Fluctuation and dissipation in memoryless open quantum evolutions

07/30/2021 ∙ by Fabricio Toscano, et al. ∙ CNRS 0

Von Neumann entropy rate for open quantum systems is, in general, written in terms of entropy production and entropy flow rates, encompassing the second law of thermodynamics. When the open-quantum-system evolution corresponds to a quantum dynamical semigroup, we find a decomposition of the infinitesimal generator of the dynamics, that allows to relate the von Neumann entropy rate with the divergence-based quantum Fisher information, at any time. Applied to quantum Gaussian channels that are dynamical semigroups, our decomposition leads to the quantum analog of the generalized classical de Bruijn identity, thus expressing the quantum fluctuation-dissipation relation in that kind of channels. Finally, from this perspective, we analyze how stationarity arises.

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I Introduction

The study on quantum communication channels that describe the input-output maps corresponding to quantum mechanical operations is at the core of quantum information theory (see e.g., [1]). Formally, these maps are given, in the Schrödinger picture, by completely positive and trace preserving operations acting on density operators, whereas, in the Heisenberg picture, by identity-preserving operations acting on observables. A physically interesting property of these maps is that their composition is also a quantum channel. Accordingly, the set of quantum channels forms a semigroup. Moreover, quantum channels have an inverse only when they describe a unitary evolution. A unitary quantum channel requires for its implementation a quantum system completely isolated from its surrounding environment. In practice, the implementation of unitary channels is a great challenge. Consequently, the most common situation corresponds to non-unitary quantum channels that lead to a degradation of information during their use. To be able to assess the degradation effects, it is important to describe the evolutionary trajectory of the system that implements the channel. The description is possible through continuous maps over time. This is the point where quantum information theory meets open quantum systems theory.

We focus in this paper on quantum channels that are members of one-parameter semigroups, for which the density operator of the evolved quantum system is solution of a time-independent Markovian master equation [2]. The set of these channels forms a quantum dynamical semigroup in the case of finite-dimensional Hilbert spaces [3] as well as in the case of Gaussian channels in continuous-variable systems [4, 2, 5], i.e., channels in infinite-dimensional separable Hilbert spaces. These quantum dynamical semigroups are suitable to describe memoryless quantum channels that are continuous in time [6], specially in continuous-variable systems [7, 2].

In general, an open system dynamics, given by the interaction between the system and its environment, is modeled by a combination of both deterministic and random effects. One of the main approaches to the systematization of these random and deterministic effects lies in the so called fluctuation-dissipation relations, both in the classical [8, 9, 10, 11, 12] and quantum domains [13, 14, 15, 16, 17, 18] (see also [19]). Roughly speaking, these are relations that connect the deterministic characteristic of a system with its fluctuating aspects, both in the equilibrium and non-equilibrium regimes. Examples of these relations are the linear and non-linear Markov fluctuation-dissipation relations [19]

. In classical theory, these relations appear within Markov processes where the probability distribution of the variables of interest satisfies a Fokker–Planck equation. In particular, in the case of a linear Fokker–Planck equation the corresponding fluctuation-dissipation relations are just a set of identities that link the correlations between the variables of interest with the intensities of the fluctuation and dissipation effects in a stationary regime 

[19].

On the other hand, the ubiquitous notion of entropy is transversal to physical and information theories, since it arises naturally as a measure of uncertainty, randomness or lack of information about the state of a system [20]. For a quantum system described by a density operator , the von Neumann entropy is defined by where denotes the trace of an operator (see e.g.,  [21]). This quantity plays a key role in different contexts as in the study of entropy production in open quantum systems in general (see e.g.,  [22]) and of quantum communication channels where different entropic quantities serve to characterize the information-processing performance of the channels (see e.g.,  [23, 24], among others). Here, we are interested in the rate of change of the von Neumann entropy in one-parameter semigroups. The entropic analysis of its temporal behavior is of crucial importance, for instance, in order to improve the communication rates of quantum channels, as well as for studying stationary situations.

In open quantum theory, the study of the rate of change of is usually oriented by a thermodynamic perspective [25, 22]. In this framework the focus is on the decomposition of the rate of change into two parts: one corresponding to the so called entropy production rate , and the other one to the entropy flux rate . From this point of view, the second law of thermodynamics is nothing but the statement . Considerable progress has been made in recent years from this perspective in particular in a general definition of for general quantum processes (see [22] and references therein). However, those approaches do not focus on the characterization of the fluctuation and dissipation aspects of the dynamics.

At this point a natural question arises: is it possible to identify generically random and deterministic effects in the master equation for the density operator and for rate of change of the von Neumann entropy in one-parameter semigroups? Here we provide a positive answer for all finite-dimensional one-parameter quantum channels and for all infinite-dimensional Gaussian one-parameter quantum channels. Both situations are described by Lindblad master equations whose solutions form quantum dynamical semigroups [3, 4, 2, 5, 26, 27]. Our approach is inspired by a well-established result in classical information theory known as de Bruijn identity [28, 29]

which quantifies the rate of change of the Shannon entropy of a classical random variable, output of an additive Gaussian noise channel. More precisely, this identity quantifies how fast the channel becomes more random in terms of the non-parametric Fisher information. The fact that a de Bruijn identity can also be formulated for quantum systems was first posed in 

[30] for the quantum diffusion process, a useful result for formulating quantum entropy power inequalities [31, 30, 32]. However, the quantum diffusion process is a special subclass of the Gaussian one-parameter quantum channels that does not have dissipation effects. Therefore, our framework is also inspired by the recently established generalization of the de Bruijn identity for more general classical channels modeled by Langevin forces described by stochastic differential equations [33, 34], capturing the trade-off between the diffusion and dissipation terms of the evolution.

First, we propose a decomposition of the non-unitary infinitesimal generator for any quantum dynamical semigroup into two terms. The first one allows us to relate the time derivative of the von Neumann entropy with the divergence-based quantum Fisher information [35]. Accordingly, this term will be associated with the fluctuations due to the noise introduced by the open quantum dynamics. In addition, the second term will be connected with the dissipative contributions of the dynamics.

Afterwards, we focus on Gaussian channels, which are among the most important channels in information processing in both classical [28, 36] and quantum [37, 38, 7, 39, 40] domains, as they provide a faithful model for attenuation and noise effects in most communication schemes, modeled by linear Lindblad master equations [2]. For these channels, we find the fully quantum counterpart of the generalization of the de Bruijn identity for Fokker–Planck channels [33, 34]. In these cases, our proposal captures in a clear manner the diffusion and dissipation contributions to the rate of change of the von Neumann entropy given by the open quantum dynamics. Finally, we analyze stationary situations within our framework.

The paper is organized as follows. In Sec. II.1 we review the classical de Bruijn identity for channels with additive Gaussian noise and its generalization for multidimensional classical channels corresponding to Ornstein–Uhlenbeck processes, i.e., modeled by Fokker–Planck equations with linear drift and constant diffusion. In Sec. II.2

we show that the diffusion term in the generalized classical de Bruijn identity can be written in terms of the Fisher information of the probability distribution under the action of Langevin forces. This provides a novel interpretation of the diffusion term as a measure of the noise introduced by these forces. We also show that the second term in the generalized classical de Bruijn identity corresponds to the average flux of the dissipative forces,

i.e., a measure of the change of the probability distribution in the directions opposite to these forces. In Sec. III.1 we present our first main result, namely a decomposition of the non-unitary infinitesimal generator for any quantum dynamical semigroup that splits the dynamics into two different parts: fluctuation and dissipation. Section III.2 is devoted to the presentation of the notion of divergence-based quantum Fisher information and some of its properties. In Sec. III.3 we present our second main contribution, namely the application of the results of Sec. III.1 in order to obtain a closed formula for the von Neumann entropy rate of change for quantum dynamical semigroups, where we discriminate the contributions of fluctuation and dissipation. In Sec. IV.1, we present the basic formalism used to describe one-parameter Gaussian channels. In Sec. IV.2, we apply our results of Sec. III.3 to obtain the quantum counterpart of the generalized classical de Bruijn identity in the case of one-parameter Gaussian channels. We also show that the diffusion and dissipation terms of the identity obtained, admit an interpretation completely analogous to that given in Sec. II.2 for the corresponding terms in the generalized classical de Bruijn identity. In Sec. IV.3 we specialize the quantum de Bruijn identity for evolving Gaussian states and in Sec. IV.4 we study the stationarity conditions in one-parameter Gaussian channels in light of our formalism. We conclude with Sec. V where we summarize our findings. For ease of reading, some auxiliary calculations and technical proofs are presented in Appendices A and B.

Ii Rate of Shannon entropy for a linear Fokker-Planck equation

ii.1 Generalized de Bruijn identity

In classical information theory [28, 36]

, the additive Gaussian noise is probably the most used noise model to describe many “natural” random processes viewed as a large sum of noise sources and due to the central limit theorem 

[41, 42, 36]. The Gaussian noise channel corresponds to , being

a normally distributed random variable with zero mean and unit variance, independent of the input random variable

(that admits a finite variance). Here, is a parameter (usually time) that controls the amount of randomness added to the system. The evolution with respect to

of the probability density function

of the output random variable is ruled by the heat equation . The de Bruijn identity plays a fundamental role in classical information theory [29, 36, 43], as it quantifies the rate of change of the Shannon entropy at the output of the channel, in terms of the Fisher information  111 is generally called nonparametric corresponding to usual parametric Fisher information when , so that differentiating in replaces differentiating in , therefore drops from . In addition, the definition holds provided that is differentiable in the closure of its support.. More precisely,

(1)

Because , the fundamental interpretation of this equality is that becomes more and more random as (time) increases, with quantifying how fast. Applications of de Bruijn identity include derivation of relevant information-theoretical inequalities [45, 46, 43, 47], definition of an entropic temperature [48], proof of monotonicity of some statistical complexity measures [49], among others [50].

De Bruijn identity extends to the multivariate case [36, 33], and its form has recently been obtained for slightly more general channels modeled by Langevin forces described by stochastic differential equations [33, 34]. A similar expression for the rate of change of Shannon entropy was previously given in the thermodynamic context [51, 52], but without pointing out its link with Fisher information. In particular, consider a multidimensional channel corresponding to an Ornstein–Uhlenbeck process. It is characterized by an

-dimensional random vector

that follows a stochastic differential equation with linear drift and constant diffusion parameter ,

(2)

Here and are real constant matrices with dimensions and () respectively, being of full rank, is an -dimensional constant real vector, and is an -dimensional standard Wiener process [53, 54, 55, 56]. For (null vector), the dimensional Gaussian additive channel is recovered. Note that the drift corresponds to the deterministic effects, whereas the diffusion term characterizes the random contributions in the dynamics.

The probability density function of the random variable satisfies the Fokker–Planck equation [54, 55, 56]:

(3)

where is the gradient with respect to and is the diffusion matrix, which is symmetric and positive semidefinite of rank . For this channel, the generalized de Bruijn identity is [34]

(4)

where  () is the Shannon entropy, denotes trace of matrices, and the Fisher information matrix is defined as

which, under regularity conditions, can be written in the form [36]

(5)

with the Hessian matrix operator. The multivariate de Bruijn identity is recovered when (null matrix) and (null vector), whereas the original (scalar) de Bruijn identity is recovered when in addition . For a continuous variable system with an underlying symplectic structure, with the number of classical modes described by the canonically conjugate variables of position and momentum ,  . Typical examples of this type are the description of Brownian motion and the electric field in a laser where and are the classical quadratures of the electric field [54, 57].

ii.2 Fluctuation-dissipation relation in the generalized de Bruijn identity

Let us rewrite the generalized de Bruijn identity (4) so as to highlight the contributions of the fluctuations and dissipation. More precisely, by expressing the Ornstein–Uhlenbeck process as , where are the drift forces and are the fluctuating or Langevin forces, the first term in the r.h.s. of (4) can be explicitly identified as induced by fluctuations due to the Langevin forces, whereas the second one by dissipation due to the drift forces.

To this end, let us first rewrite the diffusion matrix under the form

(6)

where are the column vectors of . From the linearity of the trace and the relation  , the first term in the r.h.s. of (4) can therefore be expressed as

(7)

where is the translated probability distribution (), and indeed is the Fisher information for the translated distribution.

We remark that the r.h.s. of (7) makes explicit the dependence on the Langevin forces, , and also on the rank of the diffusion matrix . From the unicity of , is unique up to an orthogonal transformation, i.e., if and only if with an arbitrary orthogonal matrix, such a transform is equivalent to use any others Langevin forces, , which are the columns vectors of . Furthermore, this also holds for any matrix of the Stiefel manifold (), i.e., such that is the identity, so that the stochastic differential equation (2) with is the equivalent Langevin equation of that with the full rank . This shows that we can also express each particular matrix with an expression like in (7) but with the () columns of a matrix . However, the Langevin forces turn out to be not all independent. This kind of situation arises for the diffusion matrices that come from linear Lindblad master equations (see Eq.(29)).

Moreover, from the expression , where

is the Kullback–Leibler divergence, or relative entropy 

[58, 36], the first term in the r.h.s. of the classical de Bruijn identity (4) is essentially a measure of the noise induced by the Langevin forces on the probability distribution .

Secondly, from the definition of , its divergence is given by , which gives, together with ,

(8)

From the fact that vanishes in the boundary of ( admits a mean), so that the integral of vanishes, one equivalently has

(9)

Note that this result can be recovered directly from the expression of , together with the identity

(10)

This is because vanishes in the boundary of ( admits a mean), so that the integral of vanishes. Therefore, the quantity can be interpreted as minus the total amount of change of the probability distribution in the directions given by the linear drift force , or as the average of the drift force flux [51]. In addition, by expressing with and , we have . Accordingly, we rewrite the drift force as with and , in order to highlight that only the symmetric part contributes to the drift force flux. Moreover, whenever , we associate with a dissipative force and with a non-dissipative one. These names are justified because in the context where the Fokker–Planck equation (3) describes a mechanical system, represents the force in the phase space of the system that does not conserve the energy and corresponds to the Hamiltonian phase space force that conserves the energy.

Finally, we collect the expressions given in Eqs. (7) and (8) to obtain

(11)

Let us observe that the first term in the r.h.s of the de Bruijn identity (4) (or equivalently (11)) is strictly positive, because each Fisher information is positive, i.e.,

. On the other hand, the second term has no definite sign due to its dependence on the sum of the eigenvalues of

, which can be negative, positive or zero. Therefore, a necessary condition for the existence of a stationary solution of the Fokker–Planck equation (3) is the entropic balance between these quantities. This can happen only if (8) is negative, which leads to a condition on the eigenvalues of . Recall that the existence of a stationary solution also requires the matrix to be asymptotically stable, that is, all its eigenvalues must strictly have negative real part. In principle, this condition has not a direct connection with respect to the one on the eigenvalues of .

In the sequel, we will obtain an analogous generalized de Bruijn identity for the rate of change of the von Neumann entropy in Gaussian channels whose Wigner function satisfies the Fokker–Planck equation (3). Before that, in the next section we will show that such a fluctuation-dissipation relation in the quantum case has its origin in a particular decomposition of the non-unitary infinitesimal generator of the evolution in quantum dynamical semigroups.

Iii Rate of von Neumann entropy for quantum dynamical semigroups

iii.1 Decomposition of the infinitesimal generators of the evolution

In the Schrödinger picture, the dynamics of an open quantum system with density operator is generally modeled by a Lindblad master equation (LME) of the form [5]

(12)

where

(13)

and

(14)

are the infinitesimal generators of the unitary and non-unitary evolution, respectively, with the Hamiltonian of the system and the Lindblad operators. When is time independent, the formal solution of (12) is , where is a quantum dynamical semigroup (QDS). At least for finite dimension, any QDS is precisely described by a LME [3]. Here, could be bounded, or unbounded as happens for QDSs in Gaussian channels [4, 2, 5].

In the Heisenberg picture, observables evolve according to , where denotes the adjoint superoperator of  [5]  222Remind that, by definition, where and the notation “” is used to avoid confusion with the adjoint “” of an operator..

QDSs are also called quantum Markov semigroups [60], and the LME (12) is also called Markovian master equation [61]. These denominations emphasize the Markov semigroup property:  [5], which is the quantum version of an analogous semigroup property in time-homogeneous classical Markov processes [62]. Accordingly, in the same way as classical Markov processes describe classical memoryless channels, QDSs are suitable for describing memoryless quantum channels.

We propose a general decomposition of , which, in the context of Gaussian channels, encompasses the notions of diffusion and dissipation, as follows

(15)

with , and real superoperators 333Remind that a superoperator is real when . given by

(16a)
(16b)
(16c)

This follows from the cartesian decomposition of the Lindblad operators via the Hermitian operators and defined by and .

Notice that is selfadjoint, i.e., , as is , while is antisymmetric, i.e., . For the unit operator, , and . Consequently, and are infinitesimal generators of QDSs in itself. Moreover, is the generator of a unital QDS [64].

It is known [5] that the LME (12) is invariant under the transformations

(17a)
(17b)

for any set of complex numbers . Under these transformations, the superoperators changes as follows

(18a)
(18b)
(18c)

iii.2 Divergence-based quantum Fisher information

As seen in Sec. II, the Fisher information is a key measure to quantify the rate of change of the Shannon entropy in the additive Gaussian channel or in the channel described by the Ornstein–Uhlenbeck process, via the classical de Bruijn identities Eqs. (1) and (4), respectively [29, 34]. The first attempt, to our knowledge, to find a quantum counterpart of (1) is given by König and Smith in Ref. [30], where the divergence-based quantum Fisher information (DQFI) is related to the rate of change of the von Neumann entropy for an evolution governed by the quantum diffusion semigroup. The DQFI is one of the forms of the Fisher information defined in the quantum domain, and is precisely that appearing in the rate of von Neumann entropy in QDSs, in general.

The DQFI is defined as the second derivative of the relative entropy [35], i.e.,

(19)

with . As noticed in [35], this DQFI is greater than the quantum Fisher information based on the symmetric logarithmic derivative [65, 66, 67], whereas the respective classical versions coincide [36].

In the particular case of the family of density operators , generated by the unitary with a generator  (where ), we have (see Appendix A)

(20)

where . It can be seen that when is independent of , the expression (20) reduces to the one given in [30].

In what follows we consider the family generated from by unitaries of the form (with ).

iii.3 The rate of change of von Neumann entropy

By exploiting the decomposition (15) with , and given in Eqs. , we obtain that the rate of change of the von Neumann entropy , can be written in terms of the divergence-based quantum Fisher information (20). Note first that . Consequently, from the expression of given by Eqs. (13) and (15)-(16), together with the selfadjoint character of and , and the antisymmetry of , we obtain

(21)

with and . Notice that we used the fact that

(22)

(this can be established by simple algebra) so that the contribution to (21) from the unitary evolution, , vanishes.

Now, from the expressions of  (16a) and of the DQFI in the form (20), we can express the first contribution in the r.h.s. of Eq. (21) in terms of the DQFI, as follows

(23)

where are generated by the unitaries (), with or being the Hermitian generators of the unitaries. From (see Appendix A or Refs. [35, 30]), we conclude that is essentially a measure of the noise induced by unitaries , with generators and , on the state . Accordingly, plays a role analogous to that given by the first term,  (7), in the r.h.s. of the classical de Bruijn identity.

In addition, let us emphasize that both quantities and are invariant under the transformations given by Eq. (17). This is due to the effects (18) induced in the superoperators together with (22) that justifies that the energy-conserving contributions coming from induce no contribution in . As a consequence, characterizes the contributions of dissipative forces to the rate of change of von Neumann entropy. Accordingly, decomposition (21) reflect a  fluctuation-dissipation relation in the rate of change of the von Neumann entropy for QDSs, analogous to the one given by Eq. (11) for the Shannon entropy rate in classical channels corresponding to Ornstein–Uhlenbeck processes.

We highlight that the fluctuation-dissipation relation (21) is a direct consequence of the decomposition (15) for the infinitesimal generators in the LME. In this respect, we can say that the decomposition (15) is itself a fluctuation-dissipation relation for QDSs in its own, where the infinitesimal generator is associated with the fluctuation part of the evolution whereas with the dissipation part.

Notice that our way to express the rate of change of the von Neumann entropy, Eq. (21), differs from the usual decomposition

(24)

given in terms of the rate of the entropy production , and the rate of the entropy flux  [22]. This decomposition is one way to write the second law of the thermodynamics. Although there is a general proposal for the form of the entropy production and the entropy flux , for a general system-environment evolution (see [22] and references herein), the entropy production, in general, depends on the evolved reduced state of the environment, which is not available in open quantum systems. However, for QDSs this problem was overcome long time ago by Spohn in [25], but only in the cases when these semigroups have an invariant state , i.e., . In this situation the entropy production rate is .

The decomposition in (24) is useful to study stationary states, not necessarily equilibrium ones, arising when . Furthermore, these states, , are equilibrium states if and only if . In the case of QDSs, is also a monotonic convex function of time as a consequence of being an increasing function of time. This determines also how the approach to the stationary state is.

Our decomposition (21), that it is still valid for QDSs without a stationary state, is also useful to study stationarity in this type of open systems but on a different perspective from that given by (24). In our framework, stationarity arises when balances , since each DQFI in (23) is always positive. As we have already established, our decomposition (21) is a fluctuation-dissipation relation, therefore the balance between and can be interpreted as a fluctuation-dissipation equilibrium. We will confirm this point of view in Sec. IV.4 for quantum Gaussian channels that admit a QDS description [2] that we call Gaussian dynamical semigroups (GDSs). This is the quantum counterpart of the fluctuation-dissipation equilibrium balance we found in the classical de Bruijn equation in Sec. II.2.

Iv Rate of von Neumann entropy for Gaussian dynamical semigroups

iv.1 Gaussian dynamical semigroups

In what follows, we focus on GDSs, being the attenuator, amplifier and additive Gaussian noise channels relevant examples [2, 68, 69]. GDSs are also useful to describe damped collective modes in deep inelastic collisions [70].

The kinematics of a quantum Gaussian channel of bosonic modes is described by a -dimensional vector of canonically conjugate operators, , such that , with the real symplectic matrix where is the identity matrix, with . The dynamics of a GDS is given by LME (12) with a Hamiltonian up to quadratic order in and linear Lindblad operators,

(25)

respectively, where (Hessian matrix), is an arbitrary -dimensional real vector, and the s are -dimensional complex vectors. Usually master equations of this type are called linear LMEs [71].

In the Weyl–Wigner representation, the symbol of is the Wigner function , where are phase space coordinates [72]. The time evolution of is given by a Fokker–Planck equation of the Ornstein–Uhlenbeck form (3), with drift and diffusion matrices given by

(26)

is the matrix that establishes the connection with the Lindblad operators given Eq. (25) and is the dissipation matrix. By definition, so that

(27)

which can be interpreted as a generalized fluctuation–dissipation relation [71]. This matrix inequality implies that , because and are symmetric and antisymmetric real matrices respectively.

A few observations and remarks are in order here. First, defining the real vectors

(28)

the diffusion matrix can be expressed as

(29)

If we compare the expressions Eqs. (6) and (29), we immediately recognize in the set Langevin forces. However, all these forces are not necessarily linear independent like those in Eq. (6). Because the complex vectors are usually linearly independents (with ), is the rank of the matrix . But the rank of the matrix expressed in Eq. (29) could range from being for example when all pairs are composed with vectors proportional to each other, and being for example when all these vectors are linearly independent. Conversely, the rank of the dissipation matrix could range from . Therefore, it is not possible in GDSs to have a dynamics without diffusion while it is possible to have a dynamics without dissipation as it is the case of a quantum diffusion process. The latter is a particular case of a GDS where and , which is precisely the GDS considered in context of quantum de Bruijn identity in [30].

Remind that in GDSs the Fokker–Plank equation Eq. (3) propagates the Wigner function , that it is a quasi-probability distribution that describes completely the quantum state of the system. However, in the classical mechanics context the same Fokker–Plank equation Eq. (3), with the drift and diffusion matrices given in (26), is also used, for example, to describe the Brownian motion in a harmonic potential. In this case the drift force in phase space can be splitted into , where and are the conservative and dissipative forces, respectively. Note, that in this case, as commented below Eq. (8), because is anti-symmetric and is symmetric, and .

Finally, using successively (i) the expression for the drift matrix in (26) so that , (ii) the quadratic classical Hamiltonian –the Weyl symbol of the quantum Hamiltonian Eq. (25)– together with so that and (judiciously) for any vector, the Fokker–Planck equation in (3) for the Wigner function of the system can be rewritten under the form

(30)

where the first term is the Poisson bracket, , between the Wigner function and the quadratic classical Hamiltonian.

Let us now write the corresponding LME of a GDS in terms of the decomposition of the infinitesimal generator given Eq. (15). More precisely, in Appendix B, we show that

(31a)
(31b)
(31c)

where we define the superoperator matrices

(32a)
(32b)

with the matrix notation: , and . The last equalities in Eqs.(32) follows from the identity given in [73]. Then, the linear LME is

(33)

For one mode (), this expression reduces to that given in [70].

Notice that (IV.1) is nothing but the Weyl-Wigner representation of (IV.1). Moreover, a direct comparison of both evolution equations follows from the correspondences

(34a)
(34b)