$\mathcal{U}_{t,\varepsilon}(c,\varphi):=\sum_{\sigma=\pm}e^{\sigma\gamma
c}\int_{\mathcal{C}_{[0,t]}}\int_{\mathbb{T}}\prod_{ij}|e^{-s+i\theta}-e^{-t_{i}^{\varepsilon}(v)+i\theta_{ij}^{\varepsilon}(v)}|^{-\gamma\sigma\alpha_{ij}}\frac{dv}{2\pi}M^{\sigma}_{\gamma}(dsd\theta),$

and
$(t_{i}^{\varepsilon}(v),\theta_{ij}^{\varepsilon}(v))=(t_{i},\theta_{ij})+\varepsilon(\cos(v),\sin(v))$.
We bound $e^{-\mu\,\mathcal{U}_{t,\varepsilon}(c,\varphi)}$ by $1$ and use the
Cauchy-Schwarz inequality to get

$\mathbb{E}[G_{\beta,\varepsilon}(c,\varphi)^{2}]\;\leqslant\;e^{2\beta|c|+2\lambda_{0}t+\frac{1}{2}\sum_{i,j}\alpha_{ij}^{2}t_{i}+2\sum\alpha_{ij}c}\mathbb{E}[\psi_{0}(c+B_{t}+\sum_{i,j}\alpha_{ij}t_{i}+\varphi_{t}+Ph_{\varepsilon}(t,\cdot))^{4}]^{1/2}\mathds{E}[\psi_{0}(c+\varphi+h_{\varepsilon})^{4}]^{1/2}.$

Note that since $t>0$, $Ph(t,\cdot):=\lim_{\varepsilon\rightarrow
0}Ph_{\varepsilon}(t,\cdot)$ is smooth. Hence, by the Cameron-Martin theorem
we have that there exists $C>0$ such that uniformly in $\varepsilon$
sufficiently small

(4.23)
$\mathbb{E}[\psi_{0}(c+B_{t}+\sum_{i,j}\alpha_{ij}t_{i}+\varphi_{t}+Ph_{\varepsilon}(t,\cdot))^{4}]\;\leqslant\;C\,\mathbb{E}[\psi_{0}(c+B_{t}+\varphi_{t})^{4}]\;\leqslant\;C\mathds{E}[e^{-t\mathbf{H}_{0}}\psi_{0}^{4}].$

Similarly, since $h_{\varepsilon}$ is unifirmly smooth as $\varepsilon\to 0$,
by the Cameron-Martin theorem

$\mathds{E}[\psi_{0}(c+\varphi+h_{\varepsilon})^{4}]\;\leqslant\;C\mathds{E}[\psi_{0}(c+\varphi)^{4}].$

Combining these estimates, Lemma 3.10 and the boundedness of $e^{-t{\bf
H}_{0}}$ on $L^{1}(H^{-s}(\mathbb{T}),\mu_{0})$ (Proposition 3.1) there exists
$C,C^{\prime}>0$, depending on $t$ such that uniformly in $\varepsilon$,

(4.24)
$\int\mathbb{E}[G_{\beta,\varepsilon}^{2}]e^{-\beta|c|}dc\;\leqslant\;C\int(e^{\beta|c|+\sum_{i,j}\alpha_{ij}c}\,\mathbb{E}[\psi_{0}^{4}]+\mathds{E}[e^{-t{\bf
H}_{0}}(\psi_{0}^{4})])dc\;\leqslant\;C^{\prime}.$

This establishes the uniform $L^{2}$ estimate (4.20).

We now turn to (4.21). Note that $G_{\beta,\varepsilon}$ converges pointwise
to $e^{|\beta|c}$ times the integrand in (4.21). Furthermore, $\eqref{eq: vc
unif l2}$ implies that $(G_{\beta,\varepsilon})$ is uniformly integrable with
respect to the finite measure $e^{-\beta|c|}\mu_{0}$. Hence by Vitali’s
convergence theorem, we obtain convergence of the corresponding integrals.
Rearranging for the factor $e^{-\beta|c|}$ yields (4.21). In order to obtain
the nontriviality, note that all the expressions in (4.21) are positive almost
everywhere _except_ for the term involving the GMC. Thus it is sufficient to
establish that

(4.25)
$e^{\sum_{ij}\alpha_{ij}c}\exp\big{(}-\mu\sum_{\sigma=\pm}e^{\sigma\gamma
c}\int_{\mathcal{C}_{[0,t]}}\prod_{ij}|e^{-s+i\theta}-e^{-t_{i}+i\theta_{ij}}|^{-\gamma\sigma\alpha_{ij}}M^{\sigma}_{\gamma}(dsd\theta)\big{)}>0\,\text{
a.e., }$

which is in turn true (i.e. the term in the exponential is finite almost
surely) if and only if $\mathcal{I}$ is $\gamma$-admissible, see [BP23, Lemma
5.19] and in particular Steps 2 and 3 there. ∎

We now turn to the proof of Theorem 1.3.

###### Proof of Theorem 1.3.

For $R=1$, the convergence, nontriviality, and explicit formula of vertex
correlations with arbitrary insertion set follows immediately from Lemma 4.10.
The generic case $R>0$, as we have argued previously, follows from the scaling
relation for GMC. We observe that for the $1$-point function, if the insertion
is at $(t_{1},\theta_{1})=(0,0)$ with weight $|\alpha|<Q$, we can let $t\to 0$
in the expression (4.21) and obtain (with
$h(\theta)=-\alpha\log|e^{i\theta}-1|$ as in (4.18)):

(4.26) $\langle V_{\alpha}(0)\rangle_{\mathcal{C}}=\|e^{\alpha
c/2}\psi_{0}(\cdot+h)\|^{2}_{\mathcal{H}}.$

We now turn towards the exponential decay of correlations for the truncated
two-point function, or covariance, of the vertex correlations. We will fix
$R=1$ for ease of notation; the case $R>0$ follows from trivial modifications.
As a preliminary calculation, let us first analyze covariances of bounded
functions of compact support. Let $\varepsilon>0$. Using the same approach as
in the proof of Proposition 4.4, for every $F_{\varepsilon}\in\mathcal{A}$
bounded and supported on $[-\varepsilon,\varepsilon]$, define the map

(4.27)
$\mathcal{B}_{F_{\varepsilon}}:\mathcal{H}\rightarrow\mathcal{H},\quad(\mathcal{B}_{F_{\varepsilon}}\psi)(c,\varphi)=\mathbb{E}_{\varphi}[F_{\varepsilon}(c+B_{\bullet+\varepsilon}+\varphi_{\bullet+\varepsilon})\psi(c+B_{2\varepsilon}+\varphi_{2\varepsilon})e^{-\mu\sum_{\sigma=\pm}e^{\sigma\gamma
c}M_{\gamma}^{\sigma}([0,2\varepsilon]\times\mathbb{T})}].$

In particular, when $\varepsilon=0$ we have that
$\mathcal{B}_{F}(\psi)=F\psi\in\mathcal{H}$. Given
$F_{\varepsilon},G_{\varepsilon}\in\mathcal{A}$ that are bounded and supported
on $[-\varepsilon,\varepsilon]$, by the domain Markov property and tower
property of conditional expectations, we have

(4.28) $\begin{split}\langle
F_{\varepsilon}\cdot\tau_{t}G_{\varepsilon}\rangle_{\mathcal{C}_{1}}=e^{\lambda_{0}(t+2\varepsilon)}\langle\mathcal{B}_{F_{\varepsilon}}e^{-(t-2\varepsilon){\bf
H}}\mathcal{B}_{G_{\varepsilon}}\psi_{0},\psi_{0}\rangle_{\mathcal{H}},\end{split}$

where we recall that $\tau_{t}G$ translates the support of $G$ by $+t$.

It is direct to check that the operator $\mathcal{B}_{F_{\varepsilon}}$ is
self-adjoint (using that $F_{\varepsilon}$ is real valued and the dynamics
$B_{s}+\varphi_{s}$ is reversible with respect to $\mu_{0}$), thus the
eigenfunction expansion in Theorem 1.1 yields (for $t>2\varepsilon)$:

$\begin{split}\langle
F_{\varepsilon}\cdot\tau_{t}G_{\varepsilon}\rangle_{\mathcal{C}_{1}}&=e^{\lambda_{0}(t+2\varepsilon)}\sum_{j\;\geqslant\;0}e^{-\lambda_{j}(t-2\varepsilon)}\langle\mathcal{B}_{F_{\varepsilon}}\psi_{0},\psi_{j}\rangle_{\mathcal{H}}\langle\mathcal{B}_{G_{\varepsilon}}\psi_{0},\psi_{j}\rangle_{\mathcal{H}}.\end{split}$

Hence since $\langle
F_{\varepsilon}\rangle_{\mathcal{C}_{1}}=e^{2\lambda_{0}\varepsilon}\langle\mathcal{B}_{F_{\varepsilon}}\psi_{0},\psi_{0}\rangle_{\mathcal{H}}$
and $\langle
G_{\varepsilon}\rangle_{\mathcal{C}_{1}}=e^{2\lambda_{0}\varepsilon}\langle\mathcal{B}_{G_{\varepsilon}}\psi_{0},\psi_{0}\rangle_{\mathcal{H}}$,
we have the representation

(4.29) ${\rm Cov}(F,\tau_{t}G):=\langle
F_{\varepsilon}\cdot\tau_{t}G_{\varepsilon}\rangle_{\mathcal{C}_{1}}-\langle
F_{\varepsilon}\rangle_{\mathcal{C}_{1}}\langle\tau_{t}G_{\varepsilon}\rangle_{\mathcal{C}_{1}}=e^{-(\lambda_{1}-\lambda_{0})t}\mathcal{R}_{t}(F_{\varepsilon},G_{\varepsilon}),$

where

(4.30)
$\mathcal{R}_{t}(F_{\varepsilon},G_{\varepsilon}):=e^{2(\lambda_{1}+\lambda_{0})\varepsilon}\sum_{j\;\geqslant\;1}e^{-(\lambda_{j}-\lambda_{1})(t-2\varepsilon)}\langle\mathcal{B}_{F_{\varepsilon}}\psi_{0},\psi_{j}\rangle_{\mathcal{H}}\langle\mathcal{B}_{G_{\varepsilon}}\psi_{0},\psi_{j}\rangle_{\mathcal{H}}$

In particular, when $F_{\varepsilon}=G_{\varepsilon}$, the covariances are
positive.

Let us now turn to vertex correlations. Fix $t>0$ without loss of generality.
Let $|\alpha_{1}|,|\alpha_{2}|<Q$ and $\theta_{1},\theta_{2}\in\mathbb{T}$.
For every $K\in\mathbb{N}^{*}$, write

(4.31) $V_{\alpha}^{\varepsilon}(s,\theta)_{K}:={\rm
min}(V_{\alpha}^{\varepsilon}(s,\theta),K)\in\mathcal{A},$

which is bounded and supported on $[-\varepsilon,\varepsilon]$. Hence by
(4.29) we have that for every $\varepsilon>0$ such that $2\varepsilon<t$ and
for every $\delta>0$,

(4.32) ${\rm
Cov}\Big{(}V_{\alpha_{1}}^{\varepsilon}(0,\theta_{1})_{K},V_{\alpha_{2}}^{\varepsilon}(t,\theta_{2})_{K}\Big{)}=e^{-(\lambda_{1}-\lambda_{0})t}e^{(\lambda_{1}-\lambda_{0})\delta}e^{-(\lambda_{1}-\lambda_{0})\delta}\mathcal{R}_{t}(V_{\alpha_{1}}^{\varepsilon}(0,\theta_{1})_{K},V_{\alpha_{2}}^{\varepsilon}(0,\theta_{2})_{K}).$

We now take $\varepsilon$ and $\delta$ such that $2\varepsilon<\delta<t$. Then
by the Cauchy-Schwarz inequality,

(4.33)
$|e^{-(\lambda_{1}-\lambda_{0})\delta}\mathcal{R}_{t}(V_{\alpha_{1}}^{\varepsilon}(0,\theta_{1})_{K},V_{\alpha_{2}}^{\varepsilon}(0,\theta_{2})_{K})|\;\leqslant\;e^{-(\lambda_{1}-\lambda_{0})\delta}\prod_{i=1,2}\mathcal{R}_{\delta}(V_{\alpha_{i}}^{\varepsilon}(0,\theta_{i})_{K},V_{\alpha_{i}}^{\varepsilon}(0,\theta_{i})_{K})^{1/2}.$

However, by (4.29) and the fact that the covariances are positive when
$F_{\varepsilon}=G_{\varepsilon}$, we obtain

$\displaystyle\limsup_{\varepsilon\rightarrow
0}\limsup_{K\rightarrow\infty}\Big{|}\prod_{i=1,2}\mathcal{R}_{\delta}(V_{\alpha_{i}}^{\varepsilon}(0,\theta_{i})_{K},V_{\alpha_{i}}^{\varepsilon}(0,\theta_{i})_{K})^{1/2}\Big{|}e^{-(\lambda_{1}-\lambda_{0})\delta}$
$\displaystyle\;\leqslant\;\lim_{\varepsilon\rightarrow
0}\lim_{K\rightarrow\infty}\prod_{i=1,2}{\rm
Cov}\Big{(}V_{\alpha_{i}}^{\varepsilon}(0,\theta_{i})_{K},V_{\alpha_{i}}^{\varepsilon}(\delta,\theta_{i})_{K}\Big{)}^{1/2}.$

But, since the covariances of vertex correlations converge, we therefore have
that

(4.34) $\Big{|}{\rm
Cov}\Big{(}V_{\alpha_{1}}(0,\theta_{1}),V_{\alpha_{2}}(t,\theta_{2})\Big{)}\Big{|}\;\leqslant\;e^{-(\lambda_{1}-\lambda_{0})t}C_{\lambda_{1}-\lambda_{0}}(\alpha_{1},\theta_{1};\alpha_{2},\theta_{2}),$

where

(4.35)
$C_{\lambda_{1}-\lambda_{0}}(\alpha_{1},\theta_{1};\alpha_{2},\theta_{2}):=e^{(\lambda_{1}-\lambda_{0})\delta}\prod_{i=1,2}{\rm
Cov}\Big{(}V_{\alpha_{i}}(0,\theta_{i}),V_{\alpha_{i}}(\delta,\theta_{i})\Big{)}^{1/2}.\qed$

##  Appendix A Heuristic for the path integral

###  A.1. Brownian motion and Ornstein-Uhlenbeck processes from the path
integral

The formal 1 dimensional path integral over the space of continuous functions
$C^{0}_{x}([0,T]):=\\{\phi\in C^{0}([0,T],\mathbb{R})\,|\,\phi(0)=x\\}$ with
$T>0$ and $x\in\mathbb{R}$

(A.1)
$\int_{C^{0}_{x}([0,T])}F(\phi)e^{-\frac{1}{2\sigma^{2}}\int_{0}^{T}|\partial_{t}\phi(t)|^{2}dt}D\phi$

can be rigorously defined using the Brownian motion $B^{\sigma}_{t}$ with
initial condition $B^{\sigma}_{0}=0$ and covariance
$\mathds{E}[B_{t}^{\sigma}B^{\sigma}_{s}]=\sigma^{2}\min(s,t)$ by

$\int_{C^{0}_{x}([0,T],\mathbb{R})}F(\phi)e^{-\frac{1}{2\sigma^{2}}\int_{0}^{T}|\partial_{t}\phi(t)|^{2}dt}D\phi:=\mathds{E}[F(x+B^{\sigma})].$

Notice that $B^{\sigma}_{t}=B_{\sigma^{2}t}$ in law, with $B_{t}=B^{1}_{t}$.
The heuristic here is first to take $x=0$, in which case one searches for a
Gaussian random variable $B$ (say, a priori, in $H^{1}$) such that
$\mathds{E}[\langle f,B\rangle_{H^{1}}\langle g,B\rangle_{H^{1}}]=\langle
f,g\rangle_{H^{1}}$ for all $f,g\in H^{1}_{0}([0,T]):=\\{f\in
H^{1}([0,T])\,|\,f(0)=0\\}$ with $\langle
f,g\rangle_{H^{1}}=\int_{0}^{T}\partial_{t}f(t)\partial_{t}g(t)dt$. A direct
calculation gives that $B$ must satisfy $\mathds{E}[B_{s}B_{t}]=\min(s,t)$ and
we thus set $B$ to be the Brownian motion (the drawback is that $B$ is only
$C^{1/2-\varepsilon}$ Hölder). Then we add $x$ to $B^{\sigma}$ to represent
the field with values $x$ at $t=0$.

We can apply the same kind of reasoning for the formal 1 dimensional path
integral over $C^{0}_{x}([0,T])$

(A.2)
$\int_{C^{0}_{x}([0,T])}F(\phi)e^{-\frac{1}{2\sigma^{2}}\int_{0}^{T}|\partial_{t}\phi(t)|^{2}+(\mu\sigma)^{2}|\phi(t)|^{2}dt}D\phi$

for $\mu,\sigma>0$. This can be rigorously defined using the Ornstein-
Uhlenbeck process $X_{t}$, which satisfies the following stochastic
differential equation

(A.3) $dX_{t}=-\mu\sigma X_{t}dt+\sigma dB_{t}.$

This is a diffusion with generator given on $\mathbb{R}$ by

$\mathcal{L}f(s)=-\frac{\sigma^{2}}{2}f^{\prime\prime}(s)+\mu\sigma
sf^{\prime}(s).$

Starting from a point $X_{0}=x$ one can explicitely solve (A.3) yielding the
formula

(A.4) $X_{t}=xe^{-\mu\sigma t}+\sigma e^{-\mu\sigma t}\int_{0}^{t}e^{\mu\sigma
s}dB_{s}=xe^{-\mu\sigma t}+\widetilde{X}_{t}$

where $\widetilde{X}_{t}$ is a Gaussian random variable with covariance
structure

$\mathds{E}[\widetilde{X}_{t}\widetilde{X}_{s}]=\frac{\sigma}{2\mu}(e^{-\mu\sigma|t-s|}-e^{-\mu\sigma(t+s)}).$

Hence at stationarity the covariance is given by
$\mathds{E}[X_{s}X_{t}]=\frac{\sigma}{2\mu}e^{-\mu\sigma|t-s|}$. The path
integral (A.2) can be given a sense as follows:

$\int_{C^{0}_{x}([0,T])}F(\phi)e^{-\frac{1}{2\sigma^{2}}\int_{0}^{T}|\partial_{t}\phi(t)|^{2}+(\mu\sigma)^{2}|\phi(t)|^{2}dt}D\phi:=\mathds{E}[F(X)]$

###  A.2. The GFF path integral

We now analyse the path integral (1.7) underlying our construction and try to
identify the associated Markov process as in the 1d case. Recall that a real
valued smooth function $\phi$ on the cylinder
$\mathcal{C}_{R}=\mathbb{R}\times\mathbb{T}_{R}$ can be decomposed using
Fourier series in $\theta$:

(A.5)
$\phi(t,\theta)=\phi_{0}(t)+\sum_{n\not=0}\phi_{n}(t)e^{in\frac{\theta}{R}}.$

The same also holds on $\mathcal{C}^{+}_{R,T}=[0,T]\times\mathbb{T}_{R}$. With
this decomposition, and writing $\phi_{n}(t)$ under the form
$\phi^{R}_{n}(t)=\frac{x_{n}(t)+iy_{n}(t)}{2\sqrt{n}}$ with
$x_{n}(t),y_{n}(t)$ real valued, a direct calculation gives

$\displaystyle\frac{1}{4\pi}\int_{\mathcal{C}^{+}_{R,T}}|\nabla\phi|^{2}dtd\theta$
$\displaystyle=\frac{1}{4\pi}\int_{\mathcal{C}^{+}_{R,T}}|\partial_{t}\phi_{0}(t)|^{2}+\sum_{n\not=0}|\partial_{t}\phi_{n}(t)|^{2}dtd\theta+\frac{1}{4\pi
R^{2}}\int_{C^{+}_{R,T}}\sum_{n\not=0}|n\phi_{n}(t)|^{2}dtd\theta$
$\displaystyle=\frac{R}{2}\int_{0}^{T}|\partial_{t}\phi_{0}(t)|^{2}dt+\sum_{n\;\geqslant\;1}\left(R\int_{0}^{T}|\partial_{t}\phi_{n}(t)|^{2}dt+\frac{n^{2}}{R}\int_{0}^{T}|\phi_{n}(t)|^{2}dt\right)$
$\displaystyle=\frac{R}{2}\int_{0}^{T}|\partial_{t}\phi_{0}|^{2}dt+\sum_{n\;\geqslant\;1}\left(\frac{R}{4n}\int_{0}^{T}|\partial_{t}x_{n}(t)|^{2}dt+\frac{n}{4R}\int_{0}^{T}|x_{n}(t)|^{2}dt\right)$
$\displaystyle+\sum_{n\;\geqslant\;1}\left(\frac{R}{4n}\int_{0}^{T}|\partial_{t}y_{n}(t)|^{2}dt+\frac{n}{4R}\int_{0}^{T}|y_{n}(t)|^{2}dt\right).$

Let us consider the formal Gaussian path integral

$\int
F(\phi)e^{-\frac{1}{4\pi}\int_{\mathcal{C}^{+}_{R,T}}|\nabla\phi|^{2}}d\phi$

on the cylinder with the condition that

$\phi(0,\theta)=c+\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\varphi_{n}e^{in\frac{\theta}{R}},\quad\varphi_{n}:=\left\\{\begin{array}[]{ll}\frac{x_{n}+iy_{n}}{2\sqrt{n}}&n>0,\\\
\frac{x_{|n|}-iy_{|n|}}{2\sqrt{|n|}}&n<0,\end{array}\right.$

for some real $x_{n},y_{n}$. Then, by the discussion above, the field $\phi$
can be represented under the form

$\phi^{R}(t,\theta)=\phi_{0}^{R}(t)+\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\varphi_{n}^{R}(t)e^{in\frac{\theta}{R}},$

where

$\phi_{0}^{R}(t):=c+B_{t/R},\quad\varphi^{R}_{n}(t):=\Bigg{\\{}\begin{array}[]{ll}\frac{x_{n}^{R}(t)+iy_{n}^{R}(t)}{2\sqrt{n}}&n>0,\\\
\frac{x_{|n|}^{R}(t)-iy^{R}_{|n|}(t)}{2\sqrt{|n|}}&n<0,\end{array}$

with $x_{n}^{R}(t),y_{n}^{R}(t)$ the Ornstein-Uhlenbeck processes with
parameters $\mu_{n}=\sqrt{\frac{n}{2R}}$ and $\sigma_{n}=\sqrt{\frac{2n}{R}}$.

We now describe how to interpret the path integral on the forward cylinder
$\mathcal{C}_{R,T}^{+}$ (we will relate it to the path integral on
$\mathcal{C}_{R,T}$ by translations below). For
$(c,\varphi)\in\mathbb{R}\times H_{0}^{-s}(\mathbb{T}_{R})$ fixed with
$\varphi=\sum_{n>0}\frac{(x_{n}+iy_{n})}{2\sqrt{n}}e^{i\frac{n}{R}\theta}+\sum_{n>0}\frac{(x_{n}-iy_{n})}{2\sqrt{n}}e^{-i\frac{n}{R}\theta}$,
let us define the following space of distributions on the cylinder with fixed
values $(c,\varphi)\in\mathbb{R}\times H_{0}^{-s}(\mathbb{T}_{R})$ at $t=0$
for some fixed $s>0$:

$\begin{split}E_{c,\varphi}(\mathcal{C}^{+}_{R,T})=\Big{\\{}\phi(t,\theta)=&x_{0}(t)+\sum_{n\not=0}\frac{x_{|n|}(t)+i{\rm
sign}(n)y_{|n|}(t)}{2\sqrt{|n|}}e^{in\frac{\theta}{R}}\in
H^{-s}(\mathcal{C}^{+}_{R,T})\\\ &\quad\,\Big{|}\,x_{0}(t)\in
C^{0}_{c}([0,T]),\forall n\;\geqslant\;1,\,x_{n}(t)\in
C^{0}_{x_{n}}([0,T]),y_{n}(t)\in C^{0}_{y_{n}}([0,T])\Big{\\}}.\end{split}$

The formal Gaussian measure on the cylinder $\mathcal{C}^{+}_{R,T}$ is defined
for e.g. bounded measurable $F$ by

$
\int_{E_{c,\varphi}(\mathcal{C}^{+}_{R,T})}F(\phi)e^{-\frac{1}{4\pi}\int_{\mathcal{C}^{+}_{R,T}}|\nabla\phi|^{2}dtd\theta}D\phi:=\mathds{E}_{\varphi}[F(\phi^{R}(t,\theta))],$

where $\phi^{R}(t,\theta)$ is the random variable in
$E_{c,\varphi}(\mathcal{C}^{+}_{R,T})$ defined by

$\phi^{R}(t,\theta):=c+B_{\frac{t}{R}}+\sum_{n=1}^{\infty}\frac{x_{n}(\frac{t}{R})+iy_{n}(\frac{t}{R})}{2\sqrt{n}}e^{in\frac{\theta}{R}}+\frac{x_{n}(\frac{t}{R})-iy_{n}(\frac{t}{R})}{2\sqrt{n}}e^{-in\frac{\theta}{R}},$

with $x_{n}^{R}(t),y_{n}^{R}(t)$ the Ornstein-Uhlenbeck processes defined
above, and the expectation is conditional on $\varphi$.

Now, to work on the strip $\mathcal{C}_{R,T}=[-T,T]\times\mathbb{T}_{R}$, we
view it as a shift by $t\mapsto t-T$ of $\mathcal{C}^{+}_{R,2T}$ and therefore
represent the path integral on $\mathcal{C}_{R,T}$ as

$\int_{E_{c,\varphi}(\mathcal{C}_{R,T})}F(\phi)e^{-\frac{1}{4\pi}\int_{\mathcal{C}_{R,T}}|\nabla\phi|^{2}dtd\theta}D\phi:=\mathds{E}[F(\phi^{R}(t,\theta))],$

where $E_{c,\varphi}(\mathcal{C}_{R,T})=\\{\phi(t+T,\theta)\,|\,\phi\in
E_{c,\varphi}(\mathcal{C}^{+}_{R,2T})\\}$ and now the random variable
$\phi^{R}$ used to define the path integral is

$\phi^{R}(t,\theta)=c+B_{\frac{t+T}{R}}+\sum_{n=1}^{\infty}\frac{x_{n}(\frac{t+T}{R})+iy_{n}(\frac{t+T}{R})}{2\sqrt{n}}e^{in\frac{\theta}{R}}+\sum_{n=1}^{\infty}\frac{x_{n}(\frac{t+T}{R})-iy_{n}(\frac{t+T}{R})}{2\sqrt{n}}e^{-in\frac{\theta}{R}}.$

## References

  * [BDV] N. Barashkov and F. De Vecchi.  Elliptic stochastic quantization of Sinh-Gordon QFT.  arXiv:2108.12664.
  * [BGK16] G. Borot, A. Guionnet, and K. Kozlowski.  Asymptotic expansion of a partition function related to the Sinh-model.  Mathematical Physics studies, 2016.
  * [BP23] N. Berestycki and E. Powell.  Gaussian Free Field and Liouville Quantum Gravity.  2023\.
  * [CRV23] B. Cerclé, R. Rhodes, and V. Vargas.  Probabilistic construction of Toda conformal field theories.  Annales Henri Lebesgue, 6(31-64), 2023.
  * [DKRV16] F. David, A. Kupiainen, R. Rhodes, and V. Vargas.  Liouville quantum gravity on the Riemann sphere.  Comm. Math. Phys., 342(3):869–907, 2016.
  * [DO94] H. Dorn and H.-J. Otto.  Two- and three-point functions in Liouville theory.  Nuclear Phys. B, 429(2):375–388, 1994.
  * [GKRV20] C. Guillarmou, A. Kupiainen, R. Rhodes, and V. Vargas.  Conformal bootstrap in Liouville Theory.  Acta Mathematica, (to appear):arXiv:2005.11530, May 2020.
  * [GKRV21] C. Guillarmou, A. Kupiainen, R. Rhodes, and V. Vargas.  Segal’s axioms and bootstrap for Liouville Theory.  arXiv e-prints, page arXiv:2112.14859, December 2021.
  * [GRV19] C. Guillarmou, R. Rhodes, and V. Vargas.  Polyakov’s formulation of $2d$ bosonic string theory.  Publ. Math. Inst. Hautes Études Sci., 130:111–185, 2019.
  * [Kah85] J-P. Kahane.  Sur le chaos multiplicatif.  Ann. Sci. Math. Québec, 9(2):105–150, 1985.
  * [KLM21] R. Konik, M. Lájer, and G. Mussardo.  Approaching the self-dual point of the sinh-Gordon model.  Journal of High Energy Physics, 2021(1):1–85, 2021.
  * [KRV20] A. Kupiainen, R. Rhodes, and V. Vargas.  Integrability of Liouville theory: proof of the DOZZ formula.  Ann. of Math. (2), 191(1):81–166, 2020.
  * [Luk01] S. Lukyanov.  Finite temperature expectation values of local fields in the sinh-Gordon model.  Nuclear Physics B, 612:391–412, 2001.
  * [LZ97] S. Lukyanov and A. Zamolodchikov.  Exact expectation values of local fields in quan- tum sine-Gordon model.  Nuclear Physics B, 493:571–587, 1997.
  * [OS99] H. Osada and H. Spohn.  Gibbs measures relative to Brownian motion.  Annals of probability, pages 1183–1207, 1999.
  * [Pol81] A. M. Polyakov.  Quantum geometry of bosonic strings.  Phys. Lett. B, 103(3):207–210, 1981.
  * [RS78] M. Reed and B. Simon.  Methods of Modern Mathematical Physics, volume IV. Analysis of operators.  Academic Press, 1978.
  * [RV10] R. Robert and V. Vargas.  Gaussian multiplicative chaos revisited.  Annals of Probability, 38(2):605–631, 2010.
  * [RV14] R. Rhodes and V. Vargas.  Gaussian multiplicative chaos and applications: a review.  Probab. Surv., 11:315–392, 2014.
  * [Szn88] A-S. Sznitman.  Brownian Motion, Obstacles and Random Media.  Springer Monographs in Mathematics. Springer-Verlag, 1988.
  * [Tes08] J. Teschner.  On the spectrum of the Sinh-Gordon model in finite volume.  Nuclear Physics B, 799:403–429, 2008.
  * [ZZ96] A. Zamolodchikov and Al. Zamolodchikov.  Conformal bootstrap in Liouville field theory.  Nuclear Phys. B, 477(2):577–605, 1996.


* [79] Zeriahi, Ahmed. Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions. Indiana Univ. Math. J. 50 (2001), no. 1, 671–703.
  * [80] Zhang, Yuguang. Collapsing of Calabi-Yau manifolds and special Lagrangian submanifolds. Univ. Iagel. Acta Math. No. 54 (2017), 53–78.


[type=editor, auid=000,bioid=1, orcid=0000-0002-7665-6562] [1]

1]organization=Space Science and Engineering Division, Southwest Research
Institute, city=San Antonio, state=Texas, country=USA
2]organization=Department of Physics, University of Oxford, city=Oxford,
country=UK 3]organization=LEISA, Observatoire de Paris, city=Paris,
country=France 4]organization=NASA Ames Research Center, city=Moffett Field,
state=California, country=USA

[1]Corresponding author

# Three-dimensional structure of thermal waves in Venus’ mesosphere from
ground-based observations

Rohini S. Giles<EMAIL_ADDRESS>Thomas K. Greathouse  Patrick G. J.
Irwin  Thérèse Encrenaz  Amanda Brecht  [  [  [  [

###### Abstract

High spectral resolution observations of Venus were obtained with the TEXES
instrument at NASA’s Infrared Telescope Facility. These observations focus on
a CO2 absorption feature at 791.4 cm-1 as the shape of this absorption feature
can be used to retrieve the vertical temperature profile in Venus’ mesosphere.
By scan-mapping the planet, we are able to build up three-dimensional
temperature maps of Venus’ atmosphere, covering one Earth-facing hemisphere
and an altitude range of 60–83 km. A temperature map from February 12, 2019
clearly shows the three-dimensional structure of a planetary-scale thermal
wave. This wave pattern appears strongest in the mid-latitudes of Venus, has a
zonal wavenumber of 2–4 and the wave fronts tilt eastward with altitude at an
angle of 8–15 degrees per km. This is consistent with a thermal tide
propagating upwards from Venus’ upper cloud decks. Ground-based observations
provide the opportunity to study Venus’ temperature structure on an ongoing
basis.

###### keywords:

Venus, atmosphere Spectroscopy Infrared observations Atmospheres, structure

Ground-based observations were used to map Venus’ three-dimensional thermal
structure

A temperature offset map from February 2019 shows a planetary-scale wave

The wave pattern is consistent with an upward propagating thermal tide

##  1 Introduction

Venus’ upper cloud layer absorbs more than half of the incident solar
radiation (Gierasch et al., 1997) and this localized energy deposit drives
large-scale gravity waves known as thermal tides (Sánchez-Lavega et al.,
2017). These tides are thought to be sun-synchronous, with an eastward phase
velocity, and propagate both upwards and downwards from the forcing region in
the upper cloud deck where the maximum amount of solar energy is absorbed.
This movement of eastward momentum vertically away from the cloud deck region
leads to the westward acceleration of atmosphere at the cloud level, which may
be an important element in maintaining Venus’ extreme cloud-top superrotation
(Fels and Lindzen, 1974; Newman and Leovy, 1992; Lebonnois et al., 2016;
Kouyama et al., 2019).

Thermal tides were first observed by Apt et al. (1980) using ground-based mid-
infrared images obtained in 1977–1979. These observations showed a strong
solar-fixed component to the thermal emission, with diurnal (zonal wavenumber
of 1), semidiurnal (zonal wavenumber of 2) and higher order components.
Further observations of the thermal tides have been obtained from instruments
on missions to Venus, including the Orbiter Infrared Radiometer (OIR) on
Pioneer Venus (Taylor et al., 1980; Schofield and Taylor, 1983) and the
Fourier Transform Spectrometer (FTS) on Venera 15 (Zasova et al., 2007), both
of which showed clear diurnal and semidiurnal components. Zasova et al. (2007)
found that semidiurnal component dominates in the region immediately above the
cloud top (58–90 km) at low latitudes, while the diurnal component dominates
at higher altitudes at low latitudes and at the cloud-top level in the polar
regions. The most recent observations of the thermal tides are from the
Akatsuki spacecraft, which is currently in orbit around Venus; global
observations from the Longwave Infrared Camera (LIR) show a clear tidal
structure that remains fixed in local time over a two year period (Kouyama et
al., 2019).

Two previous studies using spacecraft observations have shown that the phase
of the thermal tides changes with altitude, which is indicative of vertical
propagation. The OIR instrument on Pioneer Venus consisted of 10 different
channels, each of which probed a different altitude in Venus’ atmosphere.
Schofield and Taylor (1983) compared the semidiurnal wave pattern observed in
five of these channels and found that the phase shifted eastward with
altitude. The LIR instrument on the Akatsuki spacecraft consists of a single
wavelength filter, but Kouyama et al. (2019) and Akiba et al. (2021) used the
fact that observations obtained at different emission angles probe slightly
different altitudes in order to similarly show that there is an eastward shift
with altitude.

We seek to build on these previous studies by using high spectral resolution
ground-based observations of Venus to produce vertically-resolved three-
dimensional temperature maps of the planet’s middle atmosphere. These maps can
be used to study a range of thermal phenomena, including the thermal tide
structure. In this paper, we present initial observations from February 12,
2019, which show a clear vertically-propagating planetary-scale wave. The
observations and radiative transfer modeling are described in Section 2, and
the wave structure is described in Section 3 and discussed in Section 4. This
initial observation forms part of an ongoing long-term monitoring campaign to
study how Venus’ temperature structure varies with time and future studies
will combine observations obtained over several years.

##  2 Observations

###  2.1 TEXES observations

In February 2019, high spectral resolution observations of Venus were carried
out using the Texas Echelon Cross Echelle Spectrograph (TEXES), mounted at
NASA’s 3-m Infrared Telescope Facility (IRTF) in Hawaii. Observations were
made on six days within a single observing run, but in this paper we focus
exclusively on February 12, which had the best observing conditions. TEXES
(Lacy et al., 2002) is a long-slit cross-dispersed grating spectrograph that
covers wavelengths of 4.5–25 µm in the mid-infrared, with a resolving power
between 4,000 and 100,000 depending on the instrument mode and wavelength. On
February 12, 2019, Venus had an apparent diameter of 17.5” and the dawn side
of the planet was visible. The observing geometry is shown in Figure 1, which
shows Venus as it appeared on the sky. The orange star marks the location of
the sub-solar point and the orange line marks the dawn terminator.

Figure 1: The geometry of the Venus observations on February 12, 2019. The
orange star marks the location of the sub-solar point and the orange line
marks the dawn terminator. The regions of the planet shown in white are on the
dayside and the regions of the planet shown in grey are on the nightside. The
TEXES slit dimensions and step direction are shown in the upper right corner.
The spatial resolution in ideal observing conditions is shown by the red
rectangle.

In order to study Venus’ mesospheric temperatures, we used TEXES to observe
the strong 11101$\leftarrow$10002 CO2 Q-branch absorption feature at 791.4
cm-1 (12.6 µm). The observations were centered at $\sim$793 cm-1, with a
spectral bandpass of $\sim$5.5 cm-1. At the center of this CO2 band, telluric
transmission is $\sim$0.5 but the Venus absorption feature is both broader and
deeper and can therefore still be observed. TEXES was used in its highest
spectral resolution mode (R=72,000 at this spectral setting), which allows
both the telluric and Venusian spectral lines to be resolved. Because the CO2
abundance in Venus’ atmosphere is well-known and homogeneous, the shape of the
absorption feature is governed by the vertical temperature profile of the
atmosphere. At the center of the band, the gaseous opacity is highest and so
higher altitudes are probed, whereas the edges of the band and the continuum,
where the absorption is weaker, probe deeper into the atmosphere. Together,
this spectral interval probes the 60–83 km altitude region. An example of the
spectrum obtained by TEXES is shown in Figure 2. The majority of the strong
absorption lines in the spectrum are from CO2 and the TEXES data prior to
telluric correction (black) shows both telluric and Venusian CO2 absorption
with a small offset between the two due to the Doppler shift.

Figure 2: An example of the TEXES observations of Venus in the 793 cm-1 CO2
spectral setting, obtained on February 12, 2019. The TEXES spectrum prior to
telluric correction is shown in black. The regularly spaced small gaps in the
spectrum (where the radiance equals zero) are due to the small coverage gaps
between spectral orders. The grey line shows the best-fit telluric
transmission spectrum obtained from fitting the telluric lines. The blue line
and associated error bars show the spectrum after it has been divided by the
best-fit telluric spectrum.

Three-dimensional spectral image cubes of Venus were obtained by using the
TEXES scan-map observing mode. The 1.4”-wide TEXES entrance slit was aligned
along the celestial north-south direction and it was scanned across the planet
from west to east, in 0.7” increments. The size of the entrance slit and the
scan direction are shown in the upper right corner of Figure 1. Allowing for
several steps of clear sky at the beginning and end of each scan, a total of
38 steps were required to cover the planet in the east-west direction. In the
scan-map observing mode, each step has a defined 2 s integration time. To
achieve a total integration time of 8 s for each position on the planet, we
repeated each scan 4 times. At 793 cm-1, we use an entrance slit that is 7.1”
long; this is the longest slit that can be used without adjacent echelon
orders overlapping on the detector. On February 12, 2019, Venus had an
apparent diameter of 17.5”, so a total of 4 different north-south offset
positions were required to obtain full coverage with some overlap between
adjacent positions. The total integration time required for the observing
sequence was therefore $38\times 2\times 4\times 4$ = 1216 s = 20.3 min.

The red square in Figure 1 shows the spatial resolution in ideal observing
conditions. The slit width of 1.4” is the maximum spatial resolution that can
be obtained in the across-slit (east-west) direction and the step size of 0.7”
was chosen to ensure Nyquist sampling. The maximum spatial resolution in the
along-slit (north-south) direction is defined by the diffraction limit at the
IRTF, which is 0.9” at 793 cm-1. The plate scale along the slit is 0.36”, so
the sampling interval is slightly better than Nyquist sampling. In reality,
the spatial resolution in both directions is also affected by the atmospheric
seeing, which can be high during the daytime hours when Venus is observable,
and telescope drift. In this paper, we focus on February 12, 2019 because the
observing conditions were good and the spatial resolution was therefore close
to ideal.

###  2.2 Data reduction

The individual TEXES Venus scans were first reduced using the data reduction
pipeline described in Lacy et al. (2002). The data were flat-fielded, sky-
subtracted and corrected for instrumental geometric optical distortions. The
observations were wavenumber-calibrated using telluric absorption lines and
were radiometrically calibrated by using the measured radiance of a room
temperature blackbody that was automatically placed in front of instrument
aperture prior to each set of observations.

Typically, the data reduction pipeline also uses the sky observations to
correct for the effect of telluric absorption. However, for the spectral
setting used in this work, the telluric absorption from CO2 is sufficiently
high that we developed an alternative method of correcting for the Earth’s
atmosphere, adapted from the method described by Rudolf et al. (2016) for
reducing VLT/X-Shooter spectra. We used the line-by-line radiative transfer
code LBLRTM (Clough et al., 2005) to model the transmission through the
Earth’s atmosphere. The reference atmosphere inputs for this code were
obtained from the Global Data Assimilation System (GDAS, Rodell et al., 2004)
which produces global maps of the Earth’s temperatures and H2O relative
humidity between sea level and an altitude of 26 km every 3 hours; we selected
the location of the observatory and the times closest to the observations.
Other atmospheric molecular abundances, along with temperatures and H2O
abundances above 26 km, were obtained from the MIPAS equatorial model
atmosphere (Remedios, 2001) and the CO2 profile was scaled to match the mean
monthly CO2 abundance measured at Mauna Loa (Tans and Keeling, 2022). This
atmospheric data was used to generate a telluric transmission spectrum using
the LBLRTM code. We assume that the CO2 abundance is well-defined, but as H2O
varies rapidly, we iteratively adjust the H2O abundance to best fit the strong
H2O absorption line at 796 cm-1, which is located in a spectral region with
minimal CO2 absorption. The Venus observations were then divided by this best-
fit telluric transmission in order to obtain a ‘clean’ spectrum. This process
can be seen in Figure 2. The spectrum prior to telluric correction is shown in
black, and the corrected spectrum is shown in blue. The insert shows a zoomed-
in portion of the spectrum. The black line shows the larger Venusian CO2
absorption lines, which have been Doppler-shifted relative to the adjacent
smaller telluric CO2 absorption lines. The combination of the two gives an
asymmetric appearance to each absorption line. In contrast, the corrected line
(blue) shows a smooth curve between each Venusian line.

The wavenumber- and radiometrically-calibrated data were then geometrically-
calibrated. The plate scale, slit orientation, step size and step direction
were used to calculate the shape of the planet’s limb. This calculated limb
was then manually adjusted to fit the actual observed limb for each scan. This
allowed each pixel to be assigned a latitude and longitude on Venus, and the
latter was converted into a local solar time. The spectrum from each pixel was
Doppler shifted into the rest frame and then projected onto a latitude-local
time grid with a bin size of 5∘ in the latitude direction and 0.5 hrs in the
local time direction. The multiple north-south offset scans were co-added to
produce complete latitudinal coverage. The final re-projected and co-added
data cube from February 12, 2019 has continuous coverage of latitudes between
60∘S and 60∘N and local times between 3:30 AM and 11:00 AM. The number of
0.36” $\times$ 1.4” (plate scale $\times$ slit width) pixels that are co-added
into each bin varies from 1 at the highest latitudes, to 25 at the low
latitudes that comprise the overlap between adjacent north-south offsets.

The error bars on the final co-added spectra were calculated by adding in
quadrature the error on the TEXES spectrum prior to telluric correction and
the error on the best-fit telluric transmission spectrum. The error on the
TEXES spectrum is calculated by taking the difference between the radiances
observed in adjacent wavenumber bins and calculating the local ($\pm$0.1 cm-1)
standard deviation of these differences; we find the region of the spectrum
where the local standard deviation is a minimum (where the spectrum most
closely approximates a straight line) and use this value as the absolute
error. The black line in Figure 2 shows the pre-telluric correction spectrum
and associated small error bars from a single pixel; when multiple spectra are
co-added the error bars would be even smaller. The telluric transmission error
is calculated by assuming a 10% error on both the CO2 and H2O abundances,
running the telluric model with these altered values and then adding the two
errors in quadrature. The blue line in Figure 2 shows the combined error bars
on the spectrum after telluric correction. In the continuum regions, the small
error from the pre-telluric correction spectrum dominates but in regions of
telluric absorption (e.g. the H2O line at 795.9 cm-1), the telluric error
dominates.

###  2.3 Spectral modeling

In order to retrieve the vertical temperature profiles, the IRTF/TEXES spectra
were modeled using NEMESIS (Non-Linear Optimal Estimator for Multivariate
Spectral Analysis, Irwin et al., 2008). NEMESIS is a radiative transfer and
retrieval tool that has been applied to a wide range of planetary atmospheres
and has previously been used to model Venus’ atmosphere using near-infrared
observations from the Visible and Infrared Thermal Imaging Spectrometer
(VIRTIS) on Venus Express (Tsang et al., 2008) and mid- to far-infrared
observations from Pioneer Venus OIR and Venera 15 FTS (Koukouli et al., 2005).
NEMESIS consists of a radiative transfer code that computes the emergent
radiation for a given atmospheric profile and an optimal estimation retrieval
algorithm which iteratively determines the best-fit atmospheric parameters for
an observed spectrum.

For this study, Venus’ atmosphere was divided into 200 levels between the
surface (92 bar) and an altitude of 150 km ($7\times 10^{-11}$ bar), equally
spaced in log(p). The vertical distribution of atmospheric gases and the a
priori vertical temperature profile were obtained from NASA’s Planetary
Science Generator (Villanueva et al., 2018), which in turn bases its profile
primarily on the Venus International Reference Atmosphere (VIRA, Seiff et al.,
1985), with updates from Ehrenreich et al. (2012), Vandaele (2020) and Bierson
and Zhang (2020). The gases which absorb within the 790–796 cm-1 spectral
range are CO2, SO2, H2O, HCl, O3, NO, ClO, OCS and H2S. The line data for
these gases were obtained from the HITRAN molecular database (Gordon et al.,
2022). For CO2, we use the self-broadened line widths and we use a sub-
Lorentzian lineshape (Perrin and Hartmann, 1989). For all other gases, we
assume a Voigt lineshape. SO2, OCS and HCl have CO2-broadened widths available
in HITRAN (Wilzewski et al., 2016) and so these values are used; for the
remaining gases, we use the air-broadened widths. Collision-induced absorption
for CO2 is minimal in the 790–796 cm-1 spectral range (Wordsworth et al.,
2010). NEMESIS makes use of the correlated-k approximation (Lacis and Oinas,
1991) in order to reduce the computation time required. The k-distributions
produced for each gas were convolved with a Gaussian of width 0.011 cm-1 in
order to match the TEXES spectral resolution and instrument function.

Although scattering effects are significant for observations of Venus in the
near-infrared windows (Tsang et al., 2008), Koukouli et al. (2005) showed that
these effects are minimal at wavenumbers less than 900 cm-1 and it is
therefore sufficient to treat Venus’ clouds as pure absorbers in the 790–796
cm-1 spectral range used in this paper. As we are using a very narrow spectral
interval, we also assume that the absorption cross-section is constant within
this interval. We use the same Venusian cloud structure as Lee et al. (2012),
which consists of a cloud with constant extinction below 60 km, above which
the extinction drops off with a constant scale height. We use a fixed scale
height of 3.8 km, which was the best-fit value at low latitudes found by Lee
et al. (2012).

Using the a priori reference atmosphere, the NEMESIS radiative transfer code
calculates a synthetic top-of-atmosphere spectrum. The retrieval segment of
the code then compares this synthetic spectrum to the observed TEXES spectrum
and the atmospheric profiles are iteratively adjusted to fit the observed
spectrum, following an optimal estimation approach (Rodgers, 2000). The code
seeks to minimize a cost function which combines departure from the
measurement and departure from the a priori atmospheric profile, weighted by
their respective errors. In this case, we allow two elements of the
atmospheric profile to vary in the retrievals: the vertical temperature
profile and the cloud opacity. The cloud opacity is allowed to vary via a
single scaling factor, which controls the overall opacity and is therefore
equivalent to the cloud top altitude parameter in Lee et al. (2012). In
contrast, the temperature profile, which is the focus of this study, is
allowed to vary continuously.

The measurement error consists of both the error on the corrected spectrum
described in Section 2.2 and a modeling error due to inaccuracies in the model
(e.g., line data), which we conservatively assign as 5% of the continuum
radiance. We assume that the a priori temperature profile has an error of 25
K, which is the maximum variability observed by Rengel et al. (2008). The
temperatures of each of the 200 atmospheric layers are allowed to vary
independently to achieve the best fit, with a correlation length of 1.5 scale
heights ($\sim$7.5 km) used to prevent unphysically sharp deviations between
adjacent levels. The a priori error on the cloud opacity scaling factor is set
to a large value (100%) in order to allow this secondary parameter to vary
freely.

Figure 3: The normalized contribution functions (temperature functional
derivatives) for the spectral interval used in this study, showing where the
sensitivity peaks as a function of wavenumber. In the center of the CO2
absorption band at 791.4 cm-1, the sensitivity peaks at $\sim$79 km, while in
the continuum regions, the sensitivity peaks deeper in the atmosphere at
$\sim$63 km.

Figure 3 shows the normalized contribution functions (temperature functional
derivatives) for the spectral interval used in this study. The contribution
functions show the extent to which a change in the temperature of a given
atmospheric layer affects the observed top-of-atmosphere radiance; for each
wavenumber, the altitude level where the contribution function is the highest
is the pressure of maximum sensitivity. In the center of the CO2 absorption
band at 791.4 cm-1, the sensitivity peaks at $\sim$79 km, while in the
continuum regions, the sensitivity peaks deeper in the atmosphere at $\sim$63
km. Overall, the combined 790–796 cm-1 spectral region is sensitive to the
60–83 km altitude range, which is within and immediately above the Venus’
cloud layer. Based on the width of the contribution functions, the vertical
resolution in this altitude range is 4–9 km.

Figure 4: An example NEMESIS retrieval of TEXES Venus data, using data
obtained on February 12, 2019. The retrieval was run three times using three
different cloud model scale heights: 1.7 km (orange), 3.8 km (red) and 5.9 km
(blue). (a) The reduced TEXES data (black) along with the best-fit modeled
spectra (red, blue and orange). The residuals show the difference between the
modeled spectra and the data. The error bars on the data combine the
measurement error and the modeling error (see Section 2.3). The small gaps in
the spectrum are due to the gaps between the spectral orders and the larger
gap at 794 cm-1 is due to the removal of a region containing a strong telluric
water line. (b) The best-fit vertical cloud structure obtained for each
retrieval. (c) The a priori vertical temperature profile (black) along with
the retrieved best-fit vertical temperature profiles (red, blue, orange). The
vertical arrows highlight the region of peak sensitivity.

Figure 4 shows an example NEMESIS retrieval for one spatial point on Venus. In
order to understand the effect of our assumed cloud model, which used a fixed
scale height of 3.8 km (the best fit value at low latitudes, Lee et al.,
2012), we also ran retrievals using a fixed scale heights of 1.7 km (the best
fit value in the polar regions, Lee et al., 2012) and 5.9 km. All three
retrievals are presented in Figure 4; the H = 3.8 km model is shown in red,
the H = 1.7 km model is shown in orange and the H = 5.9 km model is shown in
blue.

Figure 4(a) shows the TEXES spectrum in black, along with the best-fit model
spectra in red, blue and orange. All three cloud models are able to produce
good fits to the data. Figure 4(b) shows the retrieved vertical cloud profile
in each case. Even though each model has a different scale height, they all
have a similar cloud top level, defined as the altitude where the optical
depth in the 790-796 cm-1 interval reaches one; the retrieved cloud top levels
are 62.1 km, 63.7 km and 64.0 km for H = 1.7 km, H = 3.8 km and H = 5.9 km
respectively. The retrieved quantity we are primarily interested in, the
vertical temperature profile, is shown in Figure 4(c). The a priori
temperature profile is shown in black, while the three retrieved temperature
profiles are shown in red, blue and orange. As noted above, the TEXES spectra
are primarily sensitive to the 60–83 km altitude range (highlighted by the
vertical arrows), and this is therefore the region where the retrieved
temperature profiles have the smallest error bars (2–6 K). Figure 4(c) shows
that the three retrieved temperature profiles are the same above $\sim$68 km
but differ slightly at greater depths. In the following discussion, we
therefore bear in mind that although the data is sensitive to down to 60 km,
the 60–68 km range is somewhat susceptible to cloud-related degeneracies.

Using our initial assumption of a cloud with a scale height of 3.8 km, we
repeated the retrieval process for each spatial point in the latitude-local
time grid, in order to build up a three-dimensional temperature map of Venus.
These maps were then converted into temperature offset maps by subtracting the
mean zonal temperature at each latitude and altitude level. These three-
dimensional temperature offset maps are discussed in Section 3.

##  3 Results

Figure 5: (a) A vertical slice through the three-dimensional temperature map
of Venus at a latitude of 35–40∘S, obtained on February 12, 2019 (b) The same
data as (a) but presented instead as a temperature offset map. For each
altitude level, the temperature offset is defined as the temperature relative
to the mean zonal temperature. The black lines highlight the gradients of the
banded structure. (c) A single altitude level from (b), showing the error bars
on the retrieved temperature offsets.

Figure 5(a) shows an example vertical ‘slice’ through the three-dimensional
retrieved temperature map of Venus, taken at a latitude of 35–40∘S. Note that
because Venus rotates from east to west, increasing local time corresponds to
further west longitudes. The overall temperature structure is similar to the a
priori vertical temperature profile shown in Figure 4(c) and any superimposed
thermal wave pattern is difficult to see. Figure 5(b) presents the same
retrieved temperatures as Figure 5(a), but in the form of a temperature offset
map. As described in Section 2.3, the temperature offset is defined as the
temperature relative to the zonally-averaged temperature at that latitude and
altitude level, i.e., the mean temperature offset for each row in Figure 5(b)
is zero. Figure 5(c) shows a single altitude level from Figure 5(b).

Figure 6: Vertical wavelength as a function of altitude for the wave structure
shown in Figure 5.

Figure 5(b) shows a clear vertical wave structure above $\sim$65 km: a banded
pattern that tilts morning-ward (eastward) with altitude. The black lines
highlight the three main wavefronts that can be seen; these three lines have
gradients of 2.0, 1.4 and 1.0 km per hour, which equate to 8, 11 and 15
degrees per km. The wavefronts do not appear to be perfectly parallel in local
time - log(pressure) space; instead, the vertical and horizontal wavelengths
appear to vary with altitude. In the horizontal direction, the gap between an
adjacent maximum and minimum is 3–5 hours, equivalent to zonal wavenumbers of
$\sim$2–4, with lower wavenumbers at higher altitudes. The variation in the
vertical wavelength is demonstrated in Figure 6, which shows the local
wavelength (twice the distance between an adjacent maximum and minimum)
plotted against altitude (the halfway point between the adjacent maximum and
minimum). Within the altitude region we are sensitive to, the vertical
wavelength ranges between 5 and 15 km. The peak to peak amplitude of the wave
pattern increases with altitude, from $\sim$3 K at 65 km to $\sim$6 K at 75
km. The actual amplitudes may be slightly higher, particularly at high
altitude, because the local time range does not necessarily include both a
local maximum and a local minimum (e.g. Figure 5(c) does not include a local
minimum).

Figure 7: Temperature offsets as a function of local time and latitude for
three different altitude levels, observed on February 12, 2019. The
temperature offset is defined as the temperature relative to the mean zonal
temperature.

Figure 5 focuses on a single latitude bin but this pattern persists across a
range of latitudes. This can be seen in Figure 7, which shows horizontal
‘slices’ through the three-dimensional temperature offset map of Venus at
three different altitude levels. The vertical wave pattern is strongest in the
mid-latitudes, at 20–55∘. At these latitudes, there is a clear phase
difference between the three plots, which matches up with the vertical slice
at 35–40∘S shown in Figure 5. At 80 km, the temperatures offsets in the mid-
latitudes peak at around 7 AM. At 75 km, they peak slightly later, at around
8:30 AM, and by 70 km the peak is at around 10:30 AM, with another peak
becoming visible in the northern mid-latitudes at 3:30 AM. In contrast, any
phase shift with latitude at the equator is less apparent. Across all three
latitudes, the equatorial temperatures peak at around 8 AM; at 80 km this is
slightly later than the mid-latitudes, while at 75 km this is slightly
earlier.

##  4 Discussion and Conclusions

The wave pattern described in Section 3 has the following characteristics:

  1. 1.

The wave fronts tilt towards the morningside (eastward) with altitude.

  2. 2.

The tilt of the wavefronts is 8–15 degrees per km.

  3. 3.

The wave pattern is visible in the mid-latitudes of the planet, at
$\sim$20–55∘.

  4. 4.

The wave has a zonal wavenumber of 2–4, with lower wavenumbers at higher
altitudes.

  5. 5.

The vertical wavelength increases with altitude, with wavelengths of 5–15 km
in the altitude region we are probing.

  6. 6.

The peak to peak amplitude increases with altitude, from $\sim$3 K at 65 km to
$\sim$6 K at 75 km.

These ground-based observations are consistent with thermal tide results that
have been previously obtained from spacecraft instruments. Schofield and
Taylor (1983) used observations from Pioneer Venus OIR to infer a vertical
temperature profile between 65 and 100 km. Their results show several
similarities with the results obtained from IRTF/TEXES. Below 100 km, they
found that the dominant zonal wavenumber was 2 and the wavenumber-2 amplitude
increased from 2 K at the cloud tops to 4–5 K at 80 K. This is consistent with
our measured amplitudes which range from $\sim$3 K at 65 km to $\sim$6 K at 75
km. They also found that the solar-fixed wavenumber-2 structure tilted
eastward with increasing altitude, at a rate of 6 degrees per km, which is
slightly steeper than the steepest wavefront shown in Figure 5 (8 degrees per
km).

More recently, Kouyama et al. (2019) and Akiba et al. (2021) used the LIR
instrument on the Akatsuki mission to image the global structure of Venus’
thermal tides. Kouyama et al. (2019) found that the semidiurnal (wavenumber of
2) component dominates at latitudes below 40∘, while the diurnal (wavenumber
of 1) component dominates at high latitudes. Because the TEXES observations
are restricted to the Earth-facing hemisphere and therefore have limited local
time coverage, it is difficult to accurately measure the zonal wavenumber;
however the gaps between adjacent maxima and minima suggest a dominant
wavenumber of $\geq$2 in the mid-latitudes ($\sim$20–55∘). At an altitude of
approximately 70$\pm$10 km, the LIR observations show a local peak at
$\sim$8–9 AM, which is similar to the results shown in Figures 5 and 7. Unlike
Pioneer Venus OIR, Akatsuki LIR consists of a single filter, which limits the
vertical information. However, by comparing observations obtained at multiple
emission angles, Akiba et al. (2021) were able to reconstruct vertical
temperature profiles at altitudes of $\sim$66–71 km and produce two-
dimensional slices similar to Figure 5(b). They found that the average tilt of
the wave pattern across the 66-71 km range is 9.4 degrees per km, but that
this decreases to 6.7 degrees per km when only considering the top half of
this range. This is consistent with our TEXES observations, which shows tilts
ranging from 15 degrees per km at $\sim$66 km to 8 degrees per km at $\sim$75
km.

This eastward-tilted wave structure that has been observed by Pioneer Venus
OIR, Akatsuki LIR, and now IRTF/TEXES is consistent with upward motion of an
eastward-propagating wave. This is what is expected from Venus’ thermal tides;
thermal tides are thought to be solar-synchronous, so move eastward along with
the sub-solar point and they propagate upwards and downwards from the forcing
region in the upper cloud deck (Sánchez-Lavega et al., 2017). As our
observations are sensitive to the region immediately above the cloud tops,
upward motion is consistent with expectations. Similar structures have also
been reproduced in numerical models of Venus’ atmosphere. Takagi et al. (2018)
used a Venus General Circulation Model to investigate the three-dimensional
structure of thermal tides and found that wave fronts tilt in opposite
directions above and below the cloud layer, indicating upward propagation
above 70 km and downward propagation below 60 km. Our observations suggest
that the eastward tilt, and therefore upwards propagation, persist down to
$\sim$65 km.

The TEXES observations show a vertical wavelength that increases from 5 km at
an altitude of 65 km to 15 km at an altitude of 75 km. These values are
slightly smaller than those obtained by Akiba et al. (2021) using Akatsuki LIR
data, who measured a vertical wavelength of $\sim$20 km for both the diurnal
and semidiurnal components of the thermal tide. They are broadly consistent
with Baker and Leovy (1987) who used a numerical model of Venus’ atmosphere to
investigate the semidiurnal tide and found that the vertical wavelength at
60–75 km could vary from 13–20 km depending on the specific model parameters
used. The apparent variation in the vertical wavelength with altitude could be
due to the presence of multiple wave components with different wavelengths, or
it could be due to the fact that the wavelength depends strongly on the zonal
wind and the static stability of the atmosphere, both of which vary
significantly with altitude (Takagi et al., 2018).

The eastward-titled wave structure with wavenumber $\geq$2 that is observed in
the TEXES data is strongest at $\sim$20–55∘; at the equator, there is no clear
phase shift with altitude. This is somewhat surprising as numerical models of
the thermal tides have suggested that the semidiurnal component of the thermal
tides should be strongest at latitudes below 30∘ (Takagi et al., 2018).
Schofield and Taylor (1983) and Akiba et al. (2021) did observe the tilted
wave structure at low latitudes, but in both cases, they averaged data taken
over a relatively large latitudinal zone of 0–30∘N. It is possible that this
effect, which appears to be confined to within 15∘ of the equator, can only be
seen in higher spatial resolution data. Future studies will explore whether
the change in wave pattern at the equator is a consistently observed
phenomenon.

This paper presents the first three-dimensional global temperature maps of
Venus obtained from ground-based observations. Ground-based observations
provide a complementary view to spacecraft instruments and have several
distinct advantages. The high spectral resolution that can be achieved with
IRTF/TEXES allows the temperature structure to be vertically resolved, while
the fact that we are able to observe an entire hemisphere of the planet gives
us the ability to obtain range of spatial and local time coverage within a
short time period. Ground-based observations also provide significant
flexibility, allowing us to continue to make similar observations several
times per year over a multi-year period. This allows complete local time
coverage to be built up and temporal trends to be observed. Future studies
will take advantage of observation dates where Venus’ angular diameter is
larger in order to obtain better spatial resolution and will combine multiple
IRTF/TEXES datasets obtained on different dates in order to study the short-
and long-term variability in the three-dimensional thermal tide structure.

## Acknowledgements

The authors were Visiting Astronomers at the Infrared Telescope Facility,
which is operated by the University of Hawaii under contract 80HQTR19D0030
with the National Aeronautics and Space Administration. This work was funded
by NASA Solar System Observations Grant 80NSSC20K0669.

## References

  * Akiba et al. (2021) Akiba, M., Taguchi, M., Fukuhara, T., Imamura, T., Kouyama, T., Sato, T.M., 2021\.  Thermal tides in the upper cloud layer of Venus as deduced from the emission angle dependence of the brightness temperature by Akatsuki/LIR.  Journal of Geophysical Research: Planets 126, e2020JE006808.
  * Apt et al. (1980) Apt, J., Brown, R.A., Goody, R.M., 1980.  The character of the thermal emission from Venus.  Journal of Geophysical Research: Space Physics 85, 7934–7940.
  * Baker and Leovy (1987) Baker, N.L., Leovy, C.B., 1987\.  Zonal winds near Venus’ cloud top level: A model study of the interaction between the zonal mean circulation and the semidiurnal tideß.  Icarus 69, 202–220.
  * Bierson and Zhang (2020) Bierson, C., Zhang, X., 2020\.  Chemical cycling in the venusian atmosphere: a full photochemical model from the surface to 110 km.  Journal of Geophysical Research: Planets 125, e2019JE006159.
  * Clough et al. (2005) Clough, S.A., Shephard, M.W., Mlawer, E.J., Delamere, J.S., Iacono, M.J., Cady-Pereira, K., Boukabara, S., Brown, P.D., 2005\.  Atmospheric radiative transfer modeling: a summary of the AER codes.  Journal of Quantitative Spectroscopy and Radiative Transfer 91, 233–244.
  * Ehrenreich et al. (2012) Ehrenreich, D., Vidal-Madjar, A., Widemann, T., Gronoff, G., Tanga, P., Barthélemy, M., Lilensten, J., Des Etangs, A.L., Arnold, L., 2012.  Transmission spectrum of Venus as a transiting exoplanet.  Astronomy & Astrophysics 537, L2.
  * Fels and Lindzen (1974) Fels, S.B., Lindzen, R.S., 1974\.  The interaction of thermally excited gravity waves with mean flows.  Geophysical and Astrophysical Fluid Dynamics 6, 149–191.
  * Gierasch et al. (1997) Gierasch, P.J., Goody, R.M., Young, R.E., Crisp, D., Edwards, C., Kahn, R., Rider, D., Del Genio, A., Greeley, R., Hou, A., et al., 1997\.  The general circulation of the Venus atmosphere: An assessment, in: Venus II – Geology, Geophysics, Atmosphere, and Solar Wind Environment. Tucson: Univ. Arizona Press, pp. 459–500.
  * Gordon et al. (2022) Gordon, I.E., Rothman, L.S., Hargreaves, R.J., Hashemi, R., Karlovets, E., Skinner, F., Conway, E., Hill, C., Kochanov, R.V., Tan, Y., et al., 2022\.  The HITRAN2020 molecular spectroscopic database.  Journal of Quantitative Spectroscopy and Radiative Transfer 277, 107949.
  * Irwin et al. (2008) Irwin, P.G.J., Teanby, N.A., de Kok, R., Fletcher, L.N., Howett, C.J.A., Tsang, C.C.C., Wilson, C.F., Calcutt, S.B., Nixon, C.A., Parrish, P.D., 2008\.  The NEMESIS planetary atmosphere radiative transfer and retrieval tool.  Journal of Quantitative Spectroscopy and Radiative Transfer 109, 1136–1150.
  * Koukouli et al. (2005) Koukouli, M., Irwin, P., Taylor, F., 2005.  Water vapor abundance in Venus’ middle atmosphere from Pioneer Venus OIR and Venera 15 FTS measurements.  Icarus 173, 84–99.
  * Kouyama et al. (2019) Kouyama, T., Taguchi, M., Fukuhara, T., Imamura, T., Horinouchi, T., Sato, T., Murakami, S., Hashimoto, G., Lee, Y., Futaguchi, M., et al., 2019\.  Global structure of thermal tides in the upper cloud layer of Venus revealed by LIR on board Akatsuki.  Geophysical Research Letters 46, 9457–9465.
  * Lacis and Oinas (1991) Lacis, A.A., Oinas, V., 1991\.  A description of the correlated k distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres.  Journal of Geophysical Research: Atmospheres 96, 9027–9063.
  * Lacy et al. (2002) Lacy, J.H., Richter, M.J., Greathouse, T.K., Jaffe, D.T., Zhu, Q., 2002.  TEXES: A sensitive high-resolution grating spectrograph for the mid-infrared.  Publications of the Astronomical Society of the Pacific 114, 153–168.
  * Lebonnois et al. (2016) Lebonnois, S., Sugimoto, N., Gilli, G., 2016.  Wave analysis in the atmosphere of Venus below 100-km altitude, simulated by the LMD Venus GCM.  Icarus 278, 38–51.
  * Lee et al. (2012) Lee, Y.J., Titov, D.V., Tellmann, S., Piccialli, A., Ignatiev, N., Pätzold, M., Häusler, B., Piccioni, G., Drossart, P., 2012.  Vertical structure of the Venus cloud top from the VeRa and VIRTIS observations onboard Venus Express.  Icarus 217, 599–609.
  * Newman and Leovy (1992) Newman, M., Leovy, C., 1992\.  Maintenance of strong rotational winds in Venus’ middle atmosphere by thermal tides.  Science 257, 647–650.
  * Perrin and Hartmann (1989) Perrin, M.Y., Hartmann, J.M., 1989\.  Temperature-dependent measurements and modeling of absorption by CO2-N2 mixtures in the far line-wings of the 4.3 $\mu$m CO2 band.  Journal of Quantitative Spectroscopy and Radiative Transfer 42, 311–317.
  * Remedios (2001) Remedios, J., 2001.  MIPAS Model Atmospheres.  URL: http://eodg.atm.ox.ac.uk/RFM/atm/.
  * Rengel et al. (2008) Rengel, M., Hartogh, P., Jarchow, C., 2008.  Mesospheric vertical thermal structure and winds on Venus from HHSMT CO spectral-line observations.  Planetary and Space Science 56, 1368–1384.
  * Rodell et al. (2004) Rodell, M., Houser, P.R., Jambor, U.E.A., Gottschalck, J., Mitchell, K., Meng, C.J., Arsenault, K., Cosgrove, B., Radakovich, J., Bosilovich, M., et al., 2004\.  The global land data assimilation system.  Bulletin of the American Meteorological Society 85, 381–394.
  * Rodgers (2000) Rodgers, C.D., 2000.  Inverse Methods for Atmospheric Sounding: Theory and Practice.  World Scientific.
  * Rudolf et al. (2016) Rudolf, N., Günther, H.M., Schneider, P.C., Schmitt, J.H.M.M., 2016\.  Modelling telluric line spectra in the optical and infrared with an application to VLT/X-Shooter spectra.  Astronomy & Astrophysics 585, A113.
  * Sánchez-Lavega et al. (2017) Sánchez-Lavega, A., Lebonnois, S., Imamura, T., Read, P., Luz, D., 2017.  The atmospheric dynamics of Venus.  Space Science Reviews 212, 1541–1616.
  * Schofield and Taylor (1983) Schofield, J.T., Taylor, F.W., 1983\.  Measurements of the mean, solar-fixed temperature and cloud structure of the middle atmosphere of Venus.  Quarterly Journal of the Royal Meteorological Society 109, 57–80.
  * Seiff et al. (1985) Seiff, A., Schofield, J.T., Kliore, A.J., Taylor, F.W., Limaye, S.S., Revercomb, H.E., Sromovsky, L.A., Kerzhanovich, V.V., Moroz, V.I., Marov, M.Y., 1985\.  Models of the structure of the atmosphere of Venus from the surface to 100 kilometers altitude.  Advances in Space Research 5, 3–58.
  * Takagi et al. (2018) Takagi, M., Sugimoto, N., Ando, H., Matsuda, Y., 2018\.  Three-dimensional structures of thermal tides simulated by a Venus GCM.  Journal of Geophysical Research: Planets 123, 335–352.
  * Tans and Keeling (2022) Tans, P., Keeling, R., 2022\.  Trends in Atmospheric Carbon Dioxide.  URL: gml.noaa.gov/ccgg/trends/.
  * Taylor et al. (1980) Taylor, F.W., Beer, R., Chahine, M.T., Diner, D.J., Elson, L.S., Haskins, R.D., McCleese, D.J., Martonchik, J.V., Reichley, P.E., Bradley, S.P., et al., 1980\.  Structure and meteorology of the middle atmosphere of Venus: Infrared remote sensing from the Pioneer Orbiter.  Journal of Geophysical Research: Space Physics 85, 7963–8006.
  * Tsang et al. (2008) Tsang, C.C.C., Irwin, P.G.J., Taylor, F.W., Wilson, C.F., 2008\.  A correlated-k model of radiative transfer in the near-infrared windows of Venus.  Journal of Quantitative Spectroscopy and Radiative Transfer 109, 1118–1135.
  * Vandaele (2020) Vandaele, A.C., 2020.  Composition and chemistry of the neutral atmosphere of Venus, in: Oxford Research Encyclopedia of Planetary Science.
  * Villanueva et al. (2018) Villanueva, G.L., Smith, M.D., Protopapa, S., Faggi, S., Mandell, A.M., 2018.  Planetary Spectrum Generator: An accurate online radiative transfer suite for atmospheres, comets, small bodies and exoplanets.  Journal of Quantitative Spectroscopy and Radiative Transfer 217, 86–104.
  * Wilzewski et al. (2016) Wilzewski, J.S., Gordon, I.E., Kochanov, R.V., Hill, C., Rothman, L.S., 2016.  H2, He, and CO2 line-broadening coefficients, pressure shifts and temperature-dependence exponents for the HITRAN database. Part 1: SO2, NH3, HF, HCl, OCS and C22H2.  Journal of Quantitative Spectroscopy and Radiative Transfer 168, 193–206.
  * Wordsworth et al. (2010) Wordsworth, R., Forget, F., Eymet, V., 2010.  Infrared collision-induced and far-line absorption in dense CO2 atmospheres.  Icarus 210, 992–997.
  * Zasova et al. (2007) Zasova, L.V., Ignatiev, N., Khatuntsev, I., Linkin, V., 2007\.  Structure of the Venus atmosphere.  Planetary and Space Science 55, 1712–1728.


# Limiting flux in quantum thermodynamics

Domingos S. P. Salazar  Unidade de Educação a Distância e Tecnologia,
Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil

###### Abstract

In quantum systems, entropy production is typically defined as the quantum
relative entropy between two states. This definition provides an upper bound
for any flux (of particles, energy, entropy, etc.) of bounded observables,
which proves especially useful near equilibrium. However, this bound tends to
be irrelevant in general nonequilibrium situations. We propose a new upper
bound for such fluxes in terms of quantum relative entropy, applicable even
far from equilibrium and in the strong coupling regime. Additionally, we
compare this bound with Monte Carlo simulations of random qubits with
coherence, as well as with a model of two interacting nuclear spins.

_Introduction -_ The concept of entropy production is pivotal in understanding
nonequilibrium phenomena Landi and Paternostro (2021); Campisi  _et al._
(2011); Seifert (2012); Batalhão  _et al._ (2015); Campisi  _et al._ (2011);
Esposito  _et al._ (2009); Ciliberto  _et al._ (2013); Batalhão  _et al._
(2014); Barato and Seifert (2015); Gingrich  _et al._ (2016); Pietzonka and
Seifert (2017); Vo  _et al._ (2020); Liu  _et al._ (2020); Horowitz and
Gingrich (2020); Koyuk and Seifert (2019); Dechant  _et al._ (2023); Mori  _et
al._ (2023); Zhang and Garcia-Millan (2023). It is typically associated with
various types of fluxes (energy, particles, entropy, etc.), especially near
equilibrium where Onsager’s relations are significant Onsager (1931). In
quantum systems, flux is often expressed using an observable $\hat{\theta}$ as
$\phi:=\text{tr}(\hat{\theta}(\rho-\sigma))$ between states $\sigma$ and
$\rho$. For bounded operators, a frequent scenario in finite size reservoirs,
the absolute flux is limited by a capacity $\phi_{L}$,

$|\phi|\leq\phi_{L}:=\theta_{max}-\theta_{min},$ (1)

where $\theta_{max}:=\sup_{n}\theta_{n}$, $\theta_{min}:=\inf_{n}\theta_{n}$
and $\\{\theta_{n}\\}$ are the eigenvalues of the bounded operator
$\hat{\theta}$, and it follows that $(\phi/\phi_{L})^{2}\leq 1$. Intuitively,
the maximum flux $\phi_{L}$ corresponds to an exchange between two pure states
$|\theta_{min}\rangle$ and $|\theta_{max}\rangle$. However, in practical
situations, this capacity is seldom reached. In fact, if $\rho$ and $\sigma$
are known, a more effective upper bound for $|\phi|/\phi_{L}$ can be derived
in terms of the quantum relative entropy
$S(\rho||\sigma):=\text{tr}(\rho(\ln\rho-\ln\sigma))$, linked to quantum
entropy production Landi and Paternostro (2021). Controlling such absolute
flux (1) through entropy production in quantum systems could serve as a
fundamental tool, similar to the Thermodynamic Uncertainty Relations (TURs)
Barato and Seifert (2015); Gingrich  _et al._ (2016); Polettini  _et al._
(2016); Pietzonka and Seifert (2017); Hasegawa and Van Vu (2019a, b); Vo  _et
al._ (2020); Vu and Hasegawa (2020); Timpanaro  _et al._ (2019); Liu  _et al._
(2020); Horowitz and Gingrich (2020); Potts and Samuelsson (2019); Proesmans
and Horowitz (2019); Salazar (2022a); Dieball and Godec (2023); Van Vu and
Saito (2023); Yoshimura  _et al._ (2023).

Interestingly, thermodynamics limits the flux by means of the entropy
production in the same way that quantum mutual information is used as a bound
for connected correlators Wolf  _et al._ (2008); Kudler-Flam (2023). To see
this, first we define the Schatten $k-$norm
$||x||_{k}:=\text{tr}[(\sqrt{x^{\dagger}x})^{k}]^{1/k}$, which results in the
trace norm for the particular case $k=1$. Then, we have
$|\phi|=|\text{tr}((\hat{\theta}-\lambda
I)(\rho-\sigma))|\leq||\hat{\theta}(\rho-\sigma)||_{1}$, where
$\lambda:=(\theta_{max}+\theta_{min})/2$ and $I$ is the identity operator. Now
using Hölder’s inequality, $||(\rho-\sigma)(\hat{\theta}-\lambda
I)||_{1}\leq||\rho-\sigma||_{1}||\hat{\theta}-\lambda I||_{\infty}$, it
finally results in

$\Big{(}\frac{\phi}{\phi_{L}}\Big{)}^{2}\leq\frac{||\rho-\sigma||_{1}^{2}}{4}\leq\frac{S(\rho||\sigma)}{2},$
(2)

after using $||\hat{\theta}-\lambda
I||_{\infty}^{2}=\sup_{n}|\theta_{n}-\lambda|^{2}=\phi_{L}^{2}/4$ and the
quantum Pinsker’s inequality,
$S(\rho||\sigma)\geq(1/2)||\rho-\sigma||_{1}^{2}$.

The connection of relation (2) with thermodynamics goes as follows. When
system (S) and the environment (E) are initially independent and prepared in
arbitrary states, $\rho_{SE}=\rho_{S}\otimes\rho_{E}$, then a unitary dynamics
$\mathcal{U}$ potentially builds classic and quantum correlation between them,
resulting in the final state
$\rho_{SE}^{\prime}=\mathcal{U}(\rho_{S}\otimes\rho_{E})\mathcal{U}^{\dagger}$.
In this notation, the quantum entropy production Esposito  _et al._ (2010);
Manzano  _et al._ (2018); Jarzynski (2011); Deffner and Lutz (2011); Goold
_et al._ (2015); Elouard and Lombard Latune (2023) is defined as
$\Sigma:=S(\rho_{SE}^{\prime}||\rho_{S}^{\prime}\otimes\rho_{E})$, where
$\rho_{S}^{\prime}:=\text{tr}_{E}(\rho_{SE}^{\prime})$ is the reduced state of
the system. In this case, we have from (2) with $\rho=\rho_{SE}^{\prime}$ and
$\sigma=\rho_{S}^{\prime}\otimes\rho_{E}$,

$\Big{(}\frac{\phi}{\phi_{L}}\Big{)}^{2}\leq\frac{\Sigma}{2}.$ (3)

Expression (3) proves particularly useful for small values of $\Sigma$,
resembling Onsager’s relation near equilibrium. Onsager’s relation, based on
linear response theory, is characterized by a quadratic form connecting fluxes
and entropy production de Groot and Mazur (1961). However, this quadratic
expression $\Sigma\propto\phi^{2}$ becomes invalid far from equilibrium, where
the relationship between entropy production and fluxes is only applicable to
specific systems Salazar and Landi (2020). Notably, the bound (3) also loses
its relevance for $\Sigma\geq 2$, a scenario indicative of being far from
equilibrium. In such a region, Pinsker’s inequality becomes inapplicable, and
a more fitting bound for (3) would be the straightforward
$(\phi/\phi_{L})^{2}\leq 1\leq\Sigma/2$, as derived from (1).

In this letter, we explore whether entropy production can still be a tool for
controlling fluxes in conditions far from equilibrium. We affirmatively answer
this with our main result,

$\Big{(}\frac{\phi}{\phi_{L}}\Big{)}^{2}\leq
B\Big{(}\frac{S(\rho||\sigma)+S(\sigma||\rho)}{2}\Big{)}\leq 1,$ (4)

where $B(x):=(x/g(x))^{2}$ and $g(x)$ is the inverse of $h(x)=x\tanh(x/2)$ for
$x>0$. Expression (4) means that, even arbitrarily far from equilibrium, the
quantum relative entropy can be used as bound for limiting fluxes. In quantum
thermodynamics, setting again $\rho=\rho_{SE}^{\prime}$ and
$\sigma=\rho_{S}^{\prime}\otimes\rho_{E}$, we get $\Sigma=S(\rho||\sigma)$ and
we define the dual
$\Sigma^{*}:=S(\sigma||\rho)=S(\mathcal{U^{\dagger}}\sigma\mathcal{U}||\rho_{S}\otimes\rho_{E})$.
Note that $\Sigma^{*}$ is uniquely defined from the same states used in
$\Sigma$ Salazar (2023). Moreover, relation (4) is specially useful for
systems arbitrarily far from equilibrium, where (3) fails to bring any
information (for $\Sigma\geq 2$) and (4) always produces a nontrivial bound.
In this case, we also obtain a relationship between the entropy production and
any flux from (4),

$\frac{\Sigma+\Sigma^{*}}{4}\geq\frac{\phi}{\phi_{L}}\tanh^{-1}\big{(}\frac{\phi}{\phi_{L}}\big{)}\geq\big{(}\frac{\phi}{\phi_{L}}\big{)}^{2},$
(5)

which is our second main result, valid for bounded operators $\hat{\theta}$,
including for the entropy flux (when $\hat{\theta}=\log\rho_{E}$).

The letter is organized as follows: First, we establish the formalism and
prove the main results (4) and (5). We also test (4) in a simulation involving
two interacting qubits, with one acting as the system and the other as the
environment. As an example, we demonstrate how the energy flux is controlled
by the quantum relative entropy in a model of two interacting nuclear
spins-1/2.

_Formalism -_ We prove our main result (4) using a recently proposed quantum
thermodynamic uncertainty relation (qTUR) Salazar (2023). For any Hermitian
operator $\hat{\omega}$ and states $\rho$, $\sigma$ (with
$\langle\hat{\omega}\rangle_{\rho}\neq\langle\hat{\omega}\rangle_{\sigma}$),
the qTUR states that

$\frac{\langle\hat{\omega}^{2}\rangle_{\rho}-\langle\hat{\omega}\rangle_{\rho}^{2}+\langle\hat{\omega}^{2}\rangle_{\sigma}-\langle\hat{\omega}\rangle_{\sigma}^{2}}{(1/2)(\langle\hat{\omega}\rangle_{\rho}-\langle\hat{\omega}\rangle_{\sigma})^{2}}\geq
f(\frac{S(\rho||\sigma)+S(\sigma||\rho)}{2}),$ (6)

where $f(x)=1/\sinh(g(x)/2)^{2}$ and the notation
$\langle\hat{\omega}\rangle_{\rho}:=\text{tr}(\hat{\omega}\rho)$. Now we
consider a specific operator $\hat{\omega}$ and calculate the averages
$\langle\hat{\omega}\rangle_{\rho,\sigma}$,
$\langle\hat{\omega}^{2}\rangle_{\rho,\sigma}$ in the lhs of (6). The idea is
to select $\hat{\omega}$ such that the trace norm $||\rho-\sigma||_{1}$
appears naturally.

First, note that $\rho-\sigma$ is Hermitian and it has a decomposition with
real eigenvalues, $\rho-\sigma=\sum_{k}w_{k}|w_{k}\rangle\langle w_{k}|$. Now
we define $\hat{\omega}$ as

$\hat{\omega}:=\sum_{k,w_{k}\neq
0}\operatorname{sign}(w_{k})|w_{k}\rangle\langle w_{k}|,$ (7)

where $\operatorname{sign}(x)=1~{}(-1)$, for $x>0~{}(x<0)$. Then, we write the
trace-norm $||\rho-\sigma||_{1}$ in terms of $\hat{\omega}$ from (7),

$\langle\hat{\omega}\rangle_{\rho}-\langle\hat{\omega}\rangle_{\sigma}=\text{tr}[\hat{\omega}(\rho-\sigma)]=\sum_{k}|w_{k}|=||\rho-\sigma||_{1},$
(8)

which meets the condition for the qTUR (6),
$\langle\hat{\omega}\rangle_{\sigma}\neq\langle\hat{\omega}\rangle_{\sigma}$,
for $\rho\neq\sigma$. Then, we observe that

$\hat{\omega}^{2}=\sum_{k,w_{k}\neq
0}\operatorname{sign}(w_{k})^{2}|w_{k}\rangle\langle w_{k}|=I-\hat{\epsilon},$
(9)

where $I$ is the identity operator and
$\hat{\epsilon}:=\sum_{k,w_{k}=0}|w_{k}\rangle\langle w_{k}|$ , with averages

$\langle\hat{\epsilon}\rangle_{\rho}=\langle\hat{\epsilon}\rangle_{\sigma}:=\epsilon,$
(10)

obtained from
$\langle\hat{\epsilon}\rangle_{\rho}-\langle\hat{\epsilon}\rangle_{\sigma}=\text{tr}[\hat{\epsilon}(\rho-\sigma)]=\sum_{k,w_{k}=0}w_{k}=0$.
We also have $0\leq\epsilon\leq 1$, because $\rho,\sigma$ are positive
definite and $\text{tr}(\rho)=\text{tr}(\sigma)=1$. From (9) and (10), we get

$\langle\hat{\omega}^{2}\rangle_{\rho}=\langle\hat{\omega}^{2}\rangle_{\sigma}=1-\epsilon.$
(11)

Using the averages (8) and (11) in the qTUR (6), we obtain

$\frac{2-2\epsilon-\langle\hat{\omega}\rangle_{\rho}^{2}-\langle\hat{\omega}\rangle_{\sigma}^{2}}{(1/2)||\rho-\sigma||_{1}^{2}}\geq
f(\tilde{S}(\rho,\sigma)),$ (12)

with the notation
$\tilde{S}(\rho||\sigma):=(S(\rho||\sigma)+S(\sigma||\rho))/2$. As third and
final ingredient, check that

$(1/2)||\rho-\sigma||_{1}^{2}\leq\langle\hat{\omega}\rangle_{\rho}^{2}+\langle\hat{\omega}\rangle_{\sigma}^{2},$
(13)

directly from (8) and the expression
$(1/2)(x-y)^{2}\leq(1/2)(x-y)^{2}+(1/2)(x+y)^{2}=x^{2}+y^{2}$, for
$x=\langle\hat{\omega}\rangle_{\rho}$,
$y=\langle\hat{\omega}\rangle_{\sigma}$. In this case, we obtain from (13),

$\frac{2-2\epsilon-(1/2)||\rho-\sigma||_{1}^{2}}{(1/2)||\rho-\sigma||_{1}^{2}}\geq\frac{2-2\epsilon-\langle\hat{\omega}\rangle_{\rho}^{2}-\langle\hat{\omega}\rangle_{\sigma}^{2}}{(1/2)||\rho-\sigma||_{1}^{2}}.$
(14)

Combining (14) and (12) results in

$\frac{2-2\epsilon-(1/2)||\rho-\sigma||_{1}^{2}}{(1/2)||\rho-\sigma||_{1}^{2}}\geq
f(\tilde{S}(\rho,\sigma)).$ (15)

Expression (15) is easily inverted to

$\frac{||\rho-\sigma||_{1}^{2}}{4}\leq\frac{(1-\epsilon)}{1+f(\tilde{S}(\rho,\sigma))}=(1-\epsilon)B\big{(}\tilde{S}(\rho,\sigma)\big{)},$
(16)

with $B(x):=(x/g(x))^{2}$. Finally, using the first inequality of (2), we
obtain $(\phi/\phi_{L})^{2}\leq(1-\epsilon)B(\tilde{S}(\rho,\sigma))\leq
B(\tilde{S}(\rho,\sigma))$, which is our main result (4). Check that
$B(x)=(x/g(x))^{2}\leq 1$, for $x>0$, because $x\tanh(x/2)=h(x)\leq
x=h(g(x))$, which makes $x\leq g(x)$, because $h$ is a increasing function.

_Discussion-_ We proved a stronger result in terms of $\epsilon$ in (16) that
reproduces the main result (4) in case $\epsilon=0$. Although we used the qTUR
(6) in the proof, a similar reasoning was used in recent classic results
Salazar (2022b); Vo  _et al._ (2022); Dechant (2022), connecting the total
variation distance (TV) and the symmetric Kullback-Leibler (KL) divergence.
Actually, if $[\rho,\sigma]=0$, relation (16) reduces to a upper bound for the
TV in terms of the symmetric KL divergence. In this sense, our quantum result
(4) generalizes previous classic results in information theory and stochastic
thermodynamics.

As in the classic case, the bound in (16) is saturated for a specific minimal
two-level system. Consider $\rho=[e^{a/2}|1\rangle\langle
1|+e^{-a/2}|0\rangle\langle 0|]/(2\cosh(a/2))$,
$\sigma=[e^{-a/2}|1\rangle\langle 1|+e^{a/2}|0\rangle\langle
0|]/(2\cosh(a/2))$. In this case, one has
$||\rho-\sigma||_{1}/2=\tilde{S}:=[S(\rho||\sigma)+S(\sigma||\rho)]/2=\tanh(|a|/2)$
and $\tilde{S}^{2}/g(\tilde{S})^{2}=\tanh(a/2)^{2}$ and $\epsilon=0$.
Therefore,
$||\rho-\sigma||_{1}^{2}/4=\tilde{S}^{2}/g(\tilde{S})^{2}=B(\tilde{S})$
saturates (16).

Before moving to the applications, it is worthy to mention that the choice
$\lambda=|\theta_{max}+\theta_{min}|/2$ in the derivation of (2) was selected
for a reason. If we introduce a shift in the operator given by
$\hat{\theta}\rightarrow\hat{\theta}-\lambda I$, where $I$ is the identity and
$\lambda\in\mathbb{R}$, we note that $\phi=\text{tr}((\hat{\theta}-\lambda
I)(\rho-\sigma))=\text{tr}(\hat{\theta}(\rho-\sigma))$. It means that the flux
is invariant under the shift. In this case, one could rewrite (4) for any
$\lambda$ and we could search the optimal value $\lambda^{*}$ for a given
observable $\hat{\theta}$ so that the lhs of (2) is tighter. In other words,
we want to minimize $||\hat{\theta}-\lambda I||_{\infty}$. Formally, we obtain

$\inf_{\lambda\in\mathbb{R}}||\hat{\theta}-\lambda
I||_{\infty}=\frac{1}{2}(\theta_{max}-\theta_{min})=\frac{\phi_{L}}{2}.$ (17)

The proof of (17) follows from $||\hat{\theta}-\lambda
I||_{\infty}=\sup_{n}|\theta_{n}-\lambda|=\max(|\theta_{max}-\lambda|;|\theta_{min}-\lambda|)$,
then using
$\inf_{\lambda}\max(|\theta_{max}-\lambda|;|\theta_{min}-\lambda|)=(\theta_{max}-\theta_{min})/2$
and the optimal value $\lambda^{*}=(\theta_{max}+\theta_{min})/2$. In this
case, we conclude that the bound (2) is actually the tightest for any shift,

$\frac{\phi^{2}}{4||\hat{\theta}-\lambda
I||_{\infty}^{2}}\leq\big{(}\frac{\phi}{\phi_{L}}\big{)}^{2}\leq\frac{||\rho-\sigma||_{1}^{2}}{4},$
(18)

for any $\lambda\in\mathbb{R}$, which makes (3) and (4) particularly useful
because they are now invariant under shifts of the operator $\hat{\theta}$.

_Simulations -_ We now test numerically our main result (4) for a very simple
system: two random qubits $\rho$, $\sigma$ including quantum coherence. For
each run, we draw random states $\rho,\sigma$ and a random operator
$\hat{\theta}$. We compute the flux
$\phi=\text{tr}(\hat{\theta}(\rho-\sigma))$ and $\phi_{L}$ using (1).

Figure 1: (Color online) Monte Carlo simulation of the the flux $\phi$ for
random observables $\hat{\theta}$ and states $\rho$, $\sigma$ as a function of
the symmetric quantum relative entropy
$\tilde{S}(\rho,\sigma)=[S(\rho||\sigma)+S(\sigma||\rho)]/2$, for $n=10^{4}$
draws of ($\hat{\theta},\rho,\sigma$). The dashed line represents the bound
$(\phi/\phi_{L})^{2}\leq\tilde{S}(\rho,\sigma)/2$ which works particularly
well near equilibrium, but it starts to depart significantly from the blue
points for $\tilde{S}\gg 0$. The solid line represents the bound
$(\phi/\phi_{L})^{2}\leq B(\tilde{S}(\rho,\sigma))$, which is our main result.
Note that the solid line manages to produce a nontrivial bound arbitrarily far
from equilibrium and, for $\tilde{S}(\rho,\sigma)=2$, the dashed line fails to
produce a meaningful bound, as it hits the maximum $(\phi/\phi_{L})^{2}=1.$

We also compute the quantum relative entropies $S(\rho||\sigma)$ and
$S(\sigma||\rho)$. Then, we plot $(\phi/\phi_{L})^{2}$ vs.
$\tilde{S}(\rho,\sigma):=[S(\rho||\sigma)+S(\sigma+\rho)]/2$. In Fig. 1, we
compare two bounds: the second inequality of (2) in the dashed line (quadratic
relationship, similar to Onsager’s) and our main result (4) in the solid line.
The draws were realized as follows Salazar (2023) . Let $X\sim I_{x}$ be a
random variable uniformly distributed in the interval $I_{x}$. We consider the
decomposition $\rho=(1-p_{1})|0\rangle\langle 0|+p_{1}|1\rangle\langle 1|$,
where $p_{1}\sim[0,1]$ for each run. We independently draw a random
$\sigma=(1-q_{1})|0\rangle\langle 0|+q_{1}|1\rangle\langle
1|+C|0\rangle\langle 1|+C^{*}|1\rangle\langle 0|$, where $q_{1}\sim[0,1]$,
with $C:=|C|\exp(\phi_{1}i)$, where $|C|^{2}\sim[0,q_{1}(1-q_{1})]$,
$\phi_{1}\sim[0,2\pi)$, so that $\sigma$ is completely positive. Finally, we
draw a random Hermitian operator $\hat{\theta}=\omega(|1\rangle\langle
1|-|0\rangle\langle 0|)+D|0\rangle\langle 1|+D^{*}|1\rangle\langle 0|$, where
$\hat{\omega}\sim[0,4]$ and $D:=|D|\exp(\phi_{2}i)$, with $|D|^{2}\sim[0,1]$,
$\phi_{2}\sim[0,2\pi)$. The pairs
[$(\phi/\phi_{L})^{2},\tilde{S}(\rho,\sigma$)] are depicted as blue points for
$n=10^{4}$ runs. Note that the dashed curve becomes innocuous for
$\tilde{S}(\rho,\sigma)=2$, where the solid line produces a useful bound
arbitrarily far from equilibrium.

_Application: local operators and entropy flux -_ Now we apply (4) for a
specific choice of local operator $\hat{\theta}$ acting on the environment.
The choice $\hat{\theta}=\log\rho_{E}$ in $(\ref{main})$ results in
$\phi=\Phi:=\text{tr}_{E}((\rho_{E}-\rho_{E}^{\prime})\log\rho_{E})$, which is
the quantum entropy flux Landi and Paternostro (2021), with
$\rho=\rho_{E}^{\prime}$ and $\sigma=\rho_{E}$. A direct application of (4)
yields

$\big{(}\frac{\Phi}{\Phi_{L}}\big{)}^{2}\leq
B\big{(}\tilde{S}(\rho_{E}^{\prime},\rho_{E})\big{)}\leq 1,$ (19)

where (19) is valid for bounded environments
$||\ln\rho_{E}||_{\infty}<\infty$. From (19), after using
$B(x)=(x/g(x))^{2}=[\tanh(g(x)/2)]^{2}$ and the data processing inequality
$S(\rho||\sigma)\geq S(\varepsilon(\rho)||\varepsilon(\sigma))$, where
$\varepsilon$ is a CPTP map, we also obtain,

Figure 2: (Color online) Simulation of two interacting qubits initially
uncorrelated. The dynamics builds correlation over time, but it preserves the
total energy of the system. An energy flux $\phi$ (where $\phi_{L}=1$) is
observed between $S$ and $E$, whose absolute value is displayed in two
different forms: quadratic ($2\phi^{2}$, in red) and our result
($2\phi\tanh^{-1}(\phi)$, in blue). We also show the symmetric quantum
relative entropy $\tilde{S}(\rho_{S}(0),\rho_{S}(t))$ in terms of the reduced
states as a function of time (dotted line). We see that the quadratic form
(red) and our result (blue) are initially close, because the flux is small.
Over time, as the flux is not negligible, the quantum relative entropy
approaches the blue line.

$\frac{\Sigma+\Sigma^{*}}{2}\geq\tilde{S}(\rho_{E},\rho_{E}^{\prime})\geq
2\frac{\Phi}{\Phi_{L}}\tanh^{-1}\big{(}\frac{\Phi}{\Phi_{L}}\big{)}\geq
2\big{(}\frac{\Phi}{\Phi_{L}}\big{)}^{2},$ (20)

a general relation between entropy production and flux that resembles
Onsager’s relations for small fluxes ($\Phi/\Phi_{L}\approx 0$, last
inequality), but it contains higher order terms in $\Phi$ for
$|\Phi|\rightarrow\Phi_{L}$. It is valid arbitrarily far from equilibrium in
the strong coupling regime and for nonthermal environments (as long as they
are bounded, $||\ln\rho_{E}||_{\infty}<\infty$). The same reasoning is valid
for any flux (not only entropy flux), thus we also have (18) in general. In
the case of thermal environments, $\rho_{E}=\exp(-\beta H_{E})/Z(\beta)$, with
$Z(\beta):=\text{tr}(\exp(-\beta H_{E}))$, then
$\Phi=\beta\text{tr}((\rho_{E}^{\prime}-\rho_{E})H_{E})$ and
$\Phi_{L}=\beta|E_{max}-E_{min}|$ in (5), where $\beta=1/(k_{B}T)$, $T$ is the
temperature, $\\{E_{n}\\}$ are the eigenvalues of $H_{E}$.

Similarly, in terms of a bounded operator $\hat{\theta}_{S}$ that acts locally
on the system, we have
$\phi=\text{tr}[\hat{\theta}_{S}(\rho_{S}(t)-\rho_{S}(0))]$ and one could
write (20) in terms of the reduced state of the system

$\tilde{S}(\rho_{S}(t),\rho_{S}(0))\geq
2\frac{\phi}{\phi_{L}}\tanh^{-1}\big{(}\frac{\phi}{\phi_{L}}\big{)}\geq
2\big{(}\frac{\phi}{\phi_{L}}\big{)}^{2},$ (21)

where $\rho_{S}(t)=\text{tr}_{E}(\rho(t))$, for $t>0$. Now let us verify (21)
in a physical system. The goal is to show that, even when the last inequality
in (21) is still useful ($\tilde{S}(\rho,\sigma)\leq 2$), the first one might
be significantly better.

_Example -_ Consider two qubits interacting as proposed in Micadei  _et al._
(2019) in a experimental realization of two interacting nuclear spins-1/2.
This system is interesting because, depending on the initial correlations, it
might exhibit a reverse heat flow. In our case, we consider the system
initially uncorrelated. The local Hamiltonians are
$H_{i}=\Omega|e\rangle\langle e|_{i}$, $i=S,E$, and the system is initially
prepared in a product state of the form
$\rho_{SE}=\rho_{S}(0)\otimes\rho_{E}(0)$, where
$\rho_{S}(0)=(1-p)|g\rangle\langle g|+p|e\rangle\langle e|$ and
$\rho_{E}(0)=(1-q)|g\rangle\langle g|+q|e\rangle\langle e|$. At $t=0$, the
system interacts with an energy preserving unitary dynamics
$\mathcal{U}=\exp\\{-igt(e^{i\omega_{0}}|g,e\rangle\langle
e,g|+e^{-i\omega_{0}}|e,g\rangle\langle g,e|)\\}$, where $g$ is the strength
of the interaction and $\omega_{0}$ is an arbitrary phase. We consider the
following absolute flux
$|\phi(t)|=|\text{tr}\\{H_{S}[\rho_{S}(t)-\rho_{S}(0)]\\}|=\sin(gt)^{2}|p-q|$
Landi and Paternostro (2021), where $\rho_{S}(t)=\text{tr}_{E}\\{\rho(t)\\}$.
In Fig. 2, we observe $\tilde{S}(\rho_{S}(t),\rho_{S}(0))$ in the dotted line
as a function of time for $p=0.9$, $q=0.1$, $\Omega=1$ (which makes
$\phi_{L}=1$), $g=2$, $\omega_{0}=0$. In this case, note that (21) is
satisfied for all $t\geq 0$. Particularly, the function
$2\phi\tanh^{-1}(\phi)$ approaches the quantum relative entropy
$\tilde{S}(\rho_{S}(0),\rho_{S}(t))$ over time, saturating the bound at
$gt=\pi/2~{}(t\approx 0.79)$, where the quadratic function $2\phi^{2}$ misses
the quantum relative entropy by some margin when the flux increases. This is
not always the case, as depending on the choices of the parameters, we could
have a very small flux $\phi\approx 0$, which makes both functions
indistinguishable, $2\phi^{2}\approx 2\phi\tanh^{-1}(\phi)$.

_Application: Correlation function -_ Consider again the framework of quantum
thermodynamics, now with a general bounded observable of the form
$\hat{\theta}=\hat{\theta}_{S}\otimes\hat{\theta}_{E}$, such that
$\hat{\theta}_{S}$ and $\hat{\theta}_{E}$ act on the system and the
environment, respectively. We are interested in the following correlation
function,

$\text{tr}(\hat{\theta}(\rho-\sigma))=C(\hat{\theta}_{S},\hat{\theta}_{E}):=\langle\hat{\theta}_{S}\otimes\hat{\theta}_{E}\rangle_{\rho_{SE}^{\prime}}-\langle\hat{\theta}_{S}\rangle_{\rho_{S}^{\prime}}\langle\hat{\theta}_{E}\rangle_{\rho_{E}},$
(22)

for $\rho=\rho_{SE}^{\prime}$ and $\sigma=\rho_{S}^{\prime}\otimes\rho_{E}$. A
proper correlation function Wolf  _et al._ (2008); Kudler-Flam (2023) would
consider $\rho_{E}^{\prime}$ instead of $\rho_{E}$, but we keep this form of
the“bath reset” protocol to match the definition of the entropy production
$\Sigma$. As intuition, check that if $\mathcal{U}=I$, then we are left with
the initial conditions, $\rho_{SE}^{\prime}=\rho_{S}\otimes\rho_{E}$ and
$\rho_{S}^{\prime}=\rho_{S}$, which makes
$C(\hat{\theta}_{S},\hat{\theta}_{E})=0$. But for a general dynamics
$\mathcal{U}$, correlation will build up over time. Correlations are important
as they are typically related to thermodynamic properties in classic systems
Ohga  _et al._ (2023). For this choice of operator, our main result (4) reads

$\frac{C(\hat{\theta}_{S},\hat{\theta}_{E})^{2}}{(\theta_{max}-\theta_{min})^{2}}\leq
B(\frac{\Sigma+\Sigma^{*}}{2})\leq 1.$ (23)

Expression (22) is also easily adjustable to account for other definitions of
entropy production (for instance, the “both reset” protocol would replace
$\rho_{S}^{\prime}$ for $\rho_{S}$ in (22)). Particularly, in the “both reset”
protocol, $\rho=\rho_{SE}^{\prime}$ and $\sigma=\rho_{S}\otimes\rho_{E}$, it
makes $\Sigma^{*}=S(\sigma||\rho)$ physically meaningful as it is the entropy
production of the backwards process.

_Conclusions -_ We have studied the flux $\phi$ of bounded operators in
quantum thermodynamics. We believe that limiting the flux in terms of
information theoretic quantities is in the same spirit of the TURs. To address
this task, we defined the maximum flux ($\phi_{L}$) and analyzed the ratio
$|\phi|/\phi_{L}$. As it turns out, this ratio is limited by the quantum
relative entropy (3) (which is associated with the entropy production in
quantum thermodynamics), as a consequence of the quantum Pinsker’s inequality,
but we argued that this bound is not very useful far from equilibrium.

In this context, we showed an upper bound (4) for $(\phi/\phi_{L})^{2}$ in
terms o the quantum relative entropy that is relevant in nonequilibrium
situations. The result was obtained as a consequence of a recently proposed
quantum TUR and it generalizes classic results from stochastic thermodynamics.
We verified the bound in Monte Carlo simulations of two random qubits and
random bounded operators. We also applied the main result for local operators,
obtaining a relation between entropy production and a local flux (for
instance, the entropy or energy fluxes). As an example, we tested the result
in a model of two interacting nuclear spins-1/2 system, where the the local
energy flux of the system is limited by a function of the (symmetric) quantum
relative entropy as expected from our main result.

Flux limitations by information theoretic quantities provide new insights into
the nature of quantum thermodynamic processes. We believe the results of this
paper might impact design and understanding of quantum systems, particularly
in contexts like quantum computing, quantum heat engines, or other areas where
energy transfer is plays a major role.

## References

  * Landi and Paternostro (2021) G. T. Landi and M. Paternostro, Rev. Mod. Phys. 93, 35008 (2021).
  * Campisi  _et al._ (2011) M. Campisi, P. Hänggi, and P. Talkner, Reviews of Modern Physics 83, 771 (2011).
  * Seifert (2012) U. Seifert, Reports on progress in physics. Physical Society (Great Britain) 75, 126001 (2012), arXiv:1205.4176v1 .
  * Batalhão  _et al._ (2015) T. B. Batalhão, A. M. Souza, R. S. Sarthour, I. S. Oliveira, M. Paternostro, E. Lutz, and R. M. Serra, Physical Review Letters 115, 190601 (2015), arXiv:1502.06704v1 .
  * Esposito  _et al._ (2009) M. Esposito, U. Harbola, and S. Mukamel, Reviews of Modern Physics 81, 1665 (2009).
  * Ciliberto  _et al._ (2013) S. Ciliberto, A. Imparato, A. Naert, and M. Tanase, Physical Review Letters 110, 180601 (2013).
  * Batalhão  _et al._ (2014) T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Physical Review Letters 113, 140601 (2014).
  * Barato and Seifert (2015) A. C. Barato and U. Seifert, Physical Review Letters 114, 158101 (2015).
  * Gingrich  _et al._ (2016) T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Physical Review Letters 116, 120601 (2016), arXiv:1512.02212 .
  * Pietzonka and Seifert (2017) P. Pietzonka and U. Seifert, Physical Review Letters 120, 190602 (2017), arXiv:1705.05817 .
  * Vo  _et al._ (2020) V. T. Vo, T. Van Vu, and Y. Hasegawa, Phys. Rev. E 102, 62132 (2020).
  * Liu  _et al._ (2020) K. Liu, Z. Gong, and M. Ueda, Phys. Rev. Lett. 125, 140602 (2020).
  * Horowitz and Gingrich (2020) J. M. Horowitz and T. R. Gingrich, Nature Physics 16, 15 (2020).
  * Koyuk and Seifert (2019) T. Koyuk and U. Seifert, Phys. Rev. Lett. 122, 230601 (2019).
  * Dechant  _et al._ (2023) A. Dechant, J. Garnier-Brun, and S.-i. Sasa, Phys. Rev. Lett. 131, 167101 (2023).
  * Mori  _et al._ (2023) F. Mori, K. S. Olsen, and S. Krishnamurthy, Phys. Rev. Res. 5, 23103 (2023).
  * Zhang and Garcia-Millan (2023) Z. Zhang and R. Garcia-Millan, Phys. Rev. Res. 5, L022033 (2023).
  * Onsager (1931) L. Onsager, Physical Review 37, 405 (1931).
  * Polettini  _et al._ (2016) M. Polettini, A. Lazarescu, and M. Esposito, Phys. Rev. E 94, 052104 (2016).
  * Hasegawa and Van Vu (2019a) Y. Hasegawa and T. Van Vu, Phys. Rev. Lett. 123, 110602 (2019a).
  * Hasegawa and Van Vu (2019b) Y. Hasegawa and T. Van Vu, Phys. Rev. E 99 (2019b).
  * Vu and Hasegawa (2020) T. V. Vu and Y. Hasegawa, Journal of Physics A: Mathematical and Theoretical 53, 75001 (2020).
  * Timpanaro  _et al._ (2019) A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Physical Review Letters 123, 090604 (2019), arXiv:1904.07574 .
  * Potts and Samuelsson (2019) P. P. Potts and P. Samuelsson, Phys. Rev. E 100, 052137 (2019), arXiv:1904.04913 .
  * Proesmans and Horowitz (2019) K. Proesmans and J. Horowitz, Journal of Statistical Mechanics: Theory and Experiment 5, 054005 (2019), arXiv:1902.07008 .
  * Salazar (2022a) D. S. P. Salazar, Phys. Rev. E 106, L062104 (2022a).
  * Dieball and Godec (2023) C. Dieball and A. c. ž. Godec, Phys. Rev. Lett. 130, 87101 (2023).
  * Van Vu and Saito (2023) T. Van Vu and K. Saito, Phys. Rev. X 13, 11013 (2023).
  * Yoshimura  _et al._ (2023) K. Yoshimura, A. Kolchinsky, A. Dechant, and S. Ito, Phys. Rev. Res. 5, 13017 (2023).
  * Wolf  _et al._ (2008) M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 70502 (2008).
  * Kudler-Flam (2023) J. Kudler-Flam, Phys. Rev. Lett. 130, 21603 (2023).
  * Esposito  _et al._ (2010) M. Esposito, K. Lindenberg, and C. Van Den Broeck, New Journal of Physics 12 (2010), 10.1088/1367-2630/12/1/013013, arXiv:0908.1125 .
  * Manzano  _et al._ (2018) G. Manzano, J. M. Horowitz, and J. M. R. Parrondo, Phys. Rev. X 8, 31037 (2018).
  * Jarzynski (2011) C. Jarzynski, Annual Review of Condensed Matter Physics 2, 329 (2011).
  * Deffner and Lutz (2011) S. Deffner and E. Lutz, Physical Review Letters 107, 140404 (2011), arXiv:1103.4775 .
  * Goold  _et al._ (2015) J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, 1505.07835 143001, 31 (2015), arXiv:1505.07835 .
  * Elouard and Lombard Latune (2023) C. Elouard and C. Lombard Latune, PRX Quantum 4, 20309 (2023).
  * de Groot and Mazur (1961) S. R. de Groot and P. Mazur,  _Non-Equilibrium Thermodynamics_ , 1st ed. (North-Holland Physics Publishing, Amsterdam, 1961) p. 536.
  * Salazar and Landi (2020) D. S. P. Salazar and G. T. Landi, Phys. Rev. Research 2, 33090 (2020).
  * Salazar (2023) D. S. P. Salazar, https://arxiv.org/pdf/2309.09990.pdf (2023).
  * Salazar (2022b) D. S. P. Salazar, Phys. Rev. E 106, L032101 (2022b).
  * Vo  _et al._ (2022) V. T. Vo, T. V. Vu, and Y. Hasegawa, Journal of Physics A: Mathematical and Theoretical 55, 405004 (2022).
  * Dechant (2022) A. Dechant, Journal of Physics A: Mathematical and Theoretical 55, 94001 (2022).
  * Micadei  _et al._ (2019) K. Micadei, J. P. S. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, T. B. Batalhão, R. M. Serra, and E. Lutz, Nature Communications 10, 2456 (2019), arXiv:1711.03323 .
  * Ohga  _et al._ (2023) N. Ohga, S. Ito, and A. Kolchinsky, Phys. Rev. Lett. 131, 77101 (2023).