# Computable Rényi mutual information: Area laws and correlations

Samuel O. Scalet  Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße
1, D-85748 Garching, Germany  Munich Center for Quantum Science and Technology
(MCQST), Schellingstr. 4, D-80799 München, Germany  Álvaro M. Alhambra  Max-
Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching,
Germany  Munich Center for Quantum Science and Technology (MCQST),
Schellingstr. 4, D-80799 München, Germany  Georgios Styliaris  Max-Planck-
Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4,
D-80799 München, Germany  J. Ignacio Cirac  Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany  Munich
Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799
München, Germany

###### Abstract

Mutual information is a measure of classical and quantum correlations of great
interest in quantum information. It is also relevant in quantum many-body
physics, by virtue of satisfying an area law for thermal states and bounding
all correlation functions. However, calculating it exactly or approximately is
often challenging in practice. Here, we consider alternative definitions based
on Rényi divergences. Their main advantage over their von Neumann counterpart
is that they can be expressed as a variational problem whose cost function can
be efficiently evaluated for families of states like matrix product operators
while preserving all desirable properties of a measure of correlations. In
particular, we show that they obey a thermal area law in great generality, and
that they upper bound all correlation functions. We also investigate their
behavior on certain tensor network states and on classical thermal
distributions.

##  I Introduction

One of the most important features of quantum systems is the nature of their
correlations, which differ from their classical counterparts, and lie behind
the complexity of many-body quantum states. It is known, however, that when
these correlations are weak and spatially localized, one can devise efficient
methods to classically simulate complex quantum states via, for instance,
tensor network methods [1, 2]. This occurs at least in gapped ground states in
1D [3, 4], Gibbs states of local Hamiltonians [5, 6, 7], and low-depth quantum
circuits [8]. Characterizing and quantifying those correlations is hence a
subject of wide interest in fields ranging from quantum computing to condensed
matter physics.

The quantum mutual information is perhaps the most widely known measure of
correlations in quantum systems. It has a number of desirable properties, such
as positivity or being nonincreasing under local operations, and a well-
defined information-theoretic meaning [9]. For a bipartite state $\rho_{AB}$,
it is given by

$I(A:B)=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (1)

where $S(\rho)=-\mathrm{Tr}[\rho\log\rho]$ is the von Neumann entropy.
Importantly, it is known to obey an area law for all quantum Gibbs states [10,
7], which implies that the correlations between adjacent subsystems scale only
like their mutual boundary and are thus spatially localized. Calculating the
mutual information in quantum systems is hence an important task in many
physically relevant scenarios. However, this is often impossible via known
analytical methods and requires numerically diagonalizing the whole density
matrix.

It is thus highly desirable to find measures of correlations that share the
appealing information-theoretic properties of $I(A:B)$, but are simpler to
compute in practice, for instance, through variational algorithms. This is
often done by replacing the entropies in Eq. (1) with the more general
$\alpha$-Rényi entropies. The resulting quantity can be computed via a variety
of numerical and analytical means and has been shown to characterize phenomena
such as quantum [11, 12] and thermal [13] phase transitions, or the
correlations in many-body localization [14]. However, it lacks a number of
important properties, which prevent it from being a sensible measure of
correlations. In particular, it can be negative [15] and can increase under
local operations.

Motivated by this, we here explore alternative definitions of the Rényi mutual
information, based on the notion of quantum Rényi divergences [16, 17]. These
are measures of distinguishability of quantum states, which play a pivotal
role in information-theoretic tasks, such as single-shot communication
protocols [18, 19], channel coding [20, 21, 22, 23, 24, 25] or hypothesis
testing [26, 27]. In principle, each of the many variants of quantum Rényi
divergences [28, 29, 20, 30, 31, 32, 33] allows us to define a mutual
information (see Appendix A). We focus on two particular cases and explain how
to compute them in practice, at least when the input state is represented via
tensor networks. We show that they satisfy desirable properties of the mutual
information, including an area law for thermal states, which constitutes our
main technical result. We also show that, like $I(A:B)$, they bound all
correlation functions, and that one of them yields by construction an area law
for a broad class of tensor network states.

The manuscript is structured as follows. In Section II, we discuss the
definitions of Rényi mutual information and their relevant variational
expressions. In Section III, we show the thermal area law and explain the
regimes in which it applies. Then, in Section IV we bound its behavior for
certain tensor network states, and in Section V, we explain that any one of
these measures upper bounds all correlation functions. The technical proofs,
as well as further details, can be found in the appendix.

##  II Definitions of Rényi mutual information

We consider a quantum system on a finite lattice
$\Lambda\subset\mathbb{Z}^{D}$ with local Hilbert spaces $\mathbb{C}^{d}$. For
a quantum state $\rho$, we denote its reduced density matrices on subsystems
$A,B\subset\Lambda$ as $\rho_{A}$ and $\rho_{B}$ respectively.

Apart from (1), the quantum mutual information is also given by the following
equivalent expressions

$\displaystyle I(A:B)$ $\displaystyle=D(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$
(2a) $\displaystyle=\min_{\sigma_{B}}D(\rho_{AB}\|\rho_{A}\otimes\sigma_{B})$
(2b)

with the Umegaki relative entropy
$D(\rho\|\sigma)=\mathrm{Tr}[\rho\log\rho-\rho\log\sigma]$. This quantity is
nonnegative and cannot increase under local operations on both $A$ and $B$.
These properties follow, respectively, from the positivity of the relative
entropy, and its contractivity under CPTP maps, i.e., the data-processing
inequality. To obtain Rényi versions of the mutual information, one can then
generalize any one of Eq. (1), (2a) and (2b). However, each of them yields a
different definition, which are no longer equivalent.

###  II.1 Rényi entropies

Starting from (1) and replacing the von Neumann entropies in the definition of
the mutual information by the Rényi entropies
$S_{\alpha}(\rho)=(1-\alpha)^{-1}\log\mathrm{Tr}[\rho^{\alpha}]$, one obtains

$I_{\alpha}(A:B)=S_{\alpha}(\rho_{A})+S_{\alpha}(\rho_{B})-S_{\alpha}(\rho_{AB}).$
(3)

For integer values of $\alpha$, this definition contains only integer powers
and traces of the density matrices. This feature makes it easily computable in
many physically relevant situations. Analytically, an important example is the
replica trick [34, 35], which has been used to calculate $S_{\alpha}$ in a
conformal field theory for integer values of $\alpha$ (see [36, 37, 38] for
calculations of $I_{\alpha}(A:B)$). It can also be calculated exactly for free
fermions [39]. Numerically, it is efficiently computable when the state $\rho$
is represented by a matrix product density operator (MPDO) [40], or by quantum
Monte Carlo methods [41, 42, 43, 44].

However, this definition lacks several of the important properties of
$I(A:B)$. For instance, it can be negative in physically relevant situations
[15]. We show in Appendix B that this can, in fact, happen in a very simple
scenario: the thermal state of a classical Ising chain with an external field.
To do so, we calculate the mutual information arising from Eq. (3)
analytically in the thermodynamic limit using transition matrices and show
that for a sufficiently weak antiferromagnetic coupling it takes negative
values. Moreover, its negativity implies that it cannot be nonincreasing under
local operations in general: tracing out the $A$ system is a local operation
that can increase $I_{\alpha}(A:B)$ from a negative value to zero.

###  II.2 Maximal Rényi divergence

A possible strategy to obtain a Rényi mutual information, which inherits the
desirable properties of $I(A:B)$, is to invoke one of its equivalent
definitions in terms of relative entropy (also known as divergence) and
directly extend the latter to the Rényi case. For applications in quantum
communication, it is common to generalize Eq. (2b), e.g., see [17]. However,
let us here start from Eq. (2a). To do so, we introduce the important maximal
Rényi divergence [45]. Given quantum states $\rho$ and $\sigma$, it is defined
as [31]

$D_{\infty}(\rho\|\sigma)=\log\inf\\{\lambda:\rho\leq\lambda\sigma\\},$ (4)

which, in turn, yields a definition of a Rényi mutual information as
$I_{\infty}(A:B)=D_{\infty}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$. The latter
is nonnegative and cannot increase under local operations, where the last
property follows from the data-processing inequality of the divergence under
CPTP maps $\mathcal{E}$ [17], i.e.,
$D_{\infty}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))\leq
D_{\infty}(\rho\|\sigma)$.

This quantity has two important features. The first is that it can be
approximated when the arguments are matrix product operators. We can rewrite
Eq. (4) as

$\displaystyle D_{\infty}(\rho\|\sigma)$
$\displaystyle=\log\inf\\{\lambda:\inf_{|\psi\rangle}\left\langle\psi\right|\lambda\sigma-\rho\left|\psi\right\rangle\geq
0\\}.$ (5)

It is then possible to approximate the braket in Eq. (5) using the DMRG
algorithm [46]. While convergence is not guaranteed, it can be checked whether
it approaches a limit with increasing bond dimension of $|\psi\rangle$,
indicating that the supremum has been well approximated. The minimal $\lambda$
can then be determined using a binary search. A potential difficulty is that
determining whether an MPO is positive is an NP-hard problem [47] and
calculating $D_{\infty}$ involves determining the positivity of the MPO
$\lambda\rho_{A}\otimes\rho_{B}-\rho$. As this problem has additional
structure it might still be efficiently solvable in practice.

The second important feature is that it upper bounds all Rényi divergences
that fulfill the data-processing inequality [16] (see Appendix A for details).
This is relevant since, as shown in Section III, it follows a thermal area
law, which automatically extends to all known divergences.

Let us briefly comment on the generalization using Eq. (2b) instead. This
yields another possible definition of Rényi mutual information, which in fact
has an operational interpretation in communication theory as the communication
cost of entanglement-assisted one-shot communication protocols [48, 19].
However, the additional optimization makes the quantity more difficult to
handle both computationally and analytically, and hence less suited for our
present purpose. Nevertheless, our area laws also apply to this quantity as it
is by definition smaller than $I_{\infty}$.

###  II.3 Measured Rényi divergence

We now present another quantity that can be efficiently computed numerically
via variational algorithms. For classical states, i.e., probability
distributions $P$ and $Q$, the Rényi divergence is defined as [49]

$D_{\alpha}(P\|Q)=\frac{1}{\alpha-1}\log\sum_{x}P(x)^{\alpha}Q(x)^{1-\alpha}.$
(6)

From this, the measured Rényi divergence is defined as [50]

$D^{\mathbb{M}}_{\alpha}(\rho\|\sigma)=\sup_{(\chi,M)}D_{\alpha}(P_{\rho,M}\|P_{\sigma,M}),$
(7)

with a supremum over all possible POVM measurements and $P_{.,M}$ the post
measurement states, i.e., the respective probability distributions over the
measurement outcomes. We denote the corresponding mutual information by
$I_{\alpha}^{\mathbb{M}}(A:B)=D^{\mathbb{M}}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$.
Like $I(A:B)$ and $I_{\infty}(A:B)$, this mutual information is positive and
does not increase under local operations. Moreover,
$D^{\mathbb{M}}_{\alpha}\leq D_{\infty}$ for all $\alpha$, so all thermal area
laws for $I_{\infty}(A:B)$ also apply to $I_{\alpha}^{\mathbb{M}}(A:B)$.

The interest of this quantity stems from the following variational expression
for $\alpha>1$ [50]

$D_{\alpha}^{\mathbb{M}}(\rho\|\sigma)=\frac{1}{\alpha-1}\log\sup_{\omega>0}\alpha\mathrm{Tr}[\rho\omega^{\alpha-1}]+(1-\alpha)\mathrm{Tr}[\sigma\omega^{\alpha}].$
(8)

Again assuming the states to be given as MPDOs, this is an optimization in
which the target function only contains products and traces of MPDOs, which
can be efficiently computed for integer values of $\alpha>1$ with DMRG-type
algorithms. In general, this can be done by optimizing over the matrix product
operators with purifications to enforce the positivity constraint.

For $\alpha=2$, we give an explicit expression for the optimizer. The
positivity constraint can be dropped as there cannot be an optimizer with
negative eigenvalues. Using a vectorized notation $|i\rangle\\!\langle
j|\to\ket{i}\ket{j}$, we then obtain (see Appendix C.1)

$\ket{\omega}=\frac{1}{2}\frac{1}{\sigma\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma}\ket{\rho}.$
(9)

A direct calculation of $\omega$ from this is inefficient due to the inverse
involved, but one can instead determine $\omega$ variationally by minimizing
$\|2(\sigma\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma)\ket{\omega}-\ket{\rho}\|_{2}^{2}$,
which is a quadratic expression in $\ket{\omega}$ and thus, again, it can be
obtained using DMRG-like algorithms.

##  III Thermal area laws

In this section we present our main technical results: area laws for the Rényi
mutual information in thermal states of local Hamiltonians. In the following,
we consider a subset of a lattice A and its complement B. The Hamiltonian is
split into three parts

$H=H_{A}+H_{B}+H_{I},$ (10)

such that $H_{A}$, $H_{B}$ have support in $A$ and $B$ respectively, and
$H_{I}$ is the interaction term. We consider thermal states $\rho=\exp(-\beta
H)/Z$, where the partition function is defined as $Z=\mathrm{Tr}[e^{-\beta
H}]$. We assume that $H$ consists of local, finite-range terms $H=\sum h_{i}$,
where each $h_{i}$ is supported on at most $k$ sites and is bounded such that
$\sup_{x}\sum_{\mathrm{supp}(h_{i})\ni x}\|h_{i}\|=J<\infty$. Consequently,
$\|H_{I}\|$, which contains only those local terms that have support in $A$
and $B$, scales with the size of the boundary. We further define $\partial
A=\\{x\in A:\exists h_{i}\text{ s.t. }x\in\mathrm{supp}(h_{i})\land
B\cap\mathrm{supp}(h_{i})\neq\emptyset\\}$ as the boundary of $A$, which is
such that there are no terms with support in both $A\setminus\partial A$ and
$B$. Notice that in 1D, $|\partial A|\propto\text{const.}$, independent of the
system size.

We first introduce a technical lemma, which allows us to prove area laws in
several special cases.

######  Lemma 1.

Let $E_{\beta}=e^{-\beta(H_{A}+H_{B})}e^{\beta H}$. For any Hamiltonian
defined as above, the maximal Rényi mutual information of a thermal state
fulfills

$I_{\infty}(A:B)\leq\beta\|H_{I}\|+\log\left(\|E_{\beta/2}\|^{2}\|E_{\beta/2}^{-1}\|^{4}\right).$
(11)

The proof is in Appendix C.2. While the first term of the RHS directly scales
with the boundary and corresponds to the result in [10], one still needs to
bound the additional second term.

In the case of a commuting Hamiltonian, we have $E_{\beta}=e^{\beta H_{I}}$
straightforwardly, and thus we find:

######  Theorem 1 (Commuting Hamiltonian).

For $[H,H_{A}+H_{B}]=0$ it holds that

$I_{\infty}(A:B)\leq 4\beta\|H_{I}\|.$ (12)

For Hamiltonians with non-commuting terms, it is no longer possible to cancel
the bulk contributions in $E_{\beta}$ directly, but the norm can still be
bounded in many cases.

In 1D systems, we use a lemma from [51] (section 2.3.1 and Theorem 3.1) based
on previous work by Araki [52]. The setting here is a finite bipartite chain
$A=[-N/2,a]$, $B=[a+1,N/2]$ with a cut after some site $a$.

######  Lemma 2.

Let $H=\sum_{i=-N/2}^{N/2}h_{i}$ be a 1D Hamiltonian with $h_{i}$ having
support on at most $l$ continuous sites. Define
$H_{A}=\sum_{\mathrm{supp}(h_{i})\subset A}h_{i}$ and $H_{B}$ analogously.
Then,

$\max\\{\|E_{\beta}\|,\|E_{\beta}^{-1}\|\\}\leq\exp\left(\frac{1}{2}f(\beta,J,l)\exp(f(\beta,J,l))\right),$
(13)

where $f(\beta,J,l)=4\beta Jl^{2}e^{1+4\beta Jl}$.

This bound is no longer linear in $\beta$ and the interaction strength, but it
still proves the following area law, as it is independent of the size of $A$
and $B$.

######  Theorem 2 (One dimension).

In 1D, the Rényi mutual information is bounded as:

$I_{\infty}(A:B)\leq 4f(\beta,J,l)\exp(f(\beta,J,l))$ (14)

In higher dimensions, we can use an imaginary-time Lieb-Robinson bound for
high temperatures [53, 54, 55, 56].

######  Lemma 3.

On a $D$-dimensional lattice, we have

$\log(\|E_{\beta}\|)\leq-\log(1-2\beta Jk)|\partial A|/(2k)$ (15)

if $2\beta Jk<1$ and the same bound holds for $\|E_{\beta}^{-1}\|$.

We give the proof in Appendix C.3. This results in our last area law based on
Lemma 1:

######  Theorem 3 (Any dimension at high temperature).

In the setting of the previous lemma and for $\beta Jk<1$, we have

$I_{\infty}(A:B)\leq-4\log(1-\beta Jk)|\partial A|/k.$ (16)

Both Lemmas 2 and 3, use an imaginary time-ordered integral to bound the norm
of $E_{\beta}$, as

$\begin{split}&||E_{\beta}||=\|e^{-\beta(H_{A}+H_{B})}e^{\beta H}\|\\\
&\leq\exp\left(\int_{0}^{\beta}\|e^{x(H_{A}+H_{B})}H_{I}e^{-x(H_{A}+H_{B})}\|\text{d}x\right),\end{split}$
(17)

which follows from the Dyson series and the triangle inequality. The same
bound holds for the inverse operator $E_{\beta}^{-1}$ with $(H_{A}+H_{B})$
replaced by $H$ on the right-hand side. The operator in the norm corresponds
to an imaginary time evolution of $H_{I}$, which can be approximated with a
Taylor series.

This technique cannot be used to extend the area law to arbitrary dimensions
and temperatures because there exists a 2D lattice Hamiltonian such that for
sufficiently small temperatures the quantity $\|e^{xH}Ae^{-xH}\|/\|A\|$
diverges in the thermodynamic limit [57, 58].

Let us now comment on the thermal area law for classical systems. For them, an
area law for the mutual information independent of temperature and energy was
shown in [10]. The idea behind it is the Markov property of classical thermal
states, which reads $P(x_{A}|x_{B})=P(x_{A}|x_{\partial B})$, where again the
boundaries are defined as above such that there is no interaction between $A$
and $B\setminus\partial B$ and vice-versa. This means that all correlations
between $A$ and $B$ are mediated through $\partial B$ and also implies that
the correlations cluster at the boundary, in the sense that $I(A:B)=I(\partial
A:\partial B)$. This leads to a bound that only depends on the dimension of
the boundary.

This latter equality also holds for the Rényi mutual information defined using
Rényi divergences, which allows for the following extension.

######  Theorem 4 (Classical temperature-independent area law).

For a classical system with local dimension $d$, we have

$I_{\alpha}^{\mathbb{M}}(A:B)\leq(|\partial A|+|\partial B|)\log d$ (18)

for $\alpha\in(0,1)\cup(1,2]$.

The proof, which uses the fact that every probability distribution majorizes
the flat distribution, is given in Appendix C.4. A challenge in the Rényi case
is that for fixed system size the mutual information is no longer bounded in
general, but only for $\alpha\leq 2$. A simple example for two bits shows that
it can be arbitrarily large for $\alpha>2$, and by extension for $I_{\infty}$
(see Appendix C.4), in which case we can only give the temperature-dependent
Theorem 1.

##  IV Rényi mutual information on tensor network states

Matrix product states and also their higher dimensional analog projected
entangled pair states (PEPS) [59], have by construction a small bipartite
entanglement entropy, bounded by the logarithm of the bond dimension $D$ times
the number of neighboring pairs across the boundary of the bipartition. The
same holds for their mutual information, as for pure states it is equal to
twice their entanglement entropy.

A natural question is whether this extends to the mutual information of
projected entangled pair density operators (PEPDOs), their mixed state analog
[2]. In [10], this question was answered positively for the mutual information
$I(A:B)$, using the additional assumption that the PEPDO has a local
purification. This means that there exists a PEPS with a physical and an
ancilla index of equal dimension on every site, whose partial trace over the
ancillas equals the PEPDO 111Notice that not all PEPDOs admit such description
[64, 65]..

This result can be extended to the measured Rényi mutual information for a
limited range of $\alpha$ (see Appendix C.5):

######  Theorem 5.

For a PEPDO with local purification, bond dimension $D$, and $|\partial A|$
the number of bonds between $A$ and $B$, it holds

$I^{\mathbb{M}}_{\alpha}(A:B)\leq 2|\partial A|\log D$ (19)

for $\alpha\in(0,1)\cup(1,2]$.

This can be proven by first noticing that the problem is equivalent to the
pure state case if one considers the purification. Then, the trace over the
ancillas of the PEPS, which yields the PEPDO, does not increase the mutual
information as it is a local operation. The remaining step is to compute the
mutual information of a pure state $\rho=|\psi\rangle\\!\langle\psi|$ and
relate it to an entropy of the subsystems. This is similar to the von Neumann
case, where $I(A:B)=2S(\rho_{A})$. The entropy of the marginal is then bounded
by the logarithm of the number of Schmidt values of the decomposition into $A$
and $B$, which yields the desired bound. We give an analog proof for the Rényi
mutual information in Appendix C.5 valid for $\alpha\leq 2$. For $\alpha>2$ we
present a simple counterexample on 2 qubits ($D=|\partial A|=2$) with
arbitrarily large measured Rényi mutual information in the appendix.

##  V Correlation functions

The mutual information quantifies both classical and quantum correlations [9].
Therefore, it seems intuitive that it should also impose a bound on
correlation functions, given by $\mathcal{C}(M_{A},M_{B}):=\langle
M_{A}M_{B}\rangle-\langle M_{A}\rangle\langle M_{B}\rangle$ for observables
$M_{A}$, $M_{B}$ supported on $A$, $B$ respectively. This was established for
$I(A:B)$ in [10]. We here extend this proof to the Rényi case, by showing that

$I_{\alpha}^{\mathbb{M}}(A:B)\geq\frac{\min\\{1,\alpha\\}\mathcal{C}(M_{A},M_{B})^{2}}{2\|M_{A}\|^{2}\|M_{B}\|^{2}}.$
(20)

The bound trivially extends to all other quantum Rényi divergences that
fulfill the data-processing inequality.

The key technical result is a generalization of the quantum Pinsker’s
inequality:

######  Lemma 4.

For $\alpha\in(0,1)\cup(1,\infty)$ we have

$\min\\{1,\alpha\\}\frac{1}{2}\|\rho-\sigma\|_{1}^{2}\leq
D_{\alpha}^{\mathbb{M}}(\rho\|\sigma).$ (21)

The proof is in Appendix C.6, and it uses the same argument as for the
relative entropy $D(\rho\|\sigma)$, where the data-processing inequality is
applied to a binary measurement [61]. The known equivalent classical result
[62] can then be applied to the measurement outcome. The bound on the
correlations Eq. (20) follows using $\|X\|_{1}\geq\mathrm{Tr}[XY]/\|Y\|$,
exactly as in [10].

##  VI Conclusion

We have given alternative definitions of the mutual information, which
measures the amount of correlations in quantum states. We have shown that, as
a measure of bipartite correlations, they satisfy a number of desirable
properties, including area laws for thermal states.

These Rényi mutual information measures differ from the widely used definition
of Eq. (3), which has nonetheless found applications in, for instance,
analyzing thermal phase transitions [13]. Our results do not rule out the
possibility that $I_{\infty}(A:B)$ has a wildly different behavior (such as a
volume law) at thermal criticality, which occurs at low temperatures when
$D>1$. It would be interesting to study whether singularities in it appear at
phase transition points or whether, on the contrary, an area law holds with
full generality.

These quantities may be a useful measure of correlations beyond thermal
states, such as in dissipative dynamics, quantum quenches, and non-equilibrium
steady states. We hope that our results motivate their study in the wider
context of quantum many-body systems. An interesting future question is
whether the smallness of any of these quantities guarantees an efficient
approximation of mixed states via tensor networks, similar to how the Rényi
entanglement entropy guarantees MPS approximations in the pure state case [3].
The Rényi entanglement of purification is known to play such a role [63], but
has the problem of being hard to calculate in practice.

## Acknowledgements

The authors acknowledge useful discussions with Ángela Capel. Á.M.A, G.S., and
J.I.C. acknowledge funding from ERC Advanced Grant QUENOCOBA under the EU
Horizon 2020 program (Grant Agreement No. 742102) and within the D-ACH Lead-
Agency Agreement through project No. 414325145 (BEYOND C).

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##  Appendix A Quantum Rényi divergences

In this section we recall various definitions and facts for the different
Rényi divergences, which will be useful for the proofs in the following
sections.

The Petz Rényi divergence [28] is perhaps the most direct quantum analogue of
the classical family of Rényi divergences, that is, for density operators that
do not, in general, commute. It is defined for $\alpha\in(0,1)\cup(1,2]$ as

$\overline{D}_{\alpha}(\rho\|\sigma)=\frac{1}{\alpha-1}\log\mathrm{Tr}[\rho^{\alpha}\sigma^{1-\alpha}]$
(22)

if $\mathrm{supp}(\rho)\subseteq\mathrm{supp}(\sigma)$ and $+\infty$
otherwise, where the inverse of $\sigma$ is understood to be a pseudo-inverse.
While this seems to be a straightforward generalization, several other
definitions that collapse to the classical definition for the commuting case
are possible.

We will invoke two main alternatives for the purposes of this work. They are
the sandwiched Rényi divergence [29, 20], which for
$\alpha\in[1/2,1)\cup(1,\infty)$ takes the form

$\widetilde{D}_{\alpha}(\rho\|\sigma)=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]$
(23)

and the geometric Rényi divergence [30], which for
$\alpha\in(0,1)\cup(1,\infty)$ reads

$\widehat{D}_{\alpha}(\rho\|\sigma)=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[\sigma\left(\sigma^{-1/2}\rho\sigma^{-1/2}\right)^{\alpha}\right].$
(24)

This geometric Rényi divergence is crucial in our proof of the thermal area
law.

These quantities share some important properties as being increasing in
$\alpha$

$D_{\alpha}(\rho\|\sigma)\leq D_{\beta}(\rho\|\sigma)\text{ for
}\alpha<\beta,$ (25)

nonnegativity

$D_{\alpha}(\rho\|\sigma)\geq 0,$ (26)

and a data-processing inequality

$D(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))\leq D(\rho\|\sigma)\text{ for any
CPTP map }\mathcal{E}.$ (27)

While for the sandwiched and Petz Rényi divergence this data-processing
inequality holds for the full range of $\alpha$ given above, for the
geometric, this is only known for $\alpha\leq 2$. Additionally, we have the
inequalities

$\displaystyle\widetilde{D}_{\alpha}(\rho\|\sigma)\leq\overline{D}_{\alpha}(\rho\|\sigma)$
$\displaystyle\text{ for }\alpha\in[1/2,1)\cup(1,2],$ (28)
$\displaystyle\overline{D}_{\alpha}(\rho\|\sigma)\leq\widehat{D}_{\alpha}(\rho\|\sigma)$
$\displaystyle\text{ for }\alpha\in(0,1)\cup(1,2],$
$\displaystyle\widetilde{D}_{\alpha}(\rho\|\sigma)\leq\widehat{D}_{\alpha}(\rho\|\sigma)$
$\displaystyle\text{ for }\alpha\in[1/2,0)\cup(1,\infty).$

The sandwiched and Petz Rényi divergences converge to the Umegaki relative
entropy in the limit $\alpha\to 1$, while the geometric converges to the so-
called Belavkin-Staszewski relative entropy [66]. For $\alpha\to\infty$, the
sandwiched and geometric Rényi divergences converge to the maximal Rényi
divergence

$D_{\infty}(\rho\|\sigma)=\log\inf\\{\lambda:\rho\leq\lambda\sigma\\}.$ (29)

This is the main reason for using the geometric Rényi divergence as a tool in
many of the following proofs: the statements for $D_{\infty}$ are then derived
by taking the limit $\alpha\to\infty$. For a more complete summary of the
properties of Rényi divergences and their proofs, see [17] and the references
therein.

As in the main text, we can now introduce Rényi mutual information arising
from Eq. (2a):

$\widehat{I}_{\alpha}(A:B)=\widehat{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$
(30)

and $\overline{I}_{\alpha}$ and $\widetilde{I}_{\alpha}$ respectively. We note
that all upper bounds for $\widehat{I}(A:B)$ are automatically also upper
bounds to $\widetilde{I}$ and $\overline{I}$ due to the inequalities (28).

In the quantum information literature the mutual information is commonly
defined in a different, inequivalent way: Instead of starting from Eq. (2a),
Eq. (2b) is taken, and the Umegaki relative entropy is being replaced by a
Rényi divergence:

$\widehat{I}_{\alpha}^{opt}(A:B)=\min_{\sigma_{B}}\widehat{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\sigma_{B})$
(31)

Again, a variety of definitions can be made by choosing the different Rényi
divergences introduced above. There are operational interpretations in
information theory for these quantities, for example in quantum hypothesis
testing [27] or entanglement-assisted single-shot communication protocols
[19]. Since they contain an additional optimization, these quantities are more
difficult to compute from a practical standpoint, and we do not use them in
this paper. However, all upper bounds (such as the area laws) trivially hold
for them as well.

##  Appendix B Negative Rényi mutual information of a classical Ising chain

In this section, we present the calculation of the Rényi mutual information
based on Rényi entropies (Eq. (3)) for a classical Ising chain in its thermal
state and show that it can become negative. We adapt the standard method of
transfer matrices used to solve the classical Ising spin chain (see, e.g.,
[67]). The model consists of $N$ spins $z_{i}$ taking values $\pm 1$ and is
placed on a ring, i.e., we use periodic boundary conditions. The Hamiltonian
is given by

$H=\sum_{i=1}^{N}hz_{i}+Jz_{i}z_{i+1},$ (32)

with the temperature included in the constants and the addition understood as
modulo $N$. A general technical problem occurring in the calculation of mutual
information is the breaking of translation symmetry due to the definition of
distinct regions $A$ and $B$. In order to deal with this problem, we put two
subsystems $A={1,\cdots,L}$ and $B={L+1,\cdots,2L}$ on a chain of length
$N>2L$ and calculate the mutual information in the limit of first taking
$N\to\infty$ and then $L\to\infty$. We define the matrix

$T=\left(\begin{array}[]{rr}e^{-J+h}&e^{J}\\\
e^{J}&e^{-J-h}\end{array}\right)=\lambda_{+}|\lambda_{+}\rangle\\!\langle\lambda_{+}|+\lambda_{-}|\lambda_{-}\rangle\\!\langle\lambda_{-}|$
(33)

with $\lambda_{+}>\lambda_{-}$ the eigenvalues and $\ket{\lambda_{+}}$,
$\ket{\lambda_{-}}$ the corresponding eigenvectors. Note that the Perron-
Frobenius theorem guarantees that $\lambda_{+}$ is the unique largest
eigenvalue (in norm) and also positive [68]. The computational basis is
denoted by $\ket{\pm 1}$. We first calculate the partition function by
rewriting it with a matrix power

$\displaystyle\mathcal{Z}$
$\displaystyle=\sum_{\sigma_{1},\cdots,\sigma_{N}\in\\{-1,1\\}}\prod_{i=1}^{N}\exp(-h\frac{\sigma_{i}+\sigma_{i+1}}{2}-J\sigma_{i}\sigma_{i+1})$
(34)
$\displaystyle=\sum_{\sigma_{1},\cdots,\sigma_{N}\in\\{-1,1\\}}\prod_{i=1}^{N}\bra{\sigma_{i}}T\ket{\sigma_{i+1}}=\mathrm{Tr}\left[T^{N}\right]=\lambda_{+}^{N}+\lambda_{-}^{N}$

and use this result to calculate probabilities as follows.

$\displaystyle P(z_{k}=\pm 1)$
$\displaystyle=\mathcal{Z}^{-1}\sum_{\sigma_{1},\cdots,\sigma_{N}\in\\{-1,1\\}}\delta_{\pm
1,\sigma_{k}}\prod_{i=1}^{N}\exp(-h\frac{\sigma_{i}+\sigma_{i+1}}{2}-J\sigma_{i}\sigma_{i+1})$
(35)
$\displaystyle=\mathcal{Z}^{-1}\sum_{\sigma_{1},\cdots,\sigma_{N}\in\\{-1,1\\}}\bra{\sigma_{1}}T\ket{\pm
1}\langle\pm
1|\sigma_{2}\rangle\prod_{i=2}^{N}\bra{\sigma_{i}}T\ket{\sigma_{i+1}}$
$\displaystyle=\mathcal{Z}^{-1}\mathrm{Tr}\left[T^{N}\ket{\pm 1}\bra{\pm
1}\right]$ $\displaystyle=\frac{\lambda_{+}^{N}\langle\lambda_{+}|\pm
1\rangle\langle\pm 1|\lambda_{+}\rangle+\lambda_{-}^{N}\langle\lambda_{-}|\pm
1\rangle\langle\pm
1|\lambda_{-}\rangle}{\lambda_{+}^{N}+\lambda_{-}^{N}}\xrightarrow{N\rightarrow\infty}\langle\lambda_{+}|\pm
1\rangle\langle\pm 1|\lambda_{+}\rangle=\langle\pm 1|\lambda_{+}\rangle^{2},$

where in the above we used translation invariance and cyclicity of the trace.
We now generalize this calculation to the probability of a configuration of
several spins and apply it to a conditional probability.

$\displaystyle P(z_{k}=\sigma_{k}$
$\displaystyle|z_{k-1}=\sigma_{k-1},\cdots,z_{1}=\sigma_{1})$ (36)
$\displaystyle=\frac{\mathrm{Tr}\left[T^{N-k+1}|\sigma_{k}\rangle\\!\langle\sigma_{k}|T\ket{\sigma_{k-1}}\cdots\bra{\sigma_{2}}T|\sigma_{1}\rangle\\!\langle\sigma_{1}|\right]}{\mathrm{Tr}\left[T^{N-k+2}|\sigma_{k-1}\rangle\\!\langle\sigma_{k-1}|T\ket{\sigma_{k-2}}\cdots\bra{\sigma_{2}}T|\sigma_{1}\rangle\\!\langle\sigma_{1}|\right]}$
$\displaystyle=\frac{\mathrm{Tr}\left[T^{N-k+1}|\sigma_{k}\rangle\\!\langle\sigma_{k}|T|\sigma_{k-1}\rangle\\!\langle\sigma_{1}|\right]}{\mathrm{Tr}\left[T^{N-k+2}|\sigma_{k-1}\rangle\\!\langle\sigma_{1}|\right]}$
$\displaystyle=\bra{\sigma_{k}}T\ket{\sigma_{k-1}}\frac{\bra{\sigma_{1}}T^{N-k+1}\ket{\sigma_{k}}}{\bra{\sigma_{1}}T^{N-k+2}\ket{\sigma_{k-1}}}$
$\displaystyle=\bra{\sigma_{k}}T\ket{\sigma_{k-1}}\frac{\langle\sigma_{1}|\lambda_{+}\rangle\langle\lambda_{+}|\sigma_{k}\rangle\lambda_{+}^{N-k+1}+\langle\sigma_{1}|\lambda_{-}\rangle\langle\lambda_{-}|\sigma_{k}\rangle\lambda_{-}^{N-k+1}}{\langle\sigma_{1}|\lambda_{+}\rangle\langle\lambda_{+}|\sigma_{k-1}\rangle\lambda_{+}^{N-k+2}+\langle\sigma_{1}|\lambda_{-}\rangle\langle\lambda_{-}|\sigma_{k-1}\rangle\lambda_{-}^{N-k+2}}$
$\displaystyle\xrightarrow{N\to\infty}\bra{\sigma_{k}}T\ket{\sigma_{k-1}}\frac{\langle\lambda_{+}|\sigma_{k}\rangle}{\langle\lambda_{+}|\sigma_{k-1}\rangle\lambda_{+}}=P(z_{k}=\sigma_{k}|z_{k-1}=\sigma_{k-1}).$

In the second equality, the cancellations within the trace explcitly verify
the Markov property of the periodic chain. In the limit, the dependency on
$\sigma_{1}$ also vanishes, which can be understood from the perspective of
correlations decaying with the distance. In the large $N$ limit the latter
have to mediate through an infinitely long region of the chain, and hence
vanish.

We are now ready to calculate the Rényi entropies

$S_{\alpha}(A)=\frac{1}{1-\alpha}\log\left(\sum_{\sigma_{1},\cdots,\sigma_{L}\in\\{-1,1\\}}(P(z_{1}=\sigma_{1})P(z_{2}=\sigma_{2}|z_{1}=\sigma_{1})...P(z_{L}=\sigma_{L}|z_{L-1}=\sigma_{L-1}))^{\alpha}\right)$
(37)

where we already used (36) to decompose the joint probability. Again, due to
translation invariance, all the conditional probabilities are the same. We
define

$T_{\alpha}=\left(\begin{array}[]{rr}P^{\alpha}(z_{k}=-1|z_{k-1}=-1)&P^{\alpha}(z_{k}=-1|z_{k-1}=1)\\\
P^{\alpha}(z_{k}=1|z_{k-1}=-1)&P^{\alpha}(z_{k}=1|z_{k-1}=1)\end{array}\right)\textrm{,
}\ket{\mathbbm{1}}=\left(\begin{array}[]{r}1\\\ 1\end{array}\right)\textrm{,
}\ket{P_{\alpha}}=\left(\begin{array}[]{r}P^{\alpha}(z_{k}=-1)\\\
P^{\alpha}(z_{k}=1)\end{array}\right),$ (38)

where $T_{\alpha}$ is no longer a stochastic matrix. With the same technique
as before, (37) becomes

$S_{\alpha}(A)=\frac{1}{1-\alpha}\log\left(\bra{\mathbbm{1}}T_{\alpha}^{L-1}\ket{P_{\alpha}}\right)$
(39)

and we obtain the Rényi mutual information

$I_{\alpha}(A:B)=\frac{1}{1-\alpha}\log\left(\frac{\bra{\mathbbm{1}}T_{\alpha}^{L-1}\ket{P_{\alpha}}^{2}}{\bra{\mathbbm{1}}T_{\alpha}^{2L-1}\ket{P_{\alpha}}}\right).$
(40)

To evaluate this expression, we use the diagonalization of $T_{\alpha}$, which
reads

$\left(\begin{array}[]{rr}t_{\alpha}^{+}&0\\\
0&t_{\alpha}^{-}\end{array}\right)=DT_{\alpha}D^{-1}$ (41)

for some invertible matrix D. The fact that $T_{\alpha}$ is diagonalizable
again results from the Perron-Frobenius theorem, which states that the largest
eigenvalue has multiplicity one if the matrix has positive entries. Therefore,
another eigenvalue exists, which proves diagonalizability. Additionally
$t_{\alpha}^{+}$, which we define to be the larger real eigenvalue, is also
strictly larger in absolute value, which allows us to calculate the limit
$L\to\infty$ of the above expression.

$\displaystyle I_{\alpha}(A:B)$
$\displaystyle=\frac{1}{1-\alpha}\log\frac{(\bra{\mathbbm{1}}D^{-1}\ket{-1}\bra{-1}D\ket{P_{\alpha}}(t_{\alpha}^{+})^{L-1}+\bra{\mathbbm{1}}D^{-1}\ket{1}\bra{1}D\ket{P_{\alpha}}(t_{\alpha}^{-})^{L-1})^{2}}{\bra{\mathbbm{1}}D^{-1}\ket{-1}\bra{-1}D\ket{P_{\alpha}}(t_{\alpha}^{+})^{2L-1}+\bra{\mathbbm{1}}D^{-1}\ket{1}\bra{1}D\ket{P_{\alpha}}(t_{\alpha}^{-})^{2L-1}}$
(42)
$\displaystyle\xrightarrow{L\to\infty}\frac{1}{1-\alpha}\log\frac{\bra{\mathbbm{1}}D^{-1}\ket{-1}\bra{-1}D\ket{P_{\alpha}}}{t_{\alpha}^{+}}$

Using the eigenvalues and eigenvectors of $T_{\alpha}$ (Eq. (38)) and $T$ (Eq.
(33)) one can easily derive an explicit analytic expression for the Rényi
mutual information $I_{\alpha}(A:B)$. However, we omit writing the exact
expression for simplicity, due to its length. Instead we numerically evaluate
the formula with the resulting plots in Figure 1, which show the existence of
a negative regime for antiferromagnetic coupling. Additionally, we find that
the mutual information is not monotonous in $\alpha$. As mentioned in the main
text, the existence of a negative regime also proves that the mutual
information violates the nonincrease under local operations, because tracing
out the $A$ system is a local quantum operation that increases
$I_{\alpha}(A:B)$ to zero.

Figure 1: Plot of the Rényi mutual information $I_{\alpha}(A:B)$ (Eq. (3)) for
a classical Ising chain in the limit $L\to\infty$ (Eq. (42)). (left) As a
function of $J$ for various values of the field $h$ with $\alpha=2$ fixed. For
sufficiently small values of $J$ the mutual information becomes negative.
(right) As a function of $\alpha$ for $J=0.6$ and $h=1$, demonstrating that
the mutual information is not a monotonous function of $\alpha$.

##  Appendix C Technical proofs of main results

###  C.1 Optimizer for the measured Rényi-divergence with $\alpha=2$

For $\alpha=2$ the expression Eq. (8) for the measured Rényi divergence
becomes

$D^{\mathbb{M}}_{2}(\rho\|\sigma)=\log\sup_{\omega>0}2\mathrm{Tr}\left[\rho\omega\right]-\mathrm{Tr}\left[\sigma\omega^{2}\right]$
(43)

with $\sigma$ a state with full support. From [50], it is known that this
expression has an optimizer. Here, we prove the explicit expression Eq. (9)
for the optimizer. We denote the target function by
$f(\omega)=2\mathrm{Tr}\left[\rho\omega\right]-\mathrm{Tr}\left[\sigma\omega^{2}\right]$.
We can extend the supremum over all hermitian operators, which does not change
its value, because for any positive semidefinite operators $\omega^{+}$,
$\omega^{-}$ we have $f(\omega^{+}-\omega^{-})\leq f(\omega^{+})$.

Given the optimizer $\omega$, the linear term of $f(\omega+\delta\omega)$ in
$\delta\omega$ must vanish as $f$ is differentiable.

$\displaystyle f(\omega+\delta\omega)-f(\omega)$
$\displaystyle=2\mathrm{Tr}[\rho\delta\omega]-\mathrm{Tr}[\sigma\omega\delta\omega+\sigma\delta\omega\omega]+\mathcal{O}(\delta\omega^{2})$
(44)
$\displaystyle=\mathrm{Tr}[(2\rho-\sigma\omega-\omega\sigma)\delta\omega]+\mathcal{O}(\delta\omega^{2})$

This vanishes for any $\delta\omega$ if and only if
$2\rho-\sigma\omega-\omega\sigma=0$. The solution to this linear equation can
either be written using the invertible superoperator
$\Phi(\omega)=\sigma\omega+\omega\sigma$ as

$\omega=\frac{1}{2}\Phi^{-1}(\rho)$ (45)

or in vectorized notation with the mapping $|i\rangle\langle
j|\mapsto|i\rangle|j\rangle$ as

$|\omega\rangle=\frac{1}{2}\frac{1}{\sigma\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma}|\rho\rangle,$
(46)

where also the invertibility of $\Phi$ for a $\sigma$ with full rank becomes
apparent.

###  C.2 Proof of Lemma 1

The strategy we will follow in order to achieve an upper bound on
$D_{\infty}$, is to first prove the following $\alpha$-independent bound for
the geometric divergence,

$\widehat{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})\leq\log\left(\|E_{\beta/2}\|^{2}\|E_{\beta/2}^{-1}\|^{4}\frac{Z}{Z_{A}Z_{B}}\right)$
(47)

with $\rho=\exp(-\beta H)/Z$, $\rho_{A}=\exp(-\beta H_{A})/Z_{A}$ and
$\rho_{B}=\exp(-\beta H_{B})/Z_{B}$. Then one can take the trivial limit
$\alpha\to\infty$ and deduce that the same bound holds for $D_{\infty}$

We start with the argument of the logarithm in $\widehat{D}$ and assume
$\alpha>1$ to be an integer:

$\displaystyle\mathrm{Tr}\left[(\rho_{A}\otimes\rho_{B})\left((\rho_{A}\otimes\rho_{B})^{-1/2}\rho_{AB}(\rho_{A}\otimes\rho_{B})^{-1/2}\right)^{\alpha}\right]$
$\displaystyle=\mathrm{Tr}\left[(\rho_{AB})\left((\rho_{AB})^{1/2}(\rho_{A}\otimes\rho_{B})^{-1}(\rho_{AB})^{1/2}\right)^{\alpha-1}\right]$
(48)
$\displaystyle\leq\left\|(\rho_{AB})^{1/2}(\rho_{A}\otimes\rho_{B})^{-1}(\rho_{AB})^{1/2}\right\|^{\alpha-1}$

We used cyclicity of the trace, submultiplicativity of the operator norm, and
the inequality

$\mathrm{Tr}[AB]=\mathrm{Tr}[B^{1/2}AB^{1/2}]=\|B^{1/2}AB^{1/2}\|_{1}\leq\|B^{1/2}\|^{2}\|A\|_{1}=\|B\|\mathrm{Tr}[A]$
(49)

using Hölder’s inequality for positive operators $A$ and $B$. From (48), we
continue with

$\displaystyle\mathrm{Tr}$
$\displaystyle\left[(\rho_{A}\otimes\rho_{B})((\rho_{A}\otimes\rho_{B})^{-1/2}\rho_{AB}(\rho_{A}\otimes\rho_{B})^{-1/2})^{\alpha}\right]$
(50) $\displaystyle\leq\|e^{-\beta
H/2}e^{\beta(H_{A}+H_{B})/2}\|^{2(\alpha-1)}\|e^{-\beta(H_{A}+H_{B})/2}(\rho_{A}\otimes\rho_{B})^{-1}\frac{1}{Z_{A}Z_{B}}e^{-\beta(H_{A}+H_{B})/2}\|^{\alpha-1}\left(\frac{Z_{A}Z_{B}}{Z}\right)^{\alpha-1}$
$\displaystyle=\|e^{-\beta
H/2}e^{\beta(H_{A}+H_{B})/2}\|^{2(\alpha-1)}\|(e^{\beta
H_{A}/2}\rho_{A}e^{\beta H_{A}/2}Z_{A})^{-1}\|^{\alpha-1}\|(e^{\beta
H_{B}/2}\rho_{B}e^{\beta
H_{B}/2}Z_{B})^{-1}\|^{\alpha-1}\left(\frac{Z_{A}Z_{B}}{Z}\right)^{\alpha-1}.$

The first norm is already of the desired form. For the second and third norms,
we write

$\displaystyle\|(e^{\beta H_{A}/2}\rho_{A}e^{\beta H_{A}/2}Z_{A})^{-1}\|$
$\displaystyle=\|\mathrm{Tr}_{B}[e^{\beta H_{A}/2}\rho e^{\beta
H_{A}/2}Z_{A}]^{-1}\|$ (51)
$\displaystyle=\|\mathrm{Tr}_{B}[e^{\beta(H_{A}+H_{B})/2}e^{-\beta
H_{AB}}e^{\beta(H_{A}+H_{B})/2}\frac{e^{-\beta
H_{B}}}{Z_{B}}\frac{Z_{A}Z_{B}}{Z}]^{-1}\|$
$\displaystyle=\|\mathrm{Tr}_{B}[e^{\beta(H_{A}+H_{B})/2}e^{-\beta
H_{AB}}e^{\beta(H_{A}+H_{B})/2}\frac{e^{-\beta
H_{B}}}{Z_{B}}]^{-1}\|\frac{Z}{Z_{A}Z_{B}},$

where we used the cyclicity of the partial trace on $B$ with respect to
operators supported only in $B$. The operator norm is just the inverse of the
smallest eigenvalue of the partial trace. Let $\ket{\psi_{A}}$ be the
corresponding eigenvector on $A$ to this eigenvalue and $p_{i}$,
$|\phi_{i}\rangle$ an eigensystem of $\exp(-\beta H_{B})/Z_{B}$. Then we get

$\displaystyle\bra{\psi_{A}}$
$\displaystyle\mathrm{Tr}_{B}[e^{\beta(H_{A}+H_{B})/2}e^{-\beta
H_{AB}}e^{(\beta H_{A}+H_{B})/2}\rho_{B}]\ket{\psi_{A}}$ (52)
$\displaystyle=\bra{\psi_{A}}\sum_{i}p_{i}\bra{\phi_{i}}e^{\beta(H_{A}+H_{B})/2}e^{-\beta
H_{AB}}e^{\beta(H_{A}+H_{B})/2}\ket{\phi_{i}}\ket{\psi_{A}}$
$\displaystyle=\sum_{i}p_{i}\bra{\psi_{A}}\bra{\phi_{i}}e^{\beta(H_{A}+H_{B})/2}e^{-\beta
H_{AB}}e^{\beta(H_{A}+H_{B})/2}\ket{\phi_{i}}\ket{\psi_{A}}$
$\displaystyle\geq\sum_{i}p_{i}\|e^{-\beta(H_{A}+H_{B})/2}e^{\beta
H_{AB}}e^{-\beta(H_{A}+H_{B})/2}\|^{-1}$
$\displaystyle=\|e^{-\beta(H_{A}+H_{B})/2}e^{\beta
H_{AB}}e^{-\beta(H_{A}+H_{B})/2}\|^{-1}$

by bounding an expectation value of a positive operator by its minimum
eigenvalue. Inserting this into (51) we get

$\|(e^{\beta H_{A}/2}\rho_{A}e^{\beta
H_{A}/2}Z_{A})^{-1}\|\leq\|e^{-\beta(H_{A}+H_{B})/2}e^{\beta
H_{AB}}e^{-\beta(H_{A}+H_{B})/2}\|^{-1}\frac{Z}{Z_{A}Z_{B}}$ (53)

and combining (50) with (53) yields

$\displaystyle\mathrm{Tr}[(\rho_{A}\otimes\rho_{B})((\rho_{A}\otimes\rho_{B})^{-1/2}\rho_{AB}(\rho_{A}\otimes\rho_{B})^{-1/2})^{\alpha}]\leq\|E_{\beta/2}\|^{2(\alpha-1)}\|(E_{\beta/2})^{-1}\|^{4(\alpha-1)}\left(\frac{Z}{Z_{A}Z_{B}}\right)^{\alpha-1},$
(54)

which completes the proof of (47) for any integer $\alpha>1$. The previous
result extends to any $\alpha>1$ by rounding up to the next integer because of
the monotonicity of $\widehat{D}_{\alpha}$ in $\alpha$. The bound (47) also
holds for $\overline{D}_{\alpha}$ in the range $\alpha\in(0,1)\cup(1,2]$ and
for $\widetilde{D}_{\alpha}$ and $\alpha\in(0,1)\cup(1,\infty)$ just by the
inequalities between these Rényi divergences.

Finally, to bound the ratio of partition functions $Z/Z_{A}Z_{B}$, we repeat a
simple proof from [69], Lemma 3.6:

$\displaystyle\left|\log\mathrm{Tr}\left[e^{-\beta(H_{A}+H_{B})}\right]-\log\mathrm{Tr}\left[e^{-\beta(H_{A}+H_{B}+H_{I})}\right]\right|$
$\displaystyle=\left|\int_{0}^{1}\frac{d}{dt}\log\mathrm{Tr}\left[e^{-\beta(H_{A}+H_{B})-t\beta
H_{I}}\right]dt\right|$ (55)
$\displaystyle\leq\int_{0}^{1}\left|\frac{\mathrm{Tr}\left[-\beta
H_{I}e^{-\beta(H_{A}+H_{B}+tH_{I})}\right]}{\mathrm{Tr}\left[e^{-\beta(H_{A}+H_{B}+tH_{I})}\right]}\right|dt\leq\beta\|H_{I}\|$

Together, (47) and (55) give the desired bound which also directly proves the
bound for $D_{\infty}$ as the right-hand side does not depend on $\alpha$ and
the limit $\alpha\rightarrow\infty$ can trivially be taken.

###  C.3 Proof of Lemma 3

We restate the lemma from the main text. It features a thermal Lieb-Robinson
bound originally due to Ruelle [70], which has previously appeared in [53, 54,
55, 56]:

Lemma 3. On a $D$-dimensional lattice, we have

$\log(\|E_{\beta}\|)\leq-\log(1-2\beta Jk)|\partial A|/(2k)$ (56)

if $2\beta Jk<1$ and the same bound holds for $\|E_{\beta}^{-1}\|$.

As explained in the main text, we start from the expression in (17) and get,
using the Baker-Campbell-Hausdorff formula,

$\|e^{x(H_{A}+H_{B})}H_{I}e^{-x(H_{A}+H_{B})}\|\leq\sum_{m=0}^{\infty}\frac{x^{m}}{m!}\text{ad}_{H_{A}+H_{B}}^{m}(H_{I})$
(57)

with the adjoint action $\text{ad}_{Y}(X)=[Y,X]$, whose powers yield nested
commutators. We bound them by Lemma 3 from Ref. [56]

$\|\text{ad}_{H_{A}+H_{B}}^{m}(H_{I})\|\leq\sum_{i\in
C}\|\text{ad}_{H_{A}+H_{B}}^{m}(h_{i})\|\leq\sum_{i\in
C}\prod_{j=1}^{m}2Jkj\|h_{i}\|\leq J|\partial A|(2Jk)^{m}m!$ (58)

with $C=\\{i:\mathrm{supp}(h_{i})\cap A\neq\emptyset,\mathrm{supp}(h_{i})\cap
B\neq\emptyset\\}$. The inequality $\sum_{i\in C}\|h_{i}\|\leq J|\partial A|$
follows from the definitions. We insert (58) into (57) and evaluate the
geometric series

$\|e^{x(H_{A}+H_{B})}H_{I}e^{-x(H_{A}+H_{B})}\|\leq\sum_{m=0}^{\infty}(2xJk)^{m}J|\partial
A|=\frac{J|\partial A|}{1-2xJk},$ (59)

which holds if $2xJk<1$. Finally, inserting this into the integral (17) we
obtain

$\log\|E_{\beta}\|\leq\int_{0}^{\beta}\frac{J|\partial
A|}{1-2xJk}dx=-\log(1-2\beta Jk)|\partial A|/(2k)$ (60)

if $2\beta Jk<1$, which finishes the proof. If we replace $E_{\beta}$ with
$E_{\beta}^{-1}$, the proof can be repeated replacing $H_{A}+H_{B}$ with
$H=H_{A}+H_{B}+H_{I}$ and $H_{I}$ with $-H_{I}$.

###  C.4 Proof of Theorem 4

The proof of the classical area law for $\alpha\leq 2$

$D_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})\leq(|\partial A|+|\partial
B|)\log d$ (61)

is split into two steps. First, we show that the mutual information between
$A$ and $B$ equals the mutual information between the boundaries and second,
we give a dimension-dependent bound for the mutual information. We denote by
$X_{A}$ the random variable of configurations on the system $A$ and use
$A^{\circ}=A\setminus\partial A$. We prove the first identity

$\displaystyle D_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{A^{\circ}},x_{\partial
A},x_{B^{\circ}},x_{\partial B}}P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial
A}=x_{\partial A},X_{B^{\circ}}=x_{B^{\circ}},X_{\partial B}=x_{\partial B})$
(62) $\displaystyle\quad\left(\frac{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial
A}=x_{\partial A},X_{B^{\circ}}=x_{B^{\circ}},X_{\partial B}=x_{\partial
B})}{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A})P(X_{B^{\circ}}=X_{B^{\circ}},X_{\partial B}=x_{\partial
B})}\right)^{\alpha-1}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{A^{\circ}},x_{\partial
A},x_{B^{\circ}},x_{\partial B}}P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial
A}=x_{\partial A},X_{B^{\circ}}=x_{B^{\circ}},X_{\partial B}=x_{\partial B})$
$\displaystyle\quad\text{
}\left(\frac{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A}|X_{B^{\circ}}=x_{B^{\circ}},X_{\partial B}=x_{\partial
B})}{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A})}\right)^{\alpha-1}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{A^{\circ}},x_{\partial
A},x_{\partial B}}P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A},X_{\partial B}=x_{\partial B})$ $\displaystyle\quad\text{
}\left(\frac{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A}|X_{\partial B}=x_{\partial B})}{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial
A}=x_{\partial A})}\right)^{\alpha-1}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{A^{\circ}},x_{\partial
A},x_{\partial B}}P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A},X_{\partial B}=x_{\partial B})$ $\displaystyle\quad\text{
}\left(\frac{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial A}=x_{\partial
A},X_{\partial B}=x_{\partial B})}{P(X_{A^{\circ}}=x_{A^{\circ}},X_{\partial
A}=x_{\partial A})P(X_{\partial B}=x_{\partial B})}\right)^{\alpha-1}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{\partial A},x_{\partial
B}}P(X_{\partial A}=x_{\partial A},X_{\partial B}=x_{\partial
B})\left(\frac{P(X_{\partial A}=x_{\partial A},X_{\partial B}=x_{\partial
B})}{P(X_{\partial A}=x_{\partial A})P(X_{\partial B}=x_{\partial
B})}\right)^{\alpha-1}$ $\displaystyle=D_{\alpha}(\rho_{\partial A\partial
B}\|\rho_{\partial A}\otimes\rho_{\partial B}),$

where the second to last line comes from repeating all previous steps on the
$A$ side.

For the next step we introduce the short-hand notation
$p(x_{A},x_{B})=P(X_{\partial A}=x_{A},X_{\partial B}=x_{B})$ and use the
dimension $D_{\partial A\partial B}$ of the system supported over $\partial
A\cup\partial B$ to find the following bound:

$\displaystyle D_{\alpha}(\rho_{\partial A\partial B}\|\rho_{\partial
A}\otimes\rho_{\partial B})$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{A},x_{B}}p(x_{A},x_{B})\left(\frac{p(x_{A},x_{B})}{p(x_{A})p(x_{B})}\right)^{\alpha-1}$
(63)
$\displaystyle\leq\frac{1}{\alpha-1}\log\sum_{x_{A},x_{B}}p(x_{A},x_{B})\left(\frac{p(x_{A},x_{B})}{p(x_{A},x_{B})p(x_{A},x_{B})}\right)^{\alpha-1}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{x_{A},x_{B}}p(x_{A},x_{B})^{2-\alpha}$
$\displaystyle=S_{2-\alpha}(\rho_{\partial A\partial B})$
$\displaystyle\leq\log D_{\partial A\partial B}=(|\partial A|+|\partial
B|)\log d$

The last inequality uses the Schur concavity of Rényi entropies and holds for
$\alpha\in(0,1)\cup(1,2]$.

At this point, it might be natural to wonder if the above area law for
classical thermal states can be extended to the case of $\alpha>2$. The
following counterexample rules out a temperature-independent area law in this
range. We take two bits $x_{A},x_{B}\in\\{0,1\\}$ and choose the probability
distribution $\mathrm{diag}(\epsilon,0,0,1-\epsilon)$ for a constant
$\epsilon$ written as a diagonal density matrix. The marginal probabilities
are then $p(0)=\epsilon$, $p(1)=1-\epsilon$ for both $A$ and $B$. We find that
the term for $x_{A}=x_{B}=0$ in the sum in the first line of (63) reads
$\epsilon^{(2-\alpha)}$ and becomes arbitrarily large for sufficiently small
$\epsilon$.

Strictly speaking, this example is no thermal state as it contains zero
probabilities, but one can choose a sequence of thermal states for every fixed
$\epsilon$ that converges to our example.

###  C.5 Rényi mutual information for pure states and proof of Theorem 5

We prove an extended version of Theorem 5 from the main text as we obtain the
additional statements directly from our proof:

######  Theorem 6.

For a PEPDO with local purification, bond dimension $D$, and $|\partial A|$
the number of bonds between $A$ and $B$, it holds

$\widetilde{I}_{\alpha}(A:B)\leq 2|\partial A|\log D$ (64)

for $\alpha\in[1/2,1)\cup(1,2]$,

$\widehat{I}_{\alpha}(A:B)\leq 2|\partial A|\log D$ (65)

for $\alpha\in(0,1)\cup(1,3/2]$ and

$I^{\mathbb{M}}_{\alpha}(A:B)\leq 2|\partial A|\log D$ (66)

for $\alpha\in(0,1)\cup(1,2]$.

As already explained in the main text, we can, by the local purification
assumption, restrict to PEPS due to the nonincrease of the mutual information
under local operations, and we prove

$\widetilde{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})=2S_{2/\alpha-1}(\rho_{A})$
(67)

for a pure state $\rho_{AB}=|\psi\rangle\\!\langle\psi|$ and
$\alpha\in(0,1)\cup(1,2)$. Since all Rényi entropies are bounded by the
logarithm of the dimension, this gives (64). The case of the measured Rényi
mutual information then follows from the data-processing inequality, which
holds for $\alpha\geq 1/2$. For the regime of $\alpha<1/2$, we use the same
argument together with the Petz-Rényi divergence, which fulfills the data-
processing inequality for $\alpha\in(0,1)\cup(1,2]$, and we prove the relation

$\overline{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})=2S_{3-2\alpha}(\rho_{A})$
(68)

for $\alpha\in(0,1)\cup(1,3/2)$. The following proofs of (67) and (68)
resemble a proof from [17], where analogous relations are given for
$I_{\alpha}^{opt}(A:B)$ (see (31)), which even show that Theorem 5 can be
extended to $I_{\infty}^{opt}(A:B)$ as they hold for all $\alpha$.

We start with the sandwiched Rényi divergence and use the Schmidt
decomposition

$|\psi\rangle=\sum_{i}\sqrt{\lambda_{i}}|e_{i}\rangle|f_{i}\rangle,$ (69)

which also yields a representation of the reduced density matrix

$\rho_{A}=\sum_{i}\lambda_{i}|e_{i}\rangle\\!\langle e_{i}|.$ (70)

Additionally, we define

$\displaystyle|\zeta\rangle$
$\displaystyle=\left(\rho_{A}^{(1-\alpha)/2\alpha}\otimes\rho_{B}^{(1-\alpha)/2\alpha}\right)|\psi\rangle$
(71)
$\displaystyle=\sum_{i,j,k}\lambda_{i}^{(1-\alpha)/2\alpha}|e_{i}\rangle\\!\langle
e_{i}|\otimes\lambda_{j}^{(1-\alpha)/2\alpha}|f_{j}\rangle\\!\langle
f_{j}|\lambda_{k}^{1/2}|e_{k}\rangle|f_{k}\rangle$
$\displaystyle=\sum_{i}\lambda_{i}^{\frac{2-\alpha}{2\alpha}}|e_{i}\rangle|f_{i}\rangle,$

which we can use to rewrite

$\displaystyle\widetilde{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$
$\displaystyle=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[\left(\left(\rho_{A}^{(1-\alpha)/2\alpha}\otimes\rho_{B}^{(1-\alpha)/2\alpha}\right)\rho_{AB}\left(\rho_{A}^{(1-\alpha)/2\alpha}\otimes\rho_{B}^{(1-\alpha)/2\alpha}\right)\right)^{\alpha}\right]$
(72)
$\displaystyle=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[|\zeta\rangle\langle\zeta|^{\alpha}\right]$
$\displaystyle=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[\left(\frac{|\zeta\rangle\langle\zeta|}{\langle\zeta|\zeta\rangle}\right)^{\alpha}\right]+\frac{\alpha}{\alpha-1}\log\langle\zeta|\zeta\rangle$
$\displaystyle=\frac{\alpha}{\alpha-1}\log\langle\zeta|\zeta\rangle$
$\displaystyle=\frac{\alpha}{\alpha-1}\log\sum\lambda_{i}^{\frac{2-\alpha}{\alpha}}$
$\displaystyle=2S_{2/\alpha-1}(\rho_{A}).$

This concludes the proof of (67).

We prove the relation for the Petz-Rényi divergence with a similar calculation
using the same Schmidt decomposition of $|\psi\rangle$ and

$\displaystyle|\eta\rangle$
$\displaystyle=\left(\rho_{A}^{(1-\alpha)/2}\otimes\rho_{B}^{(1-\alpha)/2}\right)|\psi\rangle$
(73)
$\displaystyle=\sum_{i,j,k}\lambda_{i}^{(1-\alpha)/2}|e_{i}\rangle\\!\langle
e_{i}|\otimes\lambda_{j}^{(1-\alpha)/2}|f_{j}\rangle\\!\langle
f_{j}|\lambda_{k}^{1/2}|e_{k}\rangle|f_{k}\rangle$
$\displaystyle=\sum_{i}\lambda_{i}^{\frac{3}{2}-\alpha}|e_{i}\rangle|f_{i}\rangle,$

and obtain

$\displaystyle\widehat{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$
$\displaystyle=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[\left(\rho_{A}^{(1-\alpha)/2}\otimes\rho_{B}^{(1-\alpha)/2}\right)\rho_{AB}^{\alpha}\left(\rho_{A}^{(1-\alpha)/2}\otimes\rho_{B}^{(1-\alpha)/2}\right)\right]$
(74)
$\displaystyle=\frac{1}{\alpha-1}\log\mathrm{Tr}\left[|\eta\rangle\langle\eta|\right]$
$\displaystyle=\frac{1}{\alpha-1}\log\langle\eta|\eta\rangle$
$\displaystyle=\frac{1}{\alpha-1}\log\sum\lambda_{i}^{3-2\alpha}$
$\displaystyle=2S_{3-2\alpha}(\rho_{A}).$

in the range $\alpha\in(0,1)\cup(1,3/2)$. This closes the proof of (68) and
thereby of theorem 5.

To show that the range of $\alpha$ cannot be extended we give a simple
counterexample of two qubits, which have bond dimension $D=2$ and local
dimension $d=2$. The state
$|\psi\rangle=\sqrt{\epsilon}\ket{0}\ket{0}+\sqrt{1-\epsilon}|1\rangle|1\rangle$
with a measurement in the computational basis gives the probability
distributions $\mathrm{diag}(\epsilon,0,0,1-\epsilon)$ and
$\mathrm{diag}(\epsilon^{2},\epsilon-\epsilon^{2},\epsilon-\epsilon^{2},(1-\epsilon)^{2})$
for the full system and the product of marginals respectively. We already
showed the divergence of the mutual information of this distribution in
Appendix C.4.

###  C.6 Proof of Lemma 4

We prove a generalization of Pinsker’s inequality by adapting a proof for the
relative entropy from [61]. We use the measured Rényi divergence, which is not
greater than any Rényi divergence in the range where the non-measured fulfills
the data-processing inequality. For two quantum states $\rho$ and $\sigma$,
define $\phi^{+}$ to be the projector on the non-negative subspace of
$\rho-\sigma$. Then, we have

$\|\rho-\sigma\|_{1}=\mathrm{Tr}[(\rho-\sigma)\phi^{+}]-\mathrm{Tr}[(\rho-\sigma)(\mathbbm{1}-\phi^{+})]=2(\mathrm{Tr}[\rho\phi^{+}]-\mathrm{Tr}[\sigma\phi^{+}]).$
(75)

On the other hand, the supremum of the measured Rényi divergence can be lower
bounded by a particular measurement, which we choose to be
$\phi^{+},\mathbbm{1}-\phi^{+}$:

$D_{\alpha}^{\mathbb{M}}(\rho\|\sigma)=\sup_{(\chi,M)}(P_{\rho,M}\|P_{\sigma,M})\geq
D_{\alpha}\big{(}(\mathrm{Tr}[\rho\phi^{+}],\mathrm{Tr}[\rho(\mathbbm{1}-\phi^{+})])\|(\mathrm{Tr}[\sigma\phi^{+}],\mathrm{Tr}[\sigma(\mathbbm{1}-\phi^{+})])\big{)},$
(76)

where the latter expression is to be understood as the Rényi divergence of the
binary probability distribution. Using the total variation distance of this
binary distribution $V=2\mathrm{Tr}[(\rho-\sigma)\phi^{+}]$ we can apply the
classical Pinsker inequality $D_{\alpha}\geq\alpha V^{2}/2$ for
$\alpha\in(0,1)$ from [62] to obtain

$D_{\alpha}^{\mathbb{M}}(\rho\|\sigma)\geq\frac{\alpha}{2}V^{2}=\frac{\alpha}{2}\|\rho-\sigma\|_{1}^{2}$
(77)

and

$D_{\alpha}^{\mathbb{M}}(\rho\|\sigma)\geq\frac{1}{2}\|\rho-\sigma\|_{1}^{2}$
(78)

for $\alpha>1$ by the monotonicity of $D_{\alpha}^{\mathbb{M}}$.


$\displaystyle\frac{\left|C_{S}\right|^{2}+\left|C_{S}^{{}^{\prime}}\right|^{2}}{2}\left(1-\vec{\beta}\cdot\hat{\nu}\right),$
(89a) $\displaystyle\frac{\Omega^{2}}{2}\sum_{\mbox{lepton
spins}}l_{0}^{V}l^{S*}$ $\displaystyle=$
$\displaystyle\pm\frac{C_{V}C_{S}^{*}+C_{V}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2}\frac{m_{e}}{E},$
(89b) $\displaystyle\frac{\Omega^{2}}{2}\sum_{\mbox{lepton
spins}}l_{0}^{A}l^{S*}$ $\displaystyle=$
$\displaystyle\pm\frac{C_{A}^{{}^{\prime}}C_{S}^{*}+C_{A}C_{S}^{{}^{\prime}*}}{2}\frac{m_{e}}{E},$
(89c) $\displaystyle-\frac{\Omega^{2}}{2}\sum_{\mbox{lepton
spins}}l_{3}^{V}l^{S*}$ $\displaystyle=$
$\displaystyle\pm\frac{C_{V}C_{S}^{*}+C_{V}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2}\left(\hat{\nu}\cdot\hat{q}\right)\frac{m_{e}}{E},$
(89d) $\displaystyle-\frac{\Omega^{2}}{2}\sum_{\mbox{lepton
spins}}l_{3}^{A}l^{S*}$ $\displaystyle=$
$\displaystyle\pm\frac{C_{A}^{{}^{\prime}}C_{S}^{*}+C_{A}C_{S}^{{}^{\prime}*}}{2}\left(\hat{\nu}\cdot\hat{q}\right)\frac{m_{e}}{E},$
(89e) $\displaystyle-\frac{\Omega^{2}}{2}\sum_{\mbox{lepton
spins}}l_{3}^{T}l^{S*}$ $\displaystyle=$ $\displaystyle\mp
i\sqrt{2}\frac{C_{T}^{{}^{\prime}}C_{S}^{*}+C_{T}C_{S}^{{}^{\prime}*}}{2}\hat{q}\cdot\left(\hat{\nu}-\vec{\beta}\right),$
(89f) $\displaystyle-\frac{\Omega^{2}}{2}\sum_{\mbox{lepton
spins}}l_{3}^{T^{\prime}}l^{S*}$ $\displaystyle=$ $\displaystyle\pm
i\sqrt{2}\frac{C_{T}C_{S}^{*}+C_{T}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2}\hat{q}\cdot\left(\hat{\nu}-\vec{\beta}\right).$
(89g)

Pseudoscalar traces, originating from the pseudoscalar leptonic current,
$j^{P}\equiv\bar{\psi}^{\prime}\left(\vec{r}\right)\left(C_{P}\gamma_{5}-C_{P}^{{}^{\prime}}\right)$
$\times\psi\left(\vec{r}\right)$ (compare to the scalar leptonic current in
Eq. (82)), with
$l^{P}=\frac{1}{\Omega}\bar{l}^{\prime}\left(\vec{k}^{\prime}\right)\left(C_{P}\gamma_{5}-C_{P}^{{}^{\prime}}\right)$
$\times l\left(\vec{k}\right)$, will be similar, replacing the coefficients
$C_{S}$ and $C_{S}^{{}^{\prime}}$ with the coefficients $-C_{P}^{{}^{\prime}}$
and $-C_{P}$, respectfully.

In order to match the lepton traces terms to the quark-level effective theory
one-nucleon matrix elements, we replace
$C_{\text{sym}}^{\left({}^{\prime}\right)}$ with the adjust coefficients
$\frac{C_{\text{sym}}^{\left({}^{\prime}\right)}}{g_{\text{sym}}}$
($\text{sym}\in\left\\{S,P,V,A,T\right\\}$), as we did before. Taking into
account also the parity selection rules, one can finally write a scalar and
pseudoscalar expression:

$\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)^{S,P}=\sum_{J=0}^{\infty}\left\\{\left(1-\vec{\beta}\cdot\hat{\nu}\right)\left[\frac{\left|C_{S}\right|^{2}+\left|C_{S}^{{}^{\prime}}\right|^{2}}{2g_{S}^{2}}\left|\left\langle\left\|\hat{C}_{J}^{S}\right\|\right\rangle\right|^{2}+\frac{\left|C_{P}\right|^{2}+\left|C_{P}^{{}^{\prime}}\right|^{2}}{2g_{P}^{2}}\left|\left\langle\left\|\hat{C}_{J}^{P}\right\|\right\rangle\right|^{2}\right]\right.\\\
\pm
2\frac{m_{e}}{E}\mathfrak{Re}\left[\frac{C_{V}C_{S}^{*}+C_{V}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2g_{V}g_{S}^{*}}\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{S}\right\|\right\rangle^{*}-\frac{C_{A}^{{}^{\prime}}C_{P}^{{}^{\prime}*}+C_{A}C_{P}^{*}}{2g_{A}g_{P}^{*}}\left\langle\left\|\hat{C}_{J}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{P}\right\|\right\rangle^{*}\right]\\\
\pm
2\frac{m_{e}}{E}\left(\hat{\nu}\cdot\hat{q}\right)\mathfrak{Re}\left[\frac{C_{V}C_{S}^{*}+C_{V}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2g_{V}g_{S}^{*}}\left\langle\left\|\hat{L}_{J}^{V}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{S}\right\|\right\rangle^{*}-\frac{C_{A}^{{}^{\prime}}C_{P}^{{}^{\prime}*}+C_{A}C_{P}^{*}}{2g_{A}g_{P}^{*}}\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{P}\right\|\right\rangle^{*}\right]\\\
\pm
2\sqrt{2}\hat{q}\cdot\left(\hat{\nu}-\vec{\beta}\right)\mathfrak{Re}\left[i\frac{C_{T}C_{S}^{*}+C_{T}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2g_{T}g_{S}^{*}}\left\langle\left\|\hat{L}_{J}^{T^{\prime}}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{S}\right\|\right\rangle^{*}\right.\\\
\left.\left.+i\frac{C_{T}^{{}^{\prime}}C_{P}^{{}^{\prime}*}+C_{T}C_{P}^{*}}{2g_{T}g_{P}^{*}}\left\langle\left\|\hat{L}_{J}^{T}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{P}\right\|\right\rangle^{*}\right]\right\\}.$
(90)

In order to complete this discussion, one needs to calculate the scalar and
pseudoscalar multipole operators. Starting from the hadronic currents:

$\displaystyle\hat{\mathcal{J}}^{S}\left(\vec{r}\right)$ $\displaystyle=$
$\displaystyle\bar{\phi}^{\prime}\left(\vec{r}\right)\phi\left(\vec{r}\right),$
(91a) $\displaystyle\hat{\mathcal{J}}^{P}\left(\vec{r}\right)$
$\displaystyle=$
$\displaystyle\bar{\phi}^{\prime}\left(\vec{r}\right)\gamma_{5}\phi\left(\vec{r}\right),$
(91b)

the general form of the single-nucleon matrix element of the scalar and
pseudoscalar parts of the charge changing weak current, are [24]:

$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\hat{\mathcal{J}}^{S}\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\Omega}g_{S}\left(q^{2}\right)\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right)+\mathcal{O}\left(\epsilon_{\text{recoil}}^{2}\right),$
(92a)
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\hat{\mathcal{J}}^{P}\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\Omega}g_{P}\left(q^{2}\right)\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\gamma_{5}\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right)+\mathcal{O}\left(\epsilon_{\text{recoil}}^{2}\right),$
(92b)

with $\epsilon_{\text{recoil}}\sim\frac{q}{m_{N}}$ ($\approx 0.002$ for an
endpoint of $\approx 2\text{MeV}$). Expanding the needed matrix elements in
the inverse mass (following what we did to the tensor matrix element at
appendix B), one can find the following non-relativistic expansion for the
matrix elements of the scalar and pseudoscalar nuclear currents:

$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\mathcal{J}^{S}\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\Omega}g_{S}\left(q^{2}\right)\chi_{\sigma^{\prime}}^{+}\eta_{\rho^{\prime}}^{+}\tau_{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\left(\epsilon_{\text{NR}}^{2}\right),$
(93a)
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\mathcal{J}^{P}\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\Omega}g_{P}\left(q^{2}\right)\chi_{\sigma^{\prime}}^{+}\eta_{\rho^{\prime}}^{+}\frac{\vec{\sigma}\cdot\vec{q}}{2m_{N}}\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\left(\epsilon_{\text{NR}}^{2}\right),$
(93b)

with $\epsilon_{\text{NR}}\sim\frac{P_{\text{fermi}}}{m_{N}}\approx 0.2$. As
we did in Appendix B, we use the definition of the (second quantization)
$\mathcal{J}\left(\vec{r}\right)$ current matrix element as a sum over first-
quantization currents $\hat{\mathcal{J}}^{\left(1\right)}$, identify the
nuclear density operators in first quantization:

$\displaystyle\hat{\mathcal{J}}^{S\left(1\right)}\left(0\right)$
$\displaystyle=$ $\displaystyle
g_{S}\left(q^{2}\right)\tau^{\pm}+\mathcal{O}\left(\epsilon_{\text{NR}}^{2}\right),$
(94a) $\displaystyle\hat{\mathcal{J}}^{P\left(1\right)}\left(0\right)$
$\displaystyle=$ $\displaystyle
g_{P}\left(q^{2}\right)\frac{\vec{\sigma}\cdot\vec{q}}{2m_{N}}\tau^{\pm}+\mathcal{O}\left(\epsilon_{\text{NR}}^{2}\right)\text{,}$
(94b)

and get the (second quantization) current densities:

$\displaystyle\mathcal{J}^{S}\left(\vec{r}\right)$ $\displaystyle=$
$\displaystyle
g_{S}\sum_{j=1}^{A}\tau_{j}^{\pm}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)+\mathcal{O}\left(\epsilon_{\text{NR}}^{2}\right),$
(95a) $\displaystyle\mathcal{J}^{P}\left(\vec{r}\right)$ $\displaystyle=$
$\displaystyle-\frac{i}{2m_{N}}g_{P}\sum_{j=1}^{A}\vec{\nabla}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{\text{NR}}^{2}\right)\text{.}$
(95b)

Using the current densities, the multipole operators (Eq. (9) in the main
text) can be written as a sum of one-body operators. The multipole operators,
calculated with the scalar and pseudoscalar symmetry contributions to the weak
nuclear current, will be:

$\displaystyle\hat{C}_{J}^{S}\left(q\right)$ $\displaystyle=$
$\displaystyle\frac{g_{S}}{g_{V}}\hat{C}_{J}^{V}\left(q\right)+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
(96a) $\displaystyle\hat{C}_{J}^{P}\left(q\right)$ $\displaystyle=$
$\displaystyle\frac{iq}{2m_{N}}g_{P}\sum_{j=1}^{A}\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
(96b)

where
$\hat{C}_{J}^{V}\left(q\right)=g_{V}\sum_{j=1}^{A}M_{J}\left(q\vec{r}_{j}\right)\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right)$
is the SM polar-vector Coulomb multipole operator. The BSM LO expression will
be:

$\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)^{S,P}=\sum_{J=0}^{\infty}\left\\{\left(1-\vec{\beta}\cdot\hat{\nu}\right)\frac{\left|C_{S}\right|^{2}+\left|C_{S}^{{}^{\prime}}\right|^{2}}{2\left|g_{V}\right|^{2}}\left|\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\right|^{2}\right.\\\
\left.\pm\frac{m_{e}}{E}2\mathfrak{Re}\left[\frac{C_{V}C_{S}^{*}+C_{V}^{{}^{\prime}}C_{S}^{{}^{\prime}*}}{2\left|g_{V}\right|^{2}}\left(\left|\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\right|^{2}+\left(\hat{\nu}\cdot\hat{q}\right)\left\langle\left\|\hat{L}_{J}^{V}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle^{*}\right)\right]\right\\}.$
(97)

This is a general result which hold for any semileptonic nuclear process,
including different types of beyond the Standard Model physics. After
substituting Eq. (56), it yields the scalar and pseudoscalar terms (including
their interference Fierz terms) presented in Sec. IV.

##  Appendix E Second class nuclear currents and multipole operators

The general form of the single-nucleon matrix element, of the vector and axial
parts of the charge changing weak current are (respectively) [24]:

$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\mathcal{J}_{\mu}^{V}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\frac{1}{\Omega}\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\left[g_{V}\left(q^{2}\right)\gamma_{\mu}-i\frac{\tilde{g}_{T\left(V\right)}\left(q^{2}\right)}{2m_{N}}\sigma_{\mu\nu}q^{\nu}+\frac{\tilde{g}_{S}\left(q^{2}\right)}{2m_{N}}q_{\mu}\right]\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right),$
(98a)
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\mathcal{J}_{\mu}^{A}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\frac{1}{\Omega}\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\left[g_{A}\left(q^{2}\right)\gamma_{\mu}-i\frac{\tilde{g}_{T\left(A\right)}\left(q^{2}\right)}{2m_{N}}\sigma_{\mu\nu}q^{\nu}+\frac{\tilde{g}_{P}\left(q^{2}\right)}{2m_{N}}q_{\mu}\right]\gamma_{5}\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right).$
(98b)

All the form factors $g_{i}\left(q^{2}\right)$ are functions of $q^{2}$. In
the Standard Model, $g_{V}=1$, up to second-order corrections in isospin
breaking [27, 28], as a result of the conservation of the vector current, and
$g_{A}\approx 1.276g_{V}$ [38, 39]. The induced charges,
$\tilde{g}_{T\left(V\right)}$, $\tilde{g}_{S}$, $\tilde{g}_{T\left(A\right)}$
and $\tilde{g}_{P}$ (not to confuse with the actual BSM charges, $g_{S}$,
$g_{P}$ and $g_{T}$, which appear in the scalar, pseudoscalar and tensor
currents), are all proportional to
$\epsilon_{\text{recoil}}\equiv\frac{q}{m_{N}}$ [24]. $\tilde{g}_{S}$ and
$\tilde{g}_{T\left(A\right)}$, known as second class currents, do not exist in
the Standard Model, $\tilde{g}_{S}$ due to current conservation, and
$\tilde{g}_{T\left(A\right)}$ because of G-parity considerations [26].

As before, we substitute the explicit form of Dirac spinors and make a non-
relativistic expansion, to find the required matrix elements:

$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\mathcal{J}_{0}^{V}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\frac{1}{\Omega}\chi_{\sigma^{\prime}}^{+}\eta_{\rho^{\prime}}^{+}\left(g_{V}+\frac{E_{0}}{2m_{N}}\tilde{g}_{S}\right)\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
(99a)
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\vec{\mathcal{J}}^{V}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\frac{1}{\Omega}\chi_{\sigma^{\prime}}^{+}\eta_{\rho^{\prime}}^{+}\frac{1}{2m_{N}}\left[g_{V}\vec{P}+\left(g_{V}+\tilde{g}_{T\left(V\right)}\right)i\vec{q}\times\vec{\sigma}+\tilde{g}_{S}\vec{q}\right]\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
(99b)
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\mathcal{J}_{0}^{A}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\frac{1}{\Omega}\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\frac{1}{2m_{N}}\left[g_{A}\vec{P}\cdot\vec{\sigma}-\left(\tilde{g}_{T\left(A\right)}-\frac{E_{0}}{2m_{N}}\tilde{g}_{P}\right)\vec{q}\cdot\vec{\sigma}\right]\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right)$
$\displaystyle+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
(99c)
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\vec{\mathcal{J}}^{A}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\frac{1}{\Omega}\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\left(g_{A}\vec{\sigma}-\tilde{g}_{T\left(A\right)}\frac{E_{0}}{2m_{N}}\vec{\sigma}\right)\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right)+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)}\text{.}$
(99d)

We use the definition of the second quantization current matrix element to
identify the following currents:

$\displaystyle\mathcal{J}_{0}^{V}\left(\vec{r}\right)$
$\displaystyle=\sum_{j=1}^{A}\left[g_{V}+\frac{E_{0}}{2m_{N}}\tilde{g}_{S}\right]\tau_{j}^{\pm}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
(100a) $\displaystyle\vec{\mathcal{J}}^{V}\left(\vec{r}\right)$
$\displaystyle=\frac{1}{2m_{N}}\sum_{j=1}^{A}\left[g_{V}\left\\{\vec{p}_{j},\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right\\}+\left(g_{V}+\tilde{g}_{T\left(V\right)}\right)\vec{\nabla}\times\vec{\sigma}_{j}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right.$
$\displaystyle\left.-i\tilde{g}_{S}\vec{\nabla}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right]\tau_{j}^{\pm}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
(100b) $\displaystyle\mathcal{J}_{0}^{A}\left(\vec{r}\right)$
$\displaystyle=\frac{1}{2m_{N}}\sum_{j=1}^{A}\left[g_{A}\left\\{\vec{p}_{j},\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right\\}+i\left(\tilde{g}_{T\left(A\right)}-\frac{E_{0}}{2m_{N}}\tilde{g}_{P}\right)\vec{\nabla}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
(100c) $\displaystyle\vec{\mathcal{J}}^{A}\left(\vec{r}\right)$
$\displaystyle=\sum_{j=1}^{A}\left[g_{A}-\frac{E_{0}}{2m_{N}}\tilde{g}_{T\left(A\right)}\right]\vec{\sigma}_{j}\tau_{j}^{\pm}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)}.$
(100d)

Positioning Eq. (100) into the multipole operators definition, leads to the
explicit expressions for the vector and axial currents multipole operators:

$\displaystyle\hat{C}_{J}^{V\text{[2c]}}\left(q\right)$
$\displaystyle=\left(g_{V}+\frac{E_{0}}{2m_{N}}\tilde{g}_{S}\right)\sum_{j=1}^{A}M_{J}\left(q\vec{r}_{j}\right)\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
(101a) $\displaystyle\hat{L}_{J}^{V\text{[2c]}}\left(q\right)$
$\displaystyle=-\frac{q}{2m_{N}}\sum_{j=1}^{A}\left\\{\left(g_{V}-\tilde{g}_{S}\right)M_{J}\left(q\vec{r}_{j}\right)-2g_{V}\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}_{j}\right)\right]\cdot\frac{1}{q}\vec{\nabla}\right\\}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
(101b) $\displaystyle\hat{E}_{J}^{V\text{[2c]}}\left(q\right)$
$\displaystyle=\frac{q}{m_{N}}\sum_{j=1}^{A}\left\\{-ig_{V}\left[\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\right]\cdot\frac{1}{q}\vec{\nabla}+\frac{g_{V}+\tilde{g}_{T\left(V\right)}}{2}\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\cdot\vec{\sigma}\right\\}\tau_{j}^{\pm}$
$\displaystyle+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
(101c) $\displaystyle\hat{M}_{J}^{V\text{[2c]}}\left(q\right)$
$\displaystyle=-\frac{iq}{m_{N}}\sum_{j=1}^{A}\left\\{g_{V}\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\cdot\frac{1}{q}\vec{\nabla}+i\frac{g_{V}+\tilde{g}_{T\left(V\right)}}{2}\left[\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}\right\\}\tau_{j}^{\pm}$
$\displaystyle+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
(101d) $\displaystyle\hat{C}_{J}^{A\text{[2c]}}\left(q\right)$
$\displaystyle=-\frac{iq}{m_{N}}\sum_{j=1}^{A}\left\\{g_{A}M_{J}\left(q\vec{r}_{j}\right)\vec{\sigma}\cdot\frac{1}{q}\vec{\nabla}+\frac{1}{2}\left(g_{A}+\tilde{g}_{T\left(A\right)}-\frac{E_{0}}{2m_{N}}\tilde{g}_{P}\right)\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}\right)\right]\cdot\vec{\sigma}\right\\}\tau_{j}^{\pm}$
$\displaystyle+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
(101e) $\displaystyle\hat{L}_{J}^{A\text{[2c]}}\left(q\right)$
$\displaystyle=i\left(g_{A}-\frac{E_{0}}{2m_{N}}\tilde{g}_{T\left(A\right)}\right)\sum_{j=1}^{A}\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
(101f) $\displaystyle\hat{E}_{J}^{A\text{[2c]}}\left(q\right)$
$\displaystyle=\left(g_{A}-\frac{E_{0}}{2m_{N}}\tilde{g}_{T\left(A\right)}\right)\sum_{j=1}^{A}\left[\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
(101g) $\displaystyle\hat{M}_{J}^{A\text{[2c]}}\left(q\right)$
$\displaystyle=\left(g_{A}-\frac{E_{0}}{2m_{N}}\tilde{g}_{T\left(A\right)}\right)\sum_{j=1}^{A}\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\cdot\vec{\sigma}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right)\text{,}$
(101h)

with $M_{J}$ and $\vec{M}_{JL1}$ defined in Eq. (10) in the main text. One can
recognize that these multipoles operators that include the second class
currents are actually the SM multipoles operators with small changes:

$\displaystyle\hat{C}_{J}^{V\text{[2c]}}\left(q\right)$
$\displaystyle=\frac{g_{V}+\frac{E_{0}}{2m_{N}}\tilde{g}_{S}}{g_{V}}\hat{C}_{J}^{V}\left(q\right),$
(102a) $\displaystyle\hat{L}_{J}^{V\text{[2c]}}\left(q\right)$
$\displaystyle=\hat{L}_{J}^{V}\left(q\right)+\frac{q}{2m_{N}}\frac{\tilde{g}_{S}}{g_{V}}\hat{C}_{J}^{V}\left(q\right),$
(102b) $\displaystyle\hat{C}_{J}^{A\text{[2c]}}\left(q\right)$
$\displaystyle=\hat{C}_{J}^{A}\left(q\right)-\frac{q}{2m_{N}}\frac{\tilde{g}_{T\left(A\right)}}{g_{A}}\hat{L}_{J}^{A}\left(q\right),$
(102c) $\displaystyle\hat{O}_{J}^{A\text{[2c]}}\left(q\right)$
$\displaystyle=\frac{g_{A}-\frac{E_{0}}{2m_{N}}\tilde{g}_{T\left(A\right)}}{g_{A}}\hat{O}_{J}^{A}\left(q\right),$
$\displaystyle\hat{O}\in\left\\{\hat{L},\hat{E},\hat{M}\right\\}\text{.}$
(102d)

The multipole operator $\hat{E}_{J}^{V}$ and $\hat{M}_{J}^{V}$ stay with no
change for their leading orders when including second class currents.

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