\section{Linear Dynamics and Stationary Conditions}\label{ldsc}
For a system of $n$ continuous degrees of freedom (\hypertarget{DF}{DF}),
the generalized coordinates together with the canonical conjugated momenta are collected
in a $2n$ column vector:
\begin{equation}                                                                         \label{vecx}
\hat x := (\hat q_1,...,\hat q_n, \hat p_1,...\hat p_n)^\dag.
\end{equation}
In this notation, the canonical commutation relation (\hypertarget{CCR}{CCR})
is written compactly as $[\hat x_j , \hat x_k ] = i \hbar \, \mathsf J_{jk}$
with ${\sf J}_{jk}$ given by the elements of the symplectic matrix
\begin{equation}                                                                         \label{comm}
 \mathsf J :=
\left(
\begin{array}{cc}
       {\bf 0}_n   & \mathsf I_n  \\
      -\mathsf I_n & {\bf 0}_n
       \end{array}
 \right).
\end{equation}

We consider the evolution of a quantum state governed by the \hy{LME}{LME} with
a quadratic Hamiltonian and linear Lindblad operators, {\it viz.},
\begin{equation}                                                                         \label{ham-lind}
\begin{aligned}
&\hat H = \frac{1}{2} \hat x \cdot \mathbf H \hat x +
\xi \cdot \mathsf J \hat x + H_0,\\
&\hat L_m = \lambda_m  \cdot \mathsf J \hat x + \mu_m,
\end{aligned}
\end{equation}
where $\xi \in \mathbb R^{2n}$ and $\lambda_m \in \mathbb C^{2n}$ are column vectors;
$H_0 \in \mathbb R$ and $\mu_m \in \mathbb C$ are constants and $m = 1,...,M$.
The Hessian of the Hamiltonian is symmetric by definition:
$\mathbf H = \mathbf H^\top \in {\rm Mat}(2n,\mathbb R)$.
Under such conditions, the evolution of the mean value vector
$\langle \hat x \rangle_t  :=  {\rm Tr}\left[ \hat x \hat\rho (t) \right]
                               \in \mathbb R^{2n}$
can be retrieved from (\ref{lindblad}) by using only the
\hy{CCR}{CCR} \cite{nicacio,wiseman}:
\begin{equation}                                                                         \label{mveq}
\frac{d\langle \hat x \rangle_t}{d t}  =
\xi - \eta + { \bf \Gamma } \langle \hat x \rangle_t ,
\end{equation}
where we have introduced
$\eta := \sum_{m = 1}^M {\rm Im}(\mu_m^\ast \, \lambda_m) \in {\mathbb C}^{2n}$,
\begin{equation}                                                                         \label{gammamat}
{\bf \Gamma }  := \mathsf J \mathbf H - {\rm Im} {\bf \Upsilon} \mathsf J
                 \in {\rm Mat}(2n,\mathbb R),
\end{equation}
and
\begin{equation}                                                                         \label{upsilon}
{\bm \Upsilon} :=
\sum_{m = 1}^M \lambda_m \lambda_m^\dagger \in {\rm Mat}(2n,\mathbb C).
\end{equation}

The natural question that arises at this point is whether
a solution of (\ref{mveq}) attains a finite asymptotic value when $t \to \infty$.
The answer is provided in the context of the Lyapunov stability theory \cite{dullerud}.
All solutions will be driven to an asymptotic point,
if the matrix $\bf \Gamma$ is {\it asymptotically stable} (\hypertarget{AS}{AS}),
{\it i.e.}, if its spectrum has negative real part.

Interestingly enough, from the Lindblad dynamics (\ref{lindblad})
with the operators in Eq.(\ref{ham-lind}),
a Lyapunov equation (\hypertarget{LE}{LE}) emerges naturally for the stationary value of
the covariance matrix (\hypertarget{CM}{CM}) of the system, as we shall see.
Defining the \hy{CM}{CM} of the state $\hat \rho$ as
${\bf V} = {\bf V}^\top \in {\rm Mat}(2n,\mathbb R) $ with elements \cite{note1}
\begin{equation}                                                                         \label{cmdef}
\mathbf V_{\! jk} =
\tfrac{1}{\hbar} {\rm Tr}\left[
                         \left\{ \hat x_j - \langle \hat x_j \rangle_t ,
                         \hat x_k - \langle \hat x_k \rangle_t \right\}
                         \hat\rho(t)
                     \right],
\end{equation}
and calculating its evolution \cite{nicacio,wiseman},
the (possible) stationary value of the \hy{CM}{CM} will be the solution of the
\hy{LE}{LE}
\begin{equation}                                                                         \label{lyapeq2}
{\bf V}{\bf \Gamma}^{\! \top} + {\bf \Gamma} {\bf V} + {\bf D} = 0,
\end{equation}
with $\bf \Gamma$ in (\ref{gammamat}) and
\begin{equation}                                                                         \label{dmat}
{\bf D} :=  2 \, {\rm Re}{\bf \Upsilon} = {\bf D}^{\!\top} \in {\rm Mat}(2n,\mathbb R),
\end{equation}
which is positive semidefinite, ${\bf D} \ge 0$, by the definition of $\bf \Upsilon$.
The Lyapunov theorem and its extensions \cite{lyapunov,horn,dullerud}
guarantee that for an \hy{AS}{AS} matrix $\bf \Gamma$,
the solution of Eq.(\ref{lyapeq2}) exists and is unique.
Furthermore, those theorems also relate the stability nature of the matrix $\bf \Gamma$
to the existence of matrices (in our case $\bf V$ and $\bf D$)
satisfying the \hy{LE}{LE} (\ref{lyapeq2}).

We stress that, in order to deduce Eqs.(\ref{mveq}) and (\ref{lyapeq2}),
we do not need any assumptions about the initial state of the system.
The derivation of such equations only uses the \hy{LME}{LME},
the particular structure of Eqs.~(\ref{ham-lind}), and the \hy{CCR}{CCR}.
Meanwhile, the \hy{LME}{LME} with the operators (\ref{ham-lind}) will always preserve the
Gaussian character of an initial Gaussian state \cite{nicacio}.
Once the \hy{CM}{CM} of a steady-state of the system is a solution of (\ref{lyapeq2}),
which is unique and does not depend on the initial state,
any initial state will end in a Gaussian steady-state.

\section{Lyapunov Equations}\label{le}
In this section we show and develop results concerning the generic Lyapunov equation
\begin{equation}                                                                         \label{lyapeq}
{\bf A}\, {\bf P} +
{\bf P} {\bf A}^{\!\dagger} + {\bf Q} = 0,
\end{equation}
and its solution.
Since our objective is to understand properties of stationary solutions
of the \hy{LME}{LME}, it is convenient to assume that
({\it i})   ${\bf A} \in {\rm Mat}(m,\mathbb C)$ is \hy{AS}{AS},
({\it ii})  ${\bf P} = {\bf P}^\dagger \in {\rm Mat}(m,\mathbb C)$ and
({\it iii}) ${\bf Q} = {\bf Q}^\dagger \in {\rm Mat}(m,\mathbb C)$.
From now on, the \hy{LE}{LE} (\ref{lyapeq}) will be represented by the triple
$\lfloor{\bf P},{\bf A},{\bf Q}\rceil$.

The first of those assumptions (${\bf A}$ is \hy{AS}{AS})
is enough to prove \cite{horn,dullerud}
that the {\it unique} solution for the \hy{LE}{LE} in (\ref{lyapeq}) is written as
\begin{equation}                                                                         \label{sol}
{\bf P}({\bf Q},{\bf A}) = \int_{0}^{\infty} \!\!\, dt
             \,\,
             {\rm e }^{{\bf A} t } \,
             {\bf Q}  \,\,
             {\rm e }^{{\bf A}^{\!\dagger} t } \, .
\end{equation}
Furthermore, for any
${\bf A} \in {\rm GL}(m,\mathbb C)$, it is true that
\begin{equation}
{\rm In} \left({\rm e }^{{\bf A}  t } \,
{\bf Q}  \,\,
{\rm e }^{{\bf A}^{\!\dagger} t }\right) =
{\rm In}\left( {\bf Q}  \right),
\end{equation}
because the expression inside the parenthesis on the LHS is a
congruence transformation of $\bf Q$.
The symbol ``$\rm In$'' refers to the {\it inertia index} of a matrix,
as defined in \hyperlink{Appendix}{Appendix I}.
On the other hand, once ${\bf A}$ is \hy{AS}{AS}, then
$\lim_{t \to \infty} {\rm e }^{{\bf A} t } = 0$,
which guarantees the convergence of the integral in (\ref{sol}).
These last arguments about the structure of Eq.(\ref{sol})
are used in the proof of the following result \cite{dullerud,horn}:
\begin{prop}                                                                            \label{prop:inert1}
Consider the solution (\ref{sol})
for the \hy{LE}{LE} in (\ref{lyapeq}).
If ${\bf Q} \ge 0$ (resp. $ {\bf Q} > 0$),  then
${\bf P} \ge 0$ (resp. ${\bf P} > 0$) .
\end{prop} %
\noindent Note that Prop.\ref{prop:inert1} does not exclude the statement
${\bf Q} \ge 0 \Longrightarrow {\bf P} > 0$, since the set of matrices ${\bf P}$
such that $ {\bf P} > 0 $ is a subset of $ {\bf P} \ge 0 $,
{\it cf.} \hyperlink{Appendix}{Appendix I}.
This is the case provided the pair $({\bf Q},{\bf A})$
is observable \cite{horn,dullerud}.
For our purposes, this statement is not necessary,
however it is for the results in \cite{koga}.

Now, let us go a bit further with the results in Prop.\ref{prop:inert1},
specializing the properties of the \hy{AS}{AS} matrix $\bf A$:
\begin{prop}                                                                             \label{prop:inert2}
Consider the \hy{LE}{LE}
$\lfloor{\bf P},{\bf A},{\bf Q}\rceil$ with
${\bf A} = {\bf A}^{\!\dagger}$,
then
${\bf P} \ge 0$ (resp. ${\bf P} > 0$)
if and only if
${\bf Q} \ge 0$ (resp. ${\bf Q} > 0$).
\end{prop}
\begin{proof}[Proof]
Since ${\bf A}$ is self-adjoint and negative
definite (\hy{AS}{AS}), it is possible to write
$ {\bf A} = - \sqrt{-{\bf A}}
              \sqrt{-{\bf A}}$,
where $\sqrt{-{\bf A}}$ is the unique self-adjoint
positive definite square root of $-{\bf A}$.
From the \hy{LE}{LE}~(\ref{lyapeq}),
\begin{eqnarray}
 \!\!\!\!\! {\rm Spec}_{\mathbb R}( {\bf Q}) & = &
- {\rm Spec}_{\mathbb R}
  \left(
        {\bf A}{\bf P} +
        {\bf P} {\bf A}
\right)                                      \\
&=& - {\rm Spec}_{\mathbb R}
\left[
      \left( {\bf P} +
            {\bf A} {\bf P} {\bf A}^{-1}
     \right) {\bf A}
\right]                                      \nonumber \\
& = & {\rm Spec}_{\mathbb R}
\left[ \sqrt{-{\bf A}}
       \left( {\bf P} +
            {\bf A} {\bf P} {\bf A}^{-1}
     \right)
     \sqrt{-{\bf A}}
\right]\!.                                   \nonumber
\end{eqnarray}
Since the sum of positive semidefinite
(resp. positive definite)
matrices is positive semidefinite
(resp. positive definite),
and since a congruence transformation does not change
the signs of the eigenvalues
[or the inertia of the matrix],
it follows that
${\bf Q} \ge 0$ (resp. ${\bf Q} > 0$) if
$ {\bf P} \ge 0$ (resp. ${\bf P} > 0$),
which proves the necessary condition.
The sufficiency is in Prop.\ref{prop:inert1}.
\end{proof}
\noindent Note that, once one statement in Prop.\ref{prop:inert2} is
${\bf Q} > 0 \Longleftrightarrow {\bf P} > 0$, then
the statement
${\bf Q} \ge 0 \Longleftrightarrow {\bf P} \ge 0$ necessarily means that
if $\bf Q$ has one null eigenvalue, then $\bf P$ will also have,
and {\it vice-versa}.

In the direction of the main task of this work,
we must develop some results concerning matrices of the form ${\bf P} + {\bf \Xi}$,
where ${\bf P}$ is the solution in Eq.(\ref{sol}) of
$\lfloor{\bf P},{\bf A},{\bf Q}\rceil$ and
${\bf \Xi} = {\bf \Xi}^\dagger \in {\rm Mat}(m,\mathbb C)$.
\begin{corol}                                                                              \label{cor:if}
${\bf P} + {\bf \Xi} \ge 0 $ \, if \,
${\bf Q}_{[\bf \Xi]} :=
{\bf Q} - {\bf \Xi A}^\dagger - {\bf A \Xi} \ge 0 $.
\end{corol}
\begin{proof}[Proof]
It is easy to see that the \hy{LE}{LE}
$\lfloor {\bf P} + {\bf \Xi},
         {\bf A},
         {\bf Q}_{[\bf \Xi]}   \rceil$
is equivalent to the \hy{LE}{LE} in (\ref{lyapeq}).
Thus, the proof follows from
Prop.\ref{prop:inert1}, since
${\bf Q}_{[\bf \Xi]}={\bf Q}_{[\bf \Xi]}^\dagger$.
\end{proof}
The converse statement of Corol.\ref{cor:if} is not true in general.
By using the restriction for the matrices $\bf A$,
as in Prop.\ref{prop:inert2}, we obtain the following corollary giving a
necessary and sufficient condition.
\begin{corol}                                                                            \label{cor:iff}
Consider ${\bf A} = {\bf A}^{\!\dagger}$ ($\bf A$
is \hy{AS}{AS}),
then  ${\bf P} +  {\bf \Xi}  \ge 0$ \,
if and only if \,
$\widetilde{\bf Q}_{[{\bf \Xi}]} := {\bf Q}
- {\pmb \{} {\bf \Xi, A} {\pmb \}}_{\!+} \ge 0 $.
\end{corol}
\begin{proof}[Proof]
As before, we construct the equivalent
Lyapunov equation
$\lfloor {\bf P} + {\bf \Xi},
         {\bf A},
         \widetilde{\bf Q}_{[\bf \Xi]}   \rceil$,
and use Prop.\ref{prop:inert2} since
$\widetilde{\bf Q}_{[\bf \Xi]}\!=\!
 \widetilde{\bf Q}_{[\bf \Xi]}^\dagger$.
\end{proof}

\noindent Note that, contrary to Props.~\ref{prop:inert1} and~\ref{prop:inert2},
we did not mention the strictly positive cases
(${\bf Q}_{[\bf \Xi]}\!>\!0$, $\widetilde{\bf Q}_{[\bf \Xi]}\!>\!0$) in
Corols.~{\ref{cor:if}} and~\ref{cor:iff}.
Actually, these cases follow the same prescription,
but they are not necessary for our next results.

In what follows, properties of the steady states, driven to equilibrium under
the linear evolution generated by (\ref{ham-lind}), will be considered from the
perspective of the results developed in this section.
\section{ Bona-Fide Relations and Steady-States }\label{ss}
Through the temporal evolution of a state by the \hy{LME}{LME} conditioned
to an \hy{AS}{AS} dynamics, the dependence on the initial condition is progressively
erased by the environmental action.
Therefore the steady-state properties must be completely determined only by
the environment.
An usual way to describe properties of continuous-variable states is given by
\textit{bona-fide} relations involving the \hy{CM}{CM} of the states.
From now on, we will assume that the \hy{CM}{CM} of a quantum state evolves with
$\bf \Gamma $ \hy{AS}{AS} and ${\bf D} \ge 0$ and attains an asymptotic value described
by the \hy{LE}{LE} (\ref{lyapeq2}) with solution
${\bf V} := {\bf P}({\bf D},{\bf \Gamma})$ in Eq.(\ref{sol}).

It is convenient to recall the definitions of the auxiliary matrices defined in
Corols.~{\ref{cor:if}} and~\ref{cor:iff}, but now for the LE in question.
For any matrix ${\bf \Xi} = {\bf \Xi}^\dagger \in {\rm Mat}(2n,\mathbb C)$, we have
\begin{equation}
{\bf D}_{[\bf \Xi]} :=  {\bf D} - {\bf \Xi \, \Gamma}{\!^\top} - {\bf \Gamma \, \Xi}, \,\,\,
\widetilde{\bf D}_{[{\bf \Xi}]} := {\bf D} - {\pmb \{} {\bf \Xi,\Gamma} {\pmb \}}_{\!+}.
\end{equation}

\subsection{Uncertainty Principle}\label{uc}
Any quantum state is subjected to constrains imposed by the uncertainty principle,
which is only a consequences of the \hy{CCR}{CCR}.
For the continuous-variables case,
this principle takes into account only the \hy{CM}{CM} (\ref{cmdef}).
A genuine physical state has a \hy{CM}{CM} such that \cite{simon}
\begin{equation}                                                                         \label{up1}
{\bf V} + i \mathsf J_{2n} \ge 0.
\end{equation}

Given a Hamiltonian and a collection of Lindblad operators
as in (\ref{ham-lind}),
what can our corollaries say about the genuineness of the steady-state
generated by the \hy{LME}{LME}?
Invoking Corol.\ref{cor:if}, the matrix $\bf V$ of a steady-state is
a bona-fide \hy{CM}{CM} if
${\bf D}_{ [i \mathsf J] } =
{\bf D} - i {\bf \Gamma} \mathsf J
        - i \mathsf J {\bf \Gamma}^{\!\top} \ge 0 $.
However, using Eqs.~(\ref{gammamat}) and~(\ref{upsilon}),
it is not difficult to show that ${\bf D}_{[i \mathsf J]} = 2 {\bf \Upsilon}^\ast $,
which is always positive semidefinite, according to the definition of
${\bf \Upsilon}$ in (\ref{upsilon}).
Tautologically, this says that all linear \hy{LME}{LMEs} with
$\bf \Gamma$ \hy{AS}{AS} and ${\bf D} \ge 0$
will drive the system to a steady-state obeying the relation (\ref{up1}), {\it i.e.},
a genuine physical state.

On the other hand, Eq.(\ref{lyapeq2}) is a consequence of the \hy{CCR}{CCR},
as mentioned before.
Accordingly, the \hy{LME}{LME} guarantees that the uncertainty principle holds
for all times, including the steady-state limit,
whereof the condition in Corol.{\ref{cor:if}} is necessary and sufficient
regardless of whether $\bf \Gamma$ is symmetric.
Before going to the next bona-fide relation, it is important to remark that this
extension of Corol.{\ref{cor:if}} to an ``if and only if'' condition is only true
for the relation in Eq.(\ref{up1}).
For all the other relations which will appear in what follows, the differences between
Corols.~\ref{cor:if} and~\ref{cor:iff} should be considered.

\

\subsection{Classical States}\label{cs}
Classical States are defined as having a positive Glauber-Sudarshan distribution
function, they are also called P-re\-pre\-sen\-table states.
This definition relies on the possibility to express a desired state as a classical
mixture of coherent states.
The necessary and sufficient condition for P-representability  of a Gaussian state
is written in terms of its \hy{CM}{CM} as the bona-fide relation
(see \hyperlink{Appendix}{Appendix II})
\begin{equation}                                                                         \label{bfclas}
{\bf V} - \mathsf I_{2n} \ge 0.
\end{equation}

The evolution that drives the system to a classical steady-state is subjected to the
sufficient condition given by Corol.\ref{cor:if}:
\begin{equation}                                                                         \label{clas}
{\bf D}_{[-\mathsf I]}=  {\bf \Gamma} +
                       {\bf \Gamma}^{\!\top} + {\bf D} \ge 0
\Longrightarrow {\bf V} - \mathsf I_{2n} \ge 0.
\end{equation}
The contrapositive of the above statement says that
if a given \hy{LME}{LME} is such that ${\bf D}_{[-\mathsf I]}$
has at least one negative eigenvalue,
it will lead the system to a nonclassical stationary state.
The matrix ${\bf V}$ is, by hypothesis, the \hy{CM}{CM} of a steady-state of
the \hy{LME}{LME}.
Once the converse statement of (\ref{clas}) is not true, one can conclude that
there are classical steady-states which can not be generated by a \hy{LME}{LME}
with dynamical matrices such that ${\bf D}_{[-\mathsf I]} \ge 0$.

If we consider only steady-states generated by a \hy{LME}{LME} with ${\bf \Gamma}$
symmetric, by Corol.\ref{cor:iff} the possible classical states will obey the
necessary and sufficient condition
\begin{equation}                                                                         \label{clas2}
\widetilde{\bf D}_{[-\mathsf I]} = 2{\bf \Gamma} + {\bf D} \ge 0
\Longleftrightarrow {\bf V} - \mathsf I_{2n} \ge 0.
\end{equation}
This means that all states generated by a \hy{LME}{LME} with
${\bf \Gamma} = {\bf \Gamma}^{\!\top}$ and $\widetilde{\bf D}_{[-\mathsf I]} \ge 0$
are classical states.
Conversely, all classical states with \hy{CM}{CM} $\bf V$ which are solutions of
a \hy{LE}{LE} with ${\bf \Gamma} = {\bf \Gamma}^{\!\top}$ are steady-states of a
\hy{LME}{LME} satisfying $\widetilde{\bf D}_{[-\mathsf I]} \ge 0$.

Since classicality is related to mixtures of coherent states,
one instructive example is given by the \hy{CM}{CM}
${\bf V} = \mathsf I_{2n}$ ---
{\it i.e.}, the \hy{CM}{CM} of any $n$-mode (or $n$-\hy{DF}{DF}) coherent state.
The simplicity of this case enables us to derive an useful
necessary and sufficient condition besides the relation in Eq.(\ref{clas2}).
Actually, we will be concerned with the slightly more general situation:
the tensor product of $n$ Gibbs-states with the same occupation number.
These states have the global \hy{CM}{CM} written as ${\bf V} = \alpha \mathsf I_{2n}$,
which is a \hy{CM}{CM} of a classical state if $\alpha \ge 1$.
\begin{corol}                                                                             \label{corol:diag}
For any $\alpha > 0$ and any $\bf \Gamma$ such that
$({\bf \Gamma} + {\bf \Gamma}^{\!\top}) \le 0$,
the matrix ${\bf V} = \alpha \, \mathsf I_{2n}$
is a solution of the \hy{LE}{LE}
$\lfloor{\bf V},{\bf \Gamma},{\bf D}\rceil$
if and only if
$ \alpha ( {\bf \Gamma} + {\bf \Gamma}^{\!\top} )
+ {\bf D} = 0$.
Furthermore,
$\alpha = \tfrac{1}{2}{\rm Tr} {\bf D}/
          {\rm Tr}({\rm Im} {\bf \Upsilon} \mathsf J)$.
\end{corol}
\begin{proof}[Proof]
The sufficient condition is trivially obtained by
constructing the \hy{LE}{LE}
$\lfloor \alpha \mathsf I_{2n},
         {\bf \Gamma}, {\bf D} \rceil$
from (\ref{lyapeq2}).
To prove the necessary condition, one defines
\begin{equation*}
\mathcal I := \int^\infty_0 \!\!\! dt\,
             {\rm e }^{{\bf \Gamma} t } \,
             {\rm e }^{{\bf \Gamma}^{\!\top}\!t }
\end{equation*}
and integrates it by parts to show that
$\mathcal I\, {\bf \Gamma}^{\!\top} \!+\!
{\bf \Gamma} \, \mathcal I =
-\mathsf{I}_{2n}$.
The solution (\ref{sol}) with
${\bf Q} = {\bf D} = -\alpha ({\bf \Gamma} +
                          {\bf \Gamma}^{\!\top})$
and ${\bf A}={\bf \Gamma}$
shows that
${\bf V} = -\alpha ({\bf \Gamma} \, \mathcal I +
           \mathcal I \, {\bf \Gamma}^{\!\top}) =
           \alpha \mathsf I$.
Since ${\bf \Gamma} \in {\rm Mat}(2n,\mathbb R)$,
its complex eigenvalues will occur in
conjugate pairs,
then ${\rm Tr} \, {\bf \Gamma} \in \mathbb R$.
Since it is also \hy{AS}{AS},
${\rm Tr}\,{\bf \Gamma} \ne 0$,
then the value of $\alpha$ holds if one considers
the definition of $\bf \Gamma$ in Eq.(\ref{gammamat})
and the fact that ${\rm Tr}(\mathsf J {\bf H}) = 0$,
since $\bf H$ is symmetric.
\end{proof}

The state studied in Corol.\ref{corol:diag} is an example of the multiplicity of the
steady-state with respect to different matrices $\bf \Gamma$ and $\bf D$.
There is an infinite number of matrices satisfying the relation
$\alpha ( {\bf \Gamma} + {\bf \Gamma}^{\!\top} ) + {\bf D} = 0$ and
giving rise to the same steady state. However, for a given pair of matrices
$\bf \Gamma$ and $\bf D$, the solution ${\bf V(\Gamma, D)}$
is uniquely given in Eq.(\ref{sol}).

\subsection{ Separable States }\label{seps}
A necessary and sufficient condition for an
$n$-mode Gaussian state to be {\it separable} with respect to one of the modes,
say the $k^{\underline{\text{th}}}$ one, is defined in terms of its \hy{CM}{CM} as
${\bf V}^{\!\top_{\!\! k}} + i \mathsf J \ge 0$, where
$ {\bf V}^{\!\top_{\!\! k}} := {\bf T}_{\!k} {\bf V} {\bf T}_{\!k} $.
The transformation
${\bf T}_{\!k} = {\bf T}_{\!k}^{-1} = {\bf T}_{\!k}^\top$
is a local time inversion on the $\hat x$ operator, {\it viz}.,
\begin{equation}
{\bf T}_{\!k} \hat x = (\hat q_1,..., \hat q_k,...,q_n,
                        \hat p_1,...,-\hat p_k,...,\hat p_n)^\top
\end{equation}
and, of course, can not be implemented unitarily.
Since ${\bf T}_{\!k}$ is orthogonal, we can express the separability condition
equivalently as \cite{simon2,werner}
\begin{equation}                                                                         \label{bfsep}
{\bf V} + i \mathsf J^{\!\top_{\!\! k}} \ge 0.
\end{equation}

The statements in Eqs.~(\ref{clas}) and~(\ref{clas2})
can be readily modified to the present case:
\begin{subequations}                                                                     \label{seprel}
\begin{eqnarray}
(\text{Corollary~\ref{cor:if}} ) \,\,\,
{\bf D}_{[i \mathsf J^{\!\top_{\!\! k}}]}  \ge 0
&\Longrightarrow& {\bf V} + i \mathsf J^{\!\top_{\!\! k}} \ge 0;                         \label{seprel:1} \\
(\text{Corollary~\ref{cor:iff}} ) \,\,\,
\widetilde{\bf D}_{[i \mathsf J^{\!\top_{\!\! k}}]}  \ge 0
&\Longleftrightarrow& {\bf V} + i \mathsf J^{\!\top_{\!\! k}} \ge 0.                     \label{seprel:2}
\end{eqnarray}
\end{subequations}
The interpretations of these conditions are also readily adap\-ted
from those in the previous subsection, it is just a question of changing
the dichotomy ``classical/nonclassical'' to ``separable/entangled''.

All classical states (not only the Gaussian ones) are separable, since they are written
as a convex sum of coherent states which are separable [see Eq.(\ref{prep})].
As a consequence, a hierarchy of the dynamics of \hy{LME}{LME}s can be established:
the set of matrices such that
$\widetilde{\bf D}_{[-\mathsf I]} \ge 0$ in (\ref{clas2}) is a subset of those
satisfying
$\widetilde{\bf D}_{[i \mathsf J^{\!\top_{\!\! k}}]} \ge 0$ in (\ref{seprel:2}).
However, this is not true for the matrices in (\ref{clas}) and (\ref{seprel:1}),
because both only give a sufficient condition.

Now, consider the following partition of the number of \hy{DF}{DF}
of a state: $ n = n_1 + n_2$, where $n_i$ is the number of
\hy{DF}{DF} of each partition. Define also the local time inversion operation as
\begin{equation}                                                                         \label{pt}
\!\!{\bf T}_{\!n_2} \hat x =\! (\hat q_1,...,\hat q_n,\hat p_1,...,\hat p_{n_1},
                         -\hat p_{n_1+1},...,-\hat p_{n_1+n_2})^\top\! .
\end{equation}
The separability criteria already exposed is a necessary and sufficient condition
only if $n_2 = 1$. For all other cases, entangled states with
${\bf V} + i {\bf T}_{\!n_2} \mathsf J {\bf T}_{\!n_2} \ge 0$
are bound entangled, {\it i.e.},
they have nondistillable entanglement \cite{werner}.
One can also relate the reservoir properties with this bona-fide relation
through the replacement
$\mathsf J^{\!\top_{\!\! k}} \to {\bf T}_{\!n_2} \mathsf J {\bf T}_{\!n_2}$
in Eqs.(\ref{seprel}).
Note that the bona-fide relation in question does not say whether the state is separable
or bound-entangled.

\subsection{ Gaussian Steerability }
Quantum Steering is a form of correlation related to the ability of one part of a system
to modify the state of a companion system when only local-measurements are performed on
the former.
More precisely, if through local measurements and classical communication
one part of the system is able to convince the other part that they share an
entangled state, the state is said to be steerable with respect to the
first part \cite{wiseman}.

As in the previous subsection,
considering the partition of the \hy{DF}{DF} as $ n = n_1 + n_2$,
a state is {\it non}-Gaussian-steerable with respect to the first part
(with $n_1$ \hy{DF}{DF}) if and only if \cite{wiseman2}
\begin{equation}                                                                         \label{gst}
{\bf V} + i \, {\bf \Pi}_{2} \ge 0,
\,\,\,
{\bf \Pi}_{2} := \tfrac{1}{2} (\mathsf J + {\bf T}_{\!n_2} \mathsf J {\bf T}_{\!n_2}),
\end{equation}
with ${\bf T}_{\!n_2}$ defined in (\ref{pt}).
In other words, it is not possible to steer the state of part 1,
making local Gaussian measurements on part 2, if the last condition holds.
The steering relation with respect to the second part
is obtained by changing the roles of $n_1$ and $n_2$.

As before, this concept can be related to the dynamical matrices,
and one can derive similar formulas by just replacing
$ \mathsf J^{\!\top_{\!\! k}} \to {\bf \Pi}_{2} $ in (\ref{seprel}).
All Gaussian steerable states are entangled \cite{wiseman},
thus the set of matrices such that
$\widetilde{\bf D}_{[i{\bf \Pi}_{2}]} \ge 0$ is a subset of the
ones satisfying
$\widetilde{\bf D}_{[i \mathsf J^{\!\top_{\!\! k}}]} \ge 0$.

\

This concludes our analysis of the bona-fide relations used across this paper.

\section{Symmetries of Steady-States} \label{ess}
In principle, symmetries of the steady-states can be associated with the symmetries
of the dynamics governed by the \hy{LME}{LME} \cite{albert,albert2}.
In the perspective developed in this work,
the relation of the steady-state symmetries
and the symmetries of the dynamical matrices ${\bf \Gamma}$ and ${\bf D}$
will be investigated.

Two \hy{LE}{LEs},
$\lfloor {\bf V},{\bf \Gamma}, {\bf D}   \rceil$ and
$\lfloor {\bf V}',{\bf \Gamma}', {\bf D}'   \rceil$,
are said to be {\it covariant} when their matrices are related by
\begin{equation}                                                                         \label{lyapcov}
{\bf V'} = \mathbf W {\bf V} \mathbf W^{\! \top } , \,\,\,
{\bf \Gamma'} = \mathbf W {\bf  \Gamma} \mathbf W^{-1}, \,\,\,
{\bf D'} = \mathbf W {\bf D} \mathbf W^{\! \top },
\end{equation}
for $ \mathbf W \in {\rm GL}(2n, \mathbb R)$.
Note that, for an orthogonal $\bf W$, all above matrices
are subjected to the same transformation.
This covariance is helpful to determine the invariance properties of
a steady-state, working directly at the level of the \hy{CM}{CM}:

\begin{prop}                                                                              \label{prop:inv}
If $\mathbf \Gamma$ and ${\bf D}$ are invariant under
${\bf W} \in {\rm GL}(2n,\mathbb R)$ [{\it i.e.},
${\bf W}  {\bf \Gamma} {\bf W}^{-1} = {\bf \Gamma}$
and
${\bf W} {\bf D} {\bf W}^{\!\top} = {\bf D}$],
then ${\bf V}$ is invariant as well [{\it i.e.},
${\bf W} {\bf V} {\bf W}^{\!\top} = {\bf V}$].
In this case, we say that $\mathbf W$ is a symmetry
transformation of the Lyapunov equation.
\end{prop}
\begin{proof}[Proof] Writing the solution
${\bf V}({\bf \Gamma},{\bf D})$ as in Eq.(\ref{sol})
for the \hy{LE}{LE}
$\lfloor {\bf V},{\bf \Gamma}, {\bf D} \rceil$,
and since $\det {\bf W} \ne 0$, it is easy to see that
${\bf V}({\bf \Gamma},{\bf D}) =
 {\bf V}({\bf W}  {\bf \Gamma} {\bf W}^{-1},
 {\bf W} {\bf D} {\bf W}^{\top})$,
{\it i.e.}, $\bf V$ remains unchanged.
\end{proof}

Let us give some particular but useful examples. Consider the transformation
\begin{equation}                                                                         \label{transf1}
\mathbf W = \hat \sigma_z \otimes \mathsf I_2 =
\left(
\begin{array}{cc}
\mathsf{I}_2 & 0_2 \\
0_2 & - \mathsf{I}_2
\end{array} \right)
\in {\rm GL}(4, \mathbb R).
\end{equation}
If $\bf \Gamma$ and $\bf D$ are invariant under this transformation,
they are necessarily written as
${\bf \Gamma} = {\bf \Gamma}_1 \oplus {\bf \Gamma}_2$ and
${\bf D} = {\bf D}_1 \oplus {\bf D}_2$, where
${\bf \Gamma}_j$, ${\bf D}_j \in {\rm Mat}(2,\mathbb R)$.
As a consequence of Prop.\ref{prop:inv}, ${\bf V} = {\bf V}_1 \oplus {\bf V}_2$,
{\it i.e.}, it will be the \hy{CM}{CM}
of a state without position-momentum correlations.
This invariance can be retrieved directly from the solution (\ref{sol}):
if $\bf A$ and $\bf Q$ are block-diagonals then $\bf P$ will also be
block diagonal.

Another example is the transformation
\begin{equation}                                                                         \label{change}
\mathbf W =
\left(
\begin{array}{cc}
0_2 & \mathsf{I}_2 \\
\mathsf{I}_2 & 0_2
\end{array} \right)
\in {\rm GL}(4, \mathbb R).
\end{equation}
If $\bf \Gamma$ and $\bf D$ are invariant under this,
the \hy{CM}{CM} will have momentum correlations equal to position correlations:
\begin{equation}                                                                         \label{invchange}
\mathbf V =
\left(
     \begin{array}{cc}
     {\bf V}_{ \! 1} & {\bf V}_{12} \\
     {\bf V}_{12} & {\bf V}_{ \! 1}
     \end{array}                     \right).
\end{equation}

Focusing on the symplectic group, {\it i.e.},
choosing ${\bf W} \in \mathrm{Sp}(2n, \mathbb R) \subset {\rm GL}(2n, \mathbb R)$,
we use the definition of $\bf \Gamma$ in (\ref{gammamat}) and the
covariance relation (\ref{lyapcov}) to write
\begin{equation}                                                                         \label{simcov}
{\bf V}' = {\bf W} {\bf V}{\bf W}^{\!\top}, \,
{\bf H}' = {\bf W}^{-\top} \! {\bf H} {\bf W}^{-1}, \,
\lambda' = {\bf W} {\lambda} .
\end{equation}
This symplectic covariance, by the Stone-von Neumann theorem \cite{gosson},
is nothing but the representation of a unitary transformation of the \hy{LME}{LME}:
\begin{equation}                                                                         \label{unicov}
\hat \rho \longrightarrow \hat {\mathrm U} {\hat \rho} \hat{ \mathrm U}^{\dagger}, \,\,\,
{\hat H}  \longrightarrow \hat {\mathrm U } {\hat H} \hat {\mathrm U}^{\dagger}, \,\,\,
{\hat L_i} \longrightarrow \hat{\mathrm U}  {\hat L}_i \hat {\mathrm U}^{\dagger},
\end{equation}
where the unitary operator $\hat {\mathrm U}$ is the Metaplectic operator
associated with ${\bf W} \in \mathrm{Sp}(2n, \mathbb R)$ \cite{nicacio,gosson}.

If we rearrange the elements of (\ref{cmdef}) consistently with the reordering
$\hat x \mapsto \check x := (\hat q_1,\hat p_1,...,\hat q_n,\hat p_n )$
of (\ref{vecx}), then the invariance under the
(nonreordered) transformation in (\ref{transf1}) implies that the steady-state of the
system, with \hy{CM}{CM} ${\bf V} = {\bf V}_1 \oplus {\bf V}_2$,
is the product state $\hat \rho = \hat \rho_1 \otimes \hat \rho_2$.
In the reordered basis, the transformation (\ref{transf1}) is symplectic.
Similarly, the (nonreordered) matrix in (\ref{change}) is also symplectic
in the reordered basis, and it realizes the exchange of the subsystems.
Consequently, the matrix in (\ref{invchange}) is the \hy{CM}{CM} of states with
same local purity (symmetric states).

As a last example, consider a symplectic rotation
$\mathsf{R} \in {\rm K}(n) := {\rm Sp}(2n,\mathbb R) \cap {\rm O}(2n)$.
As a consequence of its symplecticity and orthogonality, it is written as \cite{simon}
\begin{equation}
\mathsf R =
\left(
       \begin{array}{cc}
          \mathbf Y  & \mathbf Z  \\
         -\mathbf Z & \mathbf Y
       \end{array}
\right)                                                                                \label{MO3}
\end{equation}
with $\mathbf Y, \mathbf Z \in {\rm Mat}(n,\mathbb R)$ satisfying the following
conditions:
\begin{equation}
\mathbf Y\mathbf Y^{\!\top} +
\mathbf Z\mathbf Z^{\!\top} = \mathsf{I}_n,  \,\,\,\,
\mathbf Y\mathbf Z^{\!\top} -
\mathbf Z\mathbf Y^{\!\top} = 0_n .                                                        \label{MO5}
\end{equation}
Any matrix written as
$
{\bf M}: =  m_1 \mathsf I_{2n} + m_2 \mathsf J \in {\rm Mat}(2n,\mathbb R),
$
with $m_1,m_2 \in \mathbb R$ is invariant under the whole group ${\rm K}(n)$.
Note that, if ${\bf M} = {\bf M}^{\!\top}$, then $m_2 = 0$.
If we consider
\begin{equation}
{\bf \Gamma} = -\gamma_1 \mathsf I_{2n} + \gamma_2 \mathsf J, \,\,\,
{\bf D} = \delta \mathsf I_{2n},
\end{equation}
{\it i.e.}, both invariant under ${\rm K}(n)$, then Prop.\ref{prop:inv} implies that
${\bf V} = \nu \, \mathsf I_{2n}$.
By the other side, Corol.\ref{corol:diag} is a necessary and sufficient condition
for this \hy{CM}{CM}, thus $\nu = \tfrac{\gamma}{2 \delta}$.
It is important to mention that the matrices $\bf \Gamma$ and $\bf D$ on
that corollary need not to be invariant.

The subgroup of local rotations in $ {\rm K}(n)$ is described
as the set of matrices (\ref{MO3}) with
\begin{equation}
\mathbf Y = {\rm Diag}(y_{1}, y_{2}, ..., y_{n}),    \,\,\,
\mathbf Z = {\rm Diag}(z_{1}, z_{2}, ..., z_{n}),
\end{equation}
which corresponds to a rotation
\begin{equation}                                                                         \label{locrot}
{\sf R}_i := \begin{pmatrix}
              y_{i} & z_{i} \\ - z_{i} & y_{i}
              \end{pmatrix}                    \in {\rm K}(1)
\end{equation}
in each respective canonical pair $(\hat q_i, \hat p_i)$.
The matrix ${\bf \Gamma} \in {\rm Mat}(2n,\mathbb R)$ is invariant under
the local rotation subgroup if it is of the following form:
\begin{equation}                                                                         \label{locgamma}
{\bf \Gamma}  = \begin{pmatrix}
                {\bf \Gamma}_1 & {\bf \Gamma}_2 \\ - {\bf \Gamma}_2 & {\bf \Gamma}_1
                \end{pmatrix}
\end{equation}
with
${\bf \Gamma}_i := {\rm Diag}(\gamma_{i1},...,\gamma_{in}) \in {\rm Mat}(n,\mathbb R)$.
Since ${\bf D}$ is symmetric, it will be invariant under the same subgroup if it is
written as
\begin{equation}                                                                         \label{locd}
{\bf D} = {\rm Diag}(d_1,...,d_n,d_1,...,d_n) \in {\rm Mat}(2n,\mathbb R).
\end{equation}

Assuming that $\bf \Gamma$ and $\bf D$ have this invariant structure,
Prop.\ref{prop:inv} guaranties that the \hy{CM}{CM} of the steady-state will be
\begin{equation}                                                                         \label{loccm}
{\bf V} = {\rm Diag}(v_1,...,v_n,v_1,...,v_n), \,\, v_i \in \mathbb R \, \forall i,
\end{equation}
which is the \hy{CM}{CM} of $n$-mode thermal state.

\section{ Examples }\label{eI}
Let us now present some examples to show the usefulness of the results presented
in this work.
\vspace{-1.5cm}
\subsection{Two Oscillators interacting with Thermal Baths }\label{to}
\vspace{-0.29cm}
Consider two coupled harmonic oscillators, each one interacting with its own
thermal bath. The frequency of the oscillators are
$\omega_1$ and $\omega_2$ and the spring constant is $\kappa$.

The Hamiltonian of the system is given by (\ref{ham-lind}) with $\xi = 0$, $H_0 = 0$
and
\begin{equation}                                                                         \label{hess1}
{\bf H} = \left[
                 \begin{array}{cc}
                 \omega_1 + \frac{\kappa}{2} & -\tfrac{\kappa}{2} \\
                 -\tfrac{\kappa}{2} & \omega_2 + \frac{\kappa}{2}
                 \end{array}
         \right]  \oplus
          \left[
                 \begin{array}{cc}
                 \omega_1 & 0 \\
                 0        & \omega_2
                 \end{array}
         \right].
\end{equation}
The coupling between a given oscillator and the respective
reservoir is described by the Lindblad operators \cite{wiseman}
\begin{equation}                                                                         \label{tb}
\!\! \hat L_k  = \sqrt{\hbar\zeta_k (\bar N_k + 1)} \, \hat a_k , \,
\hat L_k' = \sqrt{\hbar\zeta_k \bar N_k} \, \hat a_k^\dagger, \, k = 1,2,
\end{equation}
where $\zeta_k \ge 0$ are the bath-oscillator couplings,
$\bar N_k \ge 0$ are thermal occupation numbers,
and $\hat a_k\! := \!(\hat q_k + i \hat p_k)/\sqrt{2\hbar}$
is the annihilation operator associated with mode $k$.
This choice for the reservoirs and Eq.(\ref{ham-lind}) allow us to identify  %
\begin{equation}                                                                         \label{lintb}
\begin{aligned}
{\lambda}_1 & =  \sqrt{\tfrac{\zeta_1}{2} (\bar N_1 + 1)} (i, 0, -1, 0 )^\top,  \\
\lambda'_1  & =  \sqrt{\tfrac{\zeta_1}{2} \bar N_1} ( i , 0 , -1, 0 )^\dagger,  \\
{\lambda}_2 & =  \sqrt{\tfrac{\zeta_2}{2} (\bar N_2 + 1)} (0, i, 0, -1 )^\top,  \\
\lambda'_2  & =  \sqrt{\tfrac{\zeta_2}{2} \bar N_2} ( 0 , i , 0 , -1 )^\dagger.
\end{aligned} \vspace{0cm}
\end{equation}
With the above vectors,
and using Eqs.(\ref{gammamat}), (\ref{upsilon}) and (\ref{dmat}), one finds
\begin{equation}                                                                         \label{dynsys}
\begin{aligned}
&\!\!{\bf D} = \pmb{D} \oplus \pmb{D}, \,
\pmb{D} := {\rm Diag}[ \zeta_1 (2\bar{N}_{1} + 1),\zeta_2 (2\bar{N}_{2} + 1)], \\
&\!\!{ \bf \Gamma }  =
\left[ \begin{array}{cccc}
        -\frac{\zeta_1}{2} & 0 & \omega_1 & 0 \\
         0                  &  -\frac{\zeta_2}{2} & 0  & \omega_2  \\
        -\omega_1 - \frac{\kappa}{2}  & \frac{\kappa}{2}  &  - \frac{\zeta_1}{2} & 0 \\
         \frac{\kappa}{2}  & - \omega_2 - \frac{\kappa}{2}&  0 &- \frac{\zeta_2}{2}
       \end{array}
\right].
\end{aligned}\vspace{0cm}
\end{equation}
For simplicity, we will consider
$ \zeta_1 = \zeta_2 = \zeta$, $\omega_1 = \omega_2 = \omega$,
and the eigenvalues of $\bf \Gamma$ become
\begin{equation}
{\rm Spec}_{\mathbb C}({\bf \Gamma}) =
  \left\{ -\tfrac{\zeta }{2} \pm i \omega,
 -\tfrac{\zeta }{2} \pm i \sqrt{\omega(\omega + \kappa)  } \right\}.
\end{equation}
As one can see, $\bf \Gamma$ is \hy{AS}{AS} since
${\rm Re}\left[ {\rm Spec}_{\mathbb C}({\bf \Gamma}) \right] < 0$
for all (positive) values of the parameters.

The separability of the steady-state will be retrieved from Eqs.(\ref{seprel}).
Since $\bf \Gamma \ne {\bf \Gamma}^{\!\top}$,
Eq.(\ref{seprel:1}) will be applied and, calculating the eigenvalues of
${\bf D}_{ [ i \mathsf J^{\!\top_{\!\! k} } ] }$, one finds
\begin{widetext}
\begin{equation}
{\rm Spec}_{\mathbb R} ({\bf D}_{ [ i \mathsf J^{\!\top_{\!\! 2} } ] } ) =
\left\{
\zeta (\bar{N}_1 + \bar{N}_2 + 1) \pm
\sqrt{ \tfrac{\kappa^2}{2} + \zeta^2 (\bar{N}_1 - \bar{N}_2)^2 + \zeta^2  \pm
\sqrt{ \tfrac{\kappa^4}{4} + \zeta^2 \kappa^2 + 4\zeta^4 (\bar{N}_1 - \bar{N}_2)^2 }}
\right\}.
\end{equation}
\end{widetext}
The steady-state is separable if this spectrum is non-negative,
or explicitly when
\begin{equation}                                                                         \label{sepex1}
\frac{\zeta}{\kappa} \ge
   \mathcal S( \bar{N}_1 , \bar{N}_2 ) := \sqrt{
   \frac{ ( 2\bar{N}_1 + 1 )( 2 \bar{N}_2 + 1 ) }
        { 16 \bar{N}_1 \bar{N}_2 (\bar{N}_1 + 1) (\bar{N}_2 + 1)}  }.
\end{equation}
In this example, states with
$ 0 \in {\rm Spec}_{\mathbb R}({\bf D}_{ [ i \mathsf J^{\!\top_{\!\! k} } ] } )$
are all lying in the surface
$\zeta/\kappa = \mathcal S( \bar{N}_1 , \bar{N}_2 ) $.
The classicality of the steady-state will be determined by Eq.(\ref{clas}) with
\begin{equation}
\begin{aligned}
&{\rm Spec}_{\mathbb R} ({\bf D}_{ [ -\mathsf I ] } ) = \\
&\left\{
\zeta (\bar{N}_1 + \bar{N}_2 ) \pm  \tfrac{\kappa}{2} \pm
 \sqrt{ \tfrac{\kappa^2}{4} + \zeta^2 (\bar{N}_1 - \bar{N}_2)^2 }
\right\},
\end{aligned}
\end{equation}
and the steady-state will be classical if
\begin{equation}                                                                          \label{clasex1}
\frac{\zeta}{\kappa} \ge \mathcal P (\bar{N}_1 , \bar{N}_2):=
\frac{\bar{N}_1 + \bar{N}_2  }{4 \bar{N}_1 \bar{N}_2   }.
\end{equation}
In Fig.\ref{fig1}
we show the functions in (\ref{sepex1}) and (\ref{clasex1}).
Since a classical state is always separable,
$\mathcal S(\bar{N}_1 , \bar{N}_2) < \mathcal P(\bar{N}_1 , \bar{N}_2)$.

\begin{figure}[!htb]
\centering
\includegraphics[width=7.2cm,trim=0 29 0 30]{fig1.pdf}
\caption{%
(Color Online) Separability surface $\mathcal S$ (bottom) and
Classicality surface $\mathcal P$ (top).
States either on or above $\mathcal S$ are separable while
states either on or above $\mathcal P$ are classical.
Between $\mathcal S$ and $\mathcal P$ there are classical
and nonclassical separable states.
Bellow $\mathcal S$, there are separable and entangled states.
The divergence of both functions for $\bar N_1 \to 0$ and for
$\bar N_2 \to 0$ are not shown in this plot.
}                                                                                \label{fig1}
\end{figure}

To understand the sufficiency of the results for the system in consideration,
as a consequence of the fact that ${\bf \Gamma} \ne {\bf \Gamma}^{\!\top}$,
we will explore the separability and classicality criteria directly applying both to the
\hy{CM}{CM} of the steady-state.
Using Eqs.~(\ref{dynsys}), we are able to obtain analytically the
solution ${\bf V} = {\bf P}({\bf \Gamma},{\bf D})$ using (\ref{sol})
or solving algebraically the \hy{LE}{LE} $\lfloor {\bf V}, {\bf \Gamma}, {\bf D} \rceil$.
For simplicity and without loss of generality, we choose $\bar N_1 = \bar N_2 = \bar N$
and the solution is
\begin{equation}                                                                         \label{sscm}
\begin{aligned}
{\bf V} &= (2\bar N +1) \mathsf{I}_{4} \, +   \\
&  \frac{(2\bar N +1)\kappa}{\zeta^2 + 4\omega(\omega+\kappa)}
\left[ \begin{array}{cc}
  \omega & \tfrac{1}{2} \zeta  \\
  \tfrac{1}{2} \zeta & (\omega+\kappa)
 \end{array} \right]\! \otimes \!
 \left[\! \begin{array}{rr}
  -1 & 1  \\
   1 & -1
 \end{array} \!\right].
\end{aligned}
\end{equation}
The steady-state is classical if and only if the above \hy{CM}{CM} obeys (\ref{bfclas}),
or working out its eigenvalues, if and only if
\begin{equation}                                                                         \label{clasex1a}
\frac{\zeta}{\kappa} \ge
   \mathcal P'( \bar{N} ) := \sqrt{ \frac{1}{4\bar{N}^{2}} -
                                    \left(\frac{2\omega}{\kappa} + 1\right)^2   }.
\end{equation}
As for separability, the steady state will be separable if and only if
the condition (\ref{bfsep}) is satisfied, which reads as
\begin{equation}                                                                         \label{sepex1a}
\frac{\zeta}{\kappa} \ge
   \mathcal S'( \bar{N} ) := \sqrt{
                                    \frac{ 1 } { 16 \bar{N}^2(\bar{N} + 1)^2} -
                                     \left(\frac{2\omega}{\kappa} + 1\right)^2  }.
\end{equation}
In Fig.~\ref{fig2}, the two functions representing the necessary and sufficient
conditions in Eqs.~(\ref{clasex1a}) and~(\ref{sepex1a}) are shown.
We also compare them with the two sufficient conditions
(\ref{sepex1}) and (\ref{clasex1}), already plotted in Fig.~\ref{fig1}.
It is clear that, for the system in question, even if the sufficient criteria become
tighter for smaller values of $\bar{N}$, they are in general unable to determine
whether a state is entangled or nonclassical.

\begin{figure}[!htbp]
\centering
\includegraphics[width=8.cm,trim=0 0 0 0]{fig2.pdf}
\caption{%
(Color Online) Necessary and sufficient conditions for classicality
and for separability of the steady-state.
We show the functions $\mathcal P'( \bar{N} )$ in (\ref{clasex1a})
and $\mathcal S'( \bar{N} )$ in (\ref{sepex1a}),
as indicated in the graph.
The regions limited by these two functions indicates the
nature of the steady-state.
We also plot the functions
$\mathcal S(\bar{N},\bar{N})$ in (\ref{sepex1}) and
$\mathcal P(\bar{N},\bar{N})$ in (\ref{clasex1})
as dashed curves, see Fig.~\ref{fig1}. In this plot $\omega/\kappa = 0.5$.
}                                                                                \label{fig2}
\end{figure}

For the Gaussian-steering property, we also consider the case $\bar N_1=\bar N_2 =\bar N$
and the sufficient condition is determined by calculating the matrix
${\bf D}_{ [ i{\bf \Pi}_{k} ]}$
[the matrix ${\bf \Pi}_{k}$ is defined in (\ref{gst}), and here ${n_1} = {n_2} = 1$],
which has the eigenvalues
\begin{eqnarray}
&&\!\!\!\!\!\!{\rm Spec}_{\mathbb R} (
{\bf D}_{ [ i{\bf \Pi}_{1}] } )  =
{\rm Spec}_{\mathbb R} (
{\bf D}_{ [ i{\bf \Pi}_{2}] } )  =       \nonumber \\
&&\!\!\!\!\!\! \left\{ ( 2 \bar{N} +1), \zeta( 2 \bar{N} +1) \pm \sqrt{4\zeta^2 + \kappa^2}
\right\}.
\end{eqnarray}
The steady-state will be {\it non}-Gaussian steerable with respect to both partitions if
$\zeta/\kappa \ge [4 (2\bar{N} + 1)^2 + 4]^{-1/2}$.

\subsection{Two oscillators interacting with thermal baths in RWA}\label{tor}
Let us consider the same system as before, but with the Hamiltonian
\begin{equation}                                                                         \label{hess2}
{\bf H} = \left[
                 \begin{array}{cc}
                 \varpi_1  & \Omega \\
                 \Omega    & \varpi_2
                 \end{array}
         \right]  \oplus
          \left[
                  \begin{array}{cc}
                 \varpi_1  & \Omega \\
                 \Omega  &   \varpi_2
                 \end{array}
         \right].
\end{equation}
This Hamiltonian is derived from the one in Eq.(\ref{hess1}) by applying
a rotating wave approximation (RWA).
The procedure and the validity of this result are carefully
discussed in the Appendix of Ref.\cite{nicacio2},
as well as the relation among the coupling constant $\Omega$ in Eq.(\ref{hess2})
with the parameters in Eq.(\ref{hess1}),
see \cite{footnote2} for details.
Considering also the same structure for the reservoirs in (\ref{lintb}), one finds
\begin{equation}                                                                         \label{dynsys2}
{ \bf \Gamma }  =
          \left[
                 \begin{array}{cccc}
                 -\tfrac{\zeta_1}{2}  & 0 & \varpi_1 & \Omega  \\
                 0 & -\tfrac{\zeta_2}{2} &  \Omega & \varpi_2 \\
                 -\varpi_1  & -\Omega   &     -\tfrac{\zeta_1}{2} & 0 \\
                 -\Omega    & -\varpi_2 & 0 & -\tfrac{\zeta_2}{2}
                 \end{array}
         \right],
\end{equation}
which is \hy{AS}{AS} with eigenvalues
\begin{equation}
\!\!\! {\rm Spec}_{\mathbb C}({\bf \Gamma}) \!= \!
\left\{
         -\tfrac{\zeta_1+\zeta_2}{4} \pm
         \tfrac{1}{4}\sqrt{(\zeta_1 - \zeta_2)^2 - 4 \Omega^2} \pm i \varpi
                                                                            \right\},
\end{equation}
where we used $\varpi_1 = \varpi_2 =: \varpi$.
Note that $\bf \Gamma$ in (\ref{dynsys2}) and $\mathbf D$ in (\ref{dynsys})
are invariant under a rotation by $\mathsf J \in {\rm K}(2)$.
Following Prop.\ref{prop:inv},
the steady-state \hy{CM}{CM} for this system will also be, {\it i.e.},
$ \mathbf V = \mathsf J \mathbf V \!\mathsf J^{\!\top}$,
which means that position-momentum correlations are antisymmetric and
position-position correlations are equal to momentum-momentum correlations.
This symmetry help us to solve algebraically the \hy{LE}{LE} (\ref{lyapeq2}),
obtaining \cite{assadian}
\begin{equation}                                                                         \label{cmex2}
{\bf V} = \left[
                 \begin{array}{cccc}
                 v_{1}    & 0        & 0       & v_{14}     \\
                 0        & v_{2}    & -v_{14} & 0          \\
                 0        & - v_{14} &  v_{1}  & 0          \\
                 v_{14}   & 0        & 0       & v_{2}      \\
                 \end{array} \right],
\end{equation}
with
\begin{eqnarray}
v_1 &=& 2 \frac{\zeta_1\bar{N}_1 + \zeta_2\bar{N}_2}{\zeta_1 + \zeta_2} +
          \frac{2(\bar{N}_1-\bar{N}_2)\zeta_1\zeta_2^2}
               {(\zeta_1 + \zeta_2)(4\Omega^2 + \zeta_1\zeta_2)}+1,                      \nonumber\\
v_2 &=&  2 \frac{\zeta_1\bar{N}_1 + \zeta_2\bar{N}_2}{\zeta_1 + \zeta_2} +
           \frac{2(\bar{N}_2-\bar{N}_1)\zeta_1^2\zeta_2}
               {(\zeta_1 + \zeta_2)(4\Omega^2 + \zeta_1\zeta_2)} + 1,                    \nonumber\\
v_{14} &=& \frac{4\zeta_1\zeta_2\Omega (\bar{N}_2 - \bar{N}_1)}
                {(\zeta_1+\zeta_2)(\zeta_1\zeta_2 + 4\Omega^2)}.
\end{eqnarray}
Note that, if $\zeta_2 = 0$, then ${\bf V} = (2\bar{N}_1 + 1) \mathsf I_4$,
which has a simple structure, but it can not be simply retrieved by symmetries of
$\bf \Gamma$ and $\bf D$.

The simple form of (\ref{cmex2}) can be used to explicitly analyze the results in
conditions in (\ref{clas}).
Considering for simplicity $ \zeta_1 = \zeta_2 = \zeta$,
the (doubly degenerate) spectrum of (\ref{cmex2}) is
\begin{equation}                                                                         \label{espex2}
{\rm Spec}_{\mathbb R} ({\bf V})  =
 \left\{  \bar{N}_1 + \bar{N}_2 + 1 \pm
          \frac{(\bar{N}_1 - \bar{N}_2)} {\sqrt{1+4\Omega^2/\zeta^2 }} \right\}.
\end{equation}
From this, it is easy to see that the state (\ref{cmex2}) is classical
for any values of the parameters, since ${\bf V} - \mathsf I_{4} \ge 0$.
On the other hand, let us calculate
\begin{equation}                                                                         \label{clasex2}
{\rm Spec}_{\mathbb R}({\bf D}_{ [ -\mathsf I ] } )  =
\left\{ 2\zeta \bar{N}_1, 2\zeta \bar{N}_2 \right\},
\end{equation}
which is non-negative for any value of $\bar{N}_1$ and $\bar{N}_2$.
The statement in (\ref{clas}) thus tells us that all steady-states of this system
belong to the set of classical states,
which was already found by means of Eq.(\ref{espex2}).
\subsection{Cascaded OPO}\label{opo}
Consider an optical parametric oscillator (OPO) coupled to the vacuum field \cite{koga}.
The Hamiltonian is written as
$\hat H = i\hbar\epsilon ({\hat a^{\dagger 2}} - {\hat a}^2)/4$,
where $\epsilon \ge 0$ denotes the effective pump intensity.
The coupling with the vacuum is described by the operator
$\hat L = \sqrt{\hbar\kappa} \hat a $, where $\kappa >0$ is
the damping cavity rate.

With the help of Eqs.~(\ref{ham-lind}), (\ref{gammamat}), (\ref{upsilon}),
and~(\ref{dmat}), one readily reaches
\begin{equation}                                                                         \label{opo1}
{ \bf \Gamma }  =
\left[ \begin{array}{cc}
        \frac{1}{2}(\epsilon -\kappa) & 0  \\
        0         & -\frac{1}{2}(\epsilon +\kappa)
       \end{array}
\right], \,\,\,
{\bf D} = \kappa \, \mathsf I_2.
\end{equation}
The matrix $\bf \Gamma = {\bf \Gamma}^{\!\top}$
will be \hy{AS}{AS} in so far as $\kappa > \epsilon$.
Also, ${\bf D} >0$, once $\kappa >0$. Following the statement in (\ref{clas2}),
since $\widetilde{\bf D}_{[i \mathsf J^{\!\top_{\!\! 2}}]} =
           {\rm Diag}\left(\epsilon, -\epsilon \right)$,
the steady-state will be nonclassical if $\epsilon \ne 0$ and
will be classical only if $\epsilon = 0$.
The \hy{LE}{LE} is trivially solved to give the \hy{CM}{CM} of the steady-state:
\begin{equation}
{\bf V} = {\rm Diag}[\kappa/( \kappa -\epsilon),\kappa/( \kappa +\epsilon)],
\end{equation}
which is a squeezed thermal state and corresponds to the coherent-state
solution in \cite{koga} if $\epsilon = 0$.

For a cascaded OPO \cite{koga},
the system is described by the Hamiltonian
$\hat H = \hat H_1 + \hat H_2 + \frac{1}{2i}(\hat L_1^\dagger \hat L_2
        - \hat L_2^\dagger \hat L_1) $
with
$\hat H_j = i\hbar\epsilon_j ({\hat a_j^{\dagger 2}} - {\hat a_j}^2)/4$
and $\hat L_j = \sqrt{\hbar\kappa} \hat a_j $.
The coupling of the system with the intracavity vacuum is
represented by $\hat L = \hat L_1 + \hat L_2$. %
Under these circumstances we write
\begin{equation}                                                                         \label{opod}
{\bf D} =
          \left[
                 \begin{array}{cc}
                 \kappa  & \kappa \\
                 \kappa  & \kappa
                 \end{array}
         \right]  \oplus
          \left[
                 \begin{array}{cc}
                 \kappa  & \kappa \\
                 \kappa  & \kappa
                 \end{array}
         \right],
\end{equation}
and
\begin{equation}                                                                         \label{opog}
{ \bf \Gamma }  =
         \left[\!\!
                 \begin{array}{cc}
                  \frac{\epsilon_1-\kappa}{2} & 0 \\
                  -\kappa      &  \frac{\epsilon_2-\kappa}{2}
                 \end{array}\!\!
         \right] \! \oplus \!
          \left[\!\!
                  \begin{array}{cc}
                   -\tfrac{\epsilon_1+\kappa}{2}  & 0   \\
                  -\kappa    & -\tfrac{\epsilon_2+\kappa}{2}
                 \end{array}\!\!
         \right]
\end{equation}
with spectrum given by
\begin{equation}
{\rm Spec}({\bf \Gamma}) =
\left\{ -\tfrac{1}{2} (\kappa \pm \epsilon_1),
        -\tfrac{1}{2} (\kappa \pm \epsilon_2) \right\}.
\end{equation}
Note that ${\bf \Gamma} \ne {\bf \Gamma}^{\!\top}$ is \hy{AS}{AS} if
$\kappa > \max\{|\epsilon_1|,|\epsilon_2|\}$.
Now the eigenvalues of ${\bf D}_{[i \mathsf J^{\!\top_{\!\! 2}}]}$
in (\ref{seprel}) can be calculated, yielding
\begin{equation}
{\rm Spec}_{\mathbb R} ({\bf D}_{[i \mathsf J^{\!\top_{\!\! 2}}]}) =
\{
( 1 \pm \sqrt{5})\kappa, 2\kappa, 0\},
\end{equation}
which shows that the state will be always entangled
following Corol.\ref{cor:if}.
The same arguments can be applied to determine the
Gaussian steerability. Calculating the matrix
${\bf D}_{ [ i{\bf \Pi}_{k} ]}$
[the matrix ${\bf \Pi}_{k}$ is defined in (\ref{gst}), and here ${n_1} = {n_2} = 1$],
which has the eigenvalues
\begin{equation}
\begin{aligned}
{\rm Spec}_{\mathbb R} ({\bf D}_{ [ i{\bf \Pi}_{1} ] } ) & =
\left\{   ( 1 \pm \sqrt{5} ) \frac{ \kappa }{ 2 },
          ( 3 \pm \sqrt{5} ) \frac{ \kappa }{ 2 }   \right\}, \\
{\rm Spec}_{\mathbb R} ({\bf D}_{ [ i{\bf \Pi}_{2} ] } ) & =
\left\{   ( 3 \pm \sqrt{17} ) \frac{ \kappa }{ 2 }, k , 0     \right\}.
\end{aligned}
\end{equation}
From these, one concludes that the state will be always Gaussian steerable with respect
to both modes since these spectra is nonpositive.

Following Prop.\ref{prop:inv}, the \hy{CM}{CM} of the steady-state of the cascaded OPO
will have the symmetry induced by (\ref{transf1}), and reads
\begin{equation}                                                                         \label{opocm}
{\bf V}  = \left[\begin{array}{cc}
                 \frac{k}{k-\epsilon_1}     & -\frac{2\kappa \epsilon_1}{g_{-}} \\
                 - \frac{2\kappa \epsilon_1}{ g_{-}} & -\frac{\kappa  h_+}{ g_-}
                 \end{array}\right] \oplus
           \left[\begin{array}{cc}
                 \frac{k}{k+\epsilon_1}     & \frac{2\kappa \epsilon_1}{ g_{+}} \\
                 \frac{2\kappa \epsilon_1}{ g_{+}}   & \frac{\kappa h_-}{ g_+}
                 \end{array}\right],
\end{equation}
where we have defined
\begin{eqnarray*}
g_{\pm} &=& (\epsilon_1 + \epsilon_2 \pm 2 \kappa)(\epsilon_1\pm \kappa) , \nonumber \\
h_{\pm} &=& (\epsilon_1^2 + \epsilon_1 \epsilon_2 \pm \epsilon_1 \kappa
             + 2 \kappa^2 \mp \kappa \epsilon_2)(\epsilon_2\mp k)^{-1}.
\end{eqnarray*}
\subsection{OPO and Thermal Baths}\label{otb}
Consider the Hamiltonian dynamics of two particles described by
\begin{equation}
\hat H = \frac{\epsilon}{4} {\pmb\{}\hat q_1,\hat p_1{\pmb\}}_{\!+} +
         \frac{\epsilon}{4} {\pmb\{}\hat q_2,\hat p_2{\pmb\}}_{\!+} +
         \frac{\kappa}{2} (\hat q_2 \hat p_1 + \hat p_2 \hat q_1).
\end{equation}
This Hamiltonian is similar to the one in the previous example,
it is basically the Hamiltonian of the cascaded OPO
with a phase change \cite{wiseman}.
If the particles are in contact with the thermal baths,
as in (\ref{tb}), the dynamical matrices become
\begin{equation}
{ \bf \Gamma }  =  {{\bf \Gamma}^{\!\top}} =
        \tfrac{1}{2} \!\left[\!\!
                 \begin{array}{cc}
                  \epsilon - \zeta & k \\
                  k      &  \epsilon-\zeta
                 \end{array}\!\!
         \right] \! \oplus
         \tfrac{1}{2}\! \left[\!\!
                  \begin{array}{cc}
                 - (\epsilon + \zeta) & k \\
                  k      & - (\epsilon+\zeta)
                 \end{array}\!\!
         \right],
\end{equation}
and $\bf D$ is as in (\ref{dynsys}),
and now we are considering $\bar N_1 = \bar N_2 = \bar N$ and $\zeta_1 = \zeta_2 = \zeta$.
The eigenvalues of $\bf \Gamma$ are
\begin{equation}
{\rm Spec}({\bf \Gamma}) =
\left\{ -\tfrac{1}{2} \zeta \pm \tfrac{1}{2}( \epsilon + \kappa),
        -\tfrac{1}{2} \zeta \pm \tfrac{1}{2}( \epsilon - \kappa) \right\},
\end{equation}
and it will be \hy{AS}{AS} if $\zeta > \epsilon + \kappa$.
The eigenvalues of ${\bf D}_{[-\mathsf I]}$ in (\ref{clas2}) are
\begin{equation}
{\rm Spec}_{\mathbb R} ({\bf D}_{ [ -\mathsf I ] } ) =
\left\{
2\zeta \bar{N} \pm (\epsilon + \kappa), 2\zeta \bar{N} \pm (\epsilon - \kappa)
\right\},
\end{equation}
which shows that the state will be classical if and only if
$2\zeta \bar{N} \ge (\epsilon + \kappa)$.
The eigenvalues of $\widetilde{\bf D}_{[i \mathsf J^{\!\top_{\!\! 2}}]}$
in (\ref{seprel:2}) are
\begin{equation}
{\rm Spec}_{\mathbb R} (\widetilde{\bf D}_{[i \mathsf J^{\!\top_{\!\! 2}}]}) =
\{
2\zeta \bar{N} \pm \kappa, 2\zeta(\bar{N} + 1) \pm \kappa \},
\end{equation}
and the steady-state will be entangled if and only if
$2\zeta \bar{N} < \kappa$.
In the interval $ (\epsilon + \kappa) > 2\zeta \bar{N} \ge \kappa$,
the state is nonclassical and separable.
The steerability of the state is determined by
\begin{eqnarray}
&&\!\!\!\!\!\!{\rm Spec}_{\mathbb R} (
\widetilde{\bf D}_{ [ i{\bf \Pi}_{1}] } )  =
{\rm Spec}_{\mathbb R} (
\widetilde{\bf D}_{ [ i{\bf \Pi}_{2}] } )  =       \nonumber \\
&&\!\!\!\!\!\! \left\{ ( 2 \bar{N} +\tfrac{1}{2}) \zeta \pm \sqrt{\zeta^2 + \kappa^2},
        ( 2 \bar{N} +\tfrac{3}{2}) \zeta \pm \sqrt{\zeta^2 + \kappa^2}    \right\}.
\end{eqnarray}
As a consequence of the chosen parameters,
the steady state is symmetric with respect to the steerings of
both partitions. This is state is steerable if and only if
$( 2 \bar{N} +\tfrac{1}{2}) \zeta < \sqrt{\zeta^2 + \kappa^2} $.

\section{Engineering Steady-States}\label{eps}
The unavoidable influence of uncontrollable degrees of freedom are usually responsible
for losses of the quantumness of a system through the procedure called decoherence.
However, a steady-state with a desired quantum property can be produced by controlling
the parameters of the system and of the environmental action.
As the examples of the last section,
systems of bosonic degrees of freedom have been extensively studied in what concerns
entanglement generation \cite{tan,houhou},
production of pure states \cite{koga,ikeda}, and
engineering of graph states \cite{koga,houhou}.
On the experimental side,
realizations of these techniques in the context of atomic ensembles were already
performed \cite{krauter}.

To develop a simple theoretical engi\-nee\-ring-sta\-te method for
bosonic degrees of freedom, we will use the results provided by the
Williamson theorem \cite{williamson,gosson,simon}:
\begin{theo}[Williamson]
Let
$\mathbf M \in {\rm Mat}(2n,\mathbb R)$
be a positive definite matrix:
$\mathbf M = \mathbf M^\top > 0$.
This matrix can be diagonalized by
a symplectic congruence, {\it i.e.}, there exists
$\mathsf S \in {\rm Sp}(2n, \mathbb R)$
such that
\begin{equation}                                                                         \label{tw1}
\mathsf S \mathbf M \mathsf S^\top
=
{\rm Diag}(\mu_1,...,\mu_n,\mu_1,...,\mu_n) =:{\bf \Lambda },
\end{equation}
where
$\mu_j \ge \mu_k > 0 \,\,\, \text{for} \,\,\, j > k. $
\end{theo}
\noindent The double-paired ordered set (or the diagonal matrix) $\bf \Lambda$
is called {\it symplectic spectrum} of $\mathbf M$, and $\mu_k$ are
its symplectic eigenvalues (\hypertarget{SE}{SE}).
These can be found from the (Euclidean) eigenvalues of
$\mathsf J \mathbf M$ \cite{gosson}, which turn out to be
\begin{equation}                                                                         \label{tw2}
{\rm Spec_{\mathbb C}}(\mathsf J \mathbf M) =
{\rm Diag}(i\mu_1, ...,i\mu_n,-i\mu_1,...,-i\mu_n).
\end{equation}

Suppose one wants to design a reservoir structure able to produce a steady-state
(with $n$ degrees of freedom) described by a $2n \times 2n$ \hy{CM}{CM} ${\bf V}'$.
If one identifies this \hy{CM}{CM} with $\bf M$ in (\ref{tw1}),
the first step is to find a suitable \hy{LE}{LE}
able to produce its corresponding symplectic spectrum $\bf \Lambda$ as a solution,
{\it i.e.}, its is necessary to find matrices ${\bf \Gamma}'$ and ${\bf D}'$
satisfying the LE $\lfloor {\bf \Lambda}, {\bf \Gamma}', {\bf D}' \rceil$.
Assuming that it is possible to design a system-reservoir structure with this LE,
one applies the symplectic covariance (\ref{simcov}) to it and finds
\begin{equation}                                                                         \label{engcov}
\left\lfloor  {\bf V}' ,
         \mathsf S^{-1} {\bf \Gamma}'\mathsf S,
         \mathsf S^{-1} {\bf D}'\mathsf S^{-\top} \right\rceil,
\end{equation}
which, by Eq.(\ref{tw1}), is the \hy{LE}{LE} with solution
${\bf V }' = \mathsf S^{-1} {\bf \Lambda }\mathsf S^{-\top}$.
Reservoir engineering can then be realized by finding some convenient
matrix $\bf \Lambda$ and a system-reservoir structure suitable to the unitary
transformations, as in (\ref{unicov}).
Following Eq.(\ref{simcov}), the engineered Hamiltonian and Lindblad operators
will be, respectively, such that
${\bf H}' = {\sf S}^{\!\top} \! {\bf H} {\sf S}$ and
$\lambda' = {\sf S}^{-1} {\lambda}$.

A peculiar example using (\ref{engcov}) appears when one is able to produce a
reservoir structure such that
${\bf D}' = \beta \bf \Lambda$ and
${\bf \Gamma}' = - \beta \mathsf I_{2n}$ with $\beta > 0$,
thus the use of (\ref{tw1}) gives that
$\left\lfloor  {\bf V}', - \beta \mathsf I_{2n}, \beta {\bf V}' \right\rceil$.
This shows that a Lindblad equation with
${\bf \Upsilon} = \tfrac{1}{2}\beta {\bf V}' + i \beta \mathsf I_{2n}$ [see Eq.(\ref{upsilon})]
will have a steady-state automatically given by a Gaussian with \hy{CM}{CM} ${\bf V}'$.

In fact, the matrix $\bf \Lambda$ in (\ref{tw1}) is like the
\hy{CM}{CM} in (\ref{loccm}),
thus any $\bf \Gamma'$ as in (\ref{locgamma}) and $\bf D'$ as in (\ref{locd}),
which are invariants under the local rotations in (\ref{locrot}),
are appropriate to the first step of the method.
It is noticeable that not all reservoir structure are suitable to produce
a diagonal matrix $\bf \Lambda$ as a \hy{CM}	{CM} of a steady-state,
since it can not have the desired invariant structure for performing the
first step.
However, it is still possible to use the covariance relation (\ref{simcov})
to design a specific steady-state from a known simple steady-state of some system,
one of these cases (the OPO in Sec.~\ref{opo}) will be analyzed at the end of this section.

The special condition presented in Corol.\ref{corol:diag} can be used to engineer a
specific and important class of states.
Explicitly, we will use the relation (\ref{simcov}) to generate a state with \hy{CM}{CM}
${\bf V}' = \alpha \mathsf{SS}^{\!\top}$, where
$\mathsf S \in {\rm Sp}(2n,\mathbb R)$ and $\alpha \ge 1$.
In other words, we want to know which matrices
$\bf \Gamma_{\rm p}$ \hy{AS}{AS} and
${\bf D}_{\rm p} \ge 0$
are necessary to construct a \hy{LE}{LE} with the solution given
by the desired \hy{CM}{CM}.
Note that the above-mentioned \hy{CM}{CM} represents a pure state if and only if
$\alpha = 1$ \cite{koga,simon,gosson}.

As stated in Corol.\ref{corol:diag}, the $n$-mode Gibbs state with
${\bf V} = \alpha \mathsf I_{2n}$ is generated by any \hy{LE}{LE} of the form
$\lfloor {\bf V}, {\bf \Gamma}', -\alpha( {\bf \Gamma}' + {\bf \Gamma}'^{\!\top} ) \rceil$,
for all matrices $\bf \Gamma'$ \hy{AS}{AS} such that
$( {\bf \Gamma}' + {\bf \Gamma}'^{\!\top} ) \le 0$.
Using the relation (\ref{simcov}), we are able to obtain a covariant \hy{LE}{LE}
$ \lfloor {\bf V}',
          {\bf \Gamma}_{\rm p}, {\bf D}_{\rm p} \rceil$
with
\begin{equation}                                                                         \label{es1}
\!\!\!{\bf V}' = \alpha \mathsf{SS}^{\!\top}, \,\,
{\bf \Gamma}_{\rm p} = \mathsf S {\bf \Gamma}' \mathsf S^{-1}, \,\,
{\bf D}_{\rm p} = -\alpha \, \mathsf S ( {\bf \Gamma}' +
                                         {\bf \Gamma}'^{\!\top} )\mathsf S^{\!\top}.
\end{equation}
It is interesting to note that, recalling Corol.\ref{corol:diag},
the value of $\alpha$ only depends on the reservoir structure
(it does not depend on the Hamiltonian of the system) used to
prepare the initial state ${\bf V} = \alpha \mathsf I_{2n}$;
conveniently the Hamiltonian at this stage can be taken as zero.
Another remarkable fact is that any one-mode Gaussian state $(n = 1)$
is included in the scheme provided by Eq.(\ref{es1}):
these states have a \hy{CM}{CM} written as ${\bf V}'$ in ({\ref{es1}})
with $\alpha = \mu_1 \ge 1$ and $\mathsf S \in {\rm Sp}(2,\mathbb R)$,
as a consequence of the Williamson theorem.

In \cite{koga}, the authors established specific conditions for a system to be driven to a
pure steady-state when evolving under the \hy{LME}{LME} subjected
to the restrictions in (\ref{ham-lind}).
Equation (\ref{es1}) with $\alpha = 1$ constitutes a simple connection with some
of their results.
\subsection*{Examples}
\paragraph{Two Mode Thermal Squeezed States (TMTSS).}
Consider the following \hy{CM}{CM}
\begin{equation}                                                                         \label{cm02}
{\bf V}' =  (2\bar n + 1) \mathsf{S}_r \mathsf{S}_r^{\!\top},
\end{equation}
with
\begin{equation}                                                                         \label{ex1}
{\mathsf S_r} := \left[
\left(\! \begin{array}{cc}
       \cosh r & \sinh r\\
       \sinh r & \cosh r\\
       \end{array} \! \right) \!  \oplus \!
\left( \! \begin{array}{cc}
       \cosh r& -\sinh r\\
       -\sinh r& \cosh r\\
       \end{array} \!\! \right)  \right],
\end{equation}
where $r \ge 0$ is the squeezing parameter and
$\bar n \ge 0$ is the mean number of thermal photons of both modes.
Note that ${\mathsf S_r} \in {\rm Sp}(4,\mathbb R)$.
Following the criteria (\ref{bfsep}) and (\ref{gst}),
the state in (\ref{cm02}) will be respectively
entangled iff $r > {\rm ln}(2\bar n + 1)$
and steerable iff $r > {\rm cosh}^{-1}(2\bar n + 1)$.

If one wants to engineer a reservoir with the steady-state given in (\ref{cm02}),
it is possible to apply the scheme in Eq.(\ref{es1}).
For this purpose, one needs a reservoir structure able to produce
a steady-state of the form ${\bf V} = \alpha \mathsf I_{2n}$ with
$\alpha = (2\bar n + 1)$.
An example of a system with this steady-state
is the one in Sec.\ref{tor} with $\bar N_1 = \bar N_2$.
The system in Sec.\ref{to} can also be used to this end,
but now, besides the condition $\bar N_1 = \bar N_2$,
one needs to take $\kappa = 0$  --- this condition is necessary
to guarantee that ${\bf \Gamma} + {\bf \Gamma}^{\!\top} \le 0$,
as required in Corol.\ref{corol:diag}.
Recalling that the Hamiltonian dynamics does not affect the value
of $\alpha$ in (\ref{es1}), one can use either
$\omega_1 = \omega_2 = \kappa = 0$ and $\zeta_1 = \zeta_2 := \zeta$
in (\ref{dynsys}), or
$\varpi_1 = \varpi_2 = \Omega = 0$ and $\zeta_1 = \zeta_2 := \zeta$
in (\ref{dynsys2}) to obtain
a reservoir structure with
${ \bf \Gamma }'  = -\frac{\zeta}{2}\mathsf I_{4}$ and
${\bf D}' = (2\bar n + 1)\zeta \, \mathsf I_{4}$.
This structure produces the steady-state
${\bf V} =  (2\bar n + 1) \mathsf I_{2n}$.

Now, one needs to apply the covariance relation (\ref{es1})
and determine the engineered reservoir with matrices
${\bf \Gamma}_{\rm p}$ and ${\bf D}_{\rm p}$ through the symplectic matrix (\ref{ex1}),
{\it i.e.},
\begin{equation}                                                                         \label{ex5}
{\bf \Gamma}_{\rm p} = -\frac{\zeta}{2}\mathsf I_{4}, \,\,\,
{\bf D}_{\rm p} = (2\bar n + 1)\zeta \,\mathsf{S}_r \mathsf{S}_r^{\!\top}.
\end{equation}
From Eq.(\ref{simcov}), one can see that
$\lambda' = \mathsf{S}_r \lambda$, and the Lindblad operators
in (\ref{ham-lind}) for $\lambda'$ are characteristics of
a squeezed thermal bath.
\paragraph{OPO Steady-States.}
The steady-state in (\ref{opocm}) is a pure steady-state iff
$\epsilon_1 = - \epsilon_2$ \cite{koga}.
Under these conditions and if we define
$\epsilon := \sqrt{( \kappa + \epsilon_2)/(\kappa-\epsilon_2)}$,
the CM of this steady-state is written as
\begin{equation}                                                                         \label{ex6}
{\bf V}' = \mathsf S_{\rm p} \mathsf S_{\rm p}^\top, \,\,\,
{\mathsf S_{\rm p}} :=
\left(\! \begin{array}{cc}
       \frac{1 + \epsilon}{2} & \frac{1 - \epsilon}{2} \\
       \frac{1 - \epsilon}{2} &  \frac{1 + \epsilon}{2}
\end{array} \! \right)
\oplus
\left(\! \begin{array}{cc}
       \frac{1 + \epsilon}{2\epsilon} & \frac{\epsilon-1}{2\epsilon} \\
       \frac{\epsilon-1}{2\epsilon} &  \frac{1 + \epsilon}{2\epsilon}
\end{array} \! \right),
\end{equation}
which is entangled for any value of $\epsilon$.
Obviously, if one wants to produce this pure state as a steady-state of an OPO,
the only step is to produce an OPO satisfying the mentioned conditions.
On the other side,
it is not possible to produce it by using the covariance rules in (\ref{es1}) for an OPO,
since Corol.\ref{corol:diag} requires ${\bf \Gamma} + {\bf \Gamma}^\top \ge 0$,
which is not the case for $\bf \Gamma$ in (\ref{opog})
with $\epsilon_1 = - \epsilon_2$.

By the Willianson theorem, the symplectic spectrum of the \hy{CM}{CM} of
the pure state in (\ref{ex6}) is the identity matrix \cite{footnote3}.
If one can choose suitable values of the parameters in (\ref{opod}) and (\ref{opog})
such that ${\bf \Gamma}'$ and $\bf D'$ satisfy the LE
$\lfloor \mathsf I_4, {\bf \Gamma}', {\bf D}' \rceil$,
then Eq.(\ref{engcov}) can be applied.
In fact, this happens when $\epsilon_1 = \epsilon_2 = 0$.
Thus, preparing an OPO system such that this last condition holds,
Eq.(\ref{engcov}) becomes
\begin{equation}                                                                         \label{ex7}
\left\lfloor \mathsf S_{\rm p} \mathsf S_{\rm p}^\top  ,
         \mathsf S_{\rm p} {\bf \Gamma}'\mathsf S_{\rm p}^{-1},
         \mathsf S_{\rm p} {\bf D}'\mathsf S_{\rm p}^{\top} \right\rceil.
\end{equation}
with ${\bf D}'$ as in (\ref{opod}) and
\begin{equation}                                                                         \label{ex8}
{ \bf \Gamma }'  =
         \left[\!\!
                 \begin{array}{cc}
                  -\frac{\kappa}{2} & 0 \\
                  -\kappa           & -\frac{\kappa}{2}
                 \end{array}\!\!
         \right] \! \oplus \!
          \left[\!\!
                  \begin{array}{cc}
                   -\tfrac{\kappa}{2}  & 0   \\
                  -\kappa   & -\tfrac{\kappa}{2}
                 \end{array}\!\!
         \right].
\end{equation}
Note that the matrices $\bf \Gamma'$ and $\bf D'$ in this example do not have the
invariant structure in (\ref{locrot}).
Note also that the same pure state can be engineered by using thermal baths.
The recipe for this case is just (\ref{ex5})
but replacing $\mathsf S_r$ by $\mathsf S_{\rm p}$ and using $\bar n = 0$.

An entangled and steerable (with respect to both partitions)
mixed state
can also be prepared by following the same recipe.
Suppose that one wants to create the bellow state as a steady-state
of an OPO-covariant-LE:
\begin{equation}
{\bf V}' = {\mathsf S}_{\rm p}^{-1}
          {\bf \Lambda}
          {\mathsf S}_{\rm p}^{-\top},
\end{equation}
with $\mathsf S_{\rm p}$ defined in (\ref{ex6}) and
\begin{equation}
{\bf \Lambda} = {\rm Diag}\left[1,(1-\epsilon)^{-1},1,(1+\epsilon)^{-1} \right].
\end{equation}
This matrix is the solution for the LE
$\lfloor {\bf \Lambda}, {\bf \Gamma}', {\bf D}' \rceil$
with $\bf D'$ in (\ref{opod}) and $\bf \Gamma'$ in (\ref{opog}) both with
$\kappa = 1$, $\epsilon_1 = 0$ and $\epsilon_2 = \epsilon$.
By the same recipe as before, the LE in (\ref{engcov}) has the above $\bf V'$
as solution if we replace $\mathsf S$ by $\mathsf S_{\rm p}$ and the mentioned
matrices $\bf \Gamma'$ and $\bf D'$.

\section{Final Remarks} \label{conc}
Symmetries and properties of the Lyapunov equation were used to classify the features of
the steady state of a \hy{LME}{LME} with a quadratic Hamiltonian and linear
Lindblad operators.
The connection with the Lyapunov equation eases the characterization of the state,
a task that is typically difficult when performed using the master equation directly.

For Gaussian steady-states, we focused on known bona-fide relations for the
covariance matrix of a state. Specifically, we considered conditions for the
classicality, separability, and steerability of Gaussian states.
We remark, however, that the extension for any other bona-fide relation is
straightforward and can be performed following the lines presented here.
For instance, we can refer to the characterization of tripartite
entanglement given in Ref.~\cite{giedke}.
We also analyze the consequences for the covariance matrix of a steady-state
when a transformation symmetry of the Lyapunov equation is performed.

We focused our examples on systems with one or two degrees of freedom,
which has enabled us to compare the results of our corollaries with
the results extracted directly from the covariance matrix of the system after
solving the Lyapunov equation.
However, our results are applicable to systems with a generic number of
degrees of freedom. For large systems, in particular in the absence of symmetries,
numerical solutions might be needed to find the covariance matrix of the
steady-state; for instance, the systems considered in Ref. \cite{nicacio3}.
In this situation, instabilities associated with the algorithms for solving
Lyapunov equations may arise \cite{hammarling}.
The robustness of the analytical results shows the advantage with respect to
either perturbations of the systems parameters or preparation imprecisions.
In other words, our results are advantageous since one does not need to solve
a Lyapunov equation to know some of the system properties or symmetries.
In addition, our results are suitable for the engineering of
(a reservoir leading to a specific) steady-state of a \hy{LME}{LME}
having suitable properties and symmetries.
\section*{Appendices}


\section{Elastohydrodynamic lubrication model}
Let us consider a thin viscous film of thickness $h_{0}$ placed over a flat horizontal substrate, and covered by a thin elastic sheet of constant thickness $d\ll h_0$. As depicted in Fig.~\ref{schema}, an external pressure field $\psi_{\text{ext}}(x-vt,y)$, moving along the horizontal direction $x$ and time $t$ with a constant speed $v\geq 0$, is applied on the elastic sheet. A resulting non-constant profile $h\left(x,y,t\right)$ of the liquid-elastic interface, with respect to the substrate, is created. We note that the total two-layer profile $h+d$ has the same spatiotemporal variations as $h$, and we thus focus on the later only -- with no loss of generality.

Invoking the incompressible Stokes equation and volume conservation, and considering no slip at both the substrate-liquid and the liquid-elastic interfaces, the following equation is yielded in the lubrication approximation (\cite{lister,stone,mahadevan,carlson}):
 \begin{eqnarray}
 \label{eqinit}
  \dfrac{\partial h}{\partial t}=\dfrac{1}{12\mu}\mathbf{\nabla}\cdot\left(  h^{3}\mathbf{\nabla} P_{\text{tot}} \right)\ ,
\end{eqnarray}
where $\mu$ is the dynamic viscosity, $\mathbf{\nabla}$ is the gradient operator in 2D Cartesian coordinates, and $P_{\text{tot}}(x,y,t)$ is the total pressure in the liquid. The latter is given by the addition of the bending stress, the hydrostatic pressure, and the external moving pressure field, and thus reads:
 \begin{eqnarray}
 \label{pinit}
P_{\text{tot}}=B\nabla^{4}h+\rho gh +\psi_{\text{ext}} \ ,
\end{eqnarray}
where $B=E d^{3}/[12(1-\nu^{2})]$ is the bending stiffness, $E$ and $\nu$ are respectively the Young's modulus and Poisson's ratio (\cite{landau}), $g$ is the acceleration of gravity, and $\rho$ is the density of the liquid. Note that we assumed the bending stresses to be dominant over the stretching ones.

Taking $h_0$ and the gravito-elastic length $\kappa^{-1}_{\text{el}}=\left[B/\left(\rho g\right)\right]^{1/4}$ as the characteristic length scales in the vertical $z$-direction and in the $xy$ plane, respectively, the time $\tau =12 \mu/\left(\rho g \kappa^2_{\text{el}} h_{0}^{3}\right)$ as the characteristic time scale, and $P_{\text{ext}}=\kappa_{\text{el}}^{\,2}\iint dx\, dy \, \psi_{\text{ext}}$ as the characteristic pressure scale, we introduce the following dimensionless variables: $X=\kappa_{\text{el}} x$, $Y=\kappa_{\text{el}}y$, $H=h/h_{0}$, $T=t/\tau$, $\Psi=\psi_{\text{ext}}/P_{\text{ext}}$, and $\Gamma_{\text{el}}=P_{\text{ext}}/\left(\rho g h_0\right)$. In the limit of weak driving where $\Gamma_{\text{el}}\Psi\ll1$, and associated small deformation where $F=H-1\ll 1$, eqs.~(\ref{eqinit}) and~(\ref{pinit}) can be linearized and lead to the dimensionless elastohydrodynamic thin-film equation (\cite{flitton}) on the field $F(X,Y,T)$:
 \begin{eqnarray}
\dfrac{\partial F}{\partial T}=\Delta^{3}F+\Delta F + \Gamma_{\text{el}}\Delta\Psi\ ,
 \label{TFE6}
\end{eqnarray}
where $\Delta $ denotes the Laplacian operator in 2D Cartesian coordinates.

We restrict ourselves to stationary surface profiles in the comoving frame. Therefore, we introduce the new variable $U=X-V\, T$, and we define $F(X,Y,T)=\zeta\left(U,Y\right)$. In this context, eq.~(\ref{TFE6}) becomes:
\begin{equation}
\left[\dfrac{\partial^2}{\partial U^2}+\dfrac{\partial^2}{\partial Y^2}\right]^3\zeta+\left[\dfrac{\partial^2}{\partial U^2}+\dfrac{\partial^2}{\partial Y^2}\right]\zeta+V\dfrac{\partial}{\partial U}\zeta=-\Gamma_{\text{el}}\left[\dfrac{\partial^2}{\partial U^2}+\dfrac{\partial^2}{\partial Y^2}\right]\Psi \ .
\label{Eq:TFEHE}
\end{equation}
Finally, let us introduce the two relevant dimensionless parameters of the problem: the elastic Bond number $B_{\text{el}}=\left(a \, \kappa_{\text{el}}\right)^2$, where $a$ denotes the characteristic horizontal size of the external pressure field,
and the reduced speed $V=\kappa_{\text{el}} \, \tau \, v$. We note that the vertical amplitude of the dimensionless disturbance field $\Gamma_{\text{el}}\Psi$ is a trivial dimensionless number in linear response theory, and we thus avoid discussing it further.

\section{Wake}
By definition, the wake is the solution $\zeta\left(U,Y\right)$ of eq.~\eqref{Eq:TFEHE}, for given disturbance field $\Gamma_{\text{el}}\Psi$ and reduced speed $V$. Invoking the two-dimensional Fourier transforms, $\widehat{\zeta}\left(K,Q\right)$ and $\widehat{\Psi}\left(K,Q\right)$, eq.~\eqref{Eq:TFEHE} becomes:
\begin{eqnarray}
\left[ iKV-(K^{2}+Q^{2})^{3}-(K^{2}+Q^{2}) \right]\widehat{\zeta}\left(K,Q\right)=\Gamma_{\text{el}}(K^{2}+Q^{2})\,\widehat{\Psi}\left(K,Q\right)\ .
\end{eqnarray}
Consequently, the solution reads:
\begin{equation}
\zeta\left(U,Y\right)=\dfrac{\Gamma_{\text{el}}}{4\pi^{2}}\iint\dfrac{(K^{2}+Q^{2})\exp\left[i(KU+QY)\right]\widehat{\Psi}\left(K,Q\right)}{iKV-(K^{2}+Q^{2})\left[1+(K^{2}+Q^{2})^{2}\right]} dKdQ \ .
\label{Eq:zeta}
\end{equation}

In order to fix ideas through a canonical example, we introduce the following axisymmetric Lorentzian pressure field and its Fourier transform:
\begin{subequations}
\begin{align}
\Psi\left(U,Y\right) &= \dfrac{\sqrt{B_{\text{el}}}}{2\pi\left(U^2+Y^2+B_{\text{el}}\right)^{3/2}} \ , \\
\widehat{\Psi}\left(K,Q\right) &= \exp\left[-\sqrt{B_{\text{el}}\left(K^2+Q^2\right)}\right] \ .
\end{align}
\label{Eq:Lorentzian}
\end{subequations}
The combination of eqs.~\eqref{Eq:zeta} and \eqref{Eq:Lorentzian} leads to the surface pattern at stake. A parametric study has been performed, sweeping a wide range of values for the reduced speed $V$ and elastic Bond number $B_{\text{el}}$, as summarized in Fig.~\ref{wake}. At low speed, the profiles are nearly symmetric and show a cavity below the the external pressure field. When the reduced speed reaches $V\sim1$, there is an accumulation of material at the front ($U>0$) and a stretching of the cavity behind the center of the pressure field ($U<0$). At high speed, comet-like shapes surrounded by undulations are observed, with an overall extent that can largely exceed the size of the pressure field.

The comparison between the main features of the present elastohydrodynamic case and the ones previously reported for the viscocapillary case (\cite{rene}) is shown in Fig.~3. In the transverse cut of Fig.~\ref{Fig:ZYComp}, although the trends are similar, we observe that the oscillations are more pronounced in the elastohydrodynamic case -- consistent with the higher-order differential operator of the governing eq.~\eqref{Eq:TFEHE}. In the longitudinal cut of Fig.~\ref{Fig:ZUComp}, it is striking to notice the existence of oscillations at the front of the pressure disturbance in the elastohydrodynamic case, together with a longer decay length at the back.

\begin{figure}
\centering
\includegraphics[scale=0.7]{XYZ_BelV}
\caption{Top view of the normalized wake created by a Lorentzian pressure field, computed from eqs.~\eqref{Eq:zeta} and \eqref{Eq:Lorentzian}, for different values of the elastic Bond number $B_{\text{el}}$ and the reduced speed $V$ as indicated. Since we use the comoving-frame variables, $U=X-V\,T$ and $Y$, the pressure field is centered at $U=0$ and the two-layer film travels at constant speed $V$ from right to left in each subfigure.}
\label{wake}
\end{figure}

\begin{figure}
\centering
\subfigure[]{
\centering
\includegraphics[width=0.45\textwidth,trim=58mm 70mm 60mm 30mm,clip]{Z_vs_Y-Compare.pdf}
\label{Fig:ZYComp}
}
\hspace*{5mm}
\subfigure[]{
\centering
\includegraphics[width=0.45\textwidth,trim=58mm 70mm 60mm 30mm,clip]{Z_vs_U-Compare.pdf}
\label{Fig:ZUComp}
}
\label{Fig:compTF46}
\caption{\subref{Fig:ZYComp} Normalized height profile as a function of the transverse coordinate $Y$, for a fixed longitudinal coordinate $U=-20$ and reduced speeds $V$ as indicated. \subref{Fig:ZUComp} Normalized height profile  as a function of the longitudinal coordinate $U=X-V\,T$, for a fixed transverse coordinate $Y=0$ and reduced speeds $V$ as indicated. In both panels, the orange curves correspond to the elastohydrodynamic case of eqs.~\eqref{Eq:zeta} and \eqref{Eq:Lorentzian} with $B_{\text{el}}=1$, whereas the blue curves correspond to the viscocapillary case (\cite{rene}) with a Lorentzian pressure distribution and $B_{\text{o}}=1$.}
\end{figure}

\section{Wave resistance}
As the pressure disturbance moves atop the elastic sheet, it generates the previously-discussed surface deformation which is intimately coupled to the motion of the underlying liquid. As a consequence of the viscous nature of the latter, a continuous dissipation of energy takes place, and the moving disturbance experiences a force opposing its motion. This so-called wave resistance $r$ is given by Havelock's formula (\cite{havelock,elie}):
\begin{eqnarray}
r=\iint \psi_{\text{ext}}\dfrac{\partial h}{\partial x} dx \, dy\ .
\end{eqnarray}
Naturally, the power $v r$ must be furnished by the operator in order to maintain a constant disturbance speed $v$ (\cite{rene}).

With the notations introduced above, the dimensionless wave resistance $R$ reads:
 \begin{equation}
R=\dfrac{r\kappa_{\text{el}}}{\rho g h_0^2}=\Gamma_{\text{el}}\iint \Psi(U,Y)\dfrac{\partial \zeta}{\partial U}\, dU \, dY \ .
\label{Eq:WaveRes}
\end{equation}
Then, the substitution of eq.~\eqref{Eq:zeta} within eq.~\eqref{Eq:WaveRes} yields:
\begin{eqnarray}
R=\dfrac{\Gamma_{\text{el}}^{2}V}{4\pi^{2} }\iint\dfrac{K^{2}(K^{2}+Q^{2})\left\lvert\widehat{\Psi}\left(K,Q\right)\right\lvert^2}{K^{2}V^{2}+(K^{2}+Q^{2})^{2}\left[1+(K^{2}+Q^{2})^{2}\right]^{2}} \, dK\, dQ \ .
\end{eqnarray}
Invoking the polar coordinates, $K=\rho\cos\theta$ and $Q=\rho\sin\theta$, and assuming an axisymmetric pressure field $\widehat\Psi(\rho)$, allow us to integrate over $\theta$ and get the expression:
\begin{eqnarray}
R=\dfrac{\Gamma_{\text{el}}^{2}}{2\pi V }\int_{0}^{\infty}\left[1-\dfrac{\rho(\rho^{4}+1)}{\sqrt{V^{2}+\rho^{2}(\rho^{4}+1)^{2}}}\right]\left\lvert\widehat\Psi(\rho)\right\lvert^{2}\rho^{3} d\rho \ .
\label{eq:waveresist}
\end{eqnarray}

Equations~\eqref{Eq:Lorentzian} and~\eqref{eq:waveresist} allow us to compute numerically the wave resistance. The results are presented in Fig.~\ref{Fig:Resistance}. At low speed, the wave resistance shows a linear dependence with $V$; at high speed, it decreases inversely proportional to $V$; and at intermediate speed, it shows a maximum at the crossover between the two aforementioned regimes. In addition, the larger $B_{\text{el}}$, \textit{i.e.} the wider the pressure field, the lower the wave resistance; and the smaller $B_{\text{el}}$, \textit{i.e.} the narrower the Lorentzian pressure field, the more the wave resistance approaches the one associated with the Dirac pressure field ($\widehat{\Psi}=1$), as expected. Interestingly, the Dirac case shows a more gentle slope of $V^{-1/5}$ at high speed. In addition, we observe that the maximal wave resistance is bounded by the one of the Dirac case. Indeed, a Dirac pressure field excites all the wavelengths, whereas a Lorentzian pressure field cuts the large wavelengths off.

\begin{figure}
\centering
\includegraphics[scale=0.5]{R_vs_V-Lorentz.pdf}
\caption{Normalized wave resistance $R/\Gamma_{\text{el}}^{2}$, as given by eqs.~\eqref{Eq:Lorentzian} and~\eqref{eq:waveresist}, as a function of the reduced speed $V$, for various elastic Bond numbers $B_{\text{el}}$ as indicated. The Dirac limit ($B_{\text{el}}\rightarrow0$) is indicated. The vertical dashed lines indicate the position of the maximal wave resistance (see eq.~\eqref{vstar}), for each value of $B_{\text{el}}$.}
\label{Fig:Resistance}
\end{figure}

\subsection{Low-speed regime}
As seen in the low-speed regime of Fig.~\ref{Fig:Resistance}, the wave resistance shows a linear rise with the speed $V$.
Indeed, using the Dirac pressure field, we get the following asymptotic expression from eq.~\eqref{eq:waveresist} as $V\rightarrow 0$:
\begin{eqnarray}
\label{dirc}
R\sim\dfrac{\Gamma_{\text{el}}^{2}V}{32 }  \ .
\end{eqnarray}
Similarly, using the Lorentzian pressure field of eq.~\eqref{Eq:Lorentzian}, the wave resistance reads in the low-speed regime:
\begin{eqnarray}
\label{rf}
R\sim\dfrac{\Gamma_{\text{el}}^{2}V}{4\pi }\mathcal F(B_{\text{el}}) \ ,
\end{eqnarray}
where:
\begin{equation}
\label{fbel}
\mathcal F(B_\text{el})=\int_0^{+\infty}\frac{\rho \exp\left(-2\sqrt{B_\text{el}}\rho\right)}{\left(\rho^4+1\right)^2}\, \text{d}\rho\ .
\end{equation}

In the limit of a wide pressure field, $B_{\text{el}}\gg 1$, one obtains:
\begin{eqnarray}
\mathcal F(B_{\text{el}})\sim\dfrac{1}{4B_{\text{el}}} \ .
\end{eqnarray}

\subsection{High-speed regime}
In the low-speed regime of Fig.~\ref{Fig:Resistance}, one has two distinct scaling behaviours with $V$.
Indeed, for the Dirac case, using the asymptotic development of eq.~\eqref{eq:waveresist} as $V\rightarrow \infty$, one gets:
\begin{eqnarray}
R\sim-\dfrac{\Gamma_{\text{el}}^{2}\Gamma(-2/5)\Gamma(9/10)}{20\pi^{3/2}}\dfrac{1}{V^{1/5}}  \ ,
\label{eq:WRHSdirac}
\end{eqnarray}
where $\Gamma(z)$ is the Gamma function, and has no relation with the dimensionless quantity $\Gamma_{\rm{el}}$, which has been defined in section 1.

In contrast, for the Lorentzian pressure field of eq.~\eqref{Eq:Lorentzian}, we obtain:
\begin{eqnarray}
R\sim\dfrac{3\Gamma_{\text{el}}^{2}}{16\pi B_{\text{el}}^{2}}\dfrac{1}{V} \ .
\label{eq:WRHSlorentz}
\end{eqnarray}
From the comparison of eqs.~\eqref{eq:WRHSdirac} and \eqref{eq:WRHSlorentz}, we remark that the wave resistance for a realistic finite-size pressure field decays much faster than for the singular Dirac pressure field.

\subsection{Maximal wave resistance}
\begin{figure}
\centering
\includegraphics[scale=0.5]{RmaxVmax_vs_Bel-Lorentz.eps}
\caption{Computed values of the normalized maximal wave resistance $R(V^{*})/\Gamma_{\text{el}}^{2}$ (blue) and the corresponding speed $V^{*}$ (red), as functions of the elastic Bond number $B_{\text{el}}$, according to eqs.~\eqref{fbel} and \eqref{vstar}. The dashed lines represent the Dirac limits given in eqs.~\eqref{fin1} and~\eqref{fin2}.}
\label{Fig:Max}
\end{figure}
As observed in Fig.~\ref{Fig:Resistance}, all the curves show a maximum for a certain speed $V^{*}(B_{\text{el}})$. An estimation of this speed can be obtained by balancing the low-speed and high-speed asymptotic expressions of the wave resistance in the Lorentzian case:
\begin{eqnarray}
\label{vstar}
V^{*}\sim\sqrt{\dfrac{3}{4\mathcal{F}(B_{\text{el}})B_{\text{el}}^{2}}} \ .
\end{eqnarray}
Therefore, at high elastic Bond number, one finds:
\begin{equation}
\label{fin3}
V^{*}\sim\sqrt{\dfrac{3}{B_{\text{el}}}} \ ,
\end{equation}
and:
\begin{equation}
\label{fin4}
R(V^{*})\sim\dfrac{\sqrt{3}\Gamma_\text{el}^2}{16\pi B_\text{el}^{3/2}} \ .
\end{equation}
The limit of the maximal wave resistance as $B_{\text{el}}\rightarrow0$ can be obtained by balancing the low-speed and high-speed asymptotic expressions of the wave resistance in the Dirac case:
\begin{eqnarray}
\label{fin1}
V^{*}\sim\left[-\dfrac{8\Gamma(-2/5)\Gamma(9/10)}{5\pi^{3/2}}\right]^{5/6} \ ,
\end{eqnarray}
and thus:
\begin{eqnarray}
\label{fin2}
R(V^{*})\sim\dfrac{\Gamma_{\text{el}}^{2}}{32 }\left[-\dfrac{8\Gamma(-2/5)\Gamma(9/10)}{5\pi^{3/2}}\right]^{5/6} \ .
\end{eqnarray}
The asymptotic behaviors for low and high elastic Bond number, given by eqs.~\eqref{fin3} and \eqref{fin1} for $V^{*}$, and eqs.~\eqref{fin4} and \eqref{fin2} for $R(V^{*})$, are also represented in Figure.~\ref{Fig:Max}.

Surprisingly, this behaviour is qualitatively different from the viscocapillary case (\cite{rene}) for which the maximal wave resistance diverges in the Dirac limit. These results are summarized in Fig.~\ref{Fig:Max}. Therein, the maximal wave resistance decreases as $B_{\text{el}}^{-3/2}$ at large $B_{\text{el}}$, and saturates to a finite value in the Dirac limit. The corresponding speed $V^*$ decreases as $B_{\text{el}}^{-1/2}$ at large $B_{\text{el}}$, and saturates as well in in the Dirac limit.

\section{Conclusion}
We presented a theoretical investigation of the effects of a moving pressure disturbance above a thin elastic sheet placed atop a narrow viscous film. From the elastohydrodynamic lubrication model, we computed both the wake and the associated wave resistance experienced by the operator. A central dimensionless parameter of this study appeared to be the so-called elastic Bond number, measuring the ratio of gravity to elastic bending. As in the viscocapillary case, the wave resistance was observed to have a global maximum -- a point it might be interesting to control for energy-saving purposes. Finally, we conclude our study by an asymptotic analysis of the low-speed and high-speed regimes. These results are relevant to a wide class of geophysical, biological, and engineering problems, and may have implications in nanorheology, as well as wave propagation in metamaterials.

\section{Acknowledgments}
The authors thank Vincent D\'emery, Kari Dalnoki-Veress, Howard Stone and Antonin Eddi for interesting discussions. They acknowledge financial support from the Global Station for Soft Matter -- a project of Global Institution for Collaborative Research and Education at Hokkaido University.

\bibliographystyle{abbrv}