The decoherence-free subalgebra of Gaussian Quantum Markov Semigroups

We demonstrate a method for finding the decoherence-subalgebra $\mathcal{N}(\mathcal{T})$ of a Gaussian quantum Markov semigroup on the von Neumann algebra $\mathcal{B}(\Gamma(\mathbb{C}^d))$ of all bounded operator on the Fock space $\Gamma(\mathbb{C}^d)$ on $\mathbb{C}^d$. We show that $\mathcal{N}(\mathcal{T})$ is a type I von Neumann algebra $L^\infty(\mathbb{R}^{d_c};\mathbb{C})\bar{\otimes}\mathcal{B}(\Gamma(\mathbb{C}^{d_f}))$ determined, up to unitary equivalence, by two natural numbers $d_c,d_f\leq d$. This result is illustrated by some applications and examples.


Introduction
Quantum channels and quantum Markov semigroups (QMS) describe the evolution of an open quantum system subject to noise because of the interaction with the surrounding environment. Couplings to external degrees of freedom typically lead to decoherence. Gaussian quantum channels and Markov semigroups play a key role because several models are based on linear couplings of bosonic systems to other bosonic systems with quadratic Hamiltonians. As a result, the time evolution is then determined by a Gaussian channel in the discrete time case and Markov semigroup in the time continuous case.
Decoherence-free subalgebras determine observables whose evolution is not affected by noise and play a fundamental role not only in the analysis of decoherence (see [2,4,7,22,26,27] and the references therein) but also in the study of the structure of QMSs (see [18]).
The case of a norm-continuous QMS (T t ) t≥0 has been extensively studied ( [13,18] and the references therein). The generator is represented in a GKLS form Gaussian QMSs arise in several relevant models and form a class with a rich structure with a number of explicit formulas ( [1,10,20,31]), yet they are not norm-continuous. However, it is known that their generators can be written in a generalized GKLS form with operators L ℓ that are linear (see (7)) and H quadratic (see (6)) in boson creation and annihilation operators a j , a † k ( [11,29]). As a consequence, operators L ℓ , L * ℓ and their iterated generalized commutators with H are linear in boson creation and annihilation operators a j , a † k . In this paper we consider Gaussian QMS on the von Neumann algebra B(Γ(C d )) of all bounded operators on the Boson Fock space on C d and characterize their decoherence-free subalgebras as generalized commutants of these iterated generalized commutators (Theorem 6). Indeed, we show that it suffices to consider iterated commutators up to the order 2d − 1. Moreover, we prove (Theorem 13) that the decoherence-free subalgebra is a type I von Neumann algebra L ∞ (R dc ; C)⊗B(Γ(C d f )) determined, up to unitary equivalence, by two natural numbers d c , d f ≤ d. This conclusion is illustrated by some examples and a detailed analysis of the case of a Gaussian QMS with a single operator L ℓ .
The symplectic structure on C d plays a fundamental role in the origin of von Neumann algebras of the type L ∞ (R dc ; C) as possible decoherence-free subalgebras determined by generalized commutants via Araki's duality theorem. Moreover, even if at a purely algebraic level Theorem 6 looks like a natural generalization of the norm continuous case, several difficulties arise from unboundedness of operators L ℓ and H and, as a consequence, unboundedness of the generator L of the QMS T . The defining property (13) of N (T ) involves an operator x and the product x * x, but, even if x belongs to the domain of the generator of the QMS T , there is no reason why x * x should (see e.g. [17]) therefore one has to work with quadratic forms. Domain problems arising from generalized commutators have to be carefully handled because one needs to make sense of generalized commutations such as x[H, L ℓ ] ⊆ [H, L ℓ ]x and e itH xe −itH L ℓ ⊆ L ℓ e itH xe −itH for x ∈ N (T ).
The decoherence-free subalgebra of a QMS with unbounded generator was also characterized in [14] with several technical assumptions that hold in the case of Gaussian semigroups and using a dilation of the QMS via quantum stochastic calculus. The proof we give here (Appendix B) is simpler because it does not appeal to these assumptions and is more direct because it does not use quantum stochastic calculus.
The paper is organized as follows. In Section 2 we introduce Gaussian QMS, present their construction by the minimal semigroup method (see [16]), well-definedness (conservativity or identity preservation) applying the sufficient condition of [9] and the explicit formula for the action on Weyl operators (Theorem 4). Proofs, that can be obtained from applications of standard methods, are collected in Appendix A. Section 3 contains the main results of the paper. We first recall the definition of decoherence-free subalgebra. Then we prove its characterization for a Gaussian QMS (Theorem 6 with proof in Appendix B). Finally we prove the structure result Theorem 13. Applications and examples are presented in Section 4.
For any g ∈ C d , define the exponential vector e g associated with g by Creation and annihilation operators with test vector v ∈ C d can also be defined on the total set of exponential vectors (see [28]) by allows one to establish the identities The above operators are obviously defined on the linear manifold D spanned by the elements (e(n 1 , . . . , n d )) n 1 ,...,n d ≥0 of the canonical orthonormal basis of h that turns out to be an essential domain for all the operators considered so far. This also happens for quadrature operators that are symmetric and essentially self-adjoint on the domain D by Nelson's theorem on analytic vectors ( [30] Th. X.39 p. 202). The linear span of exponential vectors also turns out to be an essential domain for operators q(u) for the same reason. If the vector u has real (resp. purely imaginary) components one finds position (resp. momentum) operators and the commutation relation [q(u), q(v)] ⊆ iℑ u, v 1l (where ℑ and ℜ denote the imaginary and real part of a complex number). Momentum operators, i.e. quadratures q(ir) with r ∈ R d are also denoted by p(r) = 1≤j≤d r j p j where p j = i(a † j − a j )/ √ 2. In a similar way we write q(r) = 1≤j≤d r j q j with q j = q(e j ) = (a † j + a j )/ √ 2. Another set of operators that will play an important role in this paper are the Weyl operators, defined on the exponential vectors via the formula W (z)e g = e − z 2 /2− z,g e z+g z, g ∈ C d .
By this definition W (z)e f , W (z)e g = e f , e g for all f, g ∈ C d , therefore W (z) extends uniquely to a unitary operator on h. Weyl operators satisfy the CCR in the exponential form, namely, for every z, z ′ ∈ C d , It is well-known that W (z) is the exponential of the anti self-adjoint operator −i √ 2 q(iz) Finally, we recall here two relevant properties that are valid on D and on suitable dense A QMS T = (T t ) t≥0 is a weakly * -continuous semigroup of completely positive, identity preserving, weakly * -continuous maps on B(h). The predual semigroup T * = (T * t ) t≥0 on the predual space of trace class operators on h is a strongly continuous contraction semigroup.
Gaussian QMSs can be defined in various equivalent ways. Here we introduced them through their generator because it is the object we are mostly concerned with. The pre-generator, or form generator, of a Gaussian QMSs can be represented in a generalized (since operators L ℓ , H are unbounded) Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form (see [29] Theorems 5.1, 5.2 and also [11,32]) where 1 ≤ m ≤ 2d, and Ω := (Ω jk ) 1≤j,k≤d = Ω * and κ := (κ jk ) 1≤j,k≤d = κ T ∈ M d (C), are d × d complex matrices with Ω Hermitian and κ symmetric, V = (v ℓk ) 1≤ℓ≤m,1≤k≤d , U = (u ℓk ) 1≤ℓ≤m,1≤k≤d ∈ M m×d (C) are m × d matrices and ζ = (ζ j ) 1≤j≤d ∈ C d . The notation v ℓ• and u ℓ• refers to vectors in C d obtained from the ℓ-th row of the corresponding matrices. We exclude the case where the pre-generator L reduces to the Hamiltonian part i[H, x] and so we suppose that one among matrices V, U is non-zero. An application of Nelson's theorem on analytic vectors ( [30] Th. X.39 p. 202) shows that H, as an operator with domain D, is essentially selfadjoint. In addition, operators L ℓ are closable therefore we will identify them with their closure.
It can be shown (see [29] Theorems 5.1, 5.2) that a QMS T is Gaussian if maps T * t of the predual semigroup T * preserve Gaussian states or, still in an equivalent way, maps T t act explicitly on Weyl operators (Theorem 4 below).
Clearly, L is well defined on the dense (not closed) sub- * -algebra of B(h) generated by rank one operators |ξ ξ ′ | with ξ, ξ ′ ∈ D because all operator compositions make sense. However, since the operators H, L ℓ are unbounded, the domain of L is not the whole of B(h). For this reason we look at it as a pre-generator and describe in detail its extension to a generator of a QMS by the minimal semigroup method (Theorem 3 below). Remark. The above generalized GKLS form is the most general with operators L ℓ which are first order polynomials in a j , a † j and the self-adjoint operator H which is a second order polynomial in a j , a † j . Indeed, in the case where L ℓ are as above plus a multiple of the identity operator, exploiting non uniqueness of GKLS representations (see [28], section 30) one can always apply a translation and reduce himself to the previous case.
We choose the minimum number of operators L ℓ (also called Kraus operators), namely the parameter m.
A GKLS representation is minimal if and only if the following condition on V and U, that will be in force throughout the paper, holds.
For all x ∈ B(h) consider the quadratic form with domain D × D We postpone to the Appendix the construction of the unique Gaussian QMS with pre-generator (5) and state here the final result.
Theorem 3 There exists a unique QMS, T = (T t ) t≥0 such that, for all x ∈ B(h) and ξ, ξ ′ ∈ D, the function t → ξ ′ , T t (x)ξ is differentiable and The domain of the generator consists of x ∈ B(h) for which the quadratic form £(x) is represented by a bounded operator.
Weyl operators do not belong to the domain of the generator of T because a straightforward computation (see, for instance, Appendix A Section 4.5) shows that the quadratic form £(x) is unbounded. In spite of this we have the following explicit formula (see [11,32]) Theorem 4 Let (T t ) t≥0 be the quantum Markov semigroup with generalized GKLS generator associated with H, L ℓ as above. For all Weyl operator W (z) we have where the real linear operators Z, C on C d are We refer to Subsection 4.5 for the proof.

Structure of N (T )
The decoherence-free subalgebra (see [2,7,13,14]) of T is the defined as This is the biggest sub von Neumann algebra of B(h) on which maps T t act as * -homorphisms by the following known facts (see e.g. Evans [15] Th. 3.1).
Proposition 5 Let T be a QMS on B(h) and let N (T ) be the set defined by (13). Then:

N (T ) is a von Neumann subalgebra of B(h).
The decoherence-free subalgebra of a QMS with a bounded generator, i.e. written in a GKLS form with bounded operators H, L ℓ instead of (7), (6) is the commutator of the set of operators δ n . Generators of Gaussian QMSs are represented in a generalized GKLS form with unbounded operators L ℓ , H, but N (T ) can be characterized in a similar way considering generalized commutatant of a set of unbounded operators. We recall that, the generalized commutant of an unbounded operator L is the set of bounded operators x for which xL ⊆ Lx, namely Lx is an extension of xL.
We begin our investigation on N (T ) by the following The decoherence-free subalgebra of a Gaussian QMS with generator in a generalized GKLS form associated with operators L ℓ , H as in (7), (6) is the generalized commutant of the following linear combinations of creation and annihilation operators where δ H (x) = [H, x] denotes the generalized commutator and δ n H denotes the n-th iterate. Moreover T t (x) = e itH x e −itH for all t ≥ 0 and x ∈ N (T ).
We defer the proof to Appendix B. Here we give an example to show that inequality n ≤ 2d − 1 is sharp.
Example 7 Consider the Gaussian QMS with only one operator L ℓ , i.e. m = 1 and In the sequel we provide a simpler characterization of N (T ) in terms of real subspaces of C d and find its structure. In order to make clear the thread of the discussion, we omit technicalities related with unbounded operators that can be easily fixed because D is an essential domain for operators involved in our computations and we concentrate on the algebraic aspect. A straightforward computation yields (with the convention of summation on repeated indexes) Therefore the set of operators of which we have to consider the generalized commutant, thanks to the CCR, is particularly simple and contains only linear combinations of creation and annihilation operators together with a multiple of the identity 1l that plays no role. Now notice that each linear combination of creation and annihilation operators is uniquely determined by a pair v, u of vectors in C d representing coefficients of annihilation and creation operators so that, for example, the operator L ℓ in (7) and its adjoint L * ℓ are determined by In a similar way, after computing commutators, Denote by À the above 2d × 2d matrix (built by four d × d matrices) and let V be the real subspace of C 2d generated by vectors with ℓ = 1, . . . , m and 0 ≤ n ≤ 2d − 1.
The above remarks allow us to associate with elements of (14) a set of vectors in C 2d and characterize the generalized commutant of (14) in a purely algebraic way.

Lemma 8 An operator x ∈ B(h) belongs to N (T ) if and only if it belongs to the generalized commutant of
Proof. By the above remarks we know that the operators in the set (14) are linear combination of annihilation and creation operators up to a multiple of the identity operator and the generalized commutant of (14) coincides with the generalized commutant of To conclude the proof we just need to show that the commutants of (16) and (18) Therefore every element of (16) is a linear combination of elements of (18) and viceversa, concluding the proof.
In order to describe the structure of the decoherence-free subalgebra we recall now some useful definitions and properties of symplectic spaces. At first note that C d equipped with the real scalar product ℜ ·, · is a real Hilbert space. Considering instead ℑ ·, · we obtain a bilinear, antisymmetric (i.e. ℑ z 1 , z 2 = −ℑ z 2 , z 1 , for all z 1 , z 2 ∈ C d ) and nondegenerate (i.e. ℑ z 1 , z 2 = 0 for all z 2 ∈ C d implies z 1 = 0) form also called a symplectic form. We now recall the following definitions.
1. M is a symplectic space if ℑ ·, · is non degenerate when restricted to elements of M.
3. Let M 1 ⊂ M be a real linear subspace. We call symplectic complement of M 1 in M, and denote it by M 1 ′ , the set

the symplectic form
ℑ ·, · is non degenerate when restricted to elements of M 1 ).
In order to fix some of the concepts in the above definition we provide the following Example 10 Consider M = C d which is a symplectic space and let (e j ) d j=1 be its canonical complex orthonormal basis. Clearly (e j , ie j ) d j=1 is an orthonormal basis for C d considered as a real Hilbert space. Consider now a vector e j for a fixed index j. It is orthogonal to all other elements of the basis with respect to the real scalar product ℜ ·, · , however it is symplectically orthogonal to all other elements of the basis except for ie j . Consider now M 1 = Lin R {e 1 , ie 1 } and M 2 = Lin R {e 2 }, which are real linear subspaces of C d . It is easy to see that In particular M 2 ⊂ M ′ 2 hence it is an isotropic subspace, while M 1 is a symplectic subspace. Eventually it is worth noticing that not all symplectic subspaces of M are also complex subspaces; from here the need to consider real vector spaces. Indeed, consider The previous example although seemingly simple is actually quite representative of what happens in the general case. In analogy with classical linear algebra most complicated situations can be simplified performing a change of basis through a homomorphism. We provide here the analogous definition for symplectic spaces.
Definition 11 Let M 1 , M 2 ⊂ C d be symplectic spaces. We say B : M 1 → M 2 is a symplectic transformation if it is a real linear map and moreover We say B is a Bogoliubov transformation or symplectomorphism if it is an invertible symplectic transformation.
Next Proposition collects all the properties of symplectic spaces we need (see [6] for a comprehensive treatment) Proposition 12 Let M ⊂ C d be a symplectic space and let (e j ) d j=1 be the canonical complex orthonormal basis of C d .
1. There exists a symplectomorphism B We give a proof in Appendix C for self-containedness. For all subset M of C d we denote by W(M) the von Neumann algebra generated by Weyl operators W (z) with z ∈ M. H. Araki's Theorem 4 p. 1358 in [3], sometimes referred to as duality for Bose fields, (see also [25] Theorem 1.3.2 (iv) for a proof with our notation), up to a constant in the symplectic from and also [21] Theorem 1.1) shows that the commutant of W(M) is W(M ′ ). Applying this result we can prove the following Theorem 13 The decoherence-free subalgebra N (T ) is the von Neumann subalgebra of B(h) generated by Weyl operators W (z) such that z belongs to the sympletic complement of (17). Moreover, up to unitary equivalence, for a pair of natural numbers d c , d f ≤ d.
Conversely, if z belongs to the symplectic complement of M, then from (4) and (1) we have W (z)q(iw)e g = q(iw)W (z)e g for all w ∈ M and g ∈ C d . Since the linear span of exponential vectors is an essential domain for q(iw), for all ξ ∈ Dom(q(iw)) there exists a sequence (ξ n ) n≥1 in E such that (q(iw)ξ n ) n≥1 converges to q(iw)ξ. It follows that (q(iw)W (z)ξ n ) n≥1 converges and, since q(iw) is closed, W (z)ξ belongs to Dom(q(iw)) and W (z)q(iw)ξ = q(iw)W (z)ξ, namely q(iw)W (z) is an extension of W (z)q(iw). Therefore W (z) belongs to the generalized commutant of all q(iw) with w ∈ M and therefore to N (T ) by Lemma 8.
In order to prove ( In particular M f is symplectically orthogonal to both M r , M c . Let d c = dim R M c and 2d f = dim R M f which is even by Proposition 12 1. Still by Proposition 12 we can find a symplectomorphism B such that is the canonical complex orthonormal basis of C dc+d f . Eventually, since symplectic transformation in finite dimensional symplectic spaces are always implemented by unitary transformations on the Fock space (see [12] Theorem 3.8), we obtain the final result.
Remark. An analogous argument to the proof of the previous theorem allows us to show that also M r is a symplectic subspace which is symplectically orthogonal to both M c and M f . If 2d r = dim R M r , in total analogy with the proof, we can always find a symplectomorphism such that where M c is the real subspace of C dc generated by {e 1 , . . . , e dc }. In particular, after the unitary transformation associated with the symplectic transformation, we have Remark. It is worth noticing here that a QMS with N (T ) as in (19) does not necessarily admit a dilation with d c classical noises because the corresponding Kraus operators L ℓ could be normal but not self-adjoint (see subsection 4.2 for an example) and so one may find obstructions to dilations with classical noises as shown in [19].

Corollary 14
The decoherence-free subalgebra N (T ) is generated by Weyl operators W (z) with z belonging to real subspaces of ker(C) that are Z-invariant.
Proof. By Theorem 13 it suffices to show that z belongs to the symplectic complement M ′ of (17) if and only if it belongs to a real subspace of ker(C) that is Z-invariant. If z belongs to M ′ then W (z) ∈ N (T ) and T t (W (z)) = e itH W (z) e −itH for all t ≥ 0. Comparison with (10) yields Unitarity of both left and right operators implies ℜ e sZ z, Ce sZ z = 0 for all s ≥ 0 and e sZ z belongs to ker(C) for all s ≥ 0, namely, in an equivalent way, z and also Zz (by differentiation) belong to ker(C). Conversely, if z belongs to a real subspace of ker(C) that is Z-invariant, then e sZ z also belongs to that subset for all s ≥ 0. The explicit formula (10) shows that T t (W (z)) = exp i t 0 ℜ ζ, e sZ z ds W e tZ z therefore T t (W (z) * )T t (W (z)) = e itH W (z) * W (z) e −itH = 1l = T (W (z) * W (z)) and, in the same way, T t (W (z))T t (W (z) * ) = T (W (z)W (z) * ). It follows that W (z) ∈ N (T ) and z belongs to the symplectic complement of (17) by Theorem 13.
The following corollary shows that we can perform a unitary transformation of the Fock space in order to reduce the number of creation and annihilation operators that appear in the Kraus' operators.

Corollary 15 There exists a unitary transformation U of the Fock space such that
Proof. It suffices to consider the transformation obtained in the Remark after Theorem 13. Indeed each Kraus operator corresponds to a vector [v, u] T ∈ V which in turn corresponds to two generators in the subspace M. Performing the symplectomorphism in the cited Remark we have M ≃ C dr ⊕ M c which has dimension 2d r + d c . In particular if U is the unitary transformation that implements this symplectomorphism UL ℓ U * will depend at most from d r + d c modes.
Example. One may wonder if H can also be written in a special form in the new representation of the CCR, for example as the sum of two self-adjoint operators, one depending only on b 1 , b † 1 , . . . , b dr+dc , b † dr +dc and the other depending only on b dr +dc+1 , b † dr+dc+1 , . . . , b d , b † d . This happens when N (T ) is a countable sum of type I factors (see [13]) but not in the case of Gaussian QMSs with d c > 0 as shows this example. Let d = 2, m = 1 and L = q 1 , H = q 1 p 2 .
Clearly, by Theorem 6, N (T ) is the algebra L ∞ (R; C)⊗B(Γ(C)) but H is the product of two operators depending on different coordinates.

Applications
In this section we present two examples to illustrate the admissible structures of decoherencefree subalgebras of a Gaussian QMS on B(Γ(C d )) with d ≥ 2 and the application to an open system of two bosons in a common environment (see Ref. [8]). We begin by considering the case of only one noise operator.

The case one L, H = N
The operators (7) and (6) are the closure of operators defined on D (either v or u is nonzero). We compute recursively for all n ≥ 0, and, in the same way, δ 2n+1 Thus M ′ is the orthogonal (for the complex scalar product) of the complex linear subspace generated by v and u, it is a complex subspace of C d and If v, u are linearly independent, then the complex dimension of M ′ is d − 2, and N (T ) is isomorphic to B(Γ(C d−2 )).

The case one L, H = 0
Let L be as in (20). If H = 0, then δ H = 0. In particular and, since both v and u cannot be zero in our framework, dim R M is either 1 or 2. If it is equal to 1 (first case), clearly M ∩ M ′ = M and M r = {0} therefore d c = 1 and d r = 0. It follows that d f = d − 1 and N (T ) is a von Neumann algebra unitarily equivalent to L ∞ (R; C)⊗B(Γ(C d−1 )). If dim R M = 2 then, since M r is a symplectic space, its real dimension must be even and so we distinguish two cases: d r = 0, d c = 2 (second case) and  C d−1 )). This classification is summarized by Table 1 in which the last column labeled "L" contains possible choices of the operator L that realize each case. Table 1: N (T ) that can arise with one L and H = 0 In the last part of the section we will characterize each case by just looking directly at the operator L instead of computing M.
Suppose L is self-adjoint. In this case V is composed of only one vector which is of the form [v, v] T . Therefore M = Lin R {iv} and d c = 1, while d r = 0 (1 st case). Consider now instead the case L normal but not self-adjoint. An explicit computation shows that 0 = [L, Moreover u = v since L is not self-adjoint, hence d c = 2 (2 nd case). If L is not even normal (i.e. v 2 = u 2 ) then by the previous calculations d c = 0 and d r = 1 (3 rd case). Summing up: the 1 st case arises when L is self-adjoint, the case 2 nd case arises when L is normal but not self-adjoint and the 3 rd case arises when L is not normal or, equivalently v 2 = u 2 . In the last case it can be shown that when v 2 > u 2 (resp. v 2 < u 2 ) there exists a Bogoliubov transformation changing L to a multiple of the annihilation operator a 1 (resp. creation operator a † 1 .

Two bosons in a common bath
The following model for the open quantum system of two bosons in a common environment has been considered in Ref. [8]. Here d = 2 and H is as in equation 6 with κ = ζ = 0.
The completely positive part of the GKLS generator L is where (γ ± jk ) j,k=1,2 are positive definite 2 × 2 matrices. Note that, by a change of phase a 1 → e iθ 1 a 1 , a † 1 → e −iθ 1 a † 1 , a 2 → e iθ 2 a 2 , a † 2 → e −iθ 2 a † 2 , we can always assume that (γ − jk ) j,k=1,2 is real symmetric. Write the spectral decomposition where the vectors ϕ − , ψ − have real components. Rewrite the first term of (21) as and write in a similar way the second term of (21) j,k=1,2 We can represent L in a generalized GKLS form with a number of Kraus operators L ℓ depending on the number of strictly positive eigenvalues among λ ± , µ ± .
Relabelling if necessary, we can always assume 0 ≤ λ − ≤ µ − and 0 ≤ λ + ≤ µ + . We begin our analysis by considering the case where H = 0. If λ − > 0 (or λ + > 0) then there are four vectors v, u in the defining set of M namely Suppose now that λ + = λ − = 0 and µ − , µ + > 0 so that there are only two Kraus operators, the above L 2 and L 4 and It follows that, if ψ − , ψ + are R-linearly independent, we have again M = C 2 whence M ′ = {0} and N (T ) = C1l. Otherwise, if ψ + is a real non-zero multiple of ψ − , then, as ψ ± and iψ ± are R-linearly independent, the real dimension of M and M ′ is two, It is not difficult to see that, in any case, the dimension of M cannot be 1 or 3 (because creation and annihilation operator always appear separately in different Kraus operators L, never in the same).
Summarizing: N (T ) is non-trivial and isomorphic to B(Γ(C)) if and only if γ + and γ − are rank-one and commute.
Finally, if we consider a non-zero H, it is clear that N (T ) is always trivial unless γ + and γ − are rank-one, commute and their one-dimensional range is an eigenvector for Ω and Ω T .

Appendix A: construction of Gaussian QMSs from the GKLS generator
In this section we outline how one can construct the minimal quantum dynamical semigroup associated with operators H, L ℓ and following [16], Section 3.3. The first step is to prove that the closure of the operator G defined on the domain D generates a strongly continuous contraction semigroup. To this end we recall the result due to Palle E.T. Jorgensen (see [23], Theorem 2).
Theorem 16 Let G be a dissipative linear operator on a Hilbert space h. Let (D n ) n≥1 be an increasing family of closed subspaces of h whose union is dense in h and contained in the domain of G and let P Dn be the orthogonal projection of h onto D n . Suppose that there exists an integer n 0 such that GD n ⊂ D n+n 0 for all n ≥ 1. Then the closure G generates a strongly continuous contraction semigroup on h and ∪ n≥1 D n is a core for G, if there exists a sequence (c n ) n≥1 in R + such that ||GP Dn − P Dn GP Dn || ≤ c n for all n and ∞ n=1 c −1 n = ∞ We are now able to prove the following proposition.

Proposition 17
The operator G is the infinitesimal generator of a strongly continuous contraction semigroup on h and D is a core for this operator.
Proof. We apply Theorem 16 with D n the linear manifold spanned by vectors e(n 1 , . . . , n d ) with n 1 + . . . + n d ≤ n. Clearly D = ∪ n≥1 D n . The operator G is obviously densely defined and dissipative. Therefore it is closable (see e.g. [5], Lemma 3.1.14) and its closure, denoted G is dissipative. Clearly, by the explicit form of the action of creation and annihilation operators on vectors e(n 1 , . . . , n d ), the operator G maps D n into D n+2 for all n ≥ 0.
Therefore we have also This means that with c > 0 a constant that does not depend on n. Since the series n≥1 (n + 2) −1 diverges we can apply Theorem 16 and the proposition is proved. A similar argument allows us to prove the following

Proposition 18
The closure Φ of the operator 1≤ℓ≤m L * ℓ L ℓ defined on the domain D is essentially self-adjoint.
In the next section we will show that the minimal semigroup is identity preserving and so it is a well defined QMS, whose predual semigroup is trace preserving.

Conservativity
We will establish conservativity by applying the Chebotarev-Fagnola sufficient condition (see [9] , [16] section 3.5). More precisely, we will apply the following result: Then the minimal quantum dynamical semigroup is Markov.
In order to check the above conditions one should proceed with computations on quadratic forms. However, these are equivalent to algebraic computations of the action of the formal generator £ on first and second order polynomials is a j , a † j therefore we will go on with algebraic computations so as to reduce the clutter of the notation.

Lemma 20 It holds
Proof. First write By the CCR one has which both lead to Using the last equality and £(a † k ) = £(a k ) * concludes the proof. The following formula is verified for any generator £ of a QMS: Proof.

Recalling the usual commutator property [H, XY ] = [H, X]Y + X[H, Y ], we find then
This completes the proof. As a final step towards proving conservativity via Theorem 19, we prove the following Proof. By Lemmas 20 we have that for some complex numbers w kj , z kj , ζ j . While, by Lemma 21, we get and, in the same way Finally, from |a † By (24), (25), and (26) We can eventually state the result on conservativity.
Proof. We apply Theorem 19 with the operator C given by a j a † j u. Recalling that Ω = Ω * , κ = κ T and from (4) one gets Using the previous results one finds that L(W (z)) = W (z)X(z) for some operator X(z) If the operators L ℓ , H are unbounded, one has to cope with several problems. The operator L is unbounded and, even if we choose x, y in the domain of L, it is not clear whether y * x belongs to the domain of L (see [17]). Multiplication of generalized commutators [L ℓ , y] [L ℓ , x] may not be defined. If we choose a "nice" y ∈ Dom(L) then it is not clear whether we can take x = y because we do not know a priori if our "nice" y belongs to N (T ).
We begin the analysis of N (T ) by a few preliminary lemmas.

Lemma 24
The following derivative exists with respect norm topology for all z ∈ C Proof. The right-hand side operator G * W (z) + m ℓ=1 L * ℓ W (z)L ℓ + W (z)G is unbounded (for z = 0) therefore W (z) does not belong to the domain of L but we can consider the quadratic form £(W (z)) on D ×D. Differentiability of functions t → ξ ′ , T t (x)ξ also holds for ξ, ξ ′ in the linear span of exponential vectors. Therefore, for all such ξ, we have (Theorem 3) Recalling that T s (W (z)) = ϕ z (s)W (e sZ z) as in (10) for a complex valued function ϕ such that lim s→0 ϕ z (s) = 1, the right-hand side integrand can be written as (ϕ z (s) − 1) ξ, £(W (e sZ z))e g + ξ, £(W (e sZ z) − W (z))e g A long but straightforward computation shows the function s → £(W (e sZ z) − W (z))e g = G * W (e sZ z) + m ℓ=1 L * ℓ W (e sZ z)L ℓ + W (e sZ z)G e g is continuous vanishing at s = 0 and the function s → £(W (e sZ z))e g is bounded with respect to the Fock space norm. Therefore, taking suprema for ξ ∈ Γ(C d ), ξ = 1, we find the inequalities The conclusion follows dividing by t and taking the limit as t → 0 + .
Proof. If x ∈ N (T ), then, for all g, f, z ∈ C d and t ≥ 0 we have By Lemma 24 means the norm limit Moreover which tends to 0 as t → 0 + by weak * continuity of T t . As a result x) e f and we get The first term in the left-hand side cancels with the third term in right-hand side and last terms in both sides cancel as well. Noting that adding the first and fourth terms in the right-hand side, we find for all g, f ∈ C d and all ℓ. Therefore, by the arbitrarity of g and the explicit action of Weyl operators on exponential vectors e w , xL ℓ e f = L * ℓ e w , xe f for all w, f ∈ C d and all ℓ. Since exponential vectors form a core for L * ℓ and L ℓ is closed, this implies that xe f belongs to the domain of L ℓ and L ℓ xe f = xL ℓ e f , namely xL ℓ ⊆ L ℓ x.
Replacing x with x * we find x * L ℓ ⊆ L ℓ x * and standard results on the adjoint of products of operators (see e.g. [24] 5.26 p. 168) lead to the inclusions It follows that x belongs to the generalised commutant of the set { L ℓ , L * ℓ | 1 ≤ ℓ ≤ m }.
coefficients. The solution of the system yields analytic functions H − , H + as blocks of the exponential of a 2d × 2d matrix.
Proof. (of Theorem 6) Let G 0 be the self-adjoint extension of − d ℓ=1 L * ℓ L ℓ /2. By Proposition 27, for all y ∈ N (T ) and all v, u ∈ Dom(N), we have so the dimensions dim C (V n ) form a non decreasing sequence of natural numbers bounded where both M 1 and M 1 ′ are symplectic spaces. Clearly if z 1 , z 2 were linearly dependent we would have z 1 = sz 2 for some s ∈ R and then ℑ z 1 , z 2 = 0 which contradicts the construction of z 2 . Again since ℑ z 1 , z 2 = 0 we have M 1 ∩ M 1 ′ = {0} and M 1 is a symplectic subspace for what we proved at the beginning. Moreover if z 1 ∈ M 1 we can write z = (z + ℑ z, z 1 z 2 − ℑ z, z 2 z 1 ) + (ℑ z, z 2 z 1 − ℑ z, z 1 z 2 ), where it holds z + ℑ z, z 1 z 2 − ℑ z, z 2 z 1 ∈ M 1 ′ , ℑ z, z 2 z 1 − ℑ z, z 1 z 2 ∈ M 1 , hence we have proved M = M 1 ⊕ M 1 ′ . Eventually M 1 ′ is a symplectic space since it holds M 1 ′′ = M 1 and M 1 ′′ ∩ M 1 ′ = {0}. Note also that dim R M 1 ′ = 1 otherwise the symplectic form would be degenerate on it. We can now repeat the same reasoning starting with the symplectic space M 1 ′ in order to obtain z 3 , z 4 ∈ M 1 ′ such that ℑ z 3 , z 4 = 1 and if M 2 = Lin R {z 3 , z 4 } we have where M 1 , M 2 , M 2 ′ are symplectic spaces with dim R M j = 2 and dim R M 2 ′ = 1. Notice that they are also pairwise symplectically orthogonal, since M 2 , M 2 ′ ⊂ M 1 ′ . Since the remainder space M j ′ has always dimension different from 1 we can iterate this process until we get M j ′ = {0}. When the procedure stops we have a sequence M 1 , . . . , M d 1 of mutually (symplectically) orthogonal symplectic spaces, with M j = Lin R {z 2j , z 2j+1 }, ℑ z 2j , z 2j+1 = 1. Clearly 2d 1 = dim R M and this concludes the first step of the proof. In order to conclude the proof of this point is sufficient to construct the symplectomorphism via Bz 2j = e j , Bz 2j+1 = ie j .
This proves 2. In order to prove 3. it suffices to define to get a symplectomorphism B.