√n ξnen−1, a†ξ=X
n≥1
√n+ 1 ξnen+1 .
It is not difficult to see that a, a†are closed operators and they are mutually
adjoint. Alternatively, they can be defined via polar decomposition a†=
S(N+ 1l)1/2,a=S∗N1/2where 1l is the identity operator and Sthe right
shift defined by Sen=en+1.
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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Annihilation and creation operators satisfy the canonical commutation
relation (CCR)
[a, a†] = 1l
on Dom(N), [ ·,·] denoting the commutator. Moreover, for all z∈Cthe
operator with domain Dom(N1/2) defined by za†−za is anti-selfadjoint an
one can define the unitary Weyl operator
W(z) = exp(za†−za).
It is not difficult to see as well that exponential vectors belong to the domain
of Nkfor all k≥0. In particular, they belong to the domain of a,a†and
a e(z) = ze(z), a†e(z) = d
dεe(z+ε)ε=0 .
In this paper we are concerned with quantum Markov semigroups (QMS)
with pre-generators in a generalized Gorini–Kossakowski–Sudarshan–Lindblad
(GKSL) form (1). The domain of Lwill be described below in Theorem 1,
after the construction of a QMS by the minimal semigroup method. The op-
erators L1, L2are defined on Dom(N1/2) and His the operator on Dom(N)
defined by (3). The operator L1, L2, also called noise or Kraus operators,
will be assumed linearly independent, namely
v1u2−v2u16= 0 ,
so as to consider a generalized GKSL representation of Lwith the minimum
number of Kraus operators. We shall consider also the special case when there
is only one Kraus operator L1but not the “reversible”, purely Hamiltonian,
case with no Kraus operator.
As shown in [13, Proposition 4.9] the operator Gdefined by closure of the
operator −iH−(L∗
1L1+L∗
2L2)/2 defined on Dby
G=−1
2|v1|2+|v2|2+|u1|2+|u2|2+ iΩa†a−1
2|u1|2+|u2|21l
−1
2(v1u1+v2u2−iκ)a†2−1
2(v1u1+v2u2+ iκ)a2−i
2ζa†+ζa
generates a strongly continuous semigroup (Pt)t≥0on h, therefore we can
construct the minimal QMS associated with operators G, L1, L2.
We briefly recall the construction (see [13, Sect. 3.3]). Let x∈ B(h) and
t≥0 and define non decreasing sequence of completely positive maps T(n)
t
on B(h) by T(0)
t(x) = P∗
txPtand
Dξ0,T(n+1)
t(x)ξE=P(t)ξ0, xP (t)ξ
+
2
X
`=1
t
Z
0DL`P(t−s)ξ0,T(n)
s(x)L`P(t−s)ξEds
2150001-5
J. Agredo, F. Fagnola, and D. Poletti
where ξ, ξ0∈D. It can be shown that one can define a weak∗continu-
ous semigroup T= (Tt)t≥0of normal completely positive maps on B(h) by
Tt(x) = supn≥0T(n)
t(x) for all xpositive and then extend to an arbitrary
x∈ B(h) by decomposition as sum of positive operators.
THEOREM 1 The minimal QMS Tis identity preserving. The domain of
its generator Lconsists of the set of x∈ B(h)for which the quadratic form
with domain D×D
£(x)[ξ0, ξ] = Gξ0, xξ+
2
X
`=1 L`ξ0, xL`ξ+ξ0, x Gξ
is bounded. Moreover, Tis the unique weak*-continuous semigroup of posi-
tive operators on B(h)such that
d
dtξ0,Tt(x)ξt=0 =£(x)[ξ0, ξ]
for all x∈ B(h),ξ0, ξ ∈Dom(G).
Proof. Tis identity preserving by Proposition 4.12 in [13]. The character-
ization of the domain of the generator Lis given in [13] Proposition 3.33.
Finally, if (T0
t)t≥0is another such semigroup then, for xpositive, we can
prove inductively that T0
t(x)≥ T (n)
t(x) for all n≥0 and t≥0, therefore
T0
t(x)≥ Tt(x). Considering the operator x=kxk1l −x, which is positive, we
have also
kxk1l − T 0
t(x) = T0
t(kxk1l −x)≥ Tt(kxk1l −x) = kxk1l − Tt(x)
therefore Tt(x) = T0
t(x).
2.1. Explicit formula on Weyl operators
An interesting known feature of Gaussian QMSs is the explicit formula for
their action on the dense subalgebra of Weyl operators (see [24, 25, 30] and
also [27, 31]). We will now present this formula and establish the relationship
with the GKSL generator (1).
We begin by recalling some useful formulae on the action of Weyl opera-
tors
W(z)e(f) = exp −|z|2/2−zf e(f+z)
A straightforward computation on exponential vectors yields the Weyl com-
mutation relations
W(z)W(z0) = exp −i=(zz0)W(z+z0)
2150001-6
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
in particular W(z)∗=W(−z). Moreover, we have
W(z)∗a W (z) = a+z1l , W (z)∗a†W(z) = a†+z1l .
[a, W (z)] = z W (z),[a†, W (z)] = z W (z).
The following Theorem gives the explicit action of maps Tton Weyl operators.
In the sequel <(z) and =(z) denote the real and imaginary part of a complex
number z.
THEOREM 2 Let (Tt)t≥0be the QMS with generalized GKSL generator as-
sociated with H, L1, L2as in (2). For all Weyl operator W(z)we have
Tt(W(z)) = exp −1
2
t
Z
0<esZ z CesZ zds+ i
t
Z
0<ζ esZ zdsWetZ z,
(4)
where Zand Care the real linear operators
Zz =iΩ +
2
X
`=1
(|u`|2− |v`|2)/2z+ iκz , (5)
Cz =
2
X
`=1 (|u`|2+|v`|2)z+ 2v`u`z.(6)
Proof. One can check [30, Theorem 3.1] the semigroup law Tt(Ts(W(z))) =
Tt+s(W(z)) for all t, s ≥0. The derivative of he(g),Tt(W(z))e(f)iat time
t= 0, namely £(W(z))[e(g), e(f)] is equal to
i (hHe(g), W (z)e(f)i−he(g), W (z)H e(f)i)
−1
2
2
X
`=1 hL∗
`L`e(g), W (z)e(f)i − 2hL`e(g), W (z)L`e(f)i
+he(g), W (z)L∗
`L`e(f)i
= i he(g),[H, W (z)]e(f)i+1
2
2
X
`=1 he(g), L∗
`[W(z), L`]e(f)i
+he(g),[L∗
`, W (z)]L`e(f)i,
where all operator compositions make sense because exponential vectors are
in the domain of any power of the number operator. Computing the commu-
tators
[W(z), L`] = −(v`z+u`z)W(z),[L∗
`, W (z)] = (u`z+v`z)W(z)
2150001-7
J. Agredo, F. Fagnola, and D. Poletti
[H, W (z)] = W(z)(Ωz+κz)a+ (Ωz+κz)a†
+Ω|z|2+κz2
2+κz2
2+ζ z
2+ζz
2W(z)
[L∗
`,[W(z), L`]] = −|u`|2+|v`|2|z|2+v`u`z2+v`u`z2W(z)
we can write £(W(z))[e(g), e(f)] as he(g), W (z)X(z)e(f)iwhere
X(z) = |u`|2− |v`|2z/2 + i (Ωz+κz)a†
−|u`|2− |v`|2z/2−i (Ωz+κz)a
−1
2(|u`|2+|v`|2)|z|2+v`u`z2+v`u`z2
+ iΩ|z|2+κz2+κz2
2+ iζ z +ζ z
2.
Since exponential vectors belong to the domain of a, a†, compute now the
derivative of W(etZ z)e(f) at time t= 0 as follows
d
dtW(etZ z)e(f)t=0 =d
dtexp −1
2etZ z2−etZzfe(etZ z+f)t=0
=−<(z Zz)−Zzfexp −1
2|z|2−zf e(z+f)
+ exp −1
2|z|2−zf d
dte(etZ z+f)t=0
=−W(z)Zz a +<(z Zz)e(f)
+ exp −1
2|z|2−zf d
dt(etZ z+f)t=0a†e(z+f).
Recalling the commutation relation [a†, W (z)] = zW (z) we find
d
dtW(etZ z)e(f)t=0 =−W(z)Zz a +<(z Zz)e(f)+(Z z)a†W(z)e(f)
=W(z)(Zz a†−Zz a − < (z Zz) + z Zz)e(f)
=W(z)(Zz a†−Zz a +1
2z Zz −Zz z)e(f).
Computing the derivative of the exponential factor in (4) at t= 0, the deriva-
tive of the scalar product of the right-hand side of (4) with two exponential
vectors e(g), e(f) can be written as well as he(g), W (z)Y(z)e(f)iwhere Y(z)
is
Y(z) = (Zz a†−Zz a) + 1
2z Zz −Zz z−1
2<(z Cz)+i<ζz.
The conclusion follows form X(z) = Y(z) for all z∈C.
2150001-8
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Remark 1 If the GKSL generator has only one Kraus operator L1formula
(4) also holds with real linear operators Zand C, formally defined in the
same way, setting v2=u2= 0 in (5) and (6).
3. Irreducibility: The Case of Two Noise Operators L1,L2
In the study of the evolution of an open quantum system irreducibility plays
a key role because it guarantees that there is no proper subsystem which is
invariant under the evolution. Therefore the system has to be regarded as a
whole and reduction to subsystems is not possible. In addition, irreducibility
is a key assumption of many results on the asymptotic behaviour of QMS
(see [16]) and irreducible subsystems constitute the building blocks in the
analysis of the structure of normal invariant states of a QMS (see [10]).
In this section we show that the Gaussian QMS with two linearly inde-
pendent noise operators L1,L2is irreducible. Gaussian QMS with only one
operator Lwill be considered in Sect. 4.
DEFINITION 1 A QMS Ton B(h) is called irreducible if there exists no
non-trivial orthogonal projection pon hsuch that Tt(p)≥pfor all t≥0.
A projection psuch that Tt(p)≥pfor all t≥0 is called subharmonic following
the terminology in use in the classical theory of Markov processes. The
following result (see Theorem III.1 in [14]) characterizes such projections.
THEOREM 3 A projection pis subharmonic for Tif and only if the range
Rg(p)of pis invariant for the operators Pt(t≥0) of the strongly continuous
contraction semigroup on hgenerated by Gand L`u=pL`u, for all u∈
Dom(G)∩Rg(p), and all `≥1.
It is worth noticing here that, by general results on strongly continuous semi-
groups (see [14] Lemma III.1), if Rg(p) is invariant for the operators Pt, then
Dom(G)∩Rg(p) is dense in Rg(p) and so conditions on the operators L`are
not reduced to the sole zero vector.
In view of this characterization of subharmonic projections, it is now
intuitively clear that, if there are two linearly independent Kraus operators,
the range of a subharmonic projection should be an invariant subspace for a
and a†and so it will be trivial by irreducibility of the Fock representation of
the CCR. However, the necessary clarifications on operator domains are now
in order.
Let G0be the closure of the operator −(L∗
1L1+L∗
2L2)/2 defined on D
which is symmetric. It is easy to check that every vector in Dis an analytic
vector for G0. Therefore an application of Nelson’s theorem on analytic
vectors shows that it is selfadjoint. The following is the key result on the
domain of the operator Gthat we need for proving irreducibility.
2150001-9
J. Agredo, F. Fagnola, and D. Poletti
THEOREM 4 If there are two linearly independent noise operator L1,L2the
domain of the operators Gand G0coincide with the domain of the number
operator N.
We defer the proof to Appendix A and proceed to the main result of this
section. Note that the property Dom(G) = Dom(G0) = Dom(N) plays a key
role in the proof.
THEOREM 5 The QMS with generalized GKSL generator associated with
Has in (3) and two linearly independent noise operator L1, L2as in (2) is
irreducible.
Proof. Let Vbe a nonzero closed subspace of hwhich is invariant for the con-
traction operators Ptof the semigroup generated by Gand L`(Dom(G)∩ V)⊆
Vfor `= 1,2.
By the linear independence of L1, L2, since Dom(G) = Dom(N) we have
also
a(Dom(N)∩ V)⊆Dom(N1/2)∩ V a†(Dom(N)∩ V)⊆Dom(N1/2)∩ V
a†a(Dom(N)∩ V)⊆ V aa†(Dom(N)∩ V)⊆ V
hence, denoting by pthe orthogonal projection onto V,
p⊥ap = 0 = pap⊥, p⊥a†p= 0 = pa†p⊥
on Dom(N)∩ V and, left multiplying by a†the first identity,
p⊥a†ap = 0 = pa†ap⊥.
It follows that, for all λ > 0, we have the commutation (λ1l + N)p=
p(λ1l + N) and, left and right multiplication by the resolvent (λ1l + N)−1
yields
p(λ1l + N)−1= (λ1l + N)−1p .
In particular, for all k > 0, considering bounded Yosida approximations
Nk=kN (k1l + N)−1of Nthat converge strongly to Non Dom(N) we have
p kN (k1l + N)−1=kN (k1l + N)−1p
and so p e−tNk=e−tNkpfor all t, k > 0. Taking the limit as k→+∞, by the
Trotter-Kato theorem [12, Th. 4.8 p. 209] we find
p e−tN =e−tN p∀t≥0.(7)
Let v∈ V,v6= 0 with expansion in the canonical basis
v=X
k≥k0
vkek,
2150001-10
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
where k0is the minimum kfor which vk6= 0. Clearly, by (7), e−tN v∈ V for
all t≥0 and so
ek0te−tN v=X
k≥k0
e−(k−k0)tvkek=vk0ek0+X
k>k0
e−(k−k0)tvkek∈ V
for all t≥0. Taking the limit at t→+∞, we find ek0∈ V. Acting on ek0
with operators aand a†we can immediately show that every vector ekof the
basis belongs to Vand the proof is complete.
4. Irreducibility: The Case of a Single Noise Operator L
In this section we study the case where there is a single operator
L=va +ua†with v6= 0 or u6= 0 .
This case is much more entangled. We begin by considering the algebraic
aspect of the problem disregarding, for the moment, domain issues that will
be considered later.
We are looking for common invariant subspaces for the operators Gand
Land so also for the commutator [L, G]. A straightforward computation
yields
−2 [L, G]=[L, L∗L+ 2iH] (8)
= [L, L∗]L+ 2i (vΩ−uκ)a−2i (uΩ−vκ)a†+ 2i vζ −uζ.
Thus the candidate subspace must be invariant for the operators
G=−1
2L∗L−iH , L =va +ua†,e
L= (vΩ−uκ)a+ (vκ −uΩ) a†.
If the operators Land e
Lare linearly independent, namely
det vΩ−uκ vκ −uΩ
v u 6= 0 ,(9)
then the candidate subspace must be invariant for aand a†and so it should
be trivial as in the case of two Kraus operators L.
In the sequel, we prove that under condition (9), which is clearly a
H¨ormander-type iterated commutator condition the QMS is irreducible. Oth-
erwise, we will see that irreducibility does not hold.
It is worth noticing here that a similar condition appears also in bilinear
control (see [11], Definition 3.6 (ii) p. 102, weak ad-condition) As a matter of
fact, if, starting from any initial non-zero vector ξ0∈hwith time evolution
one can reach a total set of vectors in hvarying the control parameter z∈C
in the differential equation ˙
ξt=Gξt+zLξt, then irreducibility holds.
2150001-11
J. Agredo, F. Fagnola, and D. Poletti
LEMMA 1 Suppose |v| 6=|u|. Then Dom(G0) = Dom(N) = Dom(G).
We defer the proof to Appendix B.
PROPOSITION 1 Suppose that condition (9) holds and, moreover, |v| 6=|u|.
Then the Gaussian QMS with
L=va +ua†, H = Ω a†a+κ
2a†2+κ
2a2+ζ
2a†+ζ
2a
is irreducible.
Proof. Knowing that Dom(G) = Dom(N) the proof essentially follows the
line of that of Theorem 5.
Let V(V 6={0}) be a subspace of hwhich is invariant for the operators
Ptand L(Dom(G)∩ V) = L(Dom(N)∩ V)⊆ V for `= 1,2. Moreover,
since L(Dom(Nm)) ⊆Dom(Nm−1/2) for all m≥1/2 and G(Dom(Nm)) ⊆
Dom(Nm−1) for all m≥1, we have also [G, L]Dom(N3/2)∩ V⊆ V and
e
LDom(N3/2)∩ V⊆ V. However, the commutator [G, L] is a first order
polynomial in a, a†, therefore the previous inclusions can be extended to
Dom(N1/2)∩ V.
By the linear independence of Land e
L, we can now follow the argument
of the proof of Theorem 5, with L2=e
L.
We study separately situations in which (9) does not hold distinguishing three
cases.
4.1. The case Lof annihilation type
We first consider the case where (9) does not hold and |v|>|u|. Act with
the unitary squeeze operator S=e(za†2−za2)/2(z6= 0, z=eiϕswith s=|z|)
so that
S∗aS = cosh(s)a+eiϕsinh(s)a†, S∗a†S= cosh(s)a†+e−iϕsinh(s)a .
Then
S∗LS =vcosh(s) + e−iϕusinh(s)a+ucosh(s) + eiϕvsinh(s)a†(10)
and, by first choosing a ϕsuch that uand eiϕvhave the same phase, and an
ssuch that
|u|cosh(s) + |v|sinh(s) = 0 ⇐⇒ tanh(s) = −|u|
|v|
we can assume that Lis a strictly positive multiple (multiplying Lby a phase
does not change the GKSL representation) of the annihilation operator, i.e.,
u= 0.
2150001-12
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Of course also Ω, κ, ζ change to Ω0, κ0, ζ0
Ω0= Ω cosh2(s) + sinh2(s)+ 2 sinh(s) cosh(s)<(e−iϕκ)
κ0=κcosh2(s) + κ e2iϕsinh2(s) + 2Ω eiϕcosh(s) sinh(s)
ζ0=ζcosh(s) + ζeiϕsinh(s)
and condition (9) does not hold if and only if vκ0= 0, i.e., by v6= 0, κ0= 0
and (up to an irrelevant multiple of the identity operator in H)
L=v0a , H = Ω0a†a+ζ0
2a†+ζ0
2a , G =−|v0|2
2+iΩ0a†a−i
2(ζ0a†+ζ0a),
where v0= (|v|2− |u|2) cosh(s)/|v|, up to a phase factor. Dropping the 0to
simplify the notation, now we apply formula (4) with
Zz =−(|v|2/2 + iΩ)z , Cz =|v|2z .
Computing esZ z=e−(|v|2/2−iΩ)szand
t
Z
0<esZ z CesZ zds=|z|2
t
Z
0|v|2e−s|v|2ds=|z|21−e−t|v|2
t
Z
0<ζesZ zds=<ζz
|v|2/2−iΩ 1−e−t(|v|2/2−iΩ).
It follows that, for all g, f ∈C,
lim
t→+∞tr(|e(f)ihe(g)|Tt(W(z))
=e−|z|2/2+i<(ζz/(|v|2/2−iΩ))lim
t→+∞e(g), W (etZ z)e(f)
=e−|z|2/2+2i=(iζz/(|v|2−2iΩ))egf .
Noting that, for all µ∈C
e−|µ|2he(µ), W (z)e(µ)i=e−|z|2/2+2i=(µz),
defining µ= iζ/(|v|2+ 2iΩ) we find
lim
t→+∞tr(|e(f)ihe(g)|Tt(W(z)) = egf e−|µ|2he(µ), W (z)e(µ)i.
In particular, e−|µ|2|e(µ)ihe(µ)|is a pure invariant state and the QMS is
not irreducible. Moreover, since linear combinations of linear functionals
|e(f)ihe(g)|are dense in the Banach space of trace class operators by totality
of exponential vectors, that the above identity also proves that any initial
state converges in trace norm to this pure invariant state.
2150001-13
J. Agredo, F. Fagnola, and D. Poletti
PROPOSITION 2 The Gaussian QMS with GKSL generator with only one
Kraus operator L=va +ua†,|v|>|u|and Hamiltonian Has in (3) is
irreducible if and only if condition (9) holds. If it is not irreducible, it has a
unique invariant state e−|µ|2|e(µ)ihe(µ)|(pure) and all initial state converges
to it in trace norm.
Clearly, after our squeeze transformation µ= iζ0/(|v0|2+ 2iΩ0).
4.2. The case Lof creation type
We consider the case where (9) does not hold and |v|<|u|. First choosing a
φsuch that uand eiφvhave the same phase, and then θsuch that tanh(θ) =
|v|/|u|in (10) we can assume v= 0 and Lmultiple of the creation operator.
Parameters Ω, κ, ζ are transformed to Ω0, κ0, ζ 0and (9) does not hold if and
only if κ0= 0. In this way the given QMS is transformed to the unitarily
equivalent QMS generated by
L=u0a†, H = Ω0a†a+ζ0
2a†+ζ0
2a , G =−|u0|2
2+iΩ0aa†−i
2(ζ0a†+ζ0a),
where u0= (|u|2−|v|2) cosh(θ)/|u|up to a phase factor. In the sequel we drop
the 0to simplify the notation. Let Vbe the range of a nonzero subharmonic
projection p. Since, by Lemma 1 the operators Gand Nhave the same
domain, by Theorem 3 we have G(Dom(N)∩ V)⊆ V L(Dom(N)∩ V)⊆ V.
Adding to Ga suitable multiple of Lwe find the operator
e
G=−|u|2
2+ iΩa a†+ηa +ηa†+|η|21l
=−|u|2
2+ iΩW(η)∗a a†W(η),
where η= iζ/(|u|2−2iΩ) such that e
G(Dom(N)∩ V)⊆ V. This property,
together with L(Dom(N)∩ V)⊆ V, is clearly equivalent to G, L invariance.
Let w∈ V with expansion w=Pk≥k0wkW(−η)ekwhere k0is the mini-
mum kfor which wk6= 0. Since e
Gis a multiple of the number operator with
strictly negative real part, arguing as in the last part of the proof of Theorem
5, we can show that W(−η)ek0∈ V. As a consequence, by the commutation
a†W(−η) = W(−η)(a†−η1l),
LW (−η)ek0=u W (−η)(a†−η1l)ek0
=upk0+ 1 W(−η)ek0+1 −u η W (−η)ek0∈ V .
Applying Lwe can show inductively that, for all k0≥0, the linear space
generated by vectors W(−η)ekwith k≥k0is an invariant subspace deter-
mining a subharmonic projection and, in this case, the QMS associated with
G, L is not irreducible.
2150001-14
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
4.3. The case Lquadrature (selfadjoint)
We consider the case where (9) does not hold and |v|=|u|so that, v=reiα,
u=reiα0with r > 0 and
L=re−iαa+reiα0a†=rei(α0−α)/2e−i(α0+α)/2a+rei(α0+α)/2a†.
Therefore, multiplying Lby a phase e−i(α0−α)/2and putting α0+α= 2θwe
get the selfadjoint operator L
L=r(e−iθa+eiθa†)
which is a positive multiple of a quadrature. We could also reduce ourselves
to the case where θis zero by applying a unitary transformation eiθN on
Γ(C), however we prefer to keep the parameter θto highlight the relationship
between the phase θin the operator Land another phase of the coefficients
κof the Hamiltonian H.
Indeed, in a similar way, putting κ=|κ|e2iφwe can write
H= Ω a†a+|κ|
2e2iφa†2+e−2iφa2+ζa +ζa†.
Considering quadratures with angle θgiven by the selfadjoint
qθ=e−iθa+eiθa†/√2
and noting that
a†=e−iθqθ−iqθ+π/2/√2a=eiθqθ+ iqθ+π/2/√2 (11)
a†a+aa†=q2
θ+q2
θ+π/2(12)
we can write Has
H=Ω + |κ|cos(2(φ−θ))
2q2
θ+Ω− |κ|cos(2(φ−θ))
2q2
θ+π/2(13)
+|κ|
2sin(2(φ−θ)) qθqθ+π/2+qθ+π/2qθ+ (ζ a +ζa†)−Ω
21l .
We can immediately see that (9) does not hold if and only if
Ω = |κ|cos(2(φ−θ)) (14)
the quadratic term q2
θ+π/2in Hvanishes and the Abelian algebra generated
by the position operator qθis invariant. Indeed, for all smooth function
2150001-15
J. Agredo, F. Fagnola, and D. Poletti
f:R→C, we have [L, f (qθ)] = [L∗, f (qθ)] = 0 and by the identities
qθ+π/2, f (qθ)=hid
dqθ
, f (qθ)i= if0(qθ)
[a, f (qθ)] = eiθ
√2qθ−iqθ+π/2, f (qθ)=eiθ
√2hd
dqθ
, f (qθ)i
=eiθ
√2f0(qθ)
[a†, f (qθ)] = e−iθ
√2qθ+ iqθ+π/2, f (qθ)=e−iθ
√2h−d
dqθ
, f (qθ)i
=−e−iθ
√2f0(qθ).
As a result, we find
L(f(qθ)) = i [H, f (qθ)] = =(ζe−iθ)/√2− |κ|sin(2(θ−φ))qθf0(qθ).
Note that, if the quadratic term q2
θ+π/2in Hdoes not vanish, then we can
not get the same conclusion.
This is the generator of a deterministic translation process with drift
(in the generic case where |κ|sin(2(θ−φ)) 6= 0) towards the point x∞:=
=(ζe−iθ)/(√2|κ|sin(2(θ−φ))) (Fig. 1 below).
Fig. 1: Deterministic translation process on the algebra generated by qθ.
The invariant density of the classical process is clearly δx∞which does
not induce a faithful normal state on B(h). However this insight turns out to
be useful to demonstrate that the QMS we are considering in this subsection
is not irreducible if (14) holds. For all c > 0 consider the projection
x7−→ 1[x∞−c,x∞+c](x)
which is a candidate subharmonic projection because the classical process,
starting from a point in the interval [x∞−c, x∞+c] does not exit for all
positive times.
To prove that this projection is indeed subharmonic, consider mollifier
ϕ, namely a C∞function ϕ:R→R+with support in the interval [−1,1],
2150001-16
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
RRϕ(x)dx= 1 and lim→0ϕ(x) = lim→0−1ϕ(x/) = δ0and, for all < c
define
f(x) =
x
Z
−∞
(ϕ(y−(x∞−c)) −ϕ(y−(x∞+c))) dy .
Note that, since RRϕ(x)dx= 1 for all > 0 we have f(x) = 0 for |x−x∞|>
c+,f(x) = 1 for |x−x∞| ≤ c−and f0
(x)≥0 for x∞−c−<x<x∞−c+,
f0
(x)≤0 for x∞+c−<x<x∞+c+. It follows that the multiplication
operator by f(qθ), which belongs to the domain of the Lindbladian Lbecause
L(f(qθ)) is bounded satisfies L(f(qθ)) ≥0 and so
Tt(f(qθ)) ≥f(qθ)
for all t≥0. Taking the limit as goes to 0, fconverges to the pro-
jection 1[x∞−c,x∞+c]in L2and almost surely, therefore Tt(1[x∞−c,x∞+c])≥
1[x∞−c,x∞+c]for all t≥0 and the QMS is not irreducible.
A similar argument applies in the case where sin(2(θ−φ)) = 0 and
x∞= +∞(resp. x∞=−∞) if =(ζe−iθ)>0 (resp. =(ζe−iθ)<0) with
projections of the form 1[c,+∞[(resp. 1]−∞,c]).
We now consider the case where |u|=|v|and condition (9) holds, namely
Ω6=|κ|cos(2(θ−φ)) and show that the QMS is irreducible. To this end
we need the following result also showing that irreducibility is equivalent to
coercivity of G2
0+H2+g2
l1l, for some constant g2
l, with respect to the graph
norm of the number operator N= (q2
θ+q2
θ+π/2−1)/2.
Intuitively, looking at formula (13), one sees that the coefficient of q2
θ+π/2
is non-zero if Ω 6=|κ|cos(2(θ−φ)). Therefore, computing H2, the coefficient
of q4
θ+π/2is non-zero. The coefficient of q4
θmay vanish but one gets an
additional term r4by addition of G2
0and strict positivity of leading terms is
restored.
THEOREM 6 If condition (9) holds, namely Ω6=|κ|cos(2(φ−θ)), there
exist constants g2>0,g2
l≥0such that
G2
0+H2≥g2(q2
θ+q2
θ+π/2)2−g2
l1l .(15)
In particular Dom(G) = Dom(N).
Proof. In this proof only, to reduce the clutter of the notation, we denote
qθby q,qθ+π/2by p,c:= cos(2(φ−θ)), s:= sin(2(φ−θ)) and by {·,·} the
anticommutator.
As a first step note that, once we show that G2
0+H2≥g2
0(q2
θ+q2
θ+π/2)2+
l.o.t. for some constant g2
0>0 then, reducing the constant g0if necessary,
we can get the conclusion. Indeed, if the lower order term is, for instance,
2150001-17
J. Agredo, F. Fagnola, and D. Poletti
{a, q2+p2}for all ξ∈Dom(N2) by the Schwartz and Young inequalities, we
haveξ, {a, q2+p2}ξ=ha†ξ , (q2+p2)ξi+(q2+p2)ξ, a ξ
≥ −ka†ξk ·
(q2+p2)ξ
− ka ξk ·
(q2+p2)ξ
≥ −
(q2+p2)ξ
2−−1(ka ξk2+ka†ξk2)
=−ξ, (q2+p2)2ξ−−1ξ, (q2+p2)ξ
for all > 0. Now, again by the Schwartz and Young inequalities we have
also
−−1ξ, (q2+p2)ξ≥ −−1kξk ·
(q2+p2)ξ
≥ −
(q2+p2)ξ
2−−3kξk2.
Therefore we find the inequality
ξ, {a, q2+p2}ξ≥ −2ξ, (q2+p2)2ξ−−3kξk2
and, choosing small enough, we can reduce the constant g2in (15), increase
g2
land get the claimed inequality. We can proceed in a similar way if there
are more lower order terms.
It is now clear that we can assume that G2
0+H2is a fourth order ho-
mogenous polynomial in p, q, or, in an equivalent way, we can proceed as if
Hhad no terms of order 1 or 0. In this case the square of 2His
(2H)2= (Ω + |κ|c)2q4+ (Ω − |κ|c)2p4+ (Ω + |κ|c) (Ω − |κ|c){q2, p2}
+|κ|2s2{q, p}2+ (Ω + |κ|c)|κ|sq2,{q, p}
+ (Ω − |κ|c)|κ|sp2,{q , p}
and write (2H)2as
q2,{q, p}, p2
(Ω + |κ|c)2(Ω + |κ|c)|κ|s(Ω + |κ|c) (Ω − |κ|c)
(Ω + |κ|c)|κ|s|κ|2s2(Ω − |κ|c)|κ|s
(Ω + |κ|c) (Ω − |κ|c) (Ω − |κ|c)|κ|s(Ω − |κ|c)2
q2
{q, p}
p2
.
We now apply Lemma 8 Appendix C on a 3 ×3 matrix as above with λ=
Ω− |κ|c, µ = Ω + |κ|c, x =|κ|s. Since L=√2rq and G0=−r2q2, the
operator (2G0)2+ (2H)2is associated with a 3 ×3 matrix as in Lemma 8
therefore is bigger than (r4becomes 4r4)
4r4q4− { p, q }2/2 + {p2, q2}+λ2w q4+ l.o.t.
Note that {p2, q2}−{p, q }2/2 = −(3/2)1l and {p2, q2} ≤ p4+q4which
implies
4(G2
0+H2)≥4r4q4+λ2w q4+ l.o.t.
≥min{2r4, λ2w/2}q4+{p2, q2}+p4+ l.o.t.
=min{2r4, λ2w/2}q2+p22+ l.o.t.
2150001-18
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
The above inequality together with (33) implies existence of constants, g, g0>
0 such that
kNξk2≤gkGξk2+g0kξk2
for all ξfinite linear combination of vectors enof the c.o.n.b. Therefore
Dom(G)⊆Dom(N). The other inclusion is trivial and the proof is complete.
Let Vbe the range of a subharmonic projections. By the previous arguments
based on a, a†invariance of V ∩ Dom(N) and Dom(G) = Dom(N) as in the
proof of Theorem 5 we can now prove the following
THEOREM 7 Let Tbe the QMS with generator in a generalized GKSL form
associated with a single Kraus operator L=va +ua†and Has in (3). The
following are equivalent:
(1) Operators Land [H, L]are linearly independent, i.e., 2Ω vu 6=v2κ+
u2κ,
(2) Tis irreducible.
Proof. (1) ⇒(2) If |u| 6=|v|, the conclusion follows from Proposition 1. If
|u|=|v|, we know from Theorem 6 that Dom(G) = Dom(N) therefore the
proof of Proposition 1 goes through again and shows that Tis irreducible.
(2) ⇒(1) We showed, in Sect. 4.1 for |u|<|v|, in Sect. 4.2 for |u|>|v|, and
in Sect. 4.3 for |u|=|v|, that if condition (1) does not hold then the QMS T
is not irreducible.
Solution to the irreducibility problem is summarized by the following decision
tree.
5. Gaussian States
As a preliminary to the study of invariant states, in this section, we recall
some basic properties of Gaussian states of a one-dimensional CCR algebra.
2150001-19
J. Agredo, F. Fagnola, and D. Poletti
DEFINITION 2 A density matrix ρis called a quantum gaussian state if
there exist ω∈Cand a real linear, symmetric, invertible operator Ssuch
that
ˆρ(z) = exp −1
2<(z Sz)−i<(ωz)∀z∈C.(16)
In that case ωis said to be the mean vector and Sthe covariance operator
and we will denote it also with ρ(ω,S).
This notation is well posed since there is a bijection between density matrices
and characteristic functions.
Let Sbe a real linear operator on Cand z=x+ iy∈C. In the following
it will be useful to identify them with a real linear operator Sacting on R2
and a vector zin R2that will be denoted with characters in boldface for the
sake of clarity. Namely, z= (x, y) and, if Sz =s1z+s2zfor every z∈C, we
have
Sz =<s1+<s2=s2− =s1
=s1+=s2<s1− <s2x
y.
Vice versa, given a linear operator on R2
S=S11 S12
S21 S22 ,
we can induce a real linear operator Son Cvia
Sz =S11 +S22
2+ iS21 −S12
2z+S11 −S22
2+ iS12 +S21
2z.
In the following we will also use Jas the linear linear operator corresponding
to the multiplication by −i, namely
Jz =−iz , J=0 1
−1 0 .
Eventually we will denote the adjoint of Swith respect to the real scalar
product (z, w)→ <(zw) with ST. This operator is explicitly given by
STz=s1z+s2z ,
while STis given by the usual matrix transposition of S.
Remark 2 For a generic real linear, symmetric, invertible operator Sto be a
suitable covariance operator of a gaussian state it also needs to satisfy
S−iJ≥0,(17)
where Sand Jare now intended as complex linear operator on C2(see
[24, Theorem 3.1] with a little warning: Jthere is the multiplication by i).
Therefore positivity is evaluated with respect to the usual complex inner
product.
2150001-20
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
LEMMA 2 The real linear operators Cand Zgiven by (5), (6) satisfy
C+ i ZTJ+J Z≥0,C≥0,
where the first inequality is intended with respect to the complex scalar product
on C2. Moreover, the first inequality holds strictly if and only if there are
exactly two linear independent Kraus operator L1, L2. The second one is
strict if and only if the parameter
γ=1
2
2
X
`=1 |v`|2− |u`|2
is non-zero.
Proof. Let us begin by observing that
C=X
`|u`+v`|22=(v`u`)
2=(v`u`)|u`−v`|2,Z=−γ− =(κ)<(κ)−Ω
<(κ)+Ω −γ+=(κ).
(18)
Now a straightforward computation leads to
C+ i ZTJ+J Z= 2 "P`|u`+v`|2
2−iγ+P`=(v`u`)
iγ+P`=(v`u`)P`|u`−v`|2
2#=: 2 Q
hence positivity of C+ i ZTJ+J Zis equivalent to positivity of Q. One
has tr(Q)>0, while
det Q=1
4X
`|u`+v`|2X
`|u`−v`|2−X
`=(u`v`)2−γ2.
Now we can use |u`±v`|2=|u`|2+|v`|2±2<(u`v`) in order to obtain
det Q=X
`|u`|2X
`|v`|2−X
`
u`v`
2
which is positive by the Cauchy-Schwarz inequality. In the case where there is
only a Kraus operator L1clearly det (Q) = 0. Conversely, if det (Q) = 0 then
the Cauchy-Schwarz inequality becomes an equality, therefore we can find λ∈
Csuch that u`=λv`for every `= 1,2 which contradicts linear independence
of L1and L2. The analysis of the first inequality is now complete.
2150001-21
J. Agredo, F. Fagnola, and D. Poletti
For the second one observe that tr(C)≥0 and, with similar computa-
tions,
det (C) = X
`|u`|2+|v`|22−4X
`
u`v`
2
≥X
`|u`|2+|v`|22−4X
`|u`|2X
`|v`|2
=X
`|u`|2− |v`|22= 4γ2≥0.
This completes the proof.
6. Invariant States
In this section we characterize Gaussian QMS with normal invariant states
in terms of the parameters in the model. We begin by the explicit formula
for the action of the predual semigroup on Gaussian states.
PROPOSITION 3 Let (Tt)t≥0be the quantum Markov Semigroup with GKSL
generator associated with H, L1, L2as in (2), (3) and let (T∗t)t≥0be its pre-
dual semigroup. If ρ=ρ(ω0,S0)is a gaussian state then ρt:= T∗t(ρ)is still a
Gaussian state for every t≥0with mean vector ωtand covariance operator
Stgiven by
ωt=etZTω0−
t
Z
0
esZTζds(19)
St=etZTS0etZ +
t
Z
0
esZTCesZ ds . (20)
Proof. Applying the explicit formula (4) of Theorem 2 we can write
ˆρt(z) = tr(ρTt(W(z)))
= exp
−1
2<*z,
t
Z
0
esZTCesZ zds+−1
2<Dz, etZ TS0etZ zE
×exp
i<*t
Z
0
esZTζds, z+−i<DetZTω0, zE
.
2150001-22
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Comparing the previous equation with (16) we find (20) and (19). Now for
Stto be a suitable covariance matrix it should hold St−iJ≥0. Indeed,
using S0−iJ≥0 and Lemma 2, one gets
St−iJ≥
t
Z
0
esZTCesZds +etZTiJetZ−iJ
=
t
Z
0
esZTC+ i ZTJ+J ZesZds≥0.
Note that all the operators in the previous inequality were considered as
complex linear and therefore commutation with i was legit.
Remark 3 One could extend Proposition 3 proving that QMSs with general-
ized GKSL generator (1) with Hamiltonian Hgiven by (3) and two, one or
none operator L`linear in a, a†as (2) form the most general class of weakly∗
continuous semigroups of completely positive, identity preserving maps on
B(Γ(C)) that preserve Gaussian states. We omit the proof because, albeit
interesting, on one hand it would be too long, and, on the other hand, it is
just a slight extension of existing results. Here we limit ourselves to mention
some bibliographic references. The proof can be done in two steps by char-
acterizing, first completely positive maps preserving Gaussian states, then
semigroups composed by such maps. The first step is quite lengthy because
it involves the extension of Theorem 4.5 [15], proved for automorphisms, to
general completely positive maps. This proof occupies the major part of the
paper, accounting for Lemmas and accessory results. An extension of The-
orem 4.5 would not be too difficult but lengthy. Indeed the authors of [8],
at the very end of the paper, claim such an extension is possible, without
spelling out the details.
We now turn our attention to finding Gaussian invariant states.
THEOREM 8 Let (Tt)t≥0be the QMS with GKSL generator associated with
H, L1, L2as in (2), (3) or with Hand a single Kraus operator. If γ > 0and
γ2+ Ω2− |κ|2>0the Gaussian state ρ=ρ(ω,S)with
ω= (ZT)−1ζ=(−γ+ iΩ)ζ−iκζ
γ2+ Ω2− |κ|2, S =
∞
Z
0
esZTCesZ ds(21)
is the unique normal invariant state for the semigroup. Moreover, for all
initial state ρ0
lim
t→∞
1
t
t
Z
0T∗s(ρ0)ds=ρ
in trace norm.
2150001-23
J. Agredo, F. Fagnola, and D. Poletti
Proof. First note that, since γ2+ Ω2−κ2>0, the matrix Zin (18) has
eigenvalues with strictly negative real part, therefore the integral in (21) is
well-defined.
We now check that ρis an invariant state. Proposition 3 implies that
ρt=T∗t(ρ) is still a Gaussian state with mean vector and covariance matrix
given by equations (19) and (20). The state ρis invariant if and only if
ωt=ωand St=Sfor every t≥0 that means
t
Z
0
esZT(ZTω−ζ)ds= 0 ,
t
Z
0
esZT(C+ZTS+SZ )esZds= 0
for all t≥0. Since both esZTand esZare invertible, the invariance of ρis
equivalent to
ζ=ZTω,ZTS+SZ =−C.(22)
Conditions on the parameters of the semigroup imply the existence of a pair
(ω,S) satisfying (22). Indeed γ2+ Ω2− |κ|26= 0 implies invertibility of ZT,
which leads to ω= (ZT)−1ζ. Furthermore
ZTS+SZ =
∞
Z
0ZTesZTCesZ+esZTCesZZds
=
∞
Z
0d
dsesZTCesZds
=hesZTCesZi∞
0=−C.
Moreover we can show as in the proof of Proposition 3 that Sis a suitable
covariance matrix by noting that
S−iJ=
∞
Z
0
esZT(C+ i ZTJ+J Z)esZds ,
which exists, since Zhas only eigenvalues with negative real part, and is
positive thanks to Lemma 2.
Uniqueness follows from irreducibility by standard results on QMS with
faithful normal invariant states (see, e.g., [16] Theorem 1 and Lemma 1),
otherwise it follows from Proposition 2 since γ > 0. Convergence towards
the invariant state follows in a similar way either from known results on QMS
with faithful normal invariant states (see, e.g., [17, Theorem 2.1]), or from
Proposition 2 since γ > 0.
2150001-24
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Fig. 2: Parameter region I(shaded) of QMS with Gaussian invariant states.
Remark 4 Condition γ > 0 indicates an overall higher rate of transitions
to lower-level states. In order to interpret the other condition we begin
by recalling that the Hamiltonian Hhas discrete spectrum and the QMS
generated by i[H, ·] has normal invariant states if and only if |κ|2<Ω2.
In the case where Ω2− |κ|2<0 the Hamiltonian Hhas only continuous
spectrum and the additional condition γ2>|κ|2−Ω2appears. This means
that transitions to lower-level states must be stronger to compensate the
effect of transitions induced by the Hamiltonian without eigenstates.
Theorem 8 shows that a faithful normal Gaussian invariant state exists and
is unique for all parameters (γ, Ω2− |κ|2) lying in the open shaded region
denoted by I(see Fig. 2). We will now show that a normal invariant state,
whether Gaussian or not, does not exist for any choice of parameters (γ, Ω2−
|κ|2) lying outside of the region I.
Equations (4) and (21) suggest that the quantity <(esZz CesZ z) plays an
important role in the existence of invariant states. Therefore we begin by the
following two Lemmas, investigating the asymptotic behaviour etZzand the
convergence of the integral (21).
LEMMA 3 For all choices of parameters γ, Ω, κ such that (γ, Ω2− |κ|2)falls
outside the region I\ {(0,0)}there exist V+, a vector subspace of R2, such
that etZzdiverges as t→ ∞ for every z∈V+\ {0}.
Proof. Recall that the matrix Zis given by (18). We can divide the re-
maining set of parameters in four subsets:
1. γ < 0 and Ω2≥ |κ|2: eigenvalues of Zare −γ±iqΩ2− |κ|2both with
strictly positive real part,
2. γ≤0 and Ω2<|κ|2: eigenvalues of Zare −γ−p|κ|2−Ω2<−γ+
p|κ|2−Ω2. At least −γ+p|κ|2−Ω2is strictly positive,
2150001-25
J. Agredo, F. Fagnola, and D. Poletti
3. γ > 0 and γ2+ Ω2− |κ|2<0 so that Ω2− |κ|2<0 : eigenvalues of Z
are −γ±p|κ|2−Ω2. Only the biggest eigenvalue −γ+p|κ|2−Ω2is
strictly positive,
4. γ= 0 and Ω = ±|κ|: the only eigenvalue of Zis 0.
In each of the first three cases there is an eigenvalue λ+with positive real part
and it is sufficient to choose as V+the subspace generated by an eigenvector
of λ+. Indeed if z0is an eigenvector of λ+we have etZz0=et<λ+|z0|.
In the fourth case we have Z6= 0 but Z2= 0. Hence etZ= 1 + tZand
there exists z0∈R2such that Zz06= 0. Therefore
etZz0=|z0+tZz0| ≥ t|Z z0|
and etZz0diverges as t→ ∞. It is then sufficient to choose V+generated
by z0.
LEMMA 4 For all choices of parameters γ, Ω, κ such that (γ , Ω2− |κ|2)be-
longs to the boundary of Iexcept the origin (0,0) there exists a vector subspace
V+of R2such that for every z∈V+\ {0}the integral
t
Z
0
zTesZTCesZzds (23)
diverges as t→ ∞.
Proof. Consider first the case γ > 0, γ=q|κ|2−Ω2.
Since γ2+Ω2−|κ|2= 0, Zhas 0 as an eigenvalue. Let z0be an associated
eigenvector and fix V+as the vector subspace generated by z0. For every
z∈V+\ {0}we have zTetZCetZz=zTC z. This quantity does not depend
on tand is also strictly positive, since Cis invertible thanks to Lemma 2.
Therefore its integral (23) diverges as t→ ∞.
Consider now the case γ= 0, Ω2>|κ|2.
For every such choice of the parameters Zhas two distinct eigenvalues,
namely λ±=±iqΩ2− |κ|2=±iδand it can be diagonalized. Let v+,v−
be two eigenvectors corresponding to λ±respectively. If z=w−v++w+v−
we have etZz=w−eitδ v++w+e−itδ v−. Now consider the quantity fz(t) :=
zTetZTCetZzwhich is non negative and periodic, from the above considera-
tions. We will show fz(t) cannot be identically zero for every z∈R2. Indeed
if it were the case then fz(0) = 0 for every z∈R2and thus z∈ker C
for every z∈R2. However ker Cis one-dimensional and there must exists
z0∈R2such that fz0(t) is not identically zero. Therefore, if V+is defined to
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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
be the vector subspace generated by z0, the integral (23) diverges, since its
argument is a non-negative, periodic function which is not identically zero.
The previous two Lemmas can now be applied to prove the non-existence of
invariant states, for some choices of parameters γ, Ω, κ.
PROPOSITION 4 If
w*-lim
t→∞ Tt(W(z)) = 0 (24)
(in weak∗operator topology) for all zin a vector subspace of R2except (0,0),
then Thas no normal invariant state. In particular, for any choice of the
parameters γ, Ω, κ such that (γ, Ω2− |κ|2)falls outside of the region Ia
normal invariant states for the QMS Tdoes not exist.
Proof. If ρis a normal invariant state then for every z6= 0 such that zis in
the subspace of R2of the hypothesis
tr(ρW (z)) = tr(ρTt(W(z)))
for all t≥0. Taking the limit as t→ ∞, by Lemma 3 and Lemma 4 and
the explicit formula (4), we get tr(ρW (z)) = 0. This is a contradiction since
z→tr(ρW (z)) is continuous and tr(ρW (0)) = 1.
Observe now that if we are also outside of the region I\ {(0,0)}we can
use Lemma 3 and fix z0∈V+\ {0}. Letf, g ∈C, thanks to equation (4) we
have
he(g), W (etZ z0)e(f)i= exp n−etZ z02
2−etZ z0f−¯getZ z0+ ¯gfo.
Since etZ z0diverges as t→ ∞, we have that W(etZ z0) converges weakly to
0. Moreover the Weyl operators are unitary, hence the set {W(etZ z0) : t∈R}
is bounded and the weak topology coincides with the weak* one. Therefore
W(etZ z0) converges weakly* to 0 and
|tr(ρTt(W(z0)))|=|ct(z0)|tr(ρW (etZ z0))≤tr(ρW (etZ z0)),
where ct(z0) is the constant multiplying the Weyl operator in (4). This means
Tt(W(z0)) converges to 0 in the weak* topology for every z0∈V+\{0}. Hence
there can be no normal invariant states.
Suppose we are now in the region ∂I\{0}. Let z0∈V+, whose existence
is given by Lemma 4. Thanks to (4) one has
|tr(ρTt(W(z0)))| ≤ exp n−1
2
t
Z
0
z0TesZTCesZz0dso,
2150001-27
J. Agredo, F. Fagnola, and D. Poletti
since |tr(ρW (z))| ≤ 1 for every z∈C. Letting t→ ∞ one has Tt(W(z0)) →0
in the weak* topology for every z0∈V+\ {0}. Hence also in this case there
are no normal invariant states.
Summarizing we proved the following complement to Theorem 8.
THEOREM 9 Let (Tt)t≥0be the QMS with GKSL generator associated with
H, L1, L2as in (2), (3) or with Hand a single Kraus operator. The QMS
Thas a normal invariant state if and only if γ > 0and γ2+ Ω2− |κ|2>0.
The normal invariant state is also unique.
Moreover, we also showed that the above invariant states are either faithful
or pure.
PROPOSITION 5 The invariant state given by Theorem 8 is pure if and
only if Ω2− |κ|2>0, there is a single Kraus operator L= ¯va +ua†and
=(uv)
|u−¯v|2==(κ)
2(Ω − <(κ)) ,γ
pΩ2− |κ|2=|u−¯v|2
2|<(κ)−Ω|.(25)
In all the other cases it is faithful.
Proof. A Gaussian state is faithful if and only if S−iJ>0 otherwise it is
pure (see [24, Sect. 2] and also [25]). As in the proof of Theorem 8 we can
use
S−iJ=
∞
Z
0
esZTC+ i ZTJ+J ZesZds , (26)
and study its kernel. Clearly, if C+ i ZTJ+JZ >0, also S−iJ>0,
since esZis invertible. This happens whenever there are two Kraus operators,
thanks to Lemma 2. So the state can be pure only if there is a single Kraus
operator. We restrict ourselves to this case. Now the kernel of S−iJis
non-trivial if and only if the argument of the integral (26). has a nontrivial
kernel. This happens whenever at least one of the eigenvectors of Zbelongs
to ker(C+i(ZTJ+J Z )). Indeed suppose there is z0∈R2\{(0,0)}such that
(C+i(ZTJ+J Z ))etZz0= 0 for all t≥0. Since ker(C+i(ZTJ+JZ)) is one-
dimensional suppose it is generated by v0∈R2, we have then etZz0=λtv0
for some λt∈R. In particular λs+tv0=e(t+s)Zz0=λsetZv0which means
v0is an eigenvector for etZand therefore it is also an eigenvector for Z. The
converse implication is trivial.
Suppose Ω 6=<(κ), via explicit calculations the eigenvectors of Zare
v±= (<(κ)−Ω,=(κ)±p|κ|2−Ω2). We have
(C+ i(ZTJ+JZ))v±
=
|u+ ¯v|2(<(κ)−Ω) + 2 =(κ)±p|κ|2−Ω2(=(uv)−iγ)
=(κ)±p|κ|2−Ω2|u−¯v|2+ 2 (<(κ)−Ω) (=(uv)+iγ)
.(27)
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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Now if this were the null vector the imaginary part of the second entry
should be zero. If |κ|2−Ω2≥0 this would imply γ(<(κ)−Ω) = 0 which is
impossible since γ > 0 and we supposed <(κ)6= Ω. Therefore, for (27) to be
zero, |κ|2−Ω2<0 and, letting δ:= pΩ2− |κ|2, this is equivalent to
(|u+ ¯v|2(<(κ)−Ω) + 2=(κ)=(uv)±2δγ =±δ=(uv)−γ=(κ) = 0
=(κ)|u−¯v|2+ 2(<(κ)−Ω)=(uv) = ±δ|u−¯v|2+ 2γ(<(κ)−Ω) = 0 .
Those equation are in turn equivalent to
=(uv)
|u−¯v|2=−=(κ)
2(<(κ)−Ω) ,γ
δ=∓|u−¯v|2
2(<(κ)−Ω) ,
that lead to , having chosen v±in order to have γ > 0. Vice versa if u, v, κ, Ω
satisfy equations then one of v±is in ker(C+ i(ZTJ+JZ)).
Suppose now Ω = <(κ). The proper eigenvectors of Zare v1= (0,1),v2=
(=(κ),<(κ)) if =(κ)6= 0 or v1= (0,1),v3= (1,0) if =(κ) = 0. One has
=((C+ i(ZTJ+JZ))v1) = −2γ
0,
=((C+ i(ZTJ+JZ))v2) = −2γ<(κ)
2γ=(κ),
=((C+ i(ZTJ+JZ))v3) = 0
2γ,
that cannot both be zero since this would require either γ= 0 or κ= 0,
which would imply v2is the null vector.
7. Examples
In this section we present the application of our results in two remarkable
cases. These also serve to illustrate the relationships we have found between
the parameters that determine the behaviour of the dynamics.
7.1. Open quantum harmonic oscillator
Let Tbe the QMS with generator in a generalized GKSL form with
L1=µ a , L2=λ a†, H = Ω a†a+κ
2a†2+κ
2a2+ζ
2a†+ζ
2a(28)
2150001-29
J. Agredo, F. Fagnola, and D. Poletti
with λ, µ ≥0, Ω ∈R,κ, ζ ∈C. The special case where κ=ζ= 0 has been
analyzed in [7] providing the full spectral analysis of the generator Lin the
L2space of the invariant state for λ<µ.
In this model γ= (µ2−λ2)/2. Moreover, in the case where both λ, µ are
strictly positive, the QMS is irreducible (Theorem 5) and admits a unique
faithful normal invariant state if and only if λ2< µ2and µ2−λ2)4 +
Ω2− |κ|2>0 (Theorem 8) with the explicit mean vector ωand covariance
operator Sas in (21).
If µ= 0 and λ > 0 we obtain a QMS which is irreducible if and only if
κλ26= 0, namely κ6= 0 and has no normal invariant state (Sect. 4.2).
Finally, in the case where λ= 0 and µ > 0 we find a QMS which is
irreducible if and only if κµ26= 0, i.e., κ6= 0. It admits invariant states if
and only if |κ|2<Ω2+λ4/4; these will be faithful if <(κ)6= 0 and pure
otherwise (Sect. 4.1). For any initial state ρ0, in both cases, t−1Rt
0T∗s(ρ0)ds
converges towards the unique invariant state by Theorem 8.
It is worth noticing here that the Hamiltonian His bounded from below
or above if and only if Ω2−|κ|2≥0, in which case it has discrete spectrum.
Therefore condition |κ|2<Ω2+λ4/4 appears as a relaxation of discreteness
of spectrum that allows existence of normal invariant states.
7.2. Quantum Fokker-Planck model
The quantum Fokker–Planck (QFP) model is an open quantum system in-
troduced to describe the quantum mechanical charge-transport including dif-
fusive effects (see [2, 21, 26] and the references therein). In this subsection
we show that a simple application of our results allows one to study the
dynamics.
The formal generator
L(x) = i
2p2+ω2q2, x+ ig{p, [q, x]}
−Dqq [p, [p, x]] −Dpp[q , [q, x]] + 2Dpq[q, [p, x]] ,
can be written in generalized GKSL form (1) with
H=1
2p2+ω2q2+g(pq +qp),
and L1, L2are the operators
L1=−2Dpq + ig
p2Dpp
p+p2Dpp q , L2=2√∆
p2Dpp
p ,
2150001-30
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
where ω2>0, Dpp >0, Dqq ≥0, Dpq ∈Rand ∆ = DppDqq −D2
pq −g2/4≥0.
Clearly, L1, L2are linearly independent if and only if ∆ >0. Moreover,
L1=−2iDpq −g
2pDpp
(a†−a) + pDpp(a†+a),
L2=i√∆
pDpp
(a†−a),
H=ω2+ 1
2aa†+ω2−1 + 2ig
4a†2+ω2−1−2ig
4a2+ω2+ 1
4
so that
v1=2iDpq +g
2pDpp
+pDpp , u1=−2iDpq +g
2pDpp
+pDpp , v2=−i√∆
pDpp
,
u2=i√∆
pDpp
,Ω = ω2+ 1
2, κ =ω2−1
2+ ig .
Compute
γ=1
2
2
X
`=1 |v`|2− |u`|2=g , Ω2−|κ|2=ω2−g2, γ2+Ω2−|κ|2=ω2.
Therefore, in the case where ∆ >0 Kraus operators L1,L2are linearly
independent, the QFP semigroup is irreducible and a Gaussian invariant
state exists if and only if g=γ > 0. This is given explicitly in Theorem 8.
Moreover, it is also faithful and it is the unique normal invariant state by
irreducibility.
The case ∆ = 0 has to be considered separately (see [2, 26]). By Theorem
(7) the QFP semigroup is irreducible if and only if 2Ωv1u1=κv2
1+κu2
1.
Taking the imaginary parts of this identity we find
gDpp =−ω2Dpq .(29)
Taking real parts we find ω24D2
pq −g2+ 4D2
pp =−8gDpq Dpp and, from
(29), we find the identity
ω24D2
pq −g2+ 4D2
pp =−8gDpq Dpp = 8ω2D2
pq
namely 4D2
pp =ω24D2
pq +g2and, by ∆ = 0 together with Dpp >0
Dpp =ω2Dqq .(30)
Note that ∆ = 0 together with (29) and (30) are equivalent to conditions
under which, for γ > 0, the Gaussian normal invariant state of the QFP
model is pure (see [2, Lemma 9.1]).
2150001-31
J. Agredo, F. Fagnola, and D. Poletti
Clearly, they are equivalent to (25). Indeed, a straightforward compu-
tation shows that the first identity is equivalent to Dpq =−gDqq and the
second one to Dqq =g/(2δ) which follows from ∆ = 0, (29) and (30) (see [2,
Lemma 9.1] for details).
Convergence towards the unique invariant state of t−1Rt
0T∗t(ρ0)dsholds
for any initial state ρ0by Theorem 8.
8. Conclusions and Outlook
We considered the most general Gaussian QMS on the one mode Fock space
Γ(C) of the regular representation of one-dimensional CCR. The GKSL gen-
erator associated with unbounded operators (2) and (3) depends on 7 param-
eters (or 5 in the case where there is only one noise operator). We presented
its construction starting form the unbounded generator and proved the known
explicit formula for the action Weyl operators. We characterized irreducibil-
ity in terms of parameters of the model. This property always holds true
when there are two linearly independent noise operators L1, L2. However, if
there is only a single noise operator L1, irreducibility holds if and only if the
operators L1and [H, L1] are linearly independent (the H¨ormander type com-
mutator condition that appears in many fields of mathematics, from partial
differential equations to control theory). Finally, still in terms of the pa-
rameters of the model, we established the necessary and sufficient condition
γ > 0 and γ2+ Ω2−|κ|2>0 for existence and uniqueness of normal invariant
states. This condition also implies, by irreducibility convergence towards the
unique invariant state.
It would be useful and interesting to extend the above results to Gaus-
sian QMS on the algebra of bounded operators on d-mode Fock spaces. The
explicit formula for the action on Weyl operators is known also in this case.
We guess that the equivalence of irreducibility with an H¨ormander type com-
mutator condition can be proved as well considering commutators of Hand
noise operators L`of order up to 2d−1. This advance seems to require a deep
study of regularity properties of Gaussian semigroups as in the classical case.
However, we think that it is a necessary step in order to establish precise
relationships between the behaviour of the infinite dimensional QMS and the
2d×2ddimensional matrices Zand Cand reduce a lot of infinite dimensional
problems on the dynamics to finite dimensional ones on matrices. Results
will be the object of a forthcoming paper.
Appendix A
In this section we prove Theorem 4. We begin with the following lemma.
2150001-32
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
LEMMA 5 For all ξ∈Dom(N2)and all θ∈Rwe have
(eiθa†+e−iθa)ξ
2≤2
(aa†+a†a)1/2ξ
2
(eiθa†2+e−iθa2)ξ
2≤
(aa†+a†a)ξ
2+ 3 kξk2.
Proof. Computations below should be done on quadratic forms defined on
the domain D×D. However, we do only the algebraic computations to
simplify the notation.
To prove the first inequality we begin by expanding
0≤ |eiθa†−e−iθa|2=a†a−e2iθa†2−e−2iθa2+aa†
which implies
e2iθa†2+e−2iθa2≤a†a+aa†.
It follows that
|eiθa†+e−iθa|2≤2(a†a+aa†)
and the first inequality is proved. To prove the second inequality, first note
that
0≤ |eiθa†2−e−iθa2|2=a2a†2−e2iθa†4−e−2iθa4+a†2a2
and so
e2iθa†4+e−2iθa4≤a2a†2+a†2a2.
Now
(eiθa†2+e−iθa2)2−(aa†+a†a)2=e2iθa†4+a†2a2+a2a†2+e−2iθa4
−(aa†)2−(a†a)2−aa†2a−a†a2a†
≤2a†2a2+ 2a2a†2−(aa†)2−(a†a)2−aa†2a−a†a2a†.
The right-hand side is equal to
2N(N−1) + 2(N+ 1)(N+ 2) −(N+ 1)2−N2−(N+ 1)N−N(N+ 1) = 3
and so
(eiθa†2+e−iθa2)2≤(aa†+a†a)2+ 3 .
The claimed inequality readily follows.
We will show that the graph norms of G, G0and Nare equivalent. To this
end need two preliminary lemmas.
2150001-33
J. Agredo, F. Fagnola, and D. Poletti
LEMMA 6 Let λ0be the smallest eigenvalue of the 2×2matrix
v1v2
u1u2·v1u1
v2u2
which is strictly positive by the linear independence of L1,L2. There ex-
ists a constant c1>0depending on v1, u1, v2, u2and uniformly bounded for
v1, u1, v2, u2in a bounded subset of C4such that
(−2G0)2≥λ2
0(a†a+a a†)2−c1(a†a+a a†).
Proof. Since −2G0=L∗
1L1+L∗
2L2,
G0=−1
2
2
X
`=1 |v`|2a†a+|u`|2aa†+v`u`a†2+v`u`a2
for all ξ∈D, thinking of (a ξ, a†ξ) as a vector in h⊕hand of product of a
row vector with a column vector as the natural scalar product in h⊕h, we
can write hξ, G0ξias follows
hξ, G0ξi=−1
2[a†ξ, a ξ]v1v2
u1u2v1u1
v2u2 a ξ
a†ξ.
This notation is typical in the study of quadratic Hamiltonians (see, for
instance, [9, 27, 28, 29]). Recall that, by linear independence of L1, L2, the
above matrices have non-zero determinant. Therefore their product is strictly
positive definite and, calling λ1its biggest eigenvalue, we have
λ1Dξ, (a†a+a a†)ξE≥ hξ, −2G0ξi ≥ λ0Dξ, (a†a+a a†)ξE.(31)
In a similar way, dropping the vector ξand denoting by l.o.t. monomials of
order 2 or less in creation and annihilation operators we have the inequalities
(−2G0)2=X
`
L∗
`(−2G0)L`+ l.o.t.
= [a†, a]v1v2
u1u2−2G00
0−2G0v1u1
v2u2 a
a†+ l.o.t.
≥λ0[a†, a]v1v2
u1u2a†a+aa†0
0a†a+aa†v1u1
v2u2 a
a†+ l.o.t.
=λ0a[a†, a]v1v2
u1u2v1u1
v2u2 a
a†a†
+λ0a†[a†, a]v1v2
u1u2v1u1
v2u2 a
a†a+ l.o.t.
=λ0a(−2G0)a†+λ0a†(−2G0)a+ l.o.t.
≥λ2
0a(a†a+aa†)a†+λ2
0a†(a†a+aa†)a+ l.o.t.
=λ2
0(a†a+aa†)2+ l.o.t.
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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
Lower order terms can be controlled in terms of (2N+ 1) = (a†a+a a†) by
Lemma 5 and the proof is complete.
LEMMA 7 The commutator [H, G0]is a second order degree polynomial in
a, a†and
|hξ, [H, G0]ξi| ≤ c2hξ, (a†a+a a†)1/2ξi
for some constant c2>0depending on all parameters in the model.
Proof. A long but straightforward computation yields (summation on `=
1,2 is implicit)
[H, G0]=i=(κ(v`u`))(a†a+aa†) + Ω (v`u`)−κ
2|v`|2+|u`|2a2
+−Ω (v`u`) + κ
2|v`|2+|u`|2a†2
+ζ
2(v`u`)−ζ
2|v`|2+|u`|2a+−ζ
2(v`u`) + ζ
2|v`|2+|u`|2a†.
The claimed inequality follows from Lemma 5 and the Schwarz inequality.
Proof of Theorem 4. Clearly Dom(N) is contained in Dom(G0) and Dom(G).
In order to prove the opposite inclusion we show that there exist constants
c3, c4such that kNξk2≤c3kG0ξk2+c4kξk2for all ξ∈D. The conclusion
follows because Dis an essential domain for G0and Gby their definition.
For all ξ∈D, > 0 by Lemma 6 and the Young’s inequality, we have the
following inequalities
kG0ξk2=ξ, G2
0ξ
≥λ2
0
4Dξ, (a†a+a a†)2ξE−c1
4Dξ, (a†a+a a†)ξE
≥λ2
0
4Dξ, (a†a+a a†)2ξE−c1
4kξk ·
a†a+a a†ξ
≥λ2
0
4Dξ, (a†a+a a†)2ξE−λ2
0
8
(a†a+a a†)ξ
2−c2
1
8λ2
0kξk2
=λ2
0
8
(a†a+a a†)ξ
2−c2
1
8λ2
0kξk2.
Since
a†a+a a†ξ
2≥4kNξk2we find the inequality
kNξk2≤2
λ2
0kG0ξk2+c2
1
4λ4
0kξk2(32)
for all ξ∈Dimplying that Dom(G0)⊆Dom(N).
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J. Agredo, F. Fagnola, and D. Poletti
In order to prove that the domain of Gis also contained in the domain
of Nnote that G=G0−iHon Dand write
kGξk2=hξ, (G0+ iH)(G0−iH)ξi=ξ, (G2
0+H2)ξ+ i hξ , [H, G0]ξi.
(33)
Now by Lemma 7, ξ, H2ξ≥0 and the previous inequality (32) we find
kGξk2≥ hξ, G2
0ξi − c2Dξ, (a†a+a a†)1/2ξE
≥λ2
0
2kNξk2−c2√2kξk·kN1/2ξk − c2
1
4kξk2.
We can now proceed as in the final part of the proof of (32) with an applica-
tion of the Young inequality to show that Dom(G)⊆Dom(N).
Appendix B
In this appendix we prove Lemma 1. We begin by noting that Dom(N)⊆
Dom(G0).
Conversely, note that for all r∈R, on the domain Dom(N) of the number
operator, in a natural matrix notation, we have
L∗L=|v|2a†a+vu a†2+vu a2+|u|2aa†
=|v|2+ra†a+vua†2+vua2+|u|2−raa†+r1l
=ha†ai|v|2+r vu
vu |u|2−r a
a†+r1l
The trace of the above 2 ×2 matrix is strictly positive and the determinant
r|u|2− |v|2−r2
if we choose r=|u|2− |v|2/2, it is equal to |u|2− |v|22/4>0 and the
lowest eigenvalue is (|v|−|u|)2/2. It follows that
L∗L≥(|v|−|u|)2
2aa†+a†a+|v|2− |u|2
21l
and, denoting by l.o.t. monomials of order 2 or less in creation and annihi-
lation operators,
(L∗L)2=L∗(LL∗)L=L∗(L∗L)L+ l.o.t.
≥1
2(|v|−|u|)2L∗aa†+a†aL+ l.o.t.
=1
2(|v|−|u|)2aL∗La†+a†L∗La+ l.o.t.
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Gaussian Quantum Markov Semigroups on a One-Mode Fock Space
≥1
4(|v|−|u|)4aaa†+a†aa†+a†aa†+a†aa+ l.o.t.
= (|v|−|u|)4a†a2+ l.o.t.
Therefore there exists a constant c > 0 such that
(|v|−|u|)4ka†a ξk2≤ kL∗L ξk2+ckξk2(34)
for all ξ∈Dand Dom(L∗L)⊆Dom(N). This shows the identity Dom(G0) =
Dom(N).
In order to prove the other one, note first that Dom(N)⊆Dom(G).
Then, for all ξ∈D, compute
kGξk2=kG0ξk2+kH ξk2+hξ, i[H, G0]ξi.
Since the commutator [H, G0] is a second order polynomial in a, a†there
exists a constant c0>0 such that hξ, i[H, G0]ξi≥−c0kN1/2ξk2. Recalling
(34), by the Young inequality, we have
kGξk2≥ kG0ξk2−c0kN1/2ξk2
≥(|v|−|u|)4
4ka†aξk2−ckξk2−(|v|−|u|)4
8ka†aξk2−4
c02(|v|−|u|)4kξk2
=(|v|−|u|)4
8ka†aξk2−c00kξk2
where c00 is another constant. Thus Dom(G)⊆Dom(N) and the proof of
Lemma 1 is complete.
Appendix C
LEMMA 8 Let µ, λ, x, y ∈Rwith λ6= 0. For all r > 0and w > 0such that
w < min{1,(2x2)−1}there exists > 0such that
µ2+r4µx λµ
µx x2λx
λµ λx λ2
≥
r40 1
0−1/2 0
1 0 λ2w
Proof. The difference of the above matrices is
µ2+r4(1 −)µx λµ −
µx x2+/2λx
λµ − λx λ2(1 −w)
,
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J. Agredo, F. Fagnola, and D. Poletti
which is positive, by the Sylvester’s criterion, if and only if all principal
minors are positive. For all > 0, the principal minor obtained by removing
the first row and column is positive if and only if w < 1 and its determinant
λ2((1 −2wx2)−w2)/2 = λ2((1 −2wx2)−w)/2
is positive. This is clearly the case if < min{1, w−1,(1 −2wx2)/w}:= 1.
The principal minor obtained by removing the second row and column,
namely µ2+r4(1 −)λµ −
λµ − λ2(1 −w)
has positive diagonal elements for 0 < < 1and determinant
λ2r4+ (2λµ −λ2µ2w−λ2r4(1 + w))+λ2r4w2
which is clearly strictly positive for all 0 < < 2for some 2< 1. Finally,
the principal minor obtained by removing the third row and column, namely
µ2+r4(1 −)µx
µx x2+/2
which has positive diagonal elements for < 1, has determinant
r4x2+ (µ2+r4−2r4x2)/2−r42/2.
This is clearly positive for all small enough if x6= 0 because it tends to
r4x26= 0 but also for x= 0 since, in this case it is equal to (µ2+r4−r4/2).
This completes the proof.
Acknowledgements
J. Agredo would like to acknowledge the support from Escuela Colombiana
de Ingenier´ıa “Julio Garavito”. The financial support of GNAMPA INdAM
2020 project “Processi stocastici quantistici e applicazioni” is gratefully ac-
knowledged.
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