& \left.\left.\left.+ \frac{1}{2} \vec{r} \cdot \left( \tilde{B}_t T^{N} \vec{r} \right)   \right) W_t^N{\vec{r} }\right] \right) \, .
\end{equation}

The proof of the lemma is obtained by considering together Eqs. l8_Himaginary, l8_HRealI, l8_HRealII.

§.§ Lemma 10

Before proceeding further, we use that the action of $\mathcal{D}_\ell $ on a local exponential can be split in two terms. We have

\mathcal{D}_\ell[W_t^N(\vec{r})]=\frac{1}{2}\sum_{k=1}^N\left(\left[\Gamma_\ell(\Delta_N^\ell) j_\ell^{\dagger\, (k)}, W_t^N(\vec{r})\right]j_{\ell}^{(k)}\Gamma_\ell(\Delta_N^\ell)+\Gamma_\ell(\Delta_N^\ell)j_\ell^{\dagger \, (k)}\left[W_t^N(\vec{r}),j_\ell^{(k)}\Gamma_\ell(\Delta_N^\ell)\right]\right)\, .

Using that $[AB,C]=A[B,C]+[A,C]B$, we rewrite this as

\begin{equation}
\eqalign{
\mathcal{D}_\ell[W_t^N(\vec{r})] =& \frac{1}{2}\sum_{k=1}^N\left(\Gamma_\ell(\Delta_N^\ell)\left[ j_\ell^{\dagger\, (k)}, W_t^N(\vec{r}) \right]j_{\ell}^{(k)}\Gamma_\ell(\Delta_N^\ell) \right. \\
& \left. +\Gamma_\ell(\Delta_N^\ell) j_\ell^{\dagger\, (k)}\left[W_t^N(\vec{r}),j_\ell^{(k)}\right]\Gamma_\ell(\Delta_N^{\ell})\right)\\
\qquad \qquad \quad +&\frac{1}{2} \sum_{k=1}^N\left(\left[\Gamma_\ell(\Delta_N^\ell), W_t^N(\vec{r})\right]j_\ell^{\dagger\, (k)}j_{\ell}^{(k)}\Gamma_\ell(\Delta_N^\ell) \right. \\
& \left. + \Gamma_\ell(\Delta_N^\ell)j_\ell^{\dagger\, (k)}j_\ell^{(k)}\left[W_t^N(\vec{r}),\Gamma_\ell(\Delta_N^\ell)\right]\right)\, ,
\label{WeylLike-diss-two-terms}
\end{equation}

and we separate the action of the dissipation in the following way. The action of the last two terms on the right-hand side of Eq. WeylLike-diss-two-terms, denoted as $D_{I}$, and the first two terms, denoted as $D_{II}$, namely

\begin{equation}\label{diss-first-term}
\eqalign{
D_I \equiv &\frac{1}{2} \sum_{k=1}^N\left(\left[\Gamma_\ell(\Delta_N^\ell), W_t^N(\vec{r})\right]j_\ell^{\dagger\, (k)}j_{\ell}^{(k)}\Gamma_\ell(\Delta_N^\ell) \right. \\
& \left. + \Gamma_\ell(\Delta_N^\ell)j_\ell^{\dagger\, (k)}j_\ell^{(k)}\left[W_t^N(\vec{r}),\Gamma_\ell(\Delta_N^\ell)\right]\right)\, ,
\end{equation}

\begin{equation}\label{diss-second-term}
\eqalign{
D_{II} \equiv & \frac{1}{2}\sum_{k=1}^N\left(\Gamma_\ell(\Delta_N^\ell)\left[ j_\ell^{\dagger\, (k)}, W_t^N(\vec{r}) \right]j_{\ell}^{(k)}\Gamma_\ell(\Delta_N^\ell) \right. \\
& \left. +\Gamma_\ell(\Delta_N^\ell) j_\ell^{\dagger\, (k)}\left[W_t^N(\vec{r}),j_\ell^{(k)}\right]\Gamma_\ell(\Delta_N^{\ell})\right)\, ,
\end{equation}

Given a dynamical generator as defined by Eqs. e0_totalHamiltonian-jumps, satisfying Assumption Gamma, the dissipation term $D_I$ defined by Eq. diss-first-term acts on local exponential $W_t^N(\vec{r}) = e^{i \vec{r}\cdot \vec{F}^N_t}$ as

\begin{equation*}
\eqalign{
& \lim_{N \rightarrow \infty} \omega_{\vec{r}_1 \vec{r}_2 }  \left( e^{t\linn} \left[ D_I \right] \right) \\
& =        \lim_{N \rightarrow \infty} \omega_{\vec{r}_1 \vec{r}_2 } \left( e^{t\linn} \left[ \left( - \frac{1}{2} (\vec{r} \cdot T^{N}\vec{r}_\ell) \left(\Gamma_\ell'(\Delta_N^{\ell}) \right)^2  \frac{1}{N} \sum_{k}  j_\ell^{\dagger (k)} j_\ell^{ (k)}  (\vec{r}_\ell \cdot T^{N}\vec{r})  \right. \right.\right. \\
& \left. \left. \left. -  \frac{1}{2} \Gamma_\ell(\Delta_N^{\ell}) \Gamma_\ell'(\Delta_N^{\ell}) \vec{r}\cdot (T^N \vec{r}_\ell) (\vec{c}_{\ell} \cdot (T^{N} \vec{r} )) \right)W_t^{N}(\vec{r})  \right] \right) \, ,}
\end{equation*}

where $\vec{r}_\ell$ is the vector with components
$r_{\ell \alpha}$, introduced by Eq. eq:Delta_lc_average as coefficients defining $\Delta_N^{\ell}$ in terms of linear combinations of average operators; $\vec{c}_\ell$ is the vector with components ${c}_{\ell \eta} = \sum_{\gamma, \gamma' } c_{\ell \gamma}^{j *} c_{\ell \gamma'}^{j } \frac{1}{2}(a_{\gamma \gamma'}^\eta + b_{\gamma \gamma'}^{\eta}) $, having introduced $j_{\ell}^{(k)} = \sum_{\gamma} c_{\ell \gamma}^{j} v_{\gamma}^{(k)}$ and $b_{\gamma \gamma'}^{\eta}$ such that $  \lbrace v_{\gamma}^{(k)}, v_{\gamma'}^{(k)}\rbrace = \sum_{\eta} b_{\gamma \gamma'}^{\eta}v_{\eta}^{(k)}$.

Looking at Eq. diss-first-term, we split $D_I$ into two parts, $D_I= D_{I}^1+ D_{I}^2$, where we have

\begin{equation}
\eqalign{
D_{I}^1&=\frac{1}{2}\sum_{k=1}^{N}\left[\Gamma_\ell(\Delta_N^\ell),W_t^N(\vec{r}) \right]\left[j_\ell^{\dagger \, (k)} j_\ell^{(k)}, \Gamma_\ell(\Delta_N^\ell)\right] \, ,\\
\left[\left[\Gamma_\ell(\Delta_N^\ell),W_t^N(\vec{r}) \right],\Gamma_\ell(\Delta_N^\ell)j_\ell^{\dagger\, (k)}j_\ell^{(k)}\right]\, .
\end{equation}

We first focus on the terms $D_I^1$. The first result that we proof is the following bound,

\begin{equation}\label{comm_localexponential}
\| \left[\Gamma_\ell(\Delta_N^\ell), W_t^N(\vec{r})\right]  \| \leq \frac{\gamma(z)}{\sqrt{N}} \, ,
\end{equation}

with $\gamma(\cdot)$ defined in Assumption Gamma, and $z=\delta_\ell (1+e^{d^4 r_{\max} a_{\max}})$. Indeed, let us consider the commutator $ \left[\Gamma_\ell(\Delta_N^\ell), W_t^N(\vec{r})\right] = \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right)W_t^{N}(\vec{r}) $. The second term in the right-hand side can be written as

\begin{equation}\label{WgammaW}
\eqalign{
W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) & = \sum_{b \geq 0 } c_\ell^b \left( W_t^N(\vec{r}) \Delta_N^\ell W_t^{N \dagger}(\vec{r}) \right)^b \\
& = \sum_{b \geq 0 } c_\ell^b \left( \Delta_N^\ell +D_{\Delta} \right)^b  \\
& =\Gamma_\ell(\Delta^{N}_\ell) +   \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b  \left( \matrix{b \cr s} \right)(\Delta_N^\ell)^{b-s} D_{\Delta}^s\, ,
\end{equation}

where in the first step we employed Assumption Gamma, and in the second step we introduced

\begin{equation}\label{S_definition}
\eqalign{
& D_{\Delta} \equiv W_t^N(\vec{r})  \Delta_N^\ell W_t^{N \dagger}(\vec{r}) - \Delta_N^\ell = \sum_{n \geq 1} \frac{i^n}{n!}\mathbb{K}^n_{\vec{r}\cdot \vec{F}_t^N}[\Delta_N^\ell] \\
& = \sum_{n \geq 1} \frac{i^n}{n!} \sum_{\nu_1 ,...,\nu_n}\sum_{\alpha} r_{\ell \alpha} r_{\nu_1}\cdots r_{\nu_n} \sqrt{N}^n [m^N_{\nu_1},\cdots,[m^N_{\nu_n},m_{\alpha}^N]\cdots] \\
& = \sum_{n \geq 1} \frac{i^n}{n!} \sum_{\nu_1 ,...,\nu_n}\sum_{\alpha} r_{\ell \alpha} r_{\nu_1}\cdots r_{\nu_n} \frac{1}{N^{n/2}}\sum_{\gamma_1, \cdots ,\gamma_n}a_{\nu_n \alpha}^{\gamma_n} a_{\nu_{n-1} \gamma_n}^{\gamma^{n-1} }\cdots a_{\nu_1 \gamma_{2}}^{\gamma_1} m_{\gamma_1}^N \, .
\end{equation}

The latter operator is normed-bounded as

\begin{equation}\label{norm_bound_S}
\| D_{\Delta} \| \leq \frac{1}{\sqrt{N}}\delta_\ell \sum_{n \geq 1} \frac{1}{n!} (r_{\max} a_{\max} d^4)^n \leq \frac{1}{\sqrt{N}} \delta_\ell \, e^{d^4 r_{\max} a_{\max}} \, ,
\end{equation}

so that for the commutator $\left[\Gamma_\ell(\Delta_N^\ell), W_t^N(\vec{r})\right]$ it is

\begin{equation}
\eqalign{
& \| \left[\Gamma_\ell(\Delta_N^\ell), W_t^N(\vec{r})\right] \| \leq \| \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell)  W_t^{N \dagger}(\vec{r}) \right)\| \| W_t^{N}(\vec{r}) \| \\
& \leq  \left\| \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s   \right\| \\
& \leq \frac{1}{\sqrt{N}}\sum_{b \geq 0} |c_\ell^b| \sum_{s=1}^b \left( \matrix{b \cr s} \right) \delta_\ell^{b-s}(\delta_\ell e^{d^4 r_{\max} a_{\max}}) ^s \\
& \leq \frac{1}{\sqrt{N}} \sum_{b \geq 0} |c_\ell^b|\delta_\ell^b(1+e^{d^4 r_{\max} a_{\max}})^b \equiv \frac{1}{\sqrt{N}}\gamma( \delta_\ell (1+e^{d^4 r_{\max} a_{\max}})) \, ,
\end{equation}

It is straightforward now to show that $D_I^1$ is a norm-bounded operator. Indeed, by exploiting Lemma lemma_commutators and the bound comm_localexponential, it is

\begin{equation}
\| D_I^1 \| \leq \frac{1}{\sqrt{N}} \|j_\ell \|^2  \delta_\ell \gamma'(\delta_\ell) \gamma( \delta_\ell (1+e^{d^4 r_{\max} a_{\max}})) \, .
\end{equation}

Let us now consider the term $D_{I}^2$, which will be in turn split in the following two contributions

\begin{equation}\label{dIsplit}
\eqalign{D_{I}^2&=\frac{1}{2}\sum_{k=1}^N
\left[\left[\Gamma_\ell(\Delta_N^\ell),W_t^N(\vec{r}) \right],\Gamma_\ell(\Delta_N^\ell)\right] j_\ell^{\dagger\, (k)}j_\ell^{(k)} \\
& + \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell)
\left[\left[\Gamma_\ell(\Delta_N^\ell),W_t^N(\vec{r}) \right],j_\ell^{\dagger\, (k)}j_\ell^{(k)}\right] \\
& = D_{I}^{21} + D_{I}^{22} \, .
\end{equation}

In order to tackle the first term of the last equation, $D_I^{21}$, let us consider the following commutator at the leading order

\begin{equation}\label{GammaWdouble}
\eqalign{
& \left[\left[\Gamma_\ell(\Delta_N^\ell),W_t^N(\vec{r}) \right],\Gamma_\ell(\Delta_N^\ell)\right]
\\ &= \left\lbrace \Gamma_\ell(\Delta_N^\ell)\left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell)  W_t^{N \dagger}(\vec{r}) \right) \right. \\
& - \left. \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell)  W_t^{N \dagger}(\vec{r}) \right)W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell)  W_t^{N \dagger}(\vec{r}) \right\rbrace W_t^{N }(\vec{r}) \\
& = \left\lbrace \Gamma_\ell(\Delta_N^\ell) \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left(  \matrix{b \cr s} \right)  (\Delta_N^\ell)^{b-s} (D_{\Delta})^s \right. \\
& - \left. \left( \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right)  (\Delta_N^\ell)^{b-s} (D_{\Delta})^s \right) W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell)  W_t^{N \dagger}(\vec{r}) \right\rbrace W_t^{N }(\vec{r}) \\
& \sim  -\left( \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s \right)^2 W_t^{N }(\vec{r})
\end{equation}

where in the second last line we used the bound comm_localexponential, so that

$$ \| [W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell)  W_t^{N \dagger}(\vec{r}), \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s ] \| \leq O(N^{3/2}) \, .$$

The squared term in the last line of Eq. GammaWdouble can be further treated. To do so, we first split it into two terms,

$$\sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s = \sum_{b \geq 0} c_{\ell}^b  b (\Delta_N^\ell)^{b-1} D_{\Delta}  + \sum_{b \geq 0} c_{\ell}^b \sum_{s=2}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s \, , $$

and we notice that, by  using Assumption Gamma and Eq. norm_bound_S one obtains

\begin{equation}\label{bound_DI222}
\eqalign{
& \|\sum_{b \geq 0} b c_{\ell}^b   (\Delta_N^\ell)^{b-1} D_{\Delta}  \|  \leq \frac{1}{\sqrt{N}} e^{d^4r_{\mathrm{max}} a_{\mathrm{max}}}\delta_{\ell}\gamma'(\delta_\ell) \,  ,
\end{equation}

as well as

\begin{equation}\label{bound_DI222_b}
\eqalign{
& \|  \sum_{b \geq 0} c_{\ell}^b \sum_{ s=2 }^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{(b-s)}   \left(D_{\Delta} \right)^s   \| \\
& \leq  \sum_{b \geq 0} |c_\ell^b| \sum_{s\geq 2} \left( \matrix{b \cr s}\right) \delta_\ell^{(b-s)} \left(\frac{1}{\sqrt{N}} \delta_\ell \, e^{d^4 r_{\max} a_{\max}} \right)^s  \\
& \leq \frac{1}{N} \sum_{b \geq 0} |c_\ell^b| \delta_{\ell}^{b}\sum_{s=0}^b \left( \matrix{b \cr s} \right)   e^{s d^4 r_{\max} a_{\max}}  \leq \frac{\gamma(\delta_\ell(1+e^{d^4 r_{\max} a_{\max}}))}{N} \, ,
\end{equation}

and also the following

\begin{equation}\label{bound_DI222_c}
\eqalign{
& \left\|  \sum_{b \geq 0} b c_{\ell}^b  (\Delta_N^\ell)^{b-1}  \right.  \\
& \left. \times  \left(\sum_{ n \geq 2
} \frac{i^n}{n!} \sum_{\nu_1 ,...,\nu_n}\sum_{\alpha} r_{\ell \alpha} r_{\nu_1}\cdots r_{\nu_n} \frac{1}{N^{n/2}}\sum_{\gamma_1, \cdots ,\gamma_n}a_{\nu_n \alpha}^{\gamma_n} a_{\nu_{n-1} \gamma_n}^{\gamma^{n-1} }\cdots a_{\nu_1 \gamma_{2}}^{\gamma_1} m_{\gamma_1}^N \right)   \right\| \\
& \leq \frac{1}{N}  \delta_\ell e^{d^4 r_{\max} a_{\max}} \sum_b b |c_\ell^b|  \delta{_\ell}^{(b-1)} \leq \frac{ \delta_\ell \gamma^{'}(\delta_\ell) e^{d^4 r_{\max} a_{\max}}}{N} \, .
\end{equation}

By means the norm bounds in Eqs. bound_DI222-bound_DI222_c it follows that

\begin{equation}
\eqalign{
& \| \left( \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s \right)^2  \\
& -
\left( \sum_{b \geq 0} c_{\ell}^b b (\Delta_N^\ell)^{b-1} i \sum_{\alpha} r_{\ell \alpha} \sum_{\nu} \frac{1}{\sqrt{N}}r_{\nu} \sum_{\gamma}a_{\nu\alpha}^{\gamma} m_{\gamma}^N \right)^2 \| \\
& \leq  \frac{1}{N^2} \underbrace{ \left[ \gamma^2(\delta_\ell(1+e^{d^4 r_{\max} a_{\max}})) + \delta_\ell^2 \gamma^{' \, 2}(\delta_\ell) e^{2 d^4 r_{\max} a_{\max}} \right]}_{C_{211}} \\ & + \frac{1}{N^{3/2}} \underbrace{2\delta_{\ell}\gamma'(\delta_\ell)e^{d^4r_{\mathrm{max}} a_{\mathrm{max}}} \left[ \gamma(\delta_\ell(1+e^{d^4 r_{\max} a_{\max}}))     + \delta_\ell \gamma'(\delta_\ell) e^{ d^4 r_{\max} a_{\max}}\right]}_{C_{212}}
\, ,}
\end{equation}

where we defined $C_{211} \equiv \gamma^2(\delta_\ell(1+e^{d^4 r_{\max} a_{\max}})) + \delta_\ell^2 \gamma^{' \, 2}(\delta_\ell) e^{2 d^4 r_{\max} a_{\max}} $, and $C_{212} = 2\delta_{\ell}\gamma'(\delta_\ell)e^{d^4r_{\mathrm{max}} a_{\mathrm{max}}} \left[ \gamma(\delta_\ell(1+e^{d^4 r_{\max} a_{\max}}))     + \delta_\ell \gamma'(\delta_\ell) e^{ d^4 r_{\max} a_{\max}}\right]$.
As a result it is, at leading order,

\begin{equation}
\eqalign{   D_{I}^{21} & \sim -\frac{1}{2} \sum_{k} \left( \sum_{b \geq 0} c_{\ell}^b b (\Delta_N^\ell)^{b-1} i \sum_{\alpha} r_{\ell \alpha} \sum_{\nu} \frac{1}{\sqrt{N}}r_{\nu} \sum_{\gamma}a_{\nu\alpha}^{\gamma} m_{\gamma}^N \right)^2 \\
& \times W_t^{N }(\vec{r}) j_\ell^{\dagger (k)} j_\ell^{ (k)} W_t^{N \dagger }(\vec{r})W_t^{N }(\vec{r})  \\
& \sim \frac{1}{2} \left(\Gamma_\ell'(\Delta_N^{\ell}) \sum_{\nu} \sum_{\ell \alpha} r_{\nu} a_{\nu \alpha}^{\gamma} r_{\ell \alpha } m_{\gamma}^N \right)^2 \frac{1}{N} \sum_{k}  j_\ell^{\dagger (k)} j_\ell^{ (k)} \\
& = \frac{1}{2} (\vec{r} \cdot T^{N}\vec{r}_\ell) \left(\Gamma_\ell'(\Delta_N^{\ell}) \right)^2  \frac{1}{N} \sum_{k}  j_\ell^{\dagger (k)} j_\ell^{ (k)}  (\vec{r} \cdot T^{N}\vec{r}_\ell) \\
& = - \frac{1}{2} (\vec{r} \cdot T^{N}\vec{r}_\ell) \left(\Gamma_\ell'(\Delta_N^{\ell}) \right)^2  \frac{1}{N} \sum_{k}  j_\ell^{\dagger (k)} j_\ell^{ (k)}  (\vec{r}_\ell \cdot T^{N}\vec{r}) \, ,
\end{equation}

where the symbol $\sim$ is identifying here the following bound

\begin{equation}
\eqalign{
&  \| D_{I}^{21} + \frac{1}{2} (\vec{r} \cdot T^{N}\vec{r}_\ell) \left(\Gamma_\ell'(\Delta_N^{\ell}) \right)^2  \frac{1}{N} \sum_{k}  j_\ell^{\dagger (k)} j_\ell^{ (k)}  (\vec{r}_\ell \cdot T^{N}\vec{r}) \| \\
&\leq \frac{1}{N^{1/2}} \frac{C_{212}}{2}\| j_\ell \|^2 + \frac{1}{N} \frac{C_{211}}{2} \| j_\ell \|^2 +\frac{1}{N^{1/2}} \| j_\ell\|^{2} e^{d^2 2 v r_{\mathrm{max} }} (\delta_\ell\gamma'(\delta_\ell) e^{d^4 r_{\mathrm{max} a_{\mathrm{max} }}})^2 \, . \\
\end{equation}

Let us then consider the term $D_I^{22}$ of Eq. dIsplit, which we write here for the sake of clarity as

\begin{equation}
\eqalign{
D_I^{22} & = \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell)
\left[\left[\Gamma_\ell(\Delta_N^\ell),W_t^N(\vec{r}) \right],j_\ell^{\dagger\, (k)}j_\ell^{(k)}\right] \\
& =  \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell) \left[ \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right)W_t^{N}(\vec{r}), j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right] \\
& = \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell)  \left[ \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right), j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right]W_t^{N}(\vec{r}) \\
& + \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell)  \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right) \left[ W_t^{N}(\vec{r}), j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right] \\
&\equiv D_I^{221} + D_{I}^{222} \, . \\
\end{equation}

We start to inspect the second-last line term, which we denoted $D_I^{221}$. Its norm-bound, by using Eq. WgammaW, can be written as

\begin{equation}
\eqalign{
& \| \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell) \left[ \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right), j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right] \| \\
&\leq  \frac{N}{2}\gamma(\delta_\ell) \sum_{b \geq 0} |c_{\ell}^b| \sum_{s=1}^b \left( \matrix{b \cr s} \right)  \left\| \left[ (\Delta_N^\ell)^{b-s} (D_{\Delta})^s ,  j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right] \right\| \\
& \leq  \frac{N}{2}\gamma(\delta_\ell) \sum_{b \geq 0} |c_{\ell}^b| \sum_{s=1}^b \left( \matrix{b \cr s} \right) \\
& \times \left( \left\| \left[ (\Delta_N^\ell)^{b-s}  ,  j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right]  \right\| \|D_{\Delta} \|^s + \| (\Delta_N^\ell)\|^{b-s} \left\| \left[  (D_{\Delta})^s ,  j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right]  \right\| \right) \\
& \equiv \|D_{I}^{221'}\| +\| D_{I}^{221''}\| \, .
\end{equation}

Here, employing norm_bound_S, the first term is norm-bounded as

\begin{equation}
\| D_{I}^{221'}\| \leq  \frac{1}{N^{1/2}}\| j_\ell\|^2 \delta_\ell  \gamma(\delta_\ell)\gamma'(\delta_\ell[1+e^{s d^4 r_{\max} a_{\max}}]) \, .
\end{equation}

With respect to the second term, $\| D_{I}^{221''}\|$, we use that $[(D_{\Delta})^{s},\cdot] = \sum_{j=0}^{s-1} (D_{\Delta})^j[D_{\Delta},\cdot](D_{\Delta})^{s-1-j}$, and that, by employing the definition of $D_{\Delta}$ in Eq. S_definition, it is

\begin{equation}
\eqalign{
& [D_{\Delta},j^{(k),\dagger}_\ell j^{(k)}_\ell]\\
&=  \sum_{n \geq 1} \frac{i^n}{n!} \sum_{\nu_1 ,...,\nu_n}\sum_{\alpha} r_{\ell \alpha} r_{\nu_1}\cdots r_{\nu_n} \frac{1}{N^{n/2}}\sum_{\gamma_1, \cdots ,\gamma_n}a_{\nu_n \alpha}^{\gamma_n} a_{\nu_{n-1} \gamma_n}^{\gamma^{n-1} }\cdots a_{\nu_1 \gamma_{2}}^{\gamma_1} [m_{\gamma_1}^N, j^{(k),\dagger}_\ell j^{(k)}_\ell] \\
&=  \sum_{n \geq 1} \frac{i^n}{n!} \sum_{\nu_1 ,...,\nu_n}\sum_{\alpha} r_{\ell \alpha} r_{\nu_1}\cdots r_{\nu_n} \frac{1}{N^{n/2}}\sum_{\gamma_1, \cdots ,\gamma_n}a_{\nu_n \alpha}^{\gamma_n} a_{\nu_{n-1} \gamma_n}^{\gamma^{n-1} }\cdots a_{\nu_1 \gamma_{2}}^{\gamma_1} \frac{O_{\ell \gamma_1}^{(k)}}{N},
\end{equation}

with $O_{\ell \gamma_1}^{(k)}$ a local operator with finite support. Therefore,

\begin{equation}
\eqalign{
\| D_{I}^{221''} \| &\leq \frac{N\gamma(\delta_\ell)}{2} \sum_{b}|c_\ell^b|\sum_{s=1}^b \left( \matrix{b \cr s} \right) \delta_\ell^{b-s} \sum_j (\frac{1}{\sqrt{N}}\delta_\ell e^{d^4 r_{\max} a_{\max}})^{s}\|O_{\ell \gamma}\|\frac{1}{N} \\
& \leq \frac{1}{2N^{1/2}}\| O_{\ell \gamma}\|  \delta_\ell e^{d^4r_{\max} a_{\max}} \gamma(\delta_\ell)\gamma'(\delta_\ell(1+e^{d^4 r_{\max} a_{\max}})) \, .
\end{equation}

As a result, we see that the contribution $D_I^{221}$ in norm bounded as

\begin{equation}
\| D_I^{221} \| \leq \frac{1}{N^{1/2}}  \delta_\ell  \gamma(\delta_\ell)\gamma'(\delta_\ell[1+e^{s d^4 r_{\max} a_{\max}}]) \left(\| j_\ell \|^2 + \frac{1}{2}\|O_{\ell \gamma} \| e^{d^4r_{\max} a_{\max}}\right) \, .
\end{equation}

In order to conclude the proof of the Lemma, we have to consider the remaining term, that is

\begin{equation*}
\eqalign{
D_I^{222} & = \frac{1}{2}\sum_{k=1}^N \Gamma_\ell(\Delta_N^\ell)  \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right) \\
& \times \left(W_t^{N}(\vec{r})j_\ell^{\dagger\, (k)}j_\ell^{(k)}W_t^{N \dagger}(\vec{r})- j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right) W_t^{N}(\vec{r}) \, ,\\
\end{equation*}

Let us first consider

\begin{equation}
\eqalign{
&  \left( \Gamma_\ell(\Delta_N^\ell)-W_t^{N}(\vec{r}) \Gamma_\ell(\Delta_N^\ell) W_t^{N \dagger}(\vec{r}) \right)
\left(W_t^{N}(\vec{r})j_\ell^{\dagger\, (k)}j_\ell^{(k)}W_t^{N \dagger}(\vec{r})- j_\ell^{\dagger\, (k)}j_\ell^{(k)} \right) \\
& = - \left(\sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta})^s\right) \left( \sum_{n\geq 1} \frac{i^n}{n!} \mathbb{K}_{\vec{r} \cdot \vec{F}_t^N }[ j_\ell^{\dagger\, (k)}j_\ell^{(k)}] \right) \\
& = - \left(\sum_{b \geq 0} c_{\ell}^b b (\Delta_N^\ell)^{b-1} \frac{i}{\sqrt{N}}  \sum_{\alpha} r_{\ell \alpha} \sum_{\nu \gamma} r_{\nu} a_{\nu \alpha}^{\gamma} m_{\gamma}^N \right) \left( i [ \vec{r} \cdot \vec{F}_t^N ,j_\ell^{\dagger\, (k)}j_\ell^{(k)}] \right) + P \\
& = \left(\Gamma' (\Delta_N^\ell)  \sum_{\alpha} r_{\ell \alpha} \sum_{\nu } r_{\nu} T_{\nu \alpha}^N \right) \left(\frac{1}{\sqrt{N}} [ \vec{r} \cdot \vec{F}_t^N ,j_\ell^{\dagger\, (k)}j_\ell^{(k)}] \right) + P  \, ,\\
\end{equation}

where we have defined

\begin{equation}
P = - \left(\sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta}|_{n \neq s \neq 1})^s \right) \left( \sum_{n\geq 2} \frac{i^n}{n!} \mathbb{K}_{\vec{r} \cdot \vec{F}_t^N }[ j_\ell^{\dagger\, (k)}j_\ell^{(k)}] \right) \, .
\end{equation}

Here, the notation $D_{\Delta}|_{n \neq s \neq 1}$ signals that when the term $s=1$ is considered, the sum over $n$ in the operator $D_{\Delta}$ [as defined by Eq. S_definition] starts from $n=2$.
By using the norm-bound

\begin{equation}
\eqalign{
\left\|
\mathbb{K}^n_{\vec{r} \cdot \vec{F}^{N}_t}[j_\ell^{\dagger\, (k)}j_\ell^{(k)}]   \right\|
\leq (d^2 r_{\max} 2 v)^n \frac{\| j_\ell\|^2}{N^{n/2}} \, ,
\end{equation}

and that, by exploiting Eqs. bound_DI222_b and bound_DI222_c, the following bound holds,

\begin{equation}
\eqalign{
& \left\| \sum_{b \geq 0} c_{\ell}^b \sum_{s=1}^b \left( \matrix{b \cr s} \right) (\Delta_N^\ell)^{b-s} (D_{\Delta}|_{n \neq s \neq 1})^s
\right\| \\
& \leq \frac{1}{N}\left[\gamma(\delta_\ell(1+ e^{ d^{4}r_{\max} a_{\max}} ))+ \delta_\ell \gamma'(\delta_\ell) e^{d^4 r_{\max} a_{\max}} \right] \, ,
\end{equation}

then it is

\begin{equation}
\| P \| \leq \frac{1}{N^2} \| j_{\ell}\|^2 e^{2 d^{2}r_{\max} v}\left[ \gamma(\delta_\ell(1+ e^{ d^{4}r_{\max} a_{\max}} )) + \delta_\ell \gamma'(\delta_\ell) e^{d^4 r_{\max} a_{\max}}\right] \, .
\end{equation}

Therefore we can conclude the proof, by using $\sum_{k}\frac{1}{\sqrt{N}} [ \vec{r} \cdot \vec{F}_t^N ,j_\ell^{\dagger\, (k)}j_\ell^{(k)}] = \sum_{\mu} \sum_{\beta} r_{\mu} c_{\ell \beta} \frac{1}{N}\sum_k[v_{\mu}^{(k)}, v_{\beta}^{(k)}] = - \sum_{\mu \beta} c_{\ell \beta} T_{\beta \mu}^{N} r_{\mu} = -\vec{c}_{\ell} \cdot (T^{N} \vec{r} )$,

\begin{equation}
\eqalign{
\| D_I^{222} - & \frac{1}{2} \Gamma(\Delta_N^{\ell}) \Gamma'(\Delta_N^{\ell}) \vec{r}\cdot (T^N \vec{r}_\ell) (-\vec{c}_{\ell} \cdot (T^{N} \vec{r} ))\|  \\
& \leq \frac{1}{2N}   \| j_{\ell}\|^2 e^{2 d^{2}r_{\max} v} \gamma(\delta_\ell) \left[ \gamma(\delta_\ell(1+ e^{ d^{4}r_{\max} a_{\max}} )) +\delta_\ell \gamma'(\delta_\ell) e^{d^4 r_{\max} a_{\max}}\right] \, ,
\end{equation}

where we further used $j_{\ell}^{(k)} = \sum_{\gamma} c_{\ell \gamma}^{j} v_{\gamma}^{(k)}$, and introduced ${c}_{\ell \beta} = \sum_{\gamma, \gamma' } c_{\ell \gamma}^{j *} c_{\ell \gamma'}^{j } \frac{1}{2}(a_{\gamma \gamma'}^\beta + b_{\gamma \gamma'}^{\beta}) $, with $\lbrace v_{\gamma}^{(k)} ,v_{\gamma'}^{(k)}\rbrace =  \sum_{\beta} b^{\beta}_{\gamma \gamma'} v_{\beta}^{(k)}$.

§.§ Lemma 11

Given a dynamical generator as defined by Eqs. e0_totalHamiltonian-jumps, satisfying Assumption Gamma, the action of the dissipation term $D_{II}$ defined by Eq. diss-second-term on local exponentials  is such that

\begin{equation*}
\eqalign{
& \lim_{N \rightarrow \infty} \omega_{\vec{r}_1 \vec{r}_2 }  \left( e^{t\linn} \left[ D_{II} \right] \right)= \\
& = \lim_{N \rightarrow \infty} \omega_{\vec{r}_1 \vec{r}_2 }  \left( e^{t\linn}\left[ i \vec{r} \cdot \left( \Gamma_\ell^2(\Delta_\ell^N)M_\ell \vec{F}_t^N \right) + i \sqrt{N} \vec{r} \cdot \left( \Gamma_\ell^2(\Delta_\ell^N) M_\ell \vec{\omega}_t^N \right) \right.\right.\\
& \left. \left. + \frac{1}{2 } \vec{r} \cdot \left( \Gamma_{\ell}^2(\Delta_\ell^N)M_\ell T^N\vec{r} \right) + \frac{1}{2}\vec{r} \cdot \left( \Gamma_\ell^2(\Delta_\ell^N) G^{N}_\ell \vec{r} \right)\right] W_{t}^N(\vec{r})
\right) \, . }
\end{equation*}

To start the proof, we define the following operators

\tilde{W}_1=\frac{1}{2}\sum_{k}\left[ j_\ell^{\dagger\, (k)}, W_t^{N} \right]j_{\ell}^{(k)}\, , \qquad  \tilde{W}_2=\frac{1}{2}\sum_{k}j_\ell^{\dagger\, (k)}\left[W_t^N , j_{\ell}^{(k)}\right]\,

which are such that $\tilde{W}_1+\tilde{W}_2=\mathcal{D}_\ell^{\rm Loc}[W_t^N]$, being the action of the local dissipator $\mathcal{D}_\ell^{\rm Loc}[\cdot]$ defined by Eq. D_loc . Through such operators we write

D_{II}=\Gamma_\ell^2(\Delta_N^\ell)\mathcal{D}_\ell^{\rm Loc}[W_t^N(\vec{r})]+\Gamma_\ell(\Delta_N^\ell)\left(\left[\tilde{W}_1,\Gamma_\ell(\Delta_N^\ell)\right]+\left[\tilde{W}_2,\Gamma_\ell(\Delta_N^\ell)\right]\right)\, ,

that allows us to split $D_{II}$ in the following terms

\begin{eqnarray}
& D_{II}^1 = \Gamma_\ell^2(\Delta_N^\ell)\mathcal{D}_\ell^{\rm Loc}[W_t^N(\vec{r})] \, ,\\
& D_{II}^2 = \Gamma_\ell(\Delta_N^\ell)\left(\left[\tilde{W}_1,\Gamma_\ell(\Delta_N^\ell)\right]+\left[\tilde{W}_2,\Gamma_\ell(\Delta_N^\ell)\right]\right) \, .
\end{eqnarray}

Let us start by considering the former term, $D_{II}^1 $, and explicitly derive $ \mathcal{D}_\ell^{\rm Loc}[W_t^N(\vec{r})]$, that is

\begin{equation}\label{D_local_W}
\eqalign{
\mathcal{D}_\ell^{\rm Loc}[W_t^N(\vec{r})] & = \frac{1}{2}\sum_{k}\left( \left[ j_\ell^{\dagger\, (k)}, W_t^{N} \right]j_{\ell}^{(k)} + j_\ell^{\dagger\, (k)}\left[W_t^N , j_{\ell}^{(k)}\right]\right) \\
& =  \frac{1}{2}\sum_{k} \left[ \left( j_\ell^{\dagger\, (k)}-  W_t^{N} j_\ell^{\dagger\, (k)}W_t^{N \dagger} \right) W_t^{N} j_\ell^{(k)}W_t^{N \dagger} \right. \\
& \left. +  j_\ell^{\dagger\, (k)} \left( W_t^{N} j_\ell^{(k)}W_t^{N \dagger} - j_\ell^{ (k)}\right)   \right] W_t^N(\vec{r}) = \\
&  = \frac{1}{2}\sum_{k} \left\lbrace \left( -\sum_{n \geq 1} \frac{i^n}{n!} \mathbb{K}_{\vec{r}\cdot \vec{F}_t^N} \left[ j_\ell^{\dagger \, (k)} \right] \right) \left( \sum_{n \geq 0} \frac{i^n}{n!} \mathbb{K}_{\vec{r}\cdot \vec{F}_t^N} \left[ j_\ell^{ (k)} \right]  \right) \right.  \\
& + \left. j_\ell^{\dagger\, (k)} \left( \sum_{n \geq 1} \frac{i^n}{n!} \mathbb{K}_{\vec{r}\cdot \vec{F}_t^N} \left[ j_\ell^{ (k)} \right] \right) \right\rbrace W_t^N(\vec{r}) = \\
&  = \frac{1}{2}\sum_{k} \left\lbrace \left( - i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{\dagger\, (k)}]+\frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{\dagger\, (k)}]\right] -\tilde{S}_3[j_{\ell}^{\dagger \, (k)}] \right) \right.\\
& \times \left. \left( j_\ell^{(k)} +  i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{ (k)}] + \tilde{S}_2[j_{\ell}^{ (k)}]   \right) \right.  \\
& + \left. j_\ell^{\dagger\, (k)} \left( + i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{(k)}] - \frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{ (k)}]\right] -\tilde{S}_3[j_{\ell}^{ (k)}] \right) \right\rbrace W_t^N(\vec{r}),
\end{equation}

where $\tilde{S}_{\bar{n}}[O^{(k)}] \equiv \sum_{n \geq \bar{n}} \frac{i^n}{n!}  \mathbb{K}^{n}_{\vec{r} \cdot \vec{F}^N_t} [O^{(k)}] $. By using the norm-bound $ \| \tilde{S}_{\bar{n}} \|  \leq \|O \| e^{2d^{2}r_{\max} v} / \sqrt{N}^{\bar{n}}$, we thus have

\begin{equation}\label{Dloc_onW}
\eqalign{
\mathcal{D}_\ell^{\rm Loc}[W_t^N(\vec{r})] & = \frac{1}{2}\sum_{k} \left( - i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{\dagger\, (k)}] j_\ell^{(k)} + j_\ell^{\dagger\, (k)}i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{(k)}] \right) \\
&   +   \frac{1}{2}\sum_{k} \left(  \frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{\dagger\, (k)}]\right] j_\ell^{(k)} - j_\ell^{\dagger\, (k)} \frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{ (k)}]\right] \right.\\
& \left. + [\vec{r}\cdot \vec{F}_t^N,  j_\ell^{\dagger\, (k)}][\vec{r}\cdot \vec{F}_t^N,  j_\ell^{(k)}] \right) + P\,
\end{equation}

\begin{equation}
\eqalign{
\| P\| &  \leq  \frac{1}{2} N \left( \left\|\tilde{S}_{1}[j_\ell^{\dagger, (k)}] \right\| \, \left\| \tilde{S}_{2}[j_\ell^{ (k)}]  \right\| +2 \left\| \tilde{S}_{3}[j_\ell^{ (k)}]  \right\| \| j_\ell^{(k)}\| \right. \\
& \left. + \|\frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{\dagger\, (k)}]\right] [\vec{r}\cdot \vec{F}_t^N,  j_\ell^{(k)}] \| \right) \\
& \leq \frac{1}{2\sqrt{N}} \|j_\ell \|^2 \left[ e^{4d^{2}r_{\max} v} + 2e^{2d^{2}r_{\max} v} + (2 d^2 r_{\max} v)^3 \right]
\end{equation}

In order to express the leading order terms of $ \mathcal{D}_\ell^{\rm Loc}[W_t^N(\vec{r})]$ in a compact form, we rewrite the first line of Eq. Dloc_onW as

\begin{equation}
\eqalign{
& \frac{1}{2}\sum_{k} \left( - i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{\dagger\, (k)}] j_\ell^{(k)} + j_\ell^{\dagger\, (k)}i[\vec{r}\cdot \vec{F}_t^N,  j_\ell^{(k)}] \right) \\
& = i \sum_{\mu} r_{\mu}  \left( \frac{1}{2}\sum_{k}  \left(  \left[   j_\ell^{\dagger\, (k)}, {F}_{\mu}^N(t) \right] j_\ell^{(k)} + j_\ell^{\dagger\, (k)} [{F}_{\mu}^N(t),  j_\ell^{(k)}]  \right) \right)\\
& =  i \sum_{\mu} r_{\mu}  \mathcal{D}_\ell^{\rm Loc}[F_{\mu}^N(t)] = i \sum_{\mu} r_{\mu}  \sum_{\nu} M_{\ell \mu}^{\nu} (F_{\nu}+ \sqrt{N}\omega_t^{\nu}) \,  .
\end{equation}

Moreover, employing

\begin{equation}
\eqalign{
&    \frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{\dagger\, (k)}]\right] j_\ell^{(k)} - j_\ell^{\dagger\, (k)} \frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{ (k)}]\right]  \\
& = -\frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N,[   j_\ell^{\dagger\, (k)}, \vec{r}\cdot \vec{F}_t^N ]j_\ell^{(k)} \right]  -  \frac{1}{2}\left[\vec{r}\cdot \vec{F}_t^N, j_\ell^{\dagger\, (k)}[\vec{r}\cdot \vec{F}_t^N,   j_\ell^{ (k)}]\right] \, ,
\end{equation}

the second line of Eq. Dloc_onW reads

\begin{equation}
\eqalign{
&\frac{1}{2} \sum_{\mu} \sum_{\nu} r_{\mu} r_{\nu} \left[ \mathcal{D}_\ell^{\rm Loc}[F_{\mu}^N(t)], F_{\nu}^{N}(t) \right] = \\
& \frac{1}{2} \sum_{\mu} \sum_{\nu} r_{\mu} r_{\nu} \sum_{\gamma} M_{\ell \mu}^{\gamma}[F_{\gamma}^{N}(t), F_{\nu}^{N}(t)] = \frac{1}{2} \sum_{\mu} \sum_{\nu} r_{\mu}  \sum_{\gamma} M_{\ell \mu}^{\gamma}T_{\gamma \nu}^{N} r_{\nu} \, ,\\
\end{equation}

and, using $j_\ell^{(k)} = \sum_{\alpha}c_{\ell \alpha}^{j}v_{\alpha}^{(k)}$, the last line can be written as

\begin{equation}
\eqalign{
& \frac{1}{2} \sum_{k }  [\vec{r}\cdot \vec{F}_t^N,  j_\ell^{\dagger\, (k)}][\vec{r}\cdot \vec{F}_t^N,  j_\ell^{(k)}] \\
&= -\frac{1}{2} \sum_{\mu \nu}r_{\mu}  r_{\nu}  \frac{1}{N} \sum_k \sum_{\alpha \beta} c^{j,*}_{\ell \alpha} c^{j}_{\ell \beta} [v_{\mu}^{(k)},v_{\alpha}^{(k)}][v_{\beta}^{(k)},v_{\nu}^{(k)}] \\
&= -\frac{1}{2} \sum_{\mu \nu}r_{\mu}  r_{\nu}  \frac{1}{N} \sum_k \sum_{\alpha \beta} c^{j,*}_{\ell \alpha} c^{j}_{\ell \beta} a_{\mu \alpha}^{\gamma} a_{\beta \nu}^{\gamma'} v_{\gamma}^{(k)} v_{\gamma'}^{(k)} \\
& = -\frac{1}{2} \sum_{\mu \nu}r_{\mu}  r_{\nu}  \sum_{\alpha \beta} \sum_{\gamma \gamma'} c^{j,*}_{\ell \alpha} c^{j}_{\ell \beta} a_{\mu \alpha}^{\gamma} a_{\beta \nu}^{\gamma'} d_{\gamma \gamma'}^{\eta} m_{\eta}^N\\
\end{equation}

where we have further used $v_{\gamma }v_{\gamma'} =( \lbrace v_{\gamma }, v_{\gamma'} \rbrace + [v_{\gamma },v_{\gamma'}])/2 = \sum_{\eta} d_{\gamma \gamma'}^{\eta}m_{\eta}^N$. Before going further on, notice that, by construction, $ \frac{1}{2} \sum_{k }  [\vec{r}\cdot \vec{F}_t^N,  j_\ell^{\dagger\, (k)}][j_\ell^{(k)}, \vec{r}\cdot \vec{F}_t^N] $ is positive definite. For the sake of a lighter notation, we can define the matrix $G_{\ell, \mu \nu}^N  \equiv   \sum_{\alpha \beta} \sum_{\gamma \gamma'} a_{\mu \alpha}^{\gamma} c^{j,*}_{\ell \alpha}d_{\gamma \gamma'}^{\eta} m_{\eta}^N c^{j}_{\ell \beta}  a_{\beta \nu}^{\gamma'} $

By all the above contributions, we can finally write

\begin{equation}
\eqalign{
& \lim_{N \rightarrow \infty} \omega_{\vec{r}_1 \vec{r}_2 }  \left( e^{t\linn} \left[ D_{II}^{1} \right] \right) = \\
&  \lim_{N \rightarrow \infty} \omega_{\vec{r}_1 \vec{r}_2 }  \left( e^{t\linn}\left[ i \vec{r} \cdot \left( \Gamma_\ell^2(\Delta_\ell^N)M_\ell \vec{F}_t^N \right) + i \sqrt{N} \vec{r} \cdot \left( \Gamma_\ell^2(\Delta_\ell^N)M_\ell \vec{\omega}_t^N \right) \right.\right.\\
& \left. \left. + \frac{1}{2 } \vec{r} \cdot \left( \Gamma_\ell^2(\Delta_\ell^N)M_\ell T^N\vec{r} \right)
-\frac{1}{2}\vec{r} \cdot( \Gamma_\ell^2(\Delta_\ell^N) G_\ell^{N}\vec{r})\right] W_{t}^N(\vec{r})
\right) \, .
\end{equation}

To proceed further and evaluating the contributions coming from $D_{II}^2$, we notice that the latter can be written as
$   D_{II}^2 = \Gamma_\ell(\Delta_N^\ell) \left[\mathcal{D}_\ell^{\mathrm{Loc}}[W_t^{N}(\vec{r})],\Gamma_\ell(\Delta_N^\ell)\right] $. Thus, by employing the expression $\mathcal{D}_\ell^{\mathrm{Loc}}[W_t^{N}(\vec{r})]$ of Eq. D_local_W it is

\begin{equation}
\eqalign{
D_{II}^2 & = \Gamma_\ell(\Delta_N^\ell) \\
& \times \left\lbrace \left[  \left( i \vec{r} \cdot \left( M_\ell \vec{F}_t^N \right) + i \sqrt{N} \vec{r} \cdot \left( M_\ell \vec{\omega}_t^N \right) \right) W_t^N(\vec{r}), \Gamma_\ell(\Delta_N^\ell)  \right] \right. \\
&  + \left[ \left( \frac{1}{2 } \vec{r} \cdot \left( M_\ell T^N\vec{r} \right) + \frac{1}{2}\vec{r} \cdot( G_\ell^{N}\vec{r}) \right)W_t^N(\vec{r}) , \Gamma_\ell(\Delta_N^\ell) \right] \\
& \left. + [P W_t^N(\vec{r}), \Gamma_\ell(\Delta_N^\ell)  ]\right\rbrace \, .
\end{equation}

The contribution in the last line can be bounded as

\begin{equation}
\eqalign{
& \| \Gamma_\ell(\Delta_N^\ell) \left[P W_t^N(\vec{r}), \Gamma_\ell(\Delta_N^\ell) \right] \|  \leq  2\gamma^2(\delta_\ell)\| P \| \\
& \leq  \frac{\gamma^2(\delta_\ell)}{\sqrt{N}} \|j_\ell \|^2 \left[ e^{4d^{2}r_{\max} v} + 2e^{2d^{2}r_{\max} v} + (2 d^2 r_{\max} v)^3 \right] \, .
\end{equation}

Similarly, by employing Lemma lemma_commutators and the bound comm_localexponential, for the second-last line it is

\begin{equation}
\eqalign{
& \left\| \Gamma_\ell(\Delta_N^\ell) \left[ \left( \frac{1}{2 } \vec{r} \cdot \left( M_\ell T^N\vec{r} \right) + \frac{1}{2}\vec{r} \cdot( G_\ell^{N}\vec{r}) \right) W_t^N(\vec{r}), \Gamma_\ell(\Delta_N^\ell) \right] \right\| \\
& \leq \gamma(\delta_\ell)r_{\max}^2 \left( | M_\ell| a_{\max} +|G_\ell|   \right) \left(\frac{1}{2} \frac{1}{\sqrt{N}}\gamma( \delta_\ell (1+e^{d^4 r_{\max} a_{\max}})) + \frac{1}{N}\delta_\ell \gamma'(\delta_\ell)  \right) \,.
\end{equation}

Finally, we are left with the following term,

\begin{equation}
\eqalign{
& \Gamma_\ell(\Delta_N^\ell)  \left[  \left( i \vec{r} \cdot \left( M_\ell \vec{F}_t^N \right)  \right) W_t^N(\vec{r}), \Gamma_\ell(\Delta_N^\ell)  \right]\\
&= i \Gamma_\ell(\Delta_N^\ell) \sum_{\mu \nu}r_{\mu}M_{\ell \mu}^{\nu}\left[F_{\nu}^{N}(t),  \Gamma_\ell(\Delta_N^\ell) \right]W_t^{N}(\vec{r}) \\
& + i \Gamma_\ell(\Delta_N^\ell) \sum_{\mu \nu}r_{\mu}M_{\ell \mu}^{\nu} F_{\nu}^{N}(t) \left[W_t^{N}(\vec{r}), \Gamma_\ell(\Delta_N^\ell) \right] \,.
\end{equation}

The second line of the latter expression can be bounded, using that

\begin{equation}
\| \left[F_{\nu}^{N}(t),  \Gamma_\ell(\Delta_N^\ell) \right] \| \leq \frac{2}{\sqrt{N}}\delta_\ell \gamma'(\delta_\ell)\, ,
\end{equation}

\begin{equation}
\eqalign{
& \|\Gamma_\ell(\Delta_N^\ell) \sum_{\mu \nu}r_{\mu}M_{\ell \mu}^{\nu}\left[F_{\nu}^{N}(t),  \Gamma_\ell(\Delta_N^\ell) \right]W_t^{N}(\vec{r}) \| \\
& \leq \frac{2}{\sqrt{N}} \delta_\ell \gamma'(\delta_\ell) \gamma(\delta_\ell) d^4 r_{\max}|M_\ell| \, .
\end{equation}

Finally, using Eq. comm_localexponential, the result statement_lemmB2, and Lemma lemma_squarefluctu_convergence it is

\begin{equation}
\eqalign{
&   \lim_{N \rightarrow} \omega_{\vec{r}_1 \vec{r}_2} \left(e^{t\linn}\left[ \Gamma_\ell(\Delta_N^{\ell})F_{\nu}^N(t) \left[W_t^{N}(\vec{r}), \Gamma_\ell(\Delta_N^\ell) \right] \right] \right) \\
& =  \lim_{N \rightarrow} \omega_{\vec{r}_1 \vec{r}_2} \left(e^{t\linn}\left[ F_{\nu}^N(t) \Gamma_\ell(\Delta_N^{\ell})\left[W_t^{N}(\vec{r}), \Gamma_\ell(\Delta_N^\ell) \right] \right] \right) = \\
& \leq \lim_{N\rightarrow +\infty} \gamma^2(\delta_\ell) \| \left[W_t^{N}(\vec{r}), \Gamma_\ell(\Delta_N^\ell) \right] \|^2 \sqrt{\omega\left( W^{N}(\vec{r}_1) \left[ \left(F_{\nu}^N(t)\right)^2 \right]  W^{N \dagger}(\vec{r}_1) \right)} = 0 \, ,
\end{equation}

which concludes the proof of the Lemma.

§ ANALYSIS OF A QUANTUM HOPFIELD NEURAL NETWORK

§.§ Mean-field equations

In this Appendix, we derive the mean-field equations for the model presented in Section <ref>. We consider a system made of $N$ spin $1/2$-particles,  which evolves according to a Markovian quantum master equation, characterized by jumps operators

J_{\pm}^{(k)} = \hat{\sigma}_{\pm}^{(k)}\Gamma_{\pm}(\Delta E), \qquad \Gamma_{\pm}(\Delta E)= \frac{ e^{\pm \frac{\beta}{2}\Delta E }}{ \sqrt{2}} \, , \qquad \Delta E= \frac{1}{N}\sum_{\mu=1}^{p}\xi^{\mu}_{k}\sum_{j } \xi_{j}^{\mu} \hat{\sigma}^{(j)}_{z} \, ,$$

and Hamiltonian

$$H=\Omega\sum_{i=1}^{N} \hat{\sigma}^{(i)}_x \, .$$

As already commented in the main text, the variables $\lbrace \xi_{k}^{\mu} \rbrace$ are i.i.d. random variables, with $\xi_{k}^{\mu} = \pm 1$ and probability distribution $\mathbb{P}(\xi_{k}^{\mu} = \pm 1) = \frac{1}{2}$. By applying the large-spin mapping described in Section <ref>, we consider the mean-field equations related to the the average operators $m^{N_s}_{\alpha, k}$, which reads

\begin{equation}\label{eq:mf-largespinHNN}
\eqalign{
& \dot{m}_{k,z}(t) = 2\Omega m_{k,y}(t)-\cosh{(\beta \Delta E_{\Lambda_k}(t))} m_{k,z}(t)+\sinh{(\beta \Delta E_{\Lambda_k}(t))} \, ,\\
& \dot{m}_{k,y}(t) = -2\Omega m_{k,z}(t)-\frac{1}{2}\cosh{(\beta \Delta E_{\Lambda_k}(t))}m_{k,y}(t) \, ,\\
& \dot{m}_{k,x}(t) =-\frac{1}{2}\cosh{(\beta \Delta E_{\Lambda_k}(t)) }m_{k,x}(t) \, .\\
\end{equation}

It is worth noticing that the equation for the variable $\dot{m}_{k,x}(t)$ decouples. In the following, without loss of generality, we will consider an initial condition $m_{k,x}(0)=0$ and focus on the equations of motion along the $y$ and $z$ directions.

We remind that we aim at characterizing, in the long time limit behaviour, the retrieval property of the system. To this purpose, it is more useful to consider the equations of motion for the overlap variables. For the sake of completeness, we introduce the latter starting from the corresponding operator formulation, that reads

$$ o^{\mu,N}_{\alpha} = \frac{1}{N}  \sum_{i=1}^{N}
\xi_{i}^{\mu}\sigma_{\alpha}^{(i)} = \frac{1}{2^p} \sum_{k=1}^{2^p}e_{k}^{\mu} \frac{S_{\alpha}^{(k)}}{N_s} \, ,$$

for $\mu=1,...,p$, having employed the large-spin mapping in the last step. It is worth noticing that the expectation value of these operators quantifies the amount of alignment of the spin $1/2$ system with respect to to the (classical) configuration defined by the $\mu$-th pattern, $(\xi_{1}^{\mu},..., \xi_{N}^{\mu})$ [or, equivalently, of the system of $2^p$ large-spins with respect to the $\mu$-th configuration $(e_{1}^{\mu},..., e_{2^p}^{\mu})$]. The equations of motion for the overlap parameters $o^{\mu}_{\alpha} = \lim_{N \rightarrow + \infty} \omega_t(o^{\mu,N}_{\alpha})$ read

\begin{equation}\label{eq_overlap}
\eqalign{
& \dot{o}_z^{\mu}(t) = 2\Omega o_{y}^{\mu} (t) - \braket{\braket{ \cosh{(\beta \vec{\xi}\cdot \vec{o}_z(t) )} }} {o}_z^{\mu}(t) + \braket{\braket{\xi^{\mu} \sinh{ (\beta \vec{\xi}\cdot \vec{o}_z(t) )}}} \, , \\
& \dot{o}_y^{\mu}(t) = -2\Omega o_{z}^{\mu}(t) - \frac{1}{2}\braket{\braket{ \cosh{(\beta \vec{\xi}\cdot \vec{o}_z(t) )} }} {o}_y^{\mu}(t)\, .
} \\
\end{equation}

The above equations have been further simplified under the self-averaging hypothesis [47, 72, 71], allowing us to employ the substitution $\frac{1}{N} \sum_{i}f(\xi_i^{\mu}) \rightarrow \frac{1}{2^p} \sum_{\mathbb{\xi}}\mathbb{P}(\xi) f{(\mathbb{\xi})} \equiv \braket{\braket{f{(\mathbb{\xi}})}}$, wit h $f(·)$ a generic function. We also employed the substitution $ω_t(σ_i^α) = ξ_i^μ o^μ_α(t)$, assuming a homogeneous distribution of the misalignment between patterns and spins [79, 70, 46]. The equations \eref{eq_overlap} are symmetric with respect to $(i)$ variable permutation $(o_z^μ,o_y^μ) ↔(o_z^ν,o_y^ν)$, and $(i)$ mode inversion $(o_z^μ,o_y^μ) ↔(-o_z^μ,-o_y^μ)$, and therefore the dynamics displays the following invariant subspaces: $(o_z^μ,o_y^μ) = (0,0)$,  $(o_z^μ,o_y^μ) = (o_z^ν,o_y^ν)$, $(o_z^μ,o_y^μ) = (-o_z^ν,-o_y^ν)$.

First of all we consider the paramagnetic solutions, $(o_z^μ,o_y^μ) = (0,0)$ $∀μ$, analyzing their stability. We linearize the EoMs \eref{eq_overlap}, with respect to the solution $o_α^μ = 0+δo^μ_α $, $α=z,y$, obtaining
\begin{equation}
\left(
\matrix{ \delta \dot{o}^{\mu}_z \cr
\delta \dot{o}^{\mu}_y }
\right) =
\left(
\matrix{
\beta -1 && 2\Omega \cr
-2 \Omega && -\frac{1}{2}
\right)
\left(
\matrix{ \delta o^{\mu}_z \cr
\delta o^{\mu}_y
\right) \, .
\end{equation}
The stability matrix coincides with the one analyzed in Ref. [47]. We summarize here the results, for the sake of completeness. The eigenvalues reads
$λ_± = 1/2[β- 3/2 ±√((β-1/2)^2 - 16Ω^2)]$, and give rise to the following different regimes:
\begin{itemize}
\item[$(V)$] $ \lbrace \beta < 1+8\Omega^2 \rbrace \cap \lbrace \beta < 3/2 \rbrace \cap \lbrace |\beta - 1/2| > 4\Omega \rbrace $,  $\lambda_{\pm} \in \mathbb{R}^-$, the fixed point is stable;
\item[$(IV)$]$ \lbrace \beta < 1+8\Omega^2 \rbrace \cap \lbrace \beta < 3/2 \rbrace \cap \lbrace |\beta - 1/2| < 4\Omega \rbrace $, $\lambda_{\pm} \in \mathbb{C}$, with $\mathrm{Re}(\lambda_{\pm}) \in \mathbb{R}^-$, the fixed point is stable and spiralizing.
\item[$(III)$] $ \lbrace \beta < 1+8\Omega^2 \rbrace \cap \lbrace \beta > 3/2 \rbrace \cap \lbrace \beta  < 4\Omega +1/2 \rbrace $, $\lambda_{\pm} \in \mathbb{C}$, with $\mathrm{Re}(\lambda_{\pm}) \in \mathbb{R}^+$, the fixed point is unstable and spiralizing.
\item[$(II)$] $ \lbrace \beta < 1+8\Omega^2 \rbrace \cap \lbrace \beta > 3/2 \rbrace \cap \lbrace \beta  > 4\Omega +1/2 \rbrace $, $\lambda_{\pm} \in \mathbb{C}$, with $\mathrm{Re}(\lambda_{\pm}) \in \mathbb{R}^+$, the fixed point is unstable.
\item[$(I)$] $  \beta > 1+8\Omega^2$, $\lambda_{\pm} \in \mathbb{R}$, with different sign, the fixed point is a saddle point.
\end{itemize}

It is worth noticing that the change in the sign of $Re[λ_±]$ between regions $(III)$ and $(IV)$ identifies a Hopf bifurcation, that signals limit cycle solutions, already found by Refs. [47, 70].

Let us now turn our attention to the ferromagnetic, or retrieval, solutions, that have the form $o^ν_α= o̅_αδ_νμ $  for a specific $μ=1,...,p$. By employing Eqs. \eref{eq_overlap}, the stationary equations for these type of solutions read
\begin{equation}\label{c:statRet}
\left[1+8\frac{\Omega^2}{\cosh^2{(\beta \bar{o}_z)}} \right] \bar{o}_z = \tanh{(\beta \bar{o}_z)} \, ,
\end{equation}
which admit finite values solution for $(1+8 Ω̃^2)<β$, where $Ω̃ = Ω/ cosh(βo̅_z)$. The stability analysis can be performed by linearizing Eqs. \eref{eq_overlap} with respect to the solutions $o^ν_α = o̅_α δ_νμ + δo_α^ν $, further taking $μ=1$ without loss of generality. The stability matrix reads in this case
\begin{equation}
\left(  \matrix{
\cosh{(\beta \bar{o}_z)}\left[ \beta\left( 1- \bar{o}_z  \tan{(\beta \bar{o}_z)} \right)-1 \right] && 2\Omega \cr
-2\Omega( 1-\beta {o}_z  \tan{(\beta \bar{o}_z)}) && -\frac{1}{2} \cosh{(\beta \bar{o}_z)} \,
\right)
\end{equation}
with eigenvalues that can be written as
\eqalign{
\lambda_{ \pm} & = \frac{\cosh{(\beta \bar{o}_z)}}{2} \left\lbrace \beta' -\frac{3}{2} \pm \sqrt{(\beta'-\frac{3}{2})^2 + 2(\beta'-1+8 \tilde{\Omega}^2(\beta-1-\beta')} \right\rbrace  \\
& = \frac{\cosh{(\beta \bar{o}_z)}}{2} \left\lbrace \beta' -\frac{3}{2} \pm \sqrt{(\beta')^2 - (1+16\tilde{\Omega}^2)\beta'+16 \tilde{\Omega}^2(\beta-1)+\frac{1}{4}} \right\rbrace \, .  \\
where we introduced $β' = β(1-o̅_z tanh(βo̅_z)).$ For the parameter regime $β> 1+8Ω̃$ we have numerically evaluated the eigenvalues $λ_±$ with respect to the retrieval solutions, obtaining $λ_± ∈ℝ^-$, and thus stable retrieval solutions.

The resulting phase diagram is illustrated in Fig.~\ref{fig:phaseD}. It displays a high-temperature paramagnetic phase, and a low-temperature retrieval phase, taking place for values of $β$ and $Ω$ such that $β>(1+8Ω̃^2)$. There exists an intermediate region where stationary configurations feature self-sustained oscillations. Such a parametric region ranges from the parameters at which the Hopf bifurcation takes place, to the parameter values corresponding to stable retrieval solutions, $β< 3/2$, $Ω> 1/4$,  $β< 1+8Ω̃^2$. We further performed numerical analysis to unveil the characteristic of this phase, which enforce the suggestion that it correspond to a limit-cycle phase. Figure \ref{fig:LC_SD}(a) shows the  standard deviation
$$o_{\mathrm{sd}} = \sqrt{\frac{1}{N_I}\sum_{i \in I}[o_z(t_i)-\bar{o}_z]^2} \, , $$
of the overlap component $o_z(t)$ with respect to the corresponding long-time average, say $o̅_z$. Here, ${o_z(t_i)}_i=1^N_I$ are obtained at times $t ∈I$, being $I$ a time-interval chosen at long times. We set $N_I=2000$, $I=[9 ×10^3, 10^4]$. The upper and lower panel in Fig.~\ref{fig:LC_SD}(b) show the parametric plot and flux diagram of the vector field $( o_z(t), o_y(t))$ at long times. In the lower panel, results are shown at $T=0.6$, $Ω=0.35$, corresponding to the retrieval phase. Here the convergence to two stable solutions is illustrated. The upper panel is obtained for values of $(T,Ω)$ in the limit-cycle phase, $T=0.6$, $Ω=0.6$, and it displays a stable orbit.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{limitcycle_pd_flux2.png}
\caption{\textbf{Limit-cycle phase}. (a) For the case $p=1$, we display the standard deviation $o_{\mathrm{sd}}$ of the overlap $o_z(t)$ with respect to the stationary solutions, at long times. We set initial conditions such that $o_{\alpha}(0) \approx 0$, for $\alpha = y, z$. In the blue region $o_{\mathrm{sd}}$ is finite, signalling a limit-cycle phase. (b) With $T=0.6$, the two panels display the parametric plot and the flux diagram of the vector field $(o_y(t),o_z(t))$, for $\Omega=0.35$ (bottom panel), and $\Omega=0.6$ (upper panel). The latter shows a closed orbit, that in general characterizes limit-cycle phases. Colors for the flux diagram identify the norm of the corresponding vector field, which increases from purple to yellow. }
\label{fig:LC_SD}
\end{figure}

\subsection{Derivation of the equation for the covariance matrix}
\label{A:covarianceM_HNN}

By exploiting Theorem \eref{theorem-mesoscopicdynamics} and the general results of Section \ref{sec4}, we derive the time evolution of the covariance matrix related to the system open quantum generalized HNN. For the sake of completeness, and for deriving some considerations (as it will be clearer at the end of the Section), we add here also an additional, all-to-all Hamiltonian of the type
$H_2= g/21/2^pN_s∑_h,k w̃_h,k S^(h)_z S^(k)_z$.
The time evolution of the covariance matrix reads
\begin{equation}
\dot{\Sigma}^{\omega} = Q\Sigma^{\omega}+\Sigma^{\omega}Q^T+\sigma S^{\mathrm{sym}}\sigma^T + \Gamma^{2}G^{\mathrm{sym}} \, ,
\end{equation}
where we assume to consider an initial state that is stationary with respect to the mean-field variables.
The matrices appearing in the equation of the covariance matrix read
$${Q} \equiv  \vec{f}(\vec{\omega}) - i\sigma(\vec{\omega}) 2ih^{\mathrm{R}} \, , $$
$$\Gamma^2 G^{\mathrm{sym}} = \sum_\ell\sum_{k'} \Gamma_\ell^2(\Delta_{\ell}^{k'}) G_\ell^{\mathrm{sym} \, k'} \, , $$
$$[G^{\mathrm{sym} \, k'}_\ell]_{(\mu h) (\nu k)} = \delta_{hk} \delta_{hk'} \left[ \sum_{\alpha \beta} c^{j,*}_{\alpha} c^{j}_{\beta} \sum_{\gamma \gamma' } a_{\mu \alpha}^{\gamma}  a_{\nu \beta }^{\gamma'} \frac{1}{2}\sum_{i}\left( \lbrace v_{\gamma}^{(i)}, v_{\gamma'}^{(i)} \rbrace + [v_{\gamma}^{(i)}, v_{\gamma'}^{(i)}] \right)\right]^{\mathrm{sym}}   \,  ,$$
[S^{\mathrm{sym}}]_{(\mu h ),(\nu k)} & =  \frac{1}{2}\sum_\ell \sum_{k'} (r_{\ell \mu}^{h k'} {r}_{\ell \nu}^{k' k} + r_{\ell \nu}^{h k'} {r}_{\ell \mu}^{k' k} ) \left(\Gamma'(\Delta^{k'}_{\ell}) \right)^2   \omega_t\left(\frac{1}{N_s}\sum_{i \in \Lambda_{k'}}  j_\ell^{\dagger (i)} j_\ell^{ (i)}   \right)  \\
& + \frac{1}{2} \sum_\ell \sum_{k'} \Gamma(\Delta_{\ell}^{k'}) \Gamma'(\Delta_{\ell}^{k'})  (r_{\ell \alpha}^{hk'} \delta_{k'k}c_{\ell \beta} + r_{\ell \beta}^{k'k}\delta_{k'h} c_{\ell \alpha}  ) \, .
We will now specify the terms of the above expression in relation to the model we are considering, starting from the matrix $Q$. We remind that the matrix form $f_α$ in the contributions $f⃗(ω⃗)$ is formally given by Lemma \ref{lemma_gen_action}. We further employ that, for the two-body interaction Hamiltonian $H_2$, it is $h_(αh)   (β, k)  = δ_αβ δ_βz g/2w̃_h,k$; the on-site jump operators reads $j_±^(k) = σ_±^(k)$; and the structure constant of the single-particle algebra read $a_αβ^γ = 2iℰ_αβγ$, with $ ℰ_αβγ$ the Levi-Civita tensor. Thus, the action of the local dissipator $𝒟_±^Loc[v_α^(i)] = ∑_β=1^d^2M_l (αh)^βk $, on elements of the basis reads
\begin{eqnarray}
& \mathcal{D}_{\pm}^{\mathrm{Loc}}[\sigma_{x}^{(i)}] = -\frac{1}{2} \sigma_{x}^{(i)} \, , \\
& \mathcal{D}_{\pm}^{\mathrm{Loc}}[\sigma_{y}^{(i)}] =  -\frac{1}{2} \sigma_{y}^{(i)} \, , \\
& \mathcal{D}_{\pm}^{\mathrm{Loc}}[\sigma_{z}^{(i)}] = \pm \mathbb{I}-\sigma_z^{(i)} \,. \\
\end{eqnarray}
This defines a real matrix $M$ such that
\begin{equation}
M_{l (\alpha h)}^{(\beta k)} = \delta_{h k}\left[ \delta_{\alpha \beta} \left(-\frac{1}{2}\delta_{x\alpha}-\frac{1}{2}\delta_{y\alpha} -\delta_{\alpha z}\right) + \ell \delta_{\alpha z} \delta_{\beta 4}  \right]\, .
\end{equation}
It is worth noticing that, as far as it concerns the contribution to the quantum fluctuations, we consider the components $α= x,y,z$, the identity playing the role of an irrelevant operator, as it commutes with the other fluctuation operators. As a consequence, the action of the mean-field equations of motion on these components can be written as
\begin{eqnarray*}
f_{(\alpha \, h) \, (\beta \, k)} & =  i  A_{(\alpha h),  (\beta,k)}  + i \sum_{\gamma, h'} B_{(\alpha h), (\gamma h') (\beta k)} m_{\gamma h'} \\
&+ \sum_{\ell}  \Gamma^{2}_{\ell}(\Delta_{\ell}^{h}) M_{\ell (\alpha h) }^{(\beta k)} \, ,  \\
\end{eqnarray*}
\begin{eqnarray*}
& A_{(\alpha \, h), (\beta \, k)} = \Omega \sum_{\beta'=x,y,z}   2i \delta_{\beta' x}\mathcal{E}_{ \beta' \alpha \beta } \delta_{k,h} \, , \\
& B_{(\alpha,h) (\gamma,h') (\beta k)} = 2i \delta_{\gamma \gamma'} \delta_{\gamma' z} \mathcal{E}_{\gamma' \alpha \beta} (g\tilde{w}_{h h'}) \delta_{hk}\, ,
\end{eqnarray*}
\begin{eqnarray*}
\sum_{\ell}  \Gamma^{2}_{\ell} (\Delta_{\ell}^{h}(t)) M_{\ell (\alpha h) }^{(\beta k)} & = \sum_{\ell=\pm}  \Gamma^{2}_{\ell} (\Delta E_{\Lambda_h}) M_{\ell (\alpha h) }^{(\beta k)} \\
& = \delta_{hk} \cosh{(\beta \Delta E_{\Lambda_h})}  \delta_{\alpha \beta} \left(-\frac{1}{2}\delta_{x\alpha}-\frac{1}{2}\delta_{y\alpha} -\delta_{\alpha z}\right) \,. \\
\end{eqnarray*}
Finally, using also
\begin{equation}
\eqalign{
-i\sum_{\gamma h'  } \sigma_{(\alpha h) \, (\gamma h')} 2ih_{(\gamma h') (\beta k)} &  =  -i\sum_{\gamma  } \sigma_{(\alpha h) \, (\gamma h')} \delta_{h h'} 2i\delta_{\gamma \beta} \delta_{\beta z}\frac{g}{2}\tilde{w}_{h' k}\\
& = + g \sigma_{\alpha \beta (h)} \delta_{\beta z} \tilde{w}_{h k} \, ,\\
\end{equation}
it is
\begin{eqnarray}
[Q]_{(\alpha \, h), (\beta \, k)} & = -2\Omega \mathcal{E}_{x\alpha \beta} \delta_{hk} -2g\mathcal{E}_{z\alpha \beta}\sum_{h'} \tilde{w}_{hh'} m_{z h'}\delta_{h k} \\
&+ g \sigma_{\alpha \beta (h)} \delta_{\beta z} \tilde{w}_{h k} \\ \nonumber
& + \delta_{hk} \cosh{(\beta \Delta E_{\Lambda_h})} \delta_{\alpha \beta} \left(-\frac{1}{2}\delta_{x\alpha}-\frac{1}{2}\delta_{y\alpha} -\delta_{\alpha z}\right) \, , \\ \nonumber
[Q]^{T}_{(\alpha \, h), (\beta \, k)} & = +2\Omega \mathcal{E}_{x\alpha \beta} \delta_{h k} + 2g\mathcal{E}_{z\alpha \beta}\sum_{h'} \tilde{w}_{hh'} m_{z h'}  \delta_{h k} \\
&- g \sigma_{\alpha \beta (h)} \delta_{\beta z} \tilde{w}_{h k} \\ \nonumber
& + \delta_{hk} \cosh{(\beta \Delta E_{\Lambda_h})} \delta_{\alpha \beta} \left(-\frac{1}{2}\delta_{x\alpha}-\frac{1}{2}\delta_{y\alpha} -\delta_{\alpha z}\right) \, .\\ \nonumber
\end{eqnarray}
By exploiting the expression of the constant of the algebra $a_αβ^γ $, and having defined the local jump operators, one obtains
\begin{equation}
\eqalign{
[\Gamma^2 G^{\mathrm{sym}} ]_{(\alpha h)(\beta k)}& = \delta_{hk} \cosh{(\beta \Delta E_{\Lambda_h})} \left\lbrace\delta_{\alpha x}\delta_{x\beta} +\delta_{\alpha y}\delta_{y\beta} \right. \\
& \left. + 2\delta_{\alpha z}\delta_{z\beta}\right[ 1-\tanh{(\beta \Delta E_{\Lambda_h})} m_{z,h}\left]  \right\rbrace \\
& - \delta_{\alpha x}\delta_{z \beta} \sinh{(\beta \Delta E_{\Lambda_h})} m_{x,h} - \delta_{\alpha y}\delta_{z \beta} \sinh{(\beta \Delta E_{\Lambda_h})} m_{y,h}\\
&- \delta_{\alpha z}\delta_{x \beta} \sinh{(\beta \Delta E_{\Lambda_h})} m_{x,h} - \delta_{\alpha z}\delta_{y \beta} \sinh{(\beta \Delta E_{\Lambda_h})} m_{y,h} \, ,
\end{equation}
We can thus derive the term $σS^sym σ^T$. To begin with, we divide it according to the two followng terms,
\begin{equation}
\eqalign{
& [S^{\mathrm{sym}}_1]_{(\mu h)(\nu k)} = \frac{1}{2}\sum_\ell \sum_{k'} (r_{\ell \mu}^{hk'} r_{\ell \nu}^{k' k}+ r_{\ell \nu}^{hk'} r_{\ell \mu}^{k' k})\left(\Gamma'(\Delta^{k'}_{\ell} ) \right)^2   \omega_t\left(\frac{1}{N_s}\sum_{i \in \Lambda_{k'}}  j_\ell^{\dagger (i)} j_\ell^{ (i)}   \right) \\
& [S^{\mathrm{sym}}_2]_{(\mu h)(\nu k)} = \frac{1}{2} \sum_\ell [\Gamma(\Delta_{\ell}^{h}) \Gamma'(\Delta_{\ell}^{h})  r_{\ell \mu}^{hk} c_{\ell \nu} + r_{\ell \nu}^{h k} c_{\ell \mu} \Gamma(\Delta_{\ell}^{k}) \Gamma'(\Delta_{\ell}^{k})  ]
\end{equation}
such that $S^sym =  S^sym_1 +  S^sym_2 $. We start from the terms $S^sym_1$, and we notice that the components of the vector $r⃗_ℓ$ read here $ r_ℓα^h k = δ_αz w̃_kh= δ_αz w̃_hk$, so that they are independent of $ℓ$ and symmetric in the large-spin index. Moreover, the average operator reads
$$\frac{1}{N_s} \sum_{i \in \Lambda_{k}} j_{\ell}^{(i)\dagger} j_{\ell}^{(i)} = \frac{1- \ell m_{z,k}^{N_s}}{2} \, ,$$
and it has to be evaluated on a clustering state, as we are considering the thermodynamic limit. Furthermore, for the derivative of the rates, $Γ_ℓ(ΔE_Λ_k)=  e^ℓβ/2ΔE_k/√(2)$, it is
\begin{equation}
\left(\Gamma_{\ell}'(\Delta E_{\Lambda_{k}}) \right)^2 = \frac{\beta^2}{4} \Gamma_{\ell}^2( \Delta E_{\Lambda_{k}} ) \, .
\end{equation}
Thus we have
\begin{equation}
\eqalign{
& [\sigma S^{\mathrm{sym}}_1 \sigma^T]_{(\alpha h), (\beta k)}  = \frac{1}{2}  \sigma_{\alpha \mu (h)} \delta_{\mu, z}\sum_{\ell, k'} \left[\Gamma_{\ell}^{'\, 2}(\Delta E_{\Lambda_{k'}})(1- \ell m_{z,k}) \right] \tilde{w}^{hk'}\tilde{w}^{k' k} \delta_{\nu z}\sigma_{ \beta \nu (k)} \\
& =  \frac{\beta^2}{8}\sigma_{\alpha z (h)}\sum_{k'} [ \cosh{(\beta \Delta E_{\Lambda_{k'}})} - m_{z k'}  \sinh{(\beta \Delta E_{\Lambda_{k'}})}]\tilde{w}^{hk'}\tilde{w}^{k' k}\sigma_{\beta z (k)} \, .
\end{equation}
Finally, we focus on the term $S^sym_2$. To this end, we notice that the elements $c_ℓη $ for $η=x, y, z$ reads here
{c}_{\ell , \eta} & = \sum_{\gamma, \gamma' } c_{\ell, \gamma}^{j *} c_{\ell, \gamma'}^{j } \frac{1}{2}(a_{\gamma \gamma'}^\eta + b_{\gamma \gamma'}^{\eta}) \\
& = \sum_{\gamma \gamma'} c_{\gamma \ell}^{j \, *}c_{\gamma' \ell}^{j \, *} [\delta_{\gamma \gamma'} + i\mathcal{E}_{\gamma \gamma' \eta}] = - \frac{1}{2}\ell \mathcal{E}_{xyz}\delta_{\eta z}
\, , }$$
where we used $j_ℓ^(k) = σ_±^(k)$, and thus $c_ℓ, γ^j = {1/2, ±i/2  ,0 ,0}$. Hence, we obtain
\begin{equation}
\eqalign{
[\sigma S^{\mathrm{sym}}_2 \sigma^T]_{(\alpha h), (\beta k)}
& = \sigma_{\alpha \mu (h)} \delta_{\mu z} \sum_{\ell} \left\lbrace -\frac{\ell}{4} \left[ \Gamma_{\ell}(\Delta E_{\Lambda_{h}}) \Gamma_{\ell}'(\Delta  E_{\Lambda_{h}})+ \right. \right. \\
& \left. \left. \Gamma_{\ell}(\Delta  E_{\Lambda_{k}}) \Gamma_{\ell}'(\Delta  E_{\Lambda_{k}} )\right] \right\rbrace \tilde{w}_{hk} \delta_{\nu z}\sigma_{\beta \nu (k)} \\
& = -\frac{\beta}{8} \sigma_{\alpha z (h)} \tilde{w}_{hk} \sum_{\ell}[\Gamma_{\ell}^2(\Delta  E_{\Lambda_{h}})+\Gamma_{\ell}^2(\Delta E_{\Lambda_{k}} ) ] \sigma_{\beta z (k)} \\
& = -\frac{\beta}{8} \sigma_{\alpha z (h)} \tilde{w}_{hk} [\cosh(\beta \Delta E_{\Lambda_{h}})+\cosh(\beta  \Delta E_{\Lambda_{k}}) ] \sigma_{\beta z (k)} \, .
\end{equation}
Collecting all the above terms together, the EoMs for the covariance matrix reads
\begin{equation}\label{C:EoM_covariancematrix}
\eqalign{
\dot{\Sigma}_{(\alpha, h), (\beta,k)}^{\omega} & = \sum_{\mu, k'} \left\lbrace -2\Omega \mathcal{E}_{x\alpha \mu} \delta_{hk'} -2g\mathcal{E}_{z\alpha \mu}\sum_{h'} \tilde{w}_{hh'} m_{z h'} \delta_{h k'} \right. \\
& + g \sigma_{\alpha \mu (h)} \delta_{\mu z} \tilde{w}_{h k'} \\
& \left. + \delta_{hk'} \cosh( \beta \Delta E_{\Lambda_{h}}) \delta_{\alpha \mu} \left(-\frac{1}{2}\delta_{x\mu}-\frac{1}{2}\delta_{y\mu} -\delta_{\mu z}\right) \right\rbrace\Sigma^{\omega}_{(\mu k') (\beta k)} \, + \\
&+\sum_{\mu k'} \Sigma^{\omega}_{(\alpha h) (\mu k')} \left\lbrace2\Omega \mathcal{E}_{x\mu \beta} \delta_{k' k} + 2g\mathcal{E}_{z\mu \beta}\sum_{h'} \tilde{w}_{k'h'} m_{z h'}(t)  \delta_{k' k} \right. \\
&- g \sigma_{\mu \beta (k')} \delta_{\beta z} \tilde{w}_{k' k} \\
& + \left. \delta_{k'k} \cosh( \beta \Delta E_{\Lambda_{k'}}) \delta_{\mu \beta} \left(-\frac{1}{2}\delta_{x\mu}-\frac{1}{2}\delta_{y\mu} -\delta_{\mu}\right) \right\rbrace+ \\
& + \delta_{hk} \cosh( \beta \Delta E_{\Lambda_{h}}) \left\lbrace\delta_{\alpha x}\delta_{x\beta} +\delta_{\alpha y}\delta_{y\beta} \right. \\
& \left. + 2\delta_{\alpha z}\delta_{z\beta}\right[ 1-\tanh( \beta \Delta E_{\Lambda_{h}} ) m_{z,h} \left]  \right\rbrace \\
& - \delta_{\alpha x}\delta_{z \beta}\sinh\left(  \beta \Delta E_{\Lambda_{h}} \right)m_{x,h} - \delta_{\alpha y}\delta_{z \beta}\sinh\left( \beta \Delta E_{\Lambda_{h}} \right)m_{y,h}\\
&- \delta_{\alpha z}\delta_{x \beta}\sinh\left(\beta \Delta E_{\Lambda_{h}} \right)m_{x,h} - \delta_{\alpha z}\delta_{y \beta}\sinh\left( \beta \Delta E_{\Lambda_{h}} \right)m_{y,h}\\
& + \frac{\beta^2}{8}\sigma_{\alpha z (h)}\sum_{k'} [ \cosh{(\beta  \Delta E_{\Lambda_{k'}})} - m_{z k'}  \sinh{(\beta  \Delta E_{\Lambda_{k'}})}]\tilde{w}^{hk'}\tilde{w}^{k' k}\sigma_{\beta z (k)} \\
&-\frac{\beta}{8} \sigma_{\alpha z (h)} \tilde{w}_{hk} [\cosh(\beta  \Delta E_{\Lambda_{h}})+\cosh(\beta  \Delta E_{\Lambda_{k}}) ] \sigma_{\beta z (k)} \, .
\end{equation}
By setting $g=0$, and $m_x,h=0$ $∀h$, from the above expression one can obtain the one in Eq.~\eref{EoM_covariancematrix}. In the following, we will however derive some considerations that go beyond the $g=0$ case. As observed in the main text, without the all-to-all interaction, i.e. at $g=0$, the matrix $Q$ does not couple the correlations amongst large-spins, that are given in terms of the off-diagonal elements $Σ^ω_(αh),(βk)$ for $h≠k$; the matrix $Γ^2 G^sym$ does not depend on $g$ and it is always diagonal with respect to the large-spin components. Finally, the term $σS^sym  σ$ presents off-diagonal terms: at $g=0$, the mean-field stationary solutions features $m_x,k = 0$ $∀k$, implying that the non-zero element are at position $(xh),(xk) $ $∀k, h$. instead, for $g ≠0$, there is more than one element for each block $(α,h) (β,k)$, thus giving a corresponding finite determinant. In general, one could thus expect stronger quantum correlations in the presence of a direct, all-to-all, interaction.

\newpage
\begin{thebibliography}{10}

[1]
H.~P. Breuer and F.~Petruccione.
\newblock {\em The theory of open quantum systems}.
\newblock Oxford University Press, Great Clarendon Street, 2002.

[2]
S.~Diehl, A.~Micheli, A.~Kantian, B.~Kraus, H.~P. B{\"u}chler, and P.~Zoller.
\newblock Quantum states and phases in driven open quantum systems with cold
\newblock {\em Nat. Phys.}, 4:878--883, 2008.

[3]
F.~Verstraete, M.~M. Wolf, and J.~I. Cirac.
\newblock Quantum computation and quantum-state engineering driven by
\newblock {\em Nature Physics}, 5(9):633--636, 2009.

[4]
H.~Weimer, M.~Müller, I.~Lesanovsky, P.~Zoller, and H.~P. Büchler.
\newblock A rydberg quantum simulator.
\newblock {\em Nature Physics}, 6(5):382–388, March 2010.

[5]
S.~Diehl, A.~Tomadin, A.~Micheli, R.~Fazio, and P.~Zoller.
\newblock {Dynamical Phase Transitions and Instabilities in Open Atomic
Many-Body Systems}.
\newblock {\em Phys. Rev. Lett.}, 105:015702, 2010.

[6]
E.~G. Dalla~Torre, E.~Demler, T.~Giamarchi, and E.~Altman.
\newblock Quantum critical states and phase transitions in the presence of
non-equilibrium noise.
\newblock {\em Nat. Phys.}, 6:806--810, 2010.

[7]
P.~Schindler, M.~M{\"u}ller, D.~Nigg, J.~T. Barreiro, E.~A. Martinez,
M.~Hennrich, T.~Monz, S.~Diehl, P.~Zoller, and R.~Blatt.
\newblock Quantum simulation of dynamical maps with trapped ions.
\newblock {\em Nat. Phys.}, 9:361--367, 2013.

[8]
U.~C. T\"auber and S.~Diehl.
\newblock {Perturbative Field-Theoretical Renormalization Group Approach to
Driven-Dissipative Bose-Einstein Criticality}.
\newblock {\em Phys. Rev. X}, 4:021010, 2014.

[9]
M.~Marcuzzi, M.~Buchhold, S.~Diehl, and I.~Lesanovsky.
\newblock {Absorbing State Phase Transition with Competing Quantum and
Classical Fluctuations}.
\newblock {\em Phys. Rev. Lett.}, 116:245701, 2016.

[10]
F.~Minganti, A.~Biella, N.~Bartolo, and C.~Ciuti.
\newblock Spectral theory of liouvillians for dissipative phase transitions.
\newblock {\em Phys. Rev. A}, 98:042118, 2018.

[11]
F.~Iemini, A.~Russomanno, J.~Keeling, M.~Schir\`o, M.~Dalmonte, and R.~Fazio.
\newblock {Boundary Time Crystals}.
\newblock {\em Phys. Rev. Lett.}, 121:035301, 2018.

[12]
F.~Carollo, E.~Gillman, H.~Weimer, and I.~Lesanovsky.
\newblock {Critical Behavior of the Quantum Contact Process in One Dimension}.
\newblock {\em Phys. Rev. Lett.}, 123:100604, 2019.

[13]
E.~Chertkov, Z.~Cheng, A.~C. Potter, S.~Gopalakrishnan, T.~M. Gatterman, J.~A.
Gerber, K.~Gilmore, D.~Gresh, A.~Hall, A.~Hankin, M.~Matheny, T.~Mengle,
D.~Hayes, B.~Neyenhuis, R.~Stutz, and M.~Foss-Feig.
\newblock Characterizing a non-equilibrium phase transition on a quantum
\newblock {\em Nature Physics}, 19(12):1799–1804, September 2023.

[14]
L.~M. Sieberer, S.~D. Huber, E.~Altman, and S.~Diehl.
\newblock {Dynamical Critical Phenomena in Driven-Dissipative Systems}.
\newblock {\em Phys. Rev. Lett.}, 110:195301, 2013.

[15]
S.~Helmrich, A.~Arias, G.~Lochead, T.~M. Wintermantel, M.~Buchhold, S.~Diehl,
and S.~Whitlock.
\newblock Signatures of self-organized criticality in an ultracold atomic gas.
\newblock {\em Nature}, 577:481--486, 2020.

[16]
M.~Jo, J.~Lee, K.~Choi, and B.~Kahng.
\newblock Absorbing phase transition with a continuously varying exponent in a
quantum contact process: A neural network approach.
\newblock {\em Phys. Rev. Research}, 3:013238, 2021.

[17]
M.~Jo and M.~Kim.
\newblock Simulating open quantum many-body systems using optimised circuits in
digital quantum simulation.
\newblock {\em arXiv:2203.14295}, 2022.

[18]
S.~P. Kelly, A.~M. Rey, and J.~Marino.
\newblock Effect of active photons on dynamical frustration in cavity qed.
\newblock {\em Phys. Rev. Lett.}, 126:133603, Apr 2021.

[19]
H.~Weimer, A.~Kshetrimayum, and R.~Or\'us.
\newblock Simulation methods for open quantum many-body systems.
\newblock {\em Rev. Mod. Phys.}, 93:015008, 2021.

[20]
O.~E.~Lanford III and D.~Ruelle.
\newblock {Observables at infinity and states with short range correlations in
statistical mechanics}.
\newblock {\em Commun. Math. Phys.}, 13:194 -- 215, 1969.

[21]
F.~Benatti, F.~Carollo, R.~Floreanini, and H.~Narnhofer.
\newblock Quantum spin chain dissipative mean-field dynamics.
\newblock {\em J. Phys. A: Math. Theor.}, 51:325001, 2018.

[22]
G.~L. Sewell and Herbert Spohn.
\newblock {Quantum Theory of Collective Phenomena}.
\newblock {\em Physics Today}, 41(1):82--82, 01 1988.

[23]
\newblock {\em Elements of Quantum Mechanics of Infinite Systems}.
\newblock WORLD SCIENTIFIC, 1985.

[24]
K.~Hepp and E.~H. Lieb.
\newblock {On the superradiant phase transition for molecules in a quantized
radiation field: the Dicke maser model}.
\newblock {\em Ann. Phys.}, 76:360--404, 1973.

[25]
R.~Alicki and J.~Messer.
\newblock Nonlinear quantum dynamical semigroups for many-body open systems.
\newblock {\em J. Stat. Phys.}, 32:299--312, 1983.

[26]
N.~Benedikter, M.~Porta, and B.~Schlein.
\newblock {\em {Effective Evolution Equations from Quantum Dynamics}}.
\newblock SpringerBriefs in Mathematical Physics. Springer International
Publishing, 2015.

[27]
M.~Merkli and A.~Rafiyi.
\newblock Mean field dynamics of some open quantum systems.
\newblock {\em Proc. R. Soc. A: Math. Phys. Eng. Sci.}, 474:20170856, 2018.

[28]
\newblock Mean field dynamics of interacting fermionic systems.
\newblock {\em Mathematical Problems in Quantum Physics}, 717:13, 2016.

[29]
\newblock A simple derivation of mean field limits for quantum systems.
\newblock {\em Lett. Math. Phys.}, 97:151--164, 2011.

[30]
F.~T. Hioe.
\newblock {Phase Transitions in Some Generalized Dicke Models of
\newblock {\em Phys. Rev. A}, 8:1440--1445, 1973.

[31]
F.~Carollo and I.~Lesanovsky.
\newblock {Exactness of Mean-Field Equations for Open Dicke Models with an
Application to Pattern Retrieval Dynamics}.
\newblock {\em Phys. Rev. Lett.}, 126:230601, 2021.

[32]
F.~Benatti, F.~Carollo, and R.~Floreanini.
\newblock Dissipative dynamics of quantum fluctuations.
\newblock {\em Annalen der Physik}, 527(9-10):639--655, 2015.

[33]
A.~F. Verbeure.
\newblock {\em Many-Body Boson Systems: Half a Century Later}.
\newblock Springer London, 2011.

[34]
D.~Goderis, A.~F. Verbeure, and P.~Vets.
\newblock Non-commutative central limits.
\newblock {\em Probability Theory and Related Fields}, 82:527--544, 1989.

[35]
D.~Goderis, A.~Verbeure, and P.~Vets.
\newblock Dynamics of fluctuations for quantum lattice systems.
\newblock {\em Communications in Mathematical Physics}, 128:533--549, 1990.

[36]
F.~Benatti, F.~Carollo, R.~Floreanini, and H.~Narnhofer.
\newblock Non-markovian mesoscopic dissipative dynamics of open quantum spin
\newblock {\em Phys. Lett. A}, 380:381--389, 2016.

[37]
F.~Benatti, F.~Carollo, R.~Floreanini, and H.~Narnhofer.
\newblock Quantum fluctuations in mesoscopic systems.
\newblock {\em J. Phys. A: Math. Theor.}, 50(42):423001, 2017.

[38]
H.~Narnhofer and W.~Thirring.
\newblock Entanglement of mesoscopic systems.
\newblock {\em Phys. Rev. A}, 66:052304, Nov 2002.

[39]
\newblock On the generators of quantum dynamical semigroups.
\newblock {\em Commun. Math. Phys.}, 48:119--130, 1976.

[40]
V.~Gorini, A.~Kossakowski, and E.~C.~G. Sudarshan.
\newblock Completely positive dynamical semigroups of n‐level systems.
\newblock {\em Journal of Mathematical Physics}, 17(5):821--825, 1976.

[41]
J.~P. Garrahan.
\newblock Aspects of non-equilibrium in classical and quantum systems: Slow
relaxation and glasses, dynamical large deviations, quantum non-ergodicity,
and open quantum dynamics.
\newblock {\em Physica A: Statistical Mechanics and its Applications},
504:130--154, 2018.
\newblock Lecture Notes of the 14th International Summer School on Fundamental
Problems in Statistical Physics.

[42]
F.~Carollo, J.~P. Garrahan, and R.~L. Jack.
\newblock {Large Deviations at Level 2.5 for Markovian Open Quantum Systems:
Quantum Jumps and Quantum State Diffusion}.
\newblock {\em J. Stat. Phys.}, 184, 2021.

[43]
E.~Fiorelli, M.~Müller, I.~Lesanovsky, and F.~Carollo.
\newblock Mean-field dynamics of open quantum systems with collective
operator-valued rates: validity and application.
\newblock {\em New Journal of Physics}, 25(8):083010, aug 2023.

[44]
J.~J. Hopfield.
\newblock Neural networks and physical systems with emergent collective
computational abilities.
\newblock {\em PNAS}, 79:2554--2558, 1982.

[45]
A.~Labay-Mora, R.~Zambrini, and G.~L. Giorgi.
\newblock Quantum associative memory with a single driven-dissipative nonlinear
\newblock {\em Phys. Rev. Lett.}, 130:190602, May 2023.

[46]
L.~B\"odeker, E.~Fiorelli, and M.~M\"uller.
\newblock Optimal storage capacity of quantum hopfield neural networks.
\newblock {\em Phys. Rev. Res.}, 5:023074, May 2023.

[47]
P.~Rotondo, M.~Marcuzzi, J.~P. Garrahan, I.~Lesanovsky, and M.~M\"uller.
\newblock {Open quantum generalisation of Hopfield neural networks}.
\newblock {\em J. Phys. A: Math. Theor.}, 51:115301, 2018.

[48]
B.~P. Marsh, Y.~Guo, R.~M. Kroeze, S.~Gopalakrishnan, S.~Ganguli, J.~Keeling,
and B.~L. Lev.
\newblock {Enhancing Associative Memory Recall and Storage Capacity Using
Confocal Cavity QED}.
\newblock {\em Phys. Rev. X}, 11:021048, 2021.

[49]
Y.~Guo, R.~M. Kroeze, V.~D. Vaidya, J.~Keeling, and B.~L. Lev.
\newblock Sign-changing photon-mediated atom interactions in multimode cavity
quantum electrodynamics.
\newblock {\em Phys. Rev. Lett.}, 122:193601, May 2019.

[50]
V.~D. Vaidya, Y.~Guo, R~M. Kroeze, Kyle~E. Ballantine, Alicia~J. Koll\'ar,
J.~Keeling, and B.~L. Lev.
\newblock Tunable-range, photon-mediated atomic interactions in multimode
cavity qed.
\newblock {\em Phys. Rev. X}, 8:011002, Jan 2018.

[51]
P.~Rotondo, M.~Cosentino~Lagomarsino, and G.~Viola.
\newblock Dicke simulators with emergent collective quantum computational
\newblock {\em Phys. Rev. Lett.}, 114:143601, Apr 2015.

[52]
C.~Luo, H.~Zhang, V.~P.~W. Koh, J.~D. Wilson, A.~Chu, M.~J. Holland, A.~M. Rey,
and J.~K. Thompson.
\newblock Cavity-mediated collective momentum-exchange interactions, 2023.

[53]
K.~Seetharam, A.~Lerose, R.~Fazio, and J.~Marino.
\newblock Correlation engineering via nonlocal dissipation.
\newblock {\em Phys. Rev. Res.}, 4:013089, Feb 2022.

[54]
M.~A. Norcia, R.~J. Lewis-Swan, J.~R.~K. Cline, B.~Zhu, A.~M. Rey, and J.~K.
\newblock Cavity-mediated collective spin-exchange interactions in a strontium
superradiant laser.
\newblock {\em Science}, 361(6399):259--262, 2018.

[55]
V.~Torggler, S.~Kr\"amer, and H.~Ritsch.
\newblock Quantum annealing with ultracold atoms in a multimode optical
\newblock {\em Phys. Rev. A}, 95:032310, Mar 2017.

[56]
O.~Bratteli and D.~W. Robinson.
\newblock {\em Operator Algebras and Quantum Statistical Mechanics II.
Equilibrium States Models in Quantum Statistical Mechanics.}
\newblock Springer Berlin, Heidelberg, 1981.

[57]
\newblock {\em Symmetry breaking}.
\newblock Springer Berlin, Heidelberg, 2021.

[58]
\newblock {\em Quantum mathematical physics: atoms, molecules and large
\newblock Springer Science \& Business Media, 2013.

[59]
G.~Grimmett and D.~Stirzaker.
\newblock {\em Probability and random processes}.
\newblock Oxford university press, 2020.

[60]
G.~H. Fredrickson and H.~C. Andersen.
\newblock {Kinetic Ising Model of the Glass Transition}.
\newblock {\em Phys. Rev. Lett.}, 53:1244--1247, 1984.

[61]
N.~Cancrini, F.~Martinelli, C.~Roberto, and C.~Toninelli.
\newblock Kinetically constrained spin models.
\newblock {\em Probab. Theory. Relat. Fields.}, 140:459--504, 2008.

[62]
J.~P Garrahan, P.~Sollich, and C.~Toninelli.
\newblock Kinetically constrained models.
\newblock {\em in "Dynamical heterogeneities in glasses, colloids, and granular
media", Eds.: L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti and W.
van Saarloos (Oxford University Press, 2011)}, 150:111--137, 2011.

[63]
R.~J. Glauber.
\newblock {Time‐Dependent Statistics of the Ising Model}.
\newblock {\em J. Math. Phys.}, 4:294--307, 1963.

[64]
J.-C. Walter and G.T. Barkema.
\newblock {An introduction to Monte Carlo methods}.
\newblock {\em Phys. A: Stat. Mech. Appl.}, 418:78--87, 2015.
\newblock Proceedings of the 13th International Summer School on Fundamental
Problems in Statistical Physics.

[65]
M.~Boneberg, I.~Lesanovsky, and F.~Carollo.
\newblock Quantum fluctuations and correlations in open quantum dicke models.
\newblock {\em Phys. Rev. A}, 106:012212, Jul 2022.

[66]
F.~Carollo and I.~Lesanovsky.
\newblock Exact solution of a boundary time-crystal phase transition:
Time-translation symmetry breaking and non-markovian dynamics of
\newblock {\em Phys. Rev. A}, 105:L040202, Apr 2022.

[67]
R.~Mattes, I.~Lesanovsky, and F.~Carollo.
\newblock Entangled time-crystal phase in an open quantum light-matter system,

[68]
D.~J. Amit.
\newblock {\em Modelling Brain Function: The World of Attractor Neural
\newblock Cambridge University Press, USA, 1st edition, 1992.

[69]
D.~J. Amit, H.~Gutfreund, and H.~Sompolinsky.
\newblock Spin-glass models of neural networks.
\newblock {\em Phys. Rev. A}, 32:1007--1018, 1985.

[70]
E.~Fiorelli, I.~Lesanovsky, and M.~M\"uller.
\newblock {Phase diagram of quantum generalized Potts-Hopfield neural
\newblock {\em New J. Phys.}, 24:033012, 2022.

[71]
H.~{Koch} and J.~{Piasko}.
\newblock {Some rigorous results on the Hopfield neural network model}.
\newblock {\em Journal of Statistical Physics}, 55(5-6):903--928, June 1989.

[72]
D.~Grensing and R.~Kuhn.
\newblock Random-site spin-glass models.
\newblock {\em Journal of Physics A: Mathematical and General}, 19(18):L1153,
dec 1986.

[73]
\newblock {Thermodynamic limit of the q-state Potts-Hopfield model with
infinitely many patterns}.
\newblock {\em J. Stat. Phys.}, 68:977--1011, 1992.

[74]
G.~Adesso and A.~Datta.
\newblock Quantum versus classical correlations in gaussian states.
\newblock {\em Phys. Rev. Lett.}, 105:030501, Jul 2010.

[75]
E.~Fiorelli, M.~Marcuzzi, P.~Rotondo, F.~Carollo, and I.~Lesanovsky.
\newblock {Signatures of Associative Memory Behavior in a Multimode Dicke
\newblock {\em Phys. Rev. Lett.}, 125:070604, 2020.

[76]
W-M. Zhang, D.~H. Feng, and R.~Gilmore.
\newblock Coherent states: Theory and some applications.
\newblock {\em Rev. Mod. Phys.}, 62:867--927, Oct 1990.

[77]
Tom Michoel and Bruno Nachtergaele.
\newblock Central limit theorems for the large-spin asymptotics of quantum
\newblock {\em Probability theory and related fields}, 130(4):493--517, 2004.

[78]
T.~H. Gronwall.
\newblock Note on the derivatives with respect to a parameter of the solutions
of a system of differential equations.
\newblock {\em Ann. Math.}, 20(4):292--296, 1919.

[79]
E.~Fiorelli, P.~Rotondo, M.~Marcuzzi, J.~P. Garrahan, and I.~Lesanovsky.
\newblock {Quantum accelerated approach to the thermal state of classical
all-to-all connected spin systems with applications to pattern retrieval in
the Hopfield neural network}.
\newblock {\em Phys. Rev. A}, 99:032126, 2019.

\end{thebibliography}
\bibliographystyle{unsrt}

\end{document}