$\mathcal{L}(x)=\mathcal{J}_{\mathfrak{A}}(x)+\mathcal{L}_{\infty}(x)=\mathcal{J}_{\mathfrak{A}}(x).$

Therefore

${\rm tr}(\hat{\sigma}\mathcal{L}(x))={\rm
tr}(\hat{\sigma}\mathcal{J}_{\mathfrak{A}}(x))=0,$

since $\mathcal{L}_{*}(\hat{\sigma})=0$. From the irreducibility assumption
(Hypothesis 2), we have that $\hat{\sigma}>0$, and so
$\mathcal{L}(x)=\mathcal{J}_{\mathfrak{A}}(x)=0$. Moreover, irreducibility
also implies that $x=\alpha{\bf 1}$ for some $\alpha\in\mathbb{R}$. Therefore
$\mathcal{J}_{\mathfrak{A}}(x)=\alpha\mathcal{J}_{\mathfrak{A}}({\bf 1})=0$.
Since $\mathcal{J}_{\mathfrak{A}}({\bf 1})$ is the sum of positive operators,
we have that $\alpha=0$, consequently $x=0$ and we reach a contradiction.

2\. Integrating over all trajectories, one can write:

$\mathbb{P}(T_{\mathfrak{A}}(k)\leq t)=\int_{t_{1}<t_{2}<\cdots<t_{k}<t}{\rm
tr}\left(e^{(t-t_{k})\mathcal{L}_{\infty*}}\mathcal{J}_{\mathfrak{A}*}\cdots
e^{(t_{2}-t_{1})\mathcal{L}_{\infty*}}\mathcal{J}_{\mathfrak{A}*}e^{t_{1}\mathcal{L}_{\infty*}}(\rho)\right)dt_{1}\cdots
dt_{k}.$

For $u<\overline{\lambda}$, one has

$-(u+\mathcal{L}_{\infty})^{-1}=\int_{0}^{+\infty}e^{(u+\mathcal{L}_{\infty})t}dt,$

hence one can write the Laplace transform of $T_{\mathfrak{A}}(k)$ as:

$\mathbb{E}_{\rho}[e^{uT_{\mathfrak{A}}(k)}]={\rm
tr}\left(\left(-\mathcal{J}_{\mathfrak{A}*}(u+\mathcal{L}_{\infty*})^{-1}\right)^{k}(\rho)\right),$

and by the definition of $\Psi$ in equation (25), we obtain the statement.

3\. The Spectral Mapping Theorem implies that ${\rm
Sp}(\mathcal{L}_{\infty}^{-1})=\\{z^{-1}:\,z\in{\rm
Sp}(\mathcal{L}_{\infty})\\}$, therefore one has that

$\left\|\mathcal{L}_{\infty}^{-1}\right\|_{\infty\rightarrow\infty}\geq
r(\mathcal{L}_{\infty}^{-1})\geq\frac{1}{\overline{\lambda}}\quad\Leftrightarrow\quad\left\|\mathcal{L}_{\infty}^{-1}\right\|_{\infty\rightarrow\infty}^{-1}\leq\overline{\lambda}.$

∎

###  C.3 Proof of Theorem 10

######  Theorem 10.

Consider a nonempty subset $\mathfrak{A}$ of the emission channels. The FPT
$T_{\mathfrak{A}}(k)/k$ satisfies a large deviation principle with good rate
function given by

$I_{\mathfrak{A}}(t):=\sup_{u\in\mathbb{R}}\\{ut-\log(r(u))\\}$

where

$r(u)=\begin{cases}r\left(\Psi_{u}\right)&\text{ if }u<\overline{\lambda}\\\
+\infty&\text{o.w.}\end{cases}$

where
$\Psi_{u}(x):=-(u+\mathcal{L}_{\infty})^{-1}\mathcal{J}_{\mathfrak{A}}(x)$ and
$\overline{\lambda}:=-\max\\{\Re(z):z\in{\rm Sp}(\mathcal{L}_{\infty})\\}.$

###### Proof.

The proof follows the same method as the proof of the classical case, Theorem
2; for completeness we write the quantum proof in full. Lemma 21 states that
in the domain $u<\overline{\lambda}$, then

$\mathbb{E}_{\rho}[e^{uT_{\mathfrak{A}}(k)}]={\rm
tr}\left(\rho\Psi_{u}^{k}({\bf 1})\right),$

where
$\Psi_{u}(x):=-(u+\mathcal{L}_{\infty})^{-1}\mathcal{J}_{\mathfrak{A}}(x)$.
Writing $\Psi_{u}$ as the integral

$\Psi_{u}(x)=\int_{0}^{\infty}e^{(u+\mathcal{L}_{\infty})t}\mathcal{J}(x)dt$

which is the composition of two completely positive maps, hence $\Psi_{u}$ is
completely positive as well. Therefore, Perron-Frobenius Theorem tells us that
$r(u):=r(\Psi_{u})$ is an eigenvalue of $\Psi_{u}$, with a positive
eigenvector $x(u)$. We can relate this operator with the generator of a
quantum dynamical semigroup:

$\Psi_{u}(x(u))=r(u)x(u)\Leftrightarrow\mathcal{L}_{s(u)}(x(u))=-ux(u),$

where $\mathcal{L}_{s(u)}:=\mathcal{L}+(e^{s(u)}-1)\mathcal{J}_{\mathfrak{A}}$
and $s(u):=-\log(r(u))$. Notice that $\mathcal{L}_{s(u)}$ has the form in Eq.
(68). Therefore, by Lemma 21 it is irreducible and $x(u)$ is in fact a unique
and strictly positive eigenvector of $\mathcal{L}_{s(u)}$ corresponding to the
eigenvalue $-u$.

We first need to show that for $u<\overline{\lambda}$

$\lim_{k\to\infty}\frac{1}{k}\log(\mathbb{E}_{\rho}[e^{uT_{\mathfrak{A}}(k)}])<\infty,$
(54)

Using Holder’s inequality, we have that ${\rm tr}\left(\rho\Psi_{u}^{k}({\bf
1})\right)\leq\|\Psi_{u}^{k}\|_{\infty\to\infty}$, hence due to Gelfand’s
formula

$\frac{1}{k}\log(\mathbb{E}_{\rho}[e^{uT_{\mathfrak{A}}(k)}])\leq\frac{1}{k}\log(\|\Psi_{u}^{k}\|_{\infty\rightarrow\infty})\rightarrow_{k\rightarrow+\infty}\log(r(u)).$

Furthermore, we can take $\|x(u)\|_{\infty\to\infty}\leq 1$ and bound from
below:

$\frac{1}{k}\log(\mathbb{E}_{\rho}[e^{uT_{\mathfrak{A}}(k)}])\geq\log(r(u))+\frac{1}{k}\log({\rm
tr}\left(\rho x(u)\right))\rightarrow_{k\rightarrow+\infty}\log(r(u)),$

which we can do since $x(u)$ is strictly positive, $\rho$ is positive
semidefinite so ${\rm tr}(\rho x(u))>0$. Therefore, we have shown that the
limit in equation (54) converges to $\log(r(u))<\infty$ in the range
$u<\overline{\lambda}$.

We now use the same version of the Gärtner-Ellis Theorem ([12, Theorem 2.3.6])
as in the proof of Theorem 2. All that remains is to show that $\log(r(u))$ is
steep, i.e. as u approaches the boundary $\overline{\lambda}$, both
$\log(r(u))$ and $\log(r(u))^{\prime}$ diverge to $+\infty$. Denote by
$\mathbf{T}$ the spectral projection of $\mathcal{L}_{\infty}$ with respect to
the eigenvalue $-\overline{\lambda}$. We can show that
$\mathcal{L}_{\infty}\mathbf{T}$ is diagonalisable, i.e. the restriction does
not feature any Jordan blocks or equivalently, the algebraic and geometric
multiplicity of $\overline{\lambda}$ coincide. To do this, assume
$\mathcal{L}_{\infty}\mathbf{T}$ is not diagonalisable. Then the map
$\mathcal{L}^{\prime}\mathbf{T}=(\mathcal{L}+\overline{\lambda}{\bf
I}_{M_{k}(\mathbb{C})})\mathbf{T}$ \- corresponding to the eigenvalue 0 - is
not diagonalisable. One can then always choose $x,y\in M_{d}(\mathbb{C})$ with
$\mathcal{L}^{\prime}(y)=0$ and $\mathcal{L}^{\prime}(x)=y$. Therefore one has

$e^{t\mathcal{L}^{\prime}}(x)=x+ty.$

The semigroup $e^{t\mathcal{L}^{\prime}}$ is in fact a contraction semigroup,
which contradicts the above equation. Therefore, we have

$\mathcal{L}_{\infty}\mathbf{T}=-\overline{\lambda}\mathbf{T}.$

Firstly, we show that $\lim_{u\uparrow\overline{\lambda}}r(u)=+\infty$. Note
that $\mathbf{T}\Psi\neq 0$ since $\mathbf{T}\Psi({\bf 1})\neq 0$. The map
$\Psi_{u}$ can be written as

$\Psi_{u}=\frac{\mathcal{L}_{\infty}}{\mathcal{L}_{\infty}+u}=\frac{\overline{\lambda}}{\overline{\lambda}-u}\mathbf{T}\Psi+({\bf
I}_{M_{k}(\mathbb{C})}-\mathbf{T})\frac{\mathcal{L}_{\infty}}{\mathcal{L}_{\infty}+u}\Psi$

and one can see it has a norm which explodes as $u\uparrow\overline{\lambda}$.
Let us assume by contradiction that $r(u)\uparrow
r(\overline{\lambda})<\infty$ as $u\uparrow\overline{\lambda}$. Then the
eigenvactor $x(u)$ can be chosen to converge to the Perron-Frobenius
eigenvector of $\mathcal{L}_{s(\overline{\lambda})}$ and $\min{\rm
Sp}(x(u))\not\rightarrow 0$. We then have for $0\leq u<\overline{\lambda}$

$\|\Psi_{u}\|_{\infty\rightarrow\infty}=\|\Psi_{u}({\bf
1})\|_{\infty}\leq\frac{1}{\min{\rm
Sp}(x(u))}\|\Psi_{u}(x(u))\|_{\infty}=r(u)\frac{\|x(u)\|_{\infty}}{\min{\rm
Sp}(x(u))}<+\infty$

which is a contradiction. Let us show that
$\lim_{u\uparrow\overline{\lambda}}\log(r(u))^{\prime}=+\infty$. For each $u$,
we can choose the left eigenvector of $\Psi_{u}$, $l(u)$, to be such that
${\rm tr}(l(u)x(u))\equiv 1$. Then we can write $r(u)$ as

$\begin{split}r(u)&=-{\rm
tr}\left(l(u)(u+\mathcal{L}_{\infty})^{-1}\mathcal{L}_{\infty}\Psi(x(u))\right)\\\
&=\underbrace{-{\rm
tr}\left(l(u)(u+\mathcal{L}_{\infty})^{-1}\mathcal{L}_{\infty}\mathbf{T}\Psi(x(u))\right)}_{(I)}+\underbrace{-{\rm
tr}\left(l(u)(u+\mathcal{L}_{\infty})^{-1}\mathcal{L}_{\infty}({\bf
I}_{M_{k}(\mathbb{C})}-\mathbf{T})\Psi(x(u))\right)}_{(II)}.\\\ \end{split}$

The term $(II)$ remains bounded, hence as $u\uparrow\overline{\lambda}$

$r(u)\asymp\frac{\overline{\lambda}}{\overline{\lambda}-u}{\rm
tr}\left(l(u)\mathbf{T}\Psi(x(u))\right)\rightarrow+\infty.$

Since $\log(r(u))^{\prime}=\frac{r^{\prime}(u)}{r(u)}$, we can differentiate
the expression for $r(u)$ to obtain

$\begin{split}\frac{r^{\prime}(u)}{r(u)}&=\frac{{\rm
tr}\left(l(u)(u+\mathcal{L}_{\infty})^{-2}\mathcal{L}_{\infty}\Psi(x(u))\right)}{r(u)}\\\
&=\underbrace{\frac{\overline{\lambda}}{(\overline{\lambda}-u)^{2}}\frac{{\rm
tr}\left(l(u)\mathbf{T}\Psi(x(u))\right)}{r(u)}}_{(I)}+\underbrace{\frac{{\rm
tr}\left(l(u)({\bf
I}_{M_{k}(\mathbb{C})}-\mathbf{T})\Psi(x(u))\right)}{r(u)}}_{(II)}.\end{split}$

We have again that part $(II)$ is bounded, but for
$u\uparrow\overline{\lambda}$, $(I)\asymp(\overline{\lambda}-u)^{-1}$ which
completes the proof.

∎

###  C.4 Proof of Theorem 11 and Corollary 12

######  Theorem 11 (Fluctuations of FPT for Total Counts).

Assume that Hypothesis 3 holds ($\Phi$ be irreducible) and let $\varepsilon$
be the absolute spectral gap of $\Phi$. Then, for every $\gamma>0$:

$\begin{split}&\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\geq\langle
t_{I}\rangle+\gamma\right)\leq
C(\rho)\exp\left(-k\frac{\gamma^{2}\varepsilon}{8c_{q}^{2}}h\left(\frac{5\gamma}{2c_{q}}\right)\right)\\\
{\rm and}\\\ &\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\leq\langle
t_{I}\rangle-\gamma\right)\leq
C(\rho)\exp\left(-k\frac{\gamma^{2}\varepsilon}{8{c_{q}^{2}}}h\left(\frac{5\gamma}{2c_{q}}\right)\right),\quad
k\in\mathbb{N},\end{split}$

where $h(x):=(\sqrt{1+x}+\frac{x}{2}+1)^{-1}$,
$C(\rho):=\left\|\sigma^{-\frac{1}{2}}\rho\sigma^{-\frac{1}{2}}\right\|_{\sigma}$
and $c_{q}$ is defined in Eq. (28).

###### Proof.

Applying Chernoff bound, we get, for $u\geq 0$:

$\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\geq\langle
t_{I}\rangle+\gamma\right)\leq e^{-uk(\langle
t_{I}\rangle+\gamma)}\mathbb{E}_{\rho}\left[e^{uT_{I}(k)}\right].$ (55)

For $u<\overline{\lambda}$, we define the tilted operator
$\Phi_{u}(x):=\mathcal{F}_{u}\Phi(x)$, with $\mathcal{F}_{u}(x):=(u{\bf
I}_{M_{k}(\mathbb{C})}+\mathcal{L}_{0})^{-1}\mathcal{L}_{0}(x)$. Using Lemma
9, for $0\leq u<\overline{\lambda}$ we can write

$\begin{split}\mathbb{E}_{\rho}[e^{uT_{I}(k)}]={\rm
tr}\left(\rho\Phi_{u}^{k}({\bf
1})\right)&=\left\langle\sigma^{-\frac{1}{2}}\rho\sigma^{-\frac{1}{2}},\Phi_{u}^{k}({\bf
1})\right\rangle_{\sigma}\\\
&\leq\underbrace{\|\sigma^{-\frac{1}{2}}\rho\sigma^{-\frac{1}{2}}\|_{\sigma}}_{:=C(\rho)}\underbrace{\|{\bf
1}\|_{\sigma}}_{=1}\|\Phi_{u}^{k}\|_{\sigma},\end{split}$

where with a small abuse of notation $\|\Phi_{u}^{k}\|_{\sigma}$ denotes the
operator norm of the map $\Phi_{u}^{k}$ with respect to the KMS inner product
associated to $\sigma$. We can further break up this operator norm:

$\|\Phi_{u}^{k}\|_{\sigma}\leq\|\Phi_{u}\|_{\sigma}^{k}=\|\mathcal{F}_{u}\Phi\|_{\sigma}^{k}.$

We now seek to upper bound $\|\mathcal{F}_{u}\Phi_{u}\|_{\sigma}$. Conversely
to the classical case, $\mathcal{F}_{u}\Phi$ is not self adjoint, but we can
upper bound its norm with an operator which is. This can be done using Lemma
24 with $\mathbf{A}=\mathcal{F}_{u},\mathbf{B}={\bf I}_{M_{k}(\mathbb{C})}$ to
get
$\|\mathcal{F}_{u}\Phi\|_{\sigma}\leq\|\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger}\|_{\sigma}^{\frac{1}{2}}$,
with $\mathcal{F}_{u}^{{\dagger}}(x)=\Gamma_{\sigma}^{-\frac{1}{2}}\circ({\bf
I}_{M_{k}(\mathbb{C})}+u\mathcal{L}_{0*}^{-1})^{-1}\circ\Gamma_{\sigma}^{\frac{1}{2}}(x)$,
cf. equation (27). Since $\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger}$
is a positive, self adjoint, irreducible map, operator Perron Frobenius theory
[67] says
$\|\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger}\|_{\sigma}=r(u)$, where
$r(u)=\sup\\{|\lambda|:\lambda\in{\rm
Sp}(\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger})\\}$ is the spectral
radius of $\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger}$. Hence, the
Laplace transform is upper bounded by:

$\mathbb{E}_{\rho}[e^{uT_{I}(k)}]\leq C(\rho)r(u)^{\frac{k}{2}}.$ (56)

For $u$ small enough, we can expand
$\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger}$ as the power series
below:

$\begin{split}\mathcal{F}_{u}\hat{\Phi}\mathcal{F}_{u}^{\dagger}&=\sum_{j\geq
0}u^{j}\left(-\frac{1}{\mathcal{L}_{0}}\right)^{j}\circ\hat{\Phi}\circ\sum_{l\geq
0}u^{l}\left(-\frac{1}{\mathcal{L}_{0}^{\dagger}}\right)^{l}\\\ &=\sum_{l\geq
0}u^{l}\underbrace{\sum_{j=0}^{l}\left(-\frac{1}{\mathcal{L}_{0}}\right)^{l-j}\circ\hat{\Phi}\circ\left(-\frac{1}{\mathcal{L}_{0}^{\dagger}}\right)^{j}}_{\Phi^{(l)}},\end{split}$

which is the quantum analogue of equation (38) from the classical case. Again
we bound each term in the power series in order to use operator perturbation
theory. Using the definition of $c_{q}$ in equation (28) and Cauchy-Schwarz,
we can upper bound $\Phi^{(l)}$ in the form:

$\|\Phi^{(l)}\|_{\sigma}\leq(l+1)c_{q}^{l}\leq(2c_{q})^{l}.$

Using perturbation theory [66], we find that for
$u<\frac{\varepsilon}{2c_{q}(2+\varepsilon)}$, the spectral radius $r(u)$ can
then be expressed as

$r(u)=1+\sum_{l=1}^{\infty}u^{l}r^{(l)},$ (57)

where

$r^{(l)}=\sum_{p=1}^{l}\frac{(-1)^{p}}{p}\sum_{\begin{subarray}{c}\nu_{1}+\dots+\nu_{p}=l,\,\nu_{i}\geq
1\\\ \mu_{1}+\dots+\mu_{p}=p-1,\,\mu_{j}\geq 0\end{subarray}}{\rm
TR}\left(\Phi^{(\nu_{1})}\mathbf{S}^{(\mu_{1})}\cdots\Phi^{(\nu_{p})}\mathbf{S}^{(\mu_{p})}\right).$
(58)

We have $\mathbf{S}^{(0)}=-\Pi$, $\mathbf{S}^{(1)}=(\hat{\Phi}-{\bf
I}_{M_{k}(\mathbb{C})}+\Pi)^{-1}-\Pi=-\epsilon^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)$, and $\mathbf{S}^{(\mu)}$ the $\mu^{\rm th}$
power of $\mathbf{S}^{(1)}$, and also
$\|\mathbf{S}^{(\mu)}\|_{\sigma}=\varepsilon^{-\mu}$ for $\mu\geq 1$. We now
want to bound the terms in the expression for $r^{(l)}$. For $p=1$:

$\displaystyle\left|{\rm
TR}\left(\Phi^{(\nu_{1})}\mathbf{S}^{(\mu_{1})}\cdots\Phi^{(\nu_{p})}\mathbf{S}^{(\mu_{p})}\right)\right|$
$\displaystyle=$ $\displaystyle\left|{\rm
TR}\left(\Phi^{(l)}\mathbf{S}^{(0)}\right)\right|=\left|\left\langle{\bf
1},\Phi^{(l)}({\bf 1})\right\rangle_{\sigma}\right|$ (59) $\displaystyle\leq$
$\displaystyle\|{\bf 1}\|_{\sigma}\|\Phi^{(l)}\|_{\sigma}\leq(2c_{q})^{l}\leq
2c_{q}\left(\frac{2c_{q}}{\varepsilon}\right)^{l-1}$

since $\varepsilon\leq 1$. For $p\geq 2$, using the fact that one of $\mu_{i}$
is zero and using trace cyclicity:

$\begin{split}\left|{\rm
TR}\left(\Phi^{(\nu_{1})}\mathbf{S}^{(\mu_{1})}\cdots\Phi^{(\nu_{p})}\mathbf{S}^{(\mu_{p})}\right)\right|&=\left|\left\langle{\bf
1},\Phi^{(\nu_{1})}\mathbf{S}^{(\mu_{1})}\cdots\mathbf{S}^{(\mu_{p-1})}\Phi^{(\nu_{p})}({\bf
1})\right\rangle_{\sigma}\right|\\\
&\leq\frac{1}{\varepsilon^{p-1}}(2c_{q})^{l}\\\
&\leq\frac{1}{\varepsilon^{l-1}}(2c_{q})^{l}\leq
2c_{q}\left(\frac{2c_{q}}{\varepsilon}\right)^{l-1}.\end{split}$ (60)

We can explicitly calculate $r^{(1)}$:

$\begin{split}r^{(1)}&=-{\rm TR}\left(\Phi^{(1)}(-\Pi)\right)=\left\langle{\bf
1},\Phi^{(1)}({\bf 1})\right\rangle_{\sigma}\\\ &=\left\langle{\bf
1},\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right)\hat{\Phi}({\bf
1})\right\rangle_{\sigma}+\left\langle{\bf 1},\hat{\Phi}\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}^{\dagger}}\right)({\bf
1})\right\rangle_{\sigma}=2\langle t_{I}\rangle.\end{split}$

The term $|r^{(2)}|$ can be bounded using equations (59) and (60):

$\begin{split}|r^{(2)}|&=\left|{\rm
TR}\left(\Phi^{(2)}\mathbf{S}^{(0)}\right)-\left\langle{\bf
1},\Phi^{(1)}\mathbf{S}^{(1)}\Phi^{(1)}({\bf
1})\right\rangle_{\sigma}\right|\\\
&\leq\frac{(2c_{q})^{2}}{\varepsilon}+\frac{(2c_{q})^{2}}{\varepsilon}=\frac{8c_{q}^{2}}{\varepsilon}.\end{split}$

Using the bound on the number of terms in (58), given by equation (41), we can
bound the $r^{(l)}$ by
$|r^{(l)}|\leq\frac{2c_{q}}{5}\left(\frac{10c_{q}}{\varepsilon}\right)^{l-1}$.
Now we have all the ingredients in place, we can finish bounding $r(u)$:

$\begin{split}r(u)&\leq 1+\sum_{l\geq 1}u^{l}|r^{(l)}|\\\ &\leq 1+2\langle
t_{I}\rangle u+\frac{8c_{q}^{2}}{\varepsilon}u^{2}+\sum_{l\geq
3}u^{l}\frac{2c_{q}}{5}\left(\frac{10c_{q}}{\varepsilon}\right)^{l-1}\\\
&=1+2\langle t_{I}\rangle
u+\frac{8c_{q}^{2}}{\varepsilon}u^{2}+u^{2}\frac{4c_{q}^{2}}{\varepsilon}\sum_{l\geq
1}u^{l}\left(\frac{10c_{q}}{\varepsilon}\right)^{l}\\\ &\leq 1+2\langle
t_{I}\rangle
u+\frac{8c_{q}^{2}}{\varepsilon}u^{2}\left(1-\frac{10c_{q}u}{\varepsilon}\right)^{-1}\\\
&\leq\exp\left({2\langle t_{I}\rangle
u+\frac{8c_{q}^{2}}{\varepsilon}u^{2}\left(1-\frac{10c_{q}u}{\varepsilon}\right)^{-1}}\right),\end{split}$

which is valid for
$u<\frac{\varepsilon}{10c_{q}}<\frac{\varepsilon}{2c_{q}(2+\varepsilon)}$.
Putting this upper bound on $r(u)$ back into equation (56) gives the upper
bound on the Laplace transform of $T_{I}(k)$:

$\mathbb{E}_{\rho}[e^{uT_{I}(k)}]\leq C(\rho)\exp\left(k\left(\langle
t_{I}\rangle
u+\frac{4c_{q}^{2}u^{2}}{\varepsilon}\left(1-\frac{10c_{q}u}{\varepsilon}\right)^{-1}\right)\right).$

Applying the Chernoff bound in equation (55) gives, $\forall
u\in[0,\frac{\varepsilon}{10c_{q}})$:

$\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\geq\langle
t_{I}\rangle+\gamma\right)\leq C(\rho)\exp\left(-k\left(\gamma
u-\frac{4c_{q}^{2}u^{2}}{\varepsilon}\left(1-\frac{10c_{q}u}{\varepsilon}\right)^{-1}\right)\right).$

Optimisation over the allowed $u$, using result (44), with
$\alpha=\frac{4c_{q}^{2}}{\varepsilon}$, $\beta=\frac{10c_{q}}{\varepsilon}$
gives the final concentration inequality for right deviations:

$\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\geq\langle
t_{I}\rangle+\gamma\right)\leq
C(\rho)\exp\left(-k\frac{\gamma^{2}\varepsilon}{8{c_{q}^{2}}}h\left(\frac{5\gamma}{2c_{q}}\right)\right).$

By considering $u\leq 0$ we can prove the bound for left deviations:

$\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\leq\langle
t_{I}\rangle-\gamma\right)\leq e^{-ku(\langle
t_{I}\rangle-\gamma)}\mathbb{E}_{\rho}[e^{uT_{I}(k)}].$

By repeating the process we get a similar bound on the Laplace transform,
valid for $0\leq|u|<\frac{\varepsilon}{10c_{q}}$:

$\mathbb{E}_{\rho}[e^{uT_{I}(k)}]\leq C(\rho)\exp\left(k\left(\langle
t_{I}\rangle
u+\frac{4c_{q}^{2}u^{2}}{\varepsilon}\left(1-\frac{10c_{q}|u|}{\varepsilon}\right)^{-1}\right)\right).$

One obtains the upper bound on left deviations, a symmetric bound to right
deviations:

$\mathbb{P}_{\rho}\left(\frac{T_{I}(k)}{k}\leq\langle
t_{I}\rangle-\gamma\right)\leq
C(\rho)\exp\left(-k\frac{\gamma^{2}\varepsilon}{8{c_{q}^{2}}}h\left(\frac{5\gamma}{2c_{q}}\right)\right).$

∎

######  Corollary 12.

The variance of the first passage time for total counts is bounded from above
by:

$\frac{{\rm
var}_{\sigma}(T_{I}(k))}{k}\leq\left(\frac{4}{\varepsilon}-(1-\varepsilon)\right)c_{q}^{2}.$

###### Proof.

Notice that for $u\geq 0$ small enough, one has

$\begin{split}\log(\mathbb{E}_{\sigma}[e^{uT_{I}(k)}])&=\langle t_{I}\rangle
ku+\frac{1}{2}{\rm var}_{\sigma}(T_{I}(k))u^{2}+o(u^{2})\\\
&\leq\frac{k}{2}\log(r(u))=\frac{k}{2}r^{\prime}(0)u+\frac{k}{4}(r^{\prime\prime}(0)-(r^{\prime}(0))^{2})u^{2}+o(u^{2})\end{split}$

where $r(u)$ is given by equation (57). We recall that

$r^{\prime}(0)=r^{(1)}=2\langle t_{I}\rangle=-2\left\langle{\bf
1},\mathcal{L}_{0}^{-1}({\bf 1})\right\rangle_{\sigma}$

and

$\begin{split}\frac{r^{\prime\prime}(0)}{2}\quad&=r^{(2)}=\langle{\bf
1},\Phi^{(2)}({\bf 1})\rangle_{\sigma}+\left\langle{\bf 1},\Phi^{(1)}({\bf
I}_{M_{k}(\mathbb{C})}-\hat{\Phi})^{-1}\Phi^{(1)}({\bf
1})\right\rangle_{\sigma}\\\ &=2\langle{\bf 1},\mathcal{L}_{0}^{-2}({\bf
1})\rangle_{\sigma}+\langle{\bf
1},\mathcal{L}_{0}^{-1}\hat{\Phi}(\mathcal{L}^{\dagger}_{0})^{-1}({\bf
1})\rangle_{\sigma}\\\ &+\langle{\bf
1},((\mathcal{L}_{0}^{\dagger})^{-1}+\mathcal{L}_{0}^{-1}\hat{\Phi})({\bf
I}_{M_{k}(\mathbb{C})}-\hat{\Phi})^{-1}(\hat{\Phi}(\mathcal{L}_{0}^{\dagger})^{-1}+\mathcal{L}_{0}^{-1})({\bf
1})\rangle_{\sigma}\\\ &=2\langle{\bf 1},\mathcal{L}_{0}^{-1}({\bf
1})\rangle^{2}_{\sigma}+2\langle{\bf 1},\mathcal{L}_{0}^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)\mathcal{L}_{0}^{-1}({\bf 1})\rangle_{\sigma}\\\
&+\langle{\bf
1},\mathcal{L}_{0}^{-1}\hat{\Phi}(\mathcal{L}_{0}^{\dagger})^{-1}({\bf
1})\rangle_{\sigma}+\varepsilon^{-1}\langle{\bf
1},(\mathcal{L}_{0}^{\dagger})^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)\mathcal{L}_{0}^{-1}({\bf 1})\rangle_{\sigma}\\\
&+(1-\varepsilon)\varepsilon^{-1}(\langle{\bf
1},(\mathcal{L}_{0}^{\dagger})^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)(\mathcal{L}_{0}^{\dagger})^{-1}({\bf
1})\rangle_{\sigma}+\langle{\bf 1},\mathcal{L}_{0}^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)\mathcal{L}_{0}^{-1}({\bf 1})\rangle_{\sigma})\\\
&+(1-\varepsilon)^{2}\varepsilon^{-1}\langle{\bf 1},\mathcal{L}_{0}^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)(\mathcal{L}_{0}^{\dagger})^{-1}({\bf
1})\rangle_{\sigma}\\\ &\leq 2\langle{\bf 1},\mathcal{L}_{0}^{-1}({\bf
1})\rangle^{2}_{\sigma}+\left(\frac{4}{\varepsilon}-(1-\varepsilon)\right)c_{q}^{2}.\end{split}$

Therefore one has

$\frac{r^{\prime\prime}(0)-(r^{\prime}(0))^{2}}{2}\leq\left(\frac{4}{\varepsilon}-(1-\varepsilon)\right)c_{q}^{2}$

and we proved the statement.

∎

###  C.5 Proof of Theorem 13 and Corollary 14

######  Theorem 13 (Fluctuations of FPT for Total Counts in Reset Processes).

Assume that Hypothesis 2 holds ($\mathcal{L}$ be irreducible) the jump
operators are of the form (29) (reset process). Let $\varepsilon$ be the
spectral gap of $\mathbf{P}^{\dagger}\mathbf{P}$. For every $\gamma>0$:

$\begin{split}&\mathbb{P}_{\nu}\left(\frac{T_{I}(k)}{k}\geq\langle
t_{I}\rangle+\gamma\right)\leq
C(\nu)\exp\left(-k\frac{\gamma^{2}\varepsilon}{4b_{r}^{2}}h\left(\frac{5c_{r}\gamma}{2b_{r}^{2}}\right)\right)\\\
{\rm and}\\\ &\mathbb{P}_{\nu}\left(\frac{T_{I}(k)}{k}\leq\langle
t_{I}\rangle-\gamma\right)\leq
C(\nu)\exp\left(-k\frac{\gamma^{2}\varepsilon}{4b_{r}^{2}}h\left(\frac{5c_{r}\gamma}{2b_{r}^{2}}\right)\right),\quad
k\in\mathbb{N},\end{split}$

where $h(x):=(\sqrt{1+x}+\frac{x}{2}+1)^{-1}$ and
$C(\nu):=(\sum_{x}\nu(x)^{2}/\pi(x))^{\frac{1}{2}}$.

###### Proof.

The general formula for the Laplace transform of $T_{\mathfrak{E}}(k)$ in the
case of reset processes reads:

$\mathbb{E}_{\nu}[e^{uT_{I}(k)}]=\left\langle\nu,(\mathbf{F}_{u}\mathbf{P})^{k}\underline{1}\right\rangle,\quad\text{
for }u<\overline{\lambda}.$

$\mathbf{P}$ is given by equation (30) and $\mathbf{F}_{u}$ is a diagonal
matrix whose entries are $(\mathbf{F}_{u})_{ii}={\rm
tr}(\ket{y_{i}}\bra{y_{i}}({\bf
I}_{M_{k}(\mathbb{C})}+u\mathcal{L}_{0}^{-1})^{-1}({\bf 1}))$,
$y_{i}\in\mathbb{C}_{d}$. $\mathbf{F}_{u}$ is self adjoint and we can use
Lemmas 23 and 24 as in the classical case to upper bound the norm
$\left\|\mathbf{P}\mathbf{F}_{u}\right\|_{\pi}$. We seek as before to expand
$\hat{\mathbf{P}}\mathbf{F}_{u}$ ($\hat{\mathbf{P}}$ given by its definition
in section 2):

$\hat{\mathbf{P}}\mathbf{F}_{u}=\hat{\mathbf{P}}+\sum_{l=1}^{\infty}u^{l}\hat{\mathbf{P}}\mathbf{D}^{(l)},$

where $\mathbf{D}^{(l)}$ is a diagonal matrix with entries ${\rm
tr}\left(\ket{y_{i}}\bra{y_{i}}\left(-\mathcal{L}_{0}^{-1}\right)^{l}({\bf
1})\right)$. Note that contrary to the classical case this is not simply some
diagonal matrix $\mathbf{D}$ raised to a power $l$. The $L_{2}(\pi)$ norm
$\|\mathbf{D}^{(l)}\|_{\pi}=\mathbf{D}^{(l)}_{\rm max}$, where
$\mathbf{D}^{(l)}_{\rm max}$ is the maximum absolute element in
$\mathbf{D}^{(l)}$. Therefore, $\mathbf{D}^{(l)}$ can be upper bounded in
$L_{2}(\pi)$ by:

$\begin{split}\|\mathbf{D}^{(l)}\|_{\pi}&=\sup_{i\in I}~{}{\rm
tr}\left(\ket{y_{\tilde{i}}}\bra{y_{\tilde{i}}}\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right)^{l}({\bf 1})\right)\\\
&\leq\left\|\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right\|_{\infty\rightarrow\infty}^{l}=c_{r}^{l}.\end{split}$
(61)

Similarly, if we set $b_{r}^{2}=\sum_{i\in
I}\pi(i)\left\|\mathcal{L}_{0*}^{-2}(\ket{y_{i}}\bra{y_{i}})\right\|_{1}$, we
can upper bound $\|\mathbf{D}^{(l)}\underline{1}\|_{\pi}$:

$\begin{split}\|\mathbf{D}^{(l)}\underline{1}\|_{\pi}^{2}&=\sum_{i\in
I}\pi(i){\rm tr}\left(\ket{y_{i}}\bra{y_{i}}\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right)^{l}({\bf 1})\right)^{2}\\\
&\leq\sum_{i\in I}\pi(i){\rm tr}\left(\ket{y_{i}}\bra{y_{i}}\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right)^{l}({\bf
1})\right)\sup_{i}\left\\{{\rm
tr}\left(\ket{y_{i}}\bra{y_{i}}\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right)^{l}({\bf
1})\right)\right\\}\\\ &\leq\sum_{i\in I}\pi(i){\rm tr}\left(\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0*}}\right)^{2}(\ket{y_{i}}\bra{y_{i}})\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{0}}\right)^{l-2}({\bf
1})\right)c_{r}^{l}\\\ &\leq b_{r}^{2}c_{r}^{2(l-1)}.\end{split}$ (62)

Therefore $\|\mathbf{D}^{(l)}\underline{1}\|_{\pi}\leq b_{r}c_{r}^{l-1}$. The
final step which differs slightly from the classical case is in calculating
the $r^{(l)}$ from equation (40). We get the same results for $r^{(0)}$,
$r^{(1)}$ and $r^{(2)}$, but for $l\geq 3$ and $p=1$:

$-{\rm
tr}(\hat{\mathbf{P}}\mathbf{D}^{(l)}(-\Pi))=\left\langle\underline{1},\mathbf{D}^{(l)}\underline{1}\right\rangle_{\pi}\leq
b_{r}^{2}c_{r}^{l-2},$

in which we used the same trick as when bounding
$\|\mathbf{D}^{(l)}\underline{1}\|_{\pi}$. For $p\geq 2$:

$\begin{split}-{\rm
tr}(\hat{\mathbf{P}}\mathbf{D}^{(\nu_{1})}\mathbf{S}^{(\mu_{1})}\cdots\hat{\mathbf{P}}\mathbf{D}^{(\nu_{p})}\mathbf{S}^{(\mu_{p})})&=\left\langle\underline{1},\mathbf{D}^{(\nu_{1})}\mathbf{S}^{(\mu_{1})}\cdots\hat{\mathbf{P}}\mathbf{D}^{(\nu_{p})}\underline{1}\right\rangle_{\pi}\\\
&\leq\|\mathbf{D}^{(\nu_{1})}\underline{1}\|_{\pi}\|\mathbf{D}^{(\nu_{p})}\underline{1}\|_{\pi}\|\mathbf{S}\|_{\pi}^{p-1}\|\mathbf{D}^{(\nu_{2})}\|_{\pi}\cdots\|\mathbf{D}^{(\nu_{p-1})}\|_{\pi}\\\
&\leq b_{r}^{2}\frac{c_{r}^{l-2}}{\varepsilon^{l-1}}.\end{split}$

From here the proof for the reset and classical process are identical and so
we obtain Theorem 13. Note that for left deviations we can indeed repeat the
classical proof.

∎

######  Corollary 14.

The variance of the first passage time for total counts is bounded from above
by:

$\frac{{\rm
var}_{\pi}(T_{I}(k))}{k}\leq\left(1+\frac{2}{\varepsilon}\right)b_{r}^{2}.$

###### Proof.

Notice that for $u\geq 0$ small enough, one has

$\begin{split}\log(\mathbb{E}_{\pi}[e^{uT_{I}(k)}])&=\langle t_{I}\rangle
ku+\frac{1}{2}{\rm var}_{\pi}(T_{I}(k))u^{2}+o(u^{2})\\\ &\leq
k\log(r(u))=kr^{\prime}(0)u+\frac{k}{2}(r^{\prime\prime}(0)-(r^{\prime}(0))^{2})u^{2}+o(u^{2})\end{split}$

where $r(u)$ is given by equation (39). We recall using the definition of
$\mathbf{D}$ from section C.5 that for reset processes

$r^{\prime}(0)=r^{(1)}=\langle
t_{I}\rangle=\langle\mathbf{D}\underline{1},\underline{1}\rangle_{\pi},\quad
r^{\prime\prime}(0)=2r^{(2)}=2\langle\mathbf{D}\underline{1},\mathbf{D}\underline{1}\rangle_{\pi}+2\left\langle\mathbf{D}\underline{1},\frac{\hat{\mathbf{P}}}{{\bf
1}-\hat{\mathbf{P}}}\mathbf{D}\underline{1}\right\rangle_{\pi}$

and remind ourselves that $\|\mathbf{D}\underline{1}\|_{\pi}\leq b_{r}$.
Therefore

$r^{\prime\prime}(0)-(r^{\prime}(0))^{2}=\langle{\bf
D}\underline{1},\underline{1}\rangle_{\pi}^{2}+2\left\langle\mathbf{D}\underline{1},\frac{{\bf
1}}{{\bf
1}-\hat{\mathbf{P}}}\mathbf{D}\underline{1}\right\rangle_{\pi}\leq\left(1+\frac{2}{\varepsilon}\right)b_{r}^{2}.$

Hence ${\rm
var}_{\pi}(T_{I}(k))\leq\left(1+\frac{2}{\varepsilon}\right)b_{r}^{2}k$. ∎

###  C.6 Proof of Theorem 15 and Corollary 16

######  Theorem 15 (Rare Fluctuations of General Quantum Counting Observable
FPTs).

Assume that Hypothesis 2 holds ($\mathcal{L}$ be irreducible), and let
$\mathfrak{A}\subseteq I$ be nonempty. For every $\gamma>\beta-\langle
t_{\mathfrak{A}}\rangle$:

$\mathbb{P}_{\nu}\left(\frac{T_{\mathfrak{A}}(k)}{k}\geq\langle
t_{\mathfrak{A}}\rangle+\gamma\right)\leq\exp\left(-k\left(\frac{\gamma+\langle
t_{\mathfrak{A}}\rangle-\beta}{\beta}-\log\left(\frac{\gamma+\langle
t_{\mathfrak{A}}\rangle}{\beta}\right)\right)\right),k\in\mathbb{N}.$

###### Proof.

We once again use Chernoff bound to upper bound the probability in terms of
the Laplace transform:

$\mathbb{P}_{\nu}\left(\frac{T_{\mathfrak{A}}(k)}{k}\geq\langle
t_{\mathfrak{A}}\rangle+\gamma\right)\leq e^{-uk(\langle
t_{\mathfrak{A}}\rangle+\gamma)}\mathbb{E}_{\nu}[e^{uT_{\mathfrak{A}}(k)}],\quad
u>0.$

Next, we write the Laplace transform in terms of the Hilbert-Schmidt inner
product:

$\begin{split}\mathbb{E}_{\rho}[e^{uT_{\mathfrak{A}}(k)}]&=\left\langle\rho,\left(\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{{\bf
I}_{M_{k}(\mathbb{C})}+\frac{u}{\mathcal{L}_{\infty}}}\Psi\right)^{k}({\bf
1})\right\rangle_{\rm HS}\\\
&=\left\langle\rho,\left(\sum_{i=0}^{\infty}u^{i}\left(-\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{\infty}}\right)\Psi\right)^{k}({\bf
1})\right\rangle_{\rm HS},\quad u<\|\mathcal{L}_{\infty*}^{-1}\|_{1\rightarrow
1}^{-1}\\\ &\leq\|\rho\|_{1\rightarrow
1}\left(\sum_{i=0}^{\infty}u^{i}\left\|\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{\mathcal{L}_{\infty*}}\right\|_{1\rightarrow
1}\|\Psi\|_{\infty\rightarrow\infty}\right)^{k}\left\|{\bf
1}\right\|_{\infty\rightarrow\infty}.\end{split}$

We can denote $\beta:=\|\mathcal{L}_{\infty*}^{-1}\|_{1\rightarrow 1}$ to
obtain:

$\mathbb{E}_{\nu}[e^{uT_{\mathfrak{A}}(k)}]\leq\left(\frac{1}{1-\beta
u}\right)^{k}=\exp\left(-k\log(1-\beta u)\right).$ (63)

From here on, the proof is identical to that of Theorem 5.

∎

######  Corollary 16.

Given any non-empty set of jumps $\mathfrak{A}$, the variance of the
corresponding first passage time at stationarity is bounded from above by:

$\frac{{\rm
var}_{\varsigma}(T_{\mathfrak{A}}(k))}{k}\leq\left(1+\frac{2}{\tilde{\varepsilon}}\right)\beta^{2},$

where

$\tilde{\varepsilon}:=1-\max\\{\|\Psi(x)\|_{\infty\to\infty}:\|x\|_{\infty\to\infty}=1,\,{\rm
tr}(\varsigma x)=0\\}.$

###### Proof.

From the proof of Lemma 18 one can see that

$\begin{split}\frac{{\rm var}_{\varsigma}[T_{\mathfrak{A}}(k)]}{k}&={\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-2}({\bf 1})\right)^{2}+2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right)\\\ &+\frac{2}{k}{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\sum_{i=2}^{k}\sum_{j=1}^{i-1}\Psi^{j}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right)\\\
&\leq\left(1+2\left(1+\frac{1}{k}\sum_{i=2}^{k}\sum_{j=1}^{i-1}(1-\tilde{\varepsilon})^{j}\right)\right)\beta^{2}\\\
&=\left(1+\frac{2}{\tilde{\varepsilon}}\right)\beta^{2}-\frac{2((1-\tilde{\varepsilon})-(1-\tilde{\varepsilon})^{k+1})}{k\tilde{\varepsilon}^{2}}\beta^{2}\\\
&\leq\left(1+\frac{2}{\tilde{\varepsilon}}\right)\beta^{2}.\\\ \end{split}$

∎

######  Lemma 18 (Variance of Quantum FPTs).

Let $\varsigma$ be the invariant state of $\Psi$, cf. section 3.1.2, and let
$\Pi_{\varsigma}$ be the map $\Pi_{\varsigma}:x\mapsto{\rm tr}\left(\varsigma
x\right){\bf 1}$. The variance of the first passage time for counting
observables $\forall k\geq 0$ is given by:

$\begin{split}\frac{{\rm
var}_{\varsigma}\left(T_{\mathfrak{A}}(k)\right)}{k}={\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}({\bf 1})\right)^{2}&+2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{{\bf I}_{M_{k}(\mathbb{C})}-\Psi}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right)\\\ &-\frac{2}{k}{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\frac{\Psi-\Psi^{k+1}}{({\bf
I}_{M_{k}(\mathbb{C})}-\Psi)^{2}}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right).\end{split}$ (64)

###### Proof.

We recall the explicit expression for the moment generating function from
Lemma 9

$\mathbb{E}_{\varsigma}[e^{uT_{\mathfrak{A}}(k)}]={\rm
tr}\left(\varsigma\left((u+\mathcal{L}_{\infty})^{-1}\mathcal{L}_{\infty}\Psi\right)^{k}({\bf
1})\right).$

We can write the first moment as

$\mathbb{E}_{\varsigma}[e^{uT_{\mathfrak{A}}(k)}]^{\prime}=-\sum_{i=1}^{k}{\rm
tr}\left(\varsigma\left(\frac{\mathcal{L}_{\infty}}{u+\mathcal{L}_{\infty}}\Psi\right)^{i-1}\left(\frac{\mathcal{L}_{\infty}}{(u+\mathcal{L}_{\infty})^{2}}\Psi\right)\left(\frac{\mathcal{L}_{\infty}}{u+\mathcal{L}_{\infty}}\Psi\right)^{k-i}({\bf
1})\right).$ (65)

At $u=0$ this gives us the form of the asymptotic mean.

$\mathbb{E}_{\varsigma}[T_{\mathfrak{A}}(k)]=-k{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}({\bf 1})\right).$ (66)

Differentiating equation (52) at $u=0$ gives us the second moment:

$\begin{split}\mathbb{E}_{\varsigma}[T_{\mathfrak{A}}(k)^{2}]&=2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-2}({\bf
1})\right)k+\sum_{i=1}^{k}{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\left(\sum_{j=1}^{i-1}\Psi^{i-j}+\sum_{j=1}^{k-i}\Psi^{j}\right)\mathcal{L}_{\infty}^{-1}({\bf
1})\right).\\\ &=2{\rm tr}\left(\varsigma\mathcal{L}_{\infty}^{-2}({\bf
1})\right)k+2{\rm tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\sum_{1\leq
j<i\leq k}\Psi^{j}\mathcal{L}_{\infty}^{-1}({\bf 1})\right)\\\ &=2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-2}({\bf
1})\right)k+2\sum_{i=2}^{k}\sum_{j=1}^{i-1}{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}({\bf 1})\right)^{2}\\\
&\qquad\qquad\qquad\qquad+2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\sum_{i=2}^{k}\sum_{j=1}^{i-1}\Psi^{j}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right)\end{split}$

where to arrive at the third line, after $\Psi^{j}$ we have inserted
$\Pi_{\varsigma}+{\bf I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma}$, where
$\Pi_{\varsigma}$ is the projection onto ${\bf 1}$. Using the fact that
$\sum_{i=2}^{k}\sum_{j=1}^{i-1}1=\frac{k}{2}(k-1)$, and recalling equation
(66) for the expression for the first moment, hence

$\begin{split}\frac{{\rm
var}_{\varsigma}\left(T_{\mathfrak{A}}(k)\right)}{k}=2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-2}({\bf 1})\right)&-{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}({\bf 1})\right)^{2}\\\ &+2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\frac{\Psi}{{\bf
I}_{M_{k}(\mathbb{C})}-\Psi}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right)\\\ &-\frac{2}{k}{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\frac{\Psi-\Psi^{k+1}}{({\bf
I}_{M_{k}(\mathbb{C})}-\Psi)^{2}}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right).\end{split}$

Finally, we can again place $\Pi_{\varsigma}+{\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma}$ in the first term, in between the two
$\mathcal{L}_{\infty}^{-1}$. Rearranging this gives the final result, $\forall
k\geq 0$

$\begin{split}\frac{{\rm
var}_{\varsigma}\left(T_{\mathfrak{A}}(k)\right)}{k}={\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}({\bf 1})\right)^{2}&+2{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\frac{{\bf
I}_{M_{k}(\mathbb{C})}}{{\bf I}_{M_{k}(\mathbb{C})}-\Psi}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right)\\\ &-\frac{2}{k}{\rm
tr}\left(\varsigma\mathcal{L}_{\infty}^{-1}\frac{\Psi-\Psi^{k+1}}{({\bf
I}_{M_{k}(\mathbb{C})}-\Psi)^{2}}({\bf
I}_{M_{k}(\mathbb{C})}-\Pi_{\varsigma})\mathcal{L}_{\infty}^{-1}({\bf
1})\right).\end{split}$

∎

###  C.7 Technical Results

####  C.7.1 Perron-Frobenius Theory

For readers’ convenience we report below the results from Perron-Frobenius
theory and, more in general, some well known results in the theory of
positivity preserving maps acting on finite dimensional functional and matrix
spaces ($C^{*}$-algebras) that we used in the proofs of this work. We refer to
Chapter XIII in [68] for the commutative case and to [69] for the general case
of $C^{*}$-algebras (which includes the cases of our interest). Let ${\cal A}$
be either $\ell^{\infty}(E)$ or $M_{n}(\mathbb{C})$ and let $\Psi$ be a
positivity preserving map acting on ${\cal A}$. $\Psi$ is said to be
irreducible if there exists no non-trivial projection $p\in{\cal A}$ such that

$\Psi(p)\leq\alpha p$ (67)

for some positive constant $\alpha$. One can show that if $\Psi$ is a
transition matrix or a quantum channel, the definition above of irreducibility
coincide with having a unique faithful invariant state.

######  Theorem 19 (Perron-Frobenius Theory).

Let $r$ be the spectral radius of $\Psi$. The following statements hold true.

  1. 1.

$r\in{\rm Sp}(\Psi)$ and there exists $x\in{\cal A}$, $x\geq 0$ such that
$\Psi(x)=rx$.

If $\Psi$ is irreducible, then one has some further results.

  1. 1.

$r$ is a geometrically simple eigenvalue;

  2. 2.

$x$ is strictly positive and is the unique positive eigenvector.

Another important result is the Russo-Dye Theorem.

######  Theorem 20 (Russo-Dye Theorem).

$\Psi$ attains its norm at the identity of ${\cal A}$.

####  C.7.2 Irreducibility of the tilted semigroup

In this section we will show that when $\mathcal{L}$ generates an irreducible
quantum Markov semigroup (Hypothesis 2), then the semigroup generated by any
perturbation of the form

$\mathcal{L}_{s}=\mathcal{L}_{0}+\sum_{i\in I}e^{a_{i}s}{\cal J},\quad
a_{i},s\in\mathbb{R}$ (68)

generates an irreducible semigroup as well, in the sense that for every $t>0$,
$e^{t\mathcal{L}_{s}}$ is irreducible according to the definition in Eq. (67).
In fact, what we will prove is even stronger: it is well known (see for
instance [67, Proposition 7.5]) that the irreducibility of $e^{t\mathcal{L}}$
for $t>0$ is equivalent to the (a priori stronger) property that
$e^{t\mathcal{L}_{*}}(\rho)>0$ for every $t>0$ and every initial state $\rho$;
we will show that such property, often called primitivity, is owned by the
semigroup generated by $\mathcal{L}_{s}$ as well.

######  Lemma 21.

If $\mathcal{L}$ satisfies Hypothesis 2, then $\mathcal{L}_{s}$ generates a
primitive semigroup.

###### Proof.

Let us consider a vector $v\in\mathbb{C}^{d}$ and a state $\rho$ such that
$v\in\ker(e^{t\mathcal{L}_{s*}}(\rho))$ for some $t>0$. Then, using the Dyson
series, one can easily see that $\langle
v,e^{t\mathcal{L}_{s*}}(\rho)v\rangle=0$ implies that $\langle
v,e^{t\mathcal{L}_{0*}}(\rho)v\rangle=0$ and for every $k\geq 1$ and
$i_{1},\dots,i_{k}\in I$

$e^{s\sum_{j=1}^{k}a_{i_{j}}}\int_{\sum_{j=1}^{k}t_{j}\leq t}\langle
v,e^{\left(t-\sum_{j=1}^{k}t_{j}\right)\mathcal{L}_{0*}}{\cal
J}_{i_{k}}\cdots{\cal J}_{i_{1}}e^{t_{1}\mathcal{L}_{0*}}(\rho)v\rangle
dt_{1}\cdots dt_{k}=0,$

which is equivalent (since $e^{s\sum_{j=1}^{k}a_{i_{j}}}>0$) to

$\int_{\sum_{j=1}^{k}t_{j}\leq t}\langle
v,e^{\left(t-\sum_{j=1}^{k}t_{j}\right)\mathcal{L}_{0*}}{\cal
J}_{i_{k}}\cdots{\cal J}_{i_{1}}e^{t_{1}\mathcal{L}_{0*}}(\rho)v\rangle
dt_{1}\cdots dt_{k}=0.$

However, using now the Dyson series for $e^{t\mathcal{L}_{0*}}(\rho)$, one
sees that equations above imply that $\langle
v,e^{t\mathcal{L}_{0*}}(\rho)v\rangle=0$. However, since $e^{t\mathcal{L}}$ is
primitive, this implies that $v=0$ and we are done. ∎

In the case of a classical Markov chain, the equivalence of irreducibility and
primitivity for a Markov chain is a consequence of Levy’s Theorem ([70,
Theorem 8]). With the same proof line, one can show the result also for the
generator $\mathbf{L}$ of a classical Markov chain and its perturbations of
the form

$\mathbf{L}_{s}=\sum_{x\neq y}e^{sa_{xy}}w_{xy}-\mathbf{R},\quad
a_{xy},s\in\mathbb{R}.$

Therefore we can state the following.

######  Lemma 22.

If $\mathbf{L}$ satisfies Hypothesis 1, then $\mathbf{L}_{s}$ generates a
primitive semigroup.

####  C.7.3 Technical Lemmas

Below we report in our notation two technical lemmas that were proved in [71]
and which are used in the proofs of some of the bounds obtained in this paper.

######  Lemma 23 (Lemma 21 (i) in [71]).

Let $\mathbf{M}_{f}:L_{\pi}^{2}(E)\rightarrow L_{\pi}^{2}(E)$ be the
multiplication operator associated to a real valued function $f$, i.e.
$\mathbf{M}_{f}g=fg$ for every $g\in L_{\pi}^{2}(E)$. Let $\hat{\bf P}$ be the
León-Perron operator defined in equation (14). Then the following statement
holds:

$\|f\|_{\pi}^{2}\leq\|\mathbf{M}_{f}\hat{\mathbf{P}}\mathbf{M}_{f}\|_{\pi}.$

The following result is a slight generalisation of Lemma 2.1 (iii) in [71]. In
the following, we will use $\Psi$ to denote either a transition matrix
$\mathbf{P}$ or a quantum channel $\Phi$, $\chi$ to indicate their invariant
state, i.e. $\pi$ and $\sigma$, and we denote by ${\cal H}$ the Hilbert space
corresponding to their invariant state, i.e. $L^{2}_{\pi}(E)$ and
$L^{2}(\sigma)$, respectively. We recall that the notation $\hat{\Psi}$ stands
for the León-Perron version of $\Psi$.

######  Lemma 24.

For any operators $\mathbf{A},\mathbf{B}$ acting on $\mathcal{H}$, the
following holds true:

$\|\mathbf{A}\Psi\mathbf{B}\|_{\chi}\leq\left\|\mathbf{B}^{\dagger}\hat{\Psi}\mathbf{B}\right\|_{\chi}^{\frac{1}{2}}\left\|\mathbf{A}\hat{\Psi}\mathbf{A}^{\dagger}\right\|_{\chi}^{\frac{1}{2}}.$

###### Proof.

Let us consider $h_{1},h_{2}\in\mathcal{H}$, then

$\begin{split}|\langle\Psi h_{1},h_{2}\rangle_{\chi}|&=|\langle({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)(\Psi-\Pi)({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{1},h_{2}\rangle_{\chi}+\langle\Pi
h_{1},h_{2}\rangle_{\chi}|\\\ &=|\langle(\Psi-\Pi)({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{1},({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{2}\rangle_{\chi}+\langle\Pi
h_{1},h_{2}\rangle_{\chi}|\\\ &\leq|\langle(\Psi-\Pi)({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{1},({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{2}\rangle_{\chi}|+|\langle\Pi
h_{1},h_{2}\rangle_{\chi}|\\\ &\leq(1-\varepsilon)\|({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{1}\|_{\chi}\|({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{2}\|_{\chi}+|\langle
h_{1},\chi\rangle\langle\chi,h_{2}\rangle|\\\
&\leq\sqrt{(1-\varepsilon)\|({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{1}\|_{\chi}^{2}+|\langle\chi,h_{1}\rangle|^{2}}\sqrt{(1-\varepsilon)\|({\bf
I}_{M_{k}(\mathbb{C})}-\Pi)h_{2}\|_{\chi}^{2}+|\langle\chi,h_{2}\rangle|^{2}}\\\
&=\langle\hat{\Psi}h_{1},h_{1}\rangle_{\chi}^{\frac{1}{2}}\langle\hat{\Psi}h_{2},h_{2}\rangle_{\chi}^{\frac{1}{2}}.\end{split}$

We can then proceed to complete the lemma:

$\begin{split}\|\mathbf{A}\Psi\mathbf{B}\|_{\chi}&=\sup_{h_{1},h_{2}:\|h_{i}\|_{\chi}=1}|\langle\mathbf{A}\Psi\mathbf{B}h_{1},h_{2}\rangle_{\chi}|\\\
&=\sup_{h_{1},h_{2}:\|h_{i}\|_{\chi}=1}|\langle\Psi\mathbf{B}h_{1},\mathbf{A}^{\dagger}h_{2}\rangle_{\chi}|\\\
&\leq\sup_{h_{1},h_{2}:\|h_{i}\|_{\chi}=1}\langle\hat{\Psi}\mathbf{B}h_{1},\mathbf{B}h_{1}\rangle_{\chi}^{\frac{1}{2}}\langle\hat{\Psi}\mathbf{A}^{\dagger}h_{2},\mathbf{A}^{\dagger}h_{2}\rangle_{\chi}^{\frac{1}{2}}\\\
&=\sup_{h_{1}:\|h_{1}\|_{\chi}=1}\langle\mathbf{B}^{\dagger}\hat{\Psi}\mathbf{B}h_{1},h_{1}\rangle_{\chi}^{\frac{1}{2}}\sup_{h_{2}:\|h_{2}\|_{\chi}=1}\langle\mathbf{A}\hat{\Psi}\mathbf{A}^{\dagger}h_{2},h_{2}\rangle_{\chi}^{\frac{1}{2}}\\\
&=\left\|\mathbf{B}^{\dagger}\hat{\Psi}\mathbf{B}\right\|_{\chi}^{\frac{1}{2}}\left\|\mathbf{A}\hat{\Psi}\mathbf{A}^{\dagger}\right\|_{\chi}^{\frac{1}{2}}.\end{split}$

∎

## References

  *   * Chandler [1987] Chandler, D.: Introduction to Modern Statistical Mechanics, (1987). https://doi.org/10.1137/1030169
  * Gardiner [2004] Gardiner, C.: Handbook of Stochastic Methods. Berlin: Springer, Berlin, Heidelberg (2004)
  * Gardiner and Zoller [2004] Gardiner, C., Zoller, P.: Quantum Noise. Springer, Berlin, Heidelberg (2004)
  * Seifert [2012] Seifert, U.: Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on Progress in Physics 75(12), 126001 (2012) https://doi.org/10.1088/0034-4885/75/12/126001
  * Limmer [2024] Limmer, D.T.: Statistical Mechanics and Stochastic Thermodynamics. Oxford University Press, Great Clarendon Street (2024)
  * Merolle et al. [2005] Merolle, M., Garrahan, J., Chandler, D.: Space-time thermodynamics of the glass transition. Proc. Natl. Acad. Sci. USA 102, 10837–10840 (2005) https://doi.org/10.1073/pnas.0504820102
  * Lecomte et al. [2007] Lecomte, V., Appert-Rolland, C., Wijland, F.: Thermodynamic formalism for systems with markov dynamics. J. Stat. Phys. 127(1), 51 (2007)
  * Garrahan et al. [2007] Garrahan, J.P., Jack, R.L., Lecomte, V., Pitard, E., Duijvendijk, K., Wijland, F.: Dynamical first-order phase transition in kinetically constrained models of glasses. Phys. Rev. Lett. 98 (2007) https://doi.org/ARTN195702
  * Maes [2020] Maes, C.: Frenesy: Time-symmetric dynamical activity in nonequilibria. Phys. Rep. 850, 1–33 (2020) https://doi.org/10.1016/j.physrep.2020.01.002
  * Plenio and Knight [1998] Plenio, M.B., Knight, P.L.: The quantum-jump approach to dissipative dynamics in quantum optics. Rev. Mod. Phys. 70, 101–144 (1998) https://doi.org/10.1103/RevModPhys.70.101
  * Touchette [2009] Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep. 478(1-3), 1 (2009)
  * Dembo and Zeitouni [2010] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability, vol. 38. Springer, Berlin, Heidelberg (2010)
  * Hedges et al. [2009] Hedges, L.O., Jack, R.L., Garrahan, J.P., Chandler, D.: Dynamic order-disorder in atomistic models of structural glass formers. Science 323(5919), 1309–1313 (2009)
  * Garrahan [2018] Garrahan, J.P.: Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics. Physica A: Statistical Mechanics and its Applications 504, 130–154 (2018) https://doi.org/10.1016/j.physa.2017.12.149
  * Jack [2019] Jack, R.L.: Ergodicity and large deviations in physical systems with stochastic dynamics (2019) arXiv:1910.09883
  * Redner [2001] Redner, S.: A Guide to First-passage Processes. Cambridge University Press, Shaftesbury Road (2001)
  * Budini et al. [2014] Budini, A.A., Turner, R.M., Garrahan, J.P.: Fluctuating observation time ensembles in the thermodynamics of trajectories. J. Stat. Mech., 03012 (2014)
  * Kiukas et al. [2015] Kiukas, J., Guta, M., Lesanovsky, I., Garrahan, J.P.: Equivalence of matrix product ensembles of trajectories in open quantum systems. Phys. Rev. E 92, 012132 (2015) https://doi.org/10.1103/PhysRevE.92.012132
  * Barato and Seifert [2015] Barato, A.C., Seifert, U.: Thermodynamic uncertainty relation for biomolecular processes. Phys. Rev. Lett. 114, 158101 (2015)
  * Gingrich et al. [2016] Gingrich, T.R., Horowitz, J.M., Perunov, N., England, J.L.: Dissipation bounds all steady-state current fluctuations. Phys. Rev. Lett. 116, 120601 (2016)
  * Garrahan [2017] Garrahan, J.P.: Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables. Phys. Rev. E 95, 032134 (2017) https://doi.org/10.1103/PhysRevE.95.032134
  * Gingrich and Horowitz [2017] Gingrich, T.R., Horowitz, J.M.: Fundamental bounds on first passage time fluctuations for currents. Phys. Rev. Lett. 119, 170601 (2017) https://doi.org/10.1103/PhysRevLett.119.170601
  * Pietzonka et al. [2016] Pietzonka, P., Barato, A.C., Seifert, U.: Universal bounds on current fluctuations. Phys. Rev. E 93, 052145 (2016) https://doi.org/10.1103/PhysRevE.93.052145
  * Polettini et al. [2016] Polettini, M., Lazarescu, A., Esposito, M.: Tightening the uncertainty principle for stochastic currents. Phys. Rev. E 94, 052104 (2016) https://doi.org/10.1103/PhysRevE.94.052104
  * Pietzonka et al. [2017] Pietzonka, P., Ritort, F., Seifert, U.: Finite-time generalization of the thermodynamic uncertainty relation. Phys. Rev. E 96, 012101 (2017) https://doi.org/10.1103/PhysRevE.96.012101
  * Horowitz and Gingrich [2017] Horowitz, J.M., Gingrich, T.R.: Proof of the finite-time thermodynamic uncertainty relation for steady-state currents. Phys. Rev. E 96, 020103 (2017)
  * Proesmans and den Broeck [2017] Proesmans, K., Broeck, C.V.: Discrete-time thermodynamic uncertainty relation. EPL 119(2), 20001 (2017)
  * Dechant and Sasa [2018] Dechant, A., Sasa, S.-i.: Entropic bounds on currents in langevin systems. Physical Review E 97(6), 062101 (2018)
  * Brandner et al. [2018] Brandner, K., Hanazato, T., Saito, K.: Thermodynamic bounds on precision in ballistic multiterminal transport. Phys. Rev. Lett. 120, 090601 (2018)
  * Macieszczak et al. [2018] Macieszczak, K., Brandner, K., Garrahan, J.P.: Unified thermodynamic uncertainty relations in linear response. Phys. Rev. Lett. 121, 130601 (2018) https://doi.org/10.1103/PhysRevLett.121.130601
  * Barato et al. [2018] Barato, A., Chetrite, R., Faggionato, A., Gabrielli, D.: A unifying picture of generalized thermodynamic uncertainty relations. arXiv:1810.11894 (2018)
  * Nardini and Touchette [2018] Nardini, C., Touchette, H.: Process interpretation of current entropic bounds. The European Physical Journal B 91(1), 1–6 (2018)
  * Carollo et al. [2019] Carollo, F., Jack, R.L., Garrahan, J.P.: Unraveling the large deviation statistics of markovian open quantum systems. Phys. Rev. Lett. 122, 130605 (2019) https://doi.org/10.1103/PhysRevLett.122.130605
  * Guarnieri et al. [2019] Guarnieri, G., Landi, G.T., Clark, S.R., Goold, J.: Thermodynamics of precision in quantum nonequilibrium steady states. Phys. Rev. Research 1, 033021 (2019)
  * Chun et al. [2019] Chun, H.-M., Fischer, L.P., Seifert, U.: Effect of a magnetic field on the thermodynamic uncertainty relation. Physical Review E 99(4), 042128 (2019)
  * Koyuk and Seifert [2019] Koyuk, T., Seifert, U.: Operationally accessible bounds on fluctuations and entropy production in periodically driven systems. Physical review letters 122(23), 230601 (2019)
  * Koyuk and Seifert [2020] Koyuk, T., Seifert, U.: Thermodynamic uncertainty relation for time-dependent driving. Physical Review Letters 125(26), 260604 (2020)
  * Liu et al. [2020] Liu, K., Gong, Z., Ueda, M.: Thermodynamic uncertainty relation for arbitrary initial states. Physical Review Letters 125(14), 140602 (2020)
  * Hasegawa [2020] Hasegawa, Y.: Quantum thermodynamic uncertainty relation for continuous measurement. Phys. Rev. Lett. 125, 050601 (2020)
  * Hasegawa [2022] Hasegawa, Y.: Thermodynamic uncertainty relation for quantum first-passage processes. Phys. Rev. E 105, 044127 (2022) https://doi.org/10.1103/PhysRevE.105.044127
  * Vo et al. [2022] Vo, V.T., Vu, T.V., Hasegawa, Y.: Unified thermodynamic–kinetic uncertainty relation. Journal of Physics A: Mathematical and Theoretical 55(40), 405004 (2022) https://doi.org/10.1088/1751-8121/ac9099
  * Horowitz and Gingrich [2020] Horowitz, J.M., Gingrich, T.R.: Thermodynamic uncertainty relations constrain non-equilibrium fluctuations. Nature Physics 16(1), 15–20 (2020) https://doi.org/10.1038/s41567-019-0702-6
  * Bakewell-Smith et al. [2023] Bakewell-Smith, G., Girotti, F., Guţă, M., Garrahan, J.P.: General upper bounds on fluctuations of trajectory observables. Phys. Rev. Lett. 131, 197101 (2023) https://doi.org/10.1103/PhysRevLett.131.197101
  * Girotti et al. [2023] Girotti, F., Garrahan, J.P., Guţă, M.: Concentration inequalities for output statistics of quantum markov processes. Annales Henri Poincaré (2023) https://doi.org/10.1007/s00023-023-01286-1
  * Bebon and Godec [2023] Bebon, R., Godec, A.: Controlling uncertainty of empirical first-passage times in the small-sample regime. arXiv:2301.08732 (2023)
  * Boucheron et al. [2013] Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Great Clarendon Street (2013)
  * Van Vu and Saito [2022] Van Vu, T., Saito, K.: Thermodynamics of precision in markovian open quantum dynamics. Phys. Rev. Lett. 128, 140602 (2022) https://doi.org/%****␣main.bbl␣Line␣675␣****10.1103/PhysRevLett.128.140602
  * Pietzonka and Coghi [2023] Pietzonka, P., Coghi, F.: Thermodynamic cost for precision of general counting observables (2023)
  * León and Perron [2004] León, C.A., Perron, F.: Optimal hoeffding bounds for discrete reversible markov chains. The Annals of Applied Probability 14(2), 958–970 (2004). Accessed 2024-04-03
  * Ellis [2006] Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Springer, Berlin, Heidelberg (2006)
  * Lezaud [1998] Lezaud, P.: Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8(3), 849–867 (1998) https://doi.org/10.1214/aoap/1028903453
  * Nigel G. Bean and Taylor [2008] Nigel G. Bean, M.F., Taylor, P.: Characterization of matrix-exponential distributions. Stochastic Models 24(3), 339–363 (2008) https://doi.org/10.1080/15326340802232186 https://doi.org/10.1080/15326340802232186
  * Nielsen and Chuang [2010] Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Shaftesbury Road (2010)
  * Gorini et al. [1976] Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of n-level systems. J. Mat. Phys. 17(5), 821–825 (1976)
  * Lindblad [1976] Lindblad, G.: On the generators of quantum dynamical semigroups. Comm. Math. Phys 48, 119 (1976)
  * Belavkin [1990] Belavkin, V.: A stochastic posterior schrödinger equation for counting nondemolition measurement. Lett. Math. Phys. 20(2), 85–89 (1990)
  * Bouten et al. [2004] Bouten, L., Guta, M., Maassen, H.: Stochastic schrödinger equations. Journal of Physics A: Mathematical and General 37(9), 3189 (2004)
  * Wiseman and Milburn [2009] Wiseman, H.M., Milburn, G.J.: Quantum Measurement and Control. Cambridge University Press, Shaftesbury Road (2009)
  * Gough and James [2009] Gough, J., James, M.R.: The series product and its application to quantum feedforward and feedback networks. IEEE Trans. Automat. Contr. 54(11), 2530–2544 (2009)
  * Garrahan and Lesanovsky [2010] Garrahan, J.P., Lesanovsky, I.: Thermodynamics of quantum jump trajectories. Phys. Rev. Lett. 104(16), 160601 (2010) https://doi.org/10.1103/PhysRevLett.104.160601
  * Nishiyama and Hasegawa [2023] Nishiyama, T., Hasegawa, Y.: Exact solution to quantum dynamical activity (2023)
  * Mori and Shirai [2022] Mori, T., Shirai, T.: Symmetrized liouvillian gap in markovian open quantum systems. Physical review letters 130 23, 230404 (2022)
  * Dechant [2023] Dechant, A.: Thermodynamic constraints on the power spectral density in and out of equilibrium. arXiv:2306.00417 (2023)
  * Carbone and Pautrat [2015] Carbone, R., Pautrat, Y.: Homogeneous open quantum random walks on a lattice. J. Stat. Phys. 160(5), 1125–1153 (2015) https://doi.org/10.1007/s10955-015-1261-6
  * Engel and Nagel [2000] Engel, K.-J., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Graduate Text in Mathematics, p. 586. Springer, New York (2000)
  * Kato [1976] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin Heidelberg New York (1976)
  * Wolf [2012] Wolf, M.: Quantum Channels & Operations Guided Tour. Online Lecture Notes (2012)
  * Gantmacher [1959] Gantmacher, F.R.: The Theory of Matrices / by F. R. Gantmacher. Chelsea Publishing Company, New York, New York (1959)
  * Evans and Høegh-Krohn [1978] Evans, D.E., Høegh-Krohn, R.: Spectral properties of positive maps on c*-algebras. Journal of The London Mathematical Society-second Series, 345–355 (1978)
  * Freedman [2012] Freedman, D.: Approximating Countable Markov Chains. Springer, Berlin, Heidelberg (2012)
  * Fan et al. [2021] Fan, J., Jiang, B., Sun, Q.: Hoeffding’s inequality for general markov chains and its applications to statistical learning. J. Mach. Learn. Res. 22(139), 1–35 (2021)