[1]Andrea Ponti

[1]Department of Economics, Management and Statistics, University of Milano-
Bicocca, Milan, 20126, Italy

# Graph data augmentation with Gromow-Wasserstein Barycenters

<EMAIL_ADDRESS>*

###### Abstract

Graphs are ubiquitous in various fields, and deep learning methods have been
successful applied in graph classification tasks. However, building large and
diverse graph datasets for training can be expensive. While augmentation
techniques exist for structured data like images or numerical data, the
augmentation of graph data remains challenging. This is primarily due to the
complex and non-Euclidean nature of graph data. In this paper, it has been
proposed a novel augmentation strategy for graphs that operates in a non-
Euclidean space. This approach leverages graphon estimation, which models the
generative mechanism of networks sequences. Computational results demonstrate
the effectiveness of the proposed augmentation framework in improving the
performance of graph classification models. Additionally, using a non-
Euclidean distance, specifically the Gromow-Wasserstein distance, results in
better approximations of the graphon. This framework also provides a means to
validate different graphon estimation approaches, particularly in real-world
scenarios where the true graphon is unknown.

###### keywords:

Barycenter, Graphon, Graph augmentation, Graph classification, Wasserstein
distance

##  1 Introduction

Networks are a powerful tool used in many fields such as social sciences or
biology, and they are becoming larger and more complex. Deep Learning methods,
like Graph Neural Networks and Graph Convolutional Networks, have been
successfully applied to perform classification on graphs. Building graph
datasets to train neural methods can be very expensive in real-word
applications. When dealing with structured data, as numerical data or images,
many augmentation techniques exist to face this problem. In the context of
graph classification this is still challenging. Existing graph data
augmentation strategies typically aim to augment graphs at a within-graph
level by either modifying edges or nodes in an individual graph, which does
not enable information exchange between different instances. The between-graph
augmentation methods are still under-explored. This is mainly due to the
complex and non-Euclidean structure of graph data [21]. Such complexity
hinders the direct application of traditional augmentation techniques used in
image, video, or text data.
One way to study large and complex networks is by modelling their generative
mechanism in order to build statistical graph models. Graphons (short for
graph functions), first introduced in [2, 14], are the limiting objects for
sequences of large, finite graphs with respect to the so-called cut metric.
Formally, a graphon is a measurable symmetric function
$W:\Omega\times\Omega\to[0,1]$, where $\Omega$ is a measurable space, that
represents a discrete network as an infinite-dimensional analytic object. A
graphon can be used to generate arbitrarily sized graphs by first sampling $N$
nodes from a uniform distribution, $v_{i}\sim\text{Uniform}(\Omega)$, and then
generating an adjacency matrix $A=[a_{ij}]$. Each element $a_{ij}\in\\{0,1\\}$
of the adjacency matrix is sampled from a Bernoulli distribution determined by
the graphon $a_{ij}\sim\text{Bernoulli}(W(v_{i},v_{j}))$. Accordingly, the
generated graph is $\mathcal{G}=(V,E)$ where
$V=\left\\{v_{1},...,v_{N}\right\\}$ and
$E=\left\\{(v_{i},v_{j})|a_{ij}=1\right\\}$. Graphons arise naturally in many
fields like random networks, quasi-random graphs, property testing of large
graphs and extremal graph theory [15, 11, 8]. They find applications in fields
like signal theory, where they have been used to overcome the limitations of
Graph Signal Processing in the case of graphs with large number of nodes or
dynamic topologies [18]. Graphon have been also used in Graph Neural Networks
(GNNs) to analyze the transferability of GNNs across graphs [17], their
stability [19], and to mitigate the over-smoothing issue via resampling from a
graphon estimate obtained from the underlying network data [10].
Many methods to learn a graphon from an observed sequence of graph are based
on the weak regularity lemma, which states that an arbitrary graphon can be
well approximated by a two-dimensional step function. Existing methods to
learn a step function typically leverage on stochastic block models [1, 3, 4]
or low-rank approximation of the observed graphs [13, 5, 7]. Most of the
methods proposed in the literature requires the observed graphs to be
generated by a single graphon and to be aligned, i.e., graphs with comparable
size and with a known correspondence between nodes. In real world graphs, this
is often not the case. Traditional learning methods involve an aligning
procedure to match the observed graph using some heuristics that often
produces failures, resulting in wrongly-aligned graphs and in poor estimation
of the underlying graphon. To overcome these limitations, the authors in [20]
proposed an approach that leverages the permutation-invariance of the Gromov-
Wasserstein (GW) distance, implicitly including the graph aligning in the
estimation phase. This approach will be further explored in Section 2.
The purpose of this paper is to propose an augmentation strategy for graphs
that works in a non-Euclidean space. Experimental results, in a graph
classification context, show an improvement in performance when using the
augmented dataset. Furthermore, results show that using a non-Euclidean
distance results in a better approximation of the graphon. The proposed
framework can also be used to validate different graphon estimation
approaches, specifically in real-word problems when the true graphon is
unknown.

##  2 Graphons as Wasserstein Barycenters

Consider a set of graphs $\\{\mathcal{G}_{m}\\}_{m=1}^{M}$. A graph with nodes
$V=\\{1,...,N\\}$ and an adjacency matrix $A=[a_{n,n^{\prime}}]$ can be
represented as a step function $G_{\mathcal{P}}$ over $N$ equitable partitions
of $\Omega$, $\mathcal{P}=\\{\mathcal{P}_{n}\\}_{n=1}^{N}$, defined as

$G_{\mathcal{P}}(x,y)=\frac{1}{N^{2}}\sum_{n,n^{\prime}=1}^{N}a_{n,n^{\prime}}\mathds{1}_{\mathcal{P}_{n}\times\mathcal{P}_{n^{\prime}}}(x,y).$
(1)

where $a_{n,n^{\prime}}\in[0,1]$ and the indicator function
$\mathds{1}_{\mathcal{P}_{n}\times\mathcal{P}_{n^{\prime}}}(x,y)$ takes value
1 if $(x,y)\in\mathcal{P}_{n}\times\mathcal{P}_{n^{\prime}}$ and 0 otherwise.
If the position of nodes is known, it is possible to derive an isomorphism of
the graph $\hat{\mathcal{G}}$ and to obtain an “oracle” step function
$\hat{G}_{\mathcal{P}}$. Using such isomorphism over the set of all graphs,
the result is a set of aligned graphs $\\{\hat{\mathcal{G}}_{m}\\}_{m=1}^{M}$
and consequently, a set of oracle step functions
$\\{\hat{G}_{m,\mathcal{P}^{m}}\\}_{m=1}^{M}$, where the number of partitions
$|\mathcal{P}^{m}|$ is equal to the number of nodes in
$\hat{\mathcal{G}}_{m}$. Then, an oracle estimator of the graphon $W$ that
generated the set of graphs $\\{\mathcal{G}_{m}\\}_{m=1}^{M}$ can be obtained
as

$W_{O}=\frac{1}{M}\sum_{m=1}^{M}\hat{G}_{m,\mathcal{P}^{m}}$ (2)

The authors in [20] have shown that the oracle estimator provides a consistent
estimation of $W$ indeed, for every $W\in\mathcal{W}$, the following
inequality is true

$\delta_{\square}(W,W_{O})\leq\frac{C}{\min_{m}|\mathcal{P}^{m}|}$ (3)

with $C$ a constant. Using Equation 3 and the triangle inequality of the cut-
distance, in [20] it has been shown that the estimation of a graphon can be
done minimizing an upper bound of the cut-distance by solving the following
optimization problem:

$\min_{W_{\mathcal{P}}}\frac{1}{M}\sum_{m=1}^{M}\delta_{1}(G_{m,\mathcal{P}^{m}},W_{\mathcal{P}}).$
(4)

To solve this problem, the authors in [20] derived a computationally-efficient
alternative of $\delta_{1}$ based on its equivalence with the 1-order GW
distance. Consider two step functions $W_{1,\mathcal{P}}$ and
$W_{2,\mathcal{Q}}$ defined on the probability spaces $(\Omega_{1},\mu_{1})$
and $(\Omega_{2},\mu_{2})$, respectively, which have equitable partitions
$\mathcal{P}=\\{\mathcal{P}_{i}\\}_{i=1}^{I}$ and
$\mathcal{P}=\\{\mathcal{Q}_{j}\\}_{j=1}^{J}$. Then, the $\delta_{1}$ distance
between the two step functions can be rewritten as the 1-order GW distance:

$\displaystyle d_{gw,1}(\textbf{W}_{1},\textbf{W}_{2})$ $\displaystyle=$ (5)
$\displaystyle\min_{\textbf{T}\in\Pi(\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2})}$
$\displaystyle\sum_{i,i^{\prime},j,j^{\prime}}|w_{1,ij}-w_{2,i^{\prime}j^{\prime}}|T_{ii^{\prime}}T_{jj^{\prime}}$

where $\textbf{W}_{1}=[w_{1,ij}]\in[0,1]^{I\times I}$ and
$\textbf{W}_{2}=[w_{2,i^{\prime}j^{\prime}}]\in[0,1]^{J\times J}$ represent
the step functions in the matrix form; vectors
$\boldsymbol{\mu}_{1}=[\mu_{1,i}]$ and $\boldsymbol{\mu}_{2}=[\mu_{2,j}]$
represent the probability measures $\mu_{1}$ and $\mu_{2}$; the matrix T is
the coupling between $\boldsymbol{\mu}_{1}$ and $\boldsymbol{\mu}_{2}$, and is
also called transport matrix. Similarly it is possible to define the squared
2-order GW distance as

$\displaystyle d_{gw,2}^{2}(\textbf{W}_{1},\textbf{W}_{2})$ (6)
$\displaystyle=\min_{\textbf{T}\in\Pi(\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2})}$
$\displaystyle\sum_{i,i^{\prime},j,j^{\prime}}|w_{1,ij}-w_{2,i^{\prime}j^{\prime}}|^{2}T_{ii^{\prime}}T_{jj^{\prime}}.$

This is usually smother and consequently easier to optimize. Then, the
learning problem becomes the estimation of a GW barycenter of the observed
graphs

$\min_{\textbf{W}\in[0,1]^{K\times
K}}\frac{1}{M}\sum_{m=1}^{M}d_{gw,2}^{2}(\textbf{A}_{m},\textbf{W})$ (7)

where $\textbf{A}_{m}$ is the adjacency matrix of the graph $\mathcal{G}_{m}$
and $\textbf{W}=[w_{kk^{\prime}}]$ is the matrix representation of the step
function $W_{\mathcal{P}}$.

##  3 Graphon for data augmentation

In the view of graphons as generative models of graph objects they become
particularly useful to augment graph datasets. This idea has been recently
explored in [9], where the authors derive a mixup graphon from the graphons of
each class and use the derived mixup graphon to generate synthetic graphs with
uncertain labels (i.e., the probability of each class is uniform). They show
that adding these synthetic graphs to the initial dataset improves the
generalization ability of GNNs and stabilize the model training. In the
current paper, a slightly different approach has been proposed. Consider the
supervised problem of classify a set of $M$ graphs
$\bm{\mathcal{G}}=\\{\mathcal{G}_{m}\\}_{m=1}^{M}$ into $N$ classes. Denote
with $\bm{\mathcal{G}}_{i}$ the set of graphs belonging to class $i$. Then,
for each class $i$, a graphon $\textbf{W}_{i}$ can be estimated from the
sequence of graphs $\bm{\mathcal{G}}_{i}$. Finally, new graphs can be sampled
from $\textbf{W}_{i}$ to augment the initial dataset.
In the computational results, reported in the following section, the datasets
have been divided into train and test set; the graphons of each class are
learned considering only the graphs in the train set. Code and data used are
available on GitHub111https://github.com/andreaponti5/graphon-data-
augmentation.

##  4 Experiments and Results

In the experiments three datasets have been considered to analyze the
performance of the proposed augmentation framework. LFR and IMDB are two
binary classification datasets of 1600 and 1000 graphs, respectively, while
ENZYMES contains 600 graphs evenly distributed into 6 classes. In addition,
five different methods to estimate graphons have been tested. SAS, SBA and LG
are based on stochastic block models while MC is based on low-rank
approximation. GB and SGB use the Gromow-Wasserstein barycenter to learn the
graphons; in particular GB is reported in Equation 7 and SGB is a smoothed
version of GB.

(a)

(b)

(c)

Figure 1: Visual representation of the graphons estimated using different
methods on the three datasets. Lighter colors means higher probability of
connection.

First, it is important to understand whether graphons are able to discriminate
the different classes in the dataset. Fig. 1 displays a visual representation
of the graphons estimated with different methods of the classes of the three
datasets.
The Wasserstein based estimations show a clear distinction for the two classes
and give some insight on the structure of graphs belonging to the different
classes. For instance, considering LFR, the graphs of class 0 appears to be
composed by many connected community, while graphs in class 1 are more likely
to be a single community with higher nodes degree. The other four baseline
estimation methods do not show a marked difference between the two classes.
Then, to evaluate the quality of the generated graphs, they have been used to
augment the initial dataset as explained in the previous section. Table
LABEL:tab:res reports the percentage of classification accuracy improvement by
augmenting the initial dataset.

Table 1: Percentage of accuracy improvement augmenting the dataset by sampling from graphons estimated using different methods. The datasets have been augmented by 1%, 5%, 10% and 25% of the number of instances in the train set. The base accuracy (percentage) is shown in square brackets next to the dataset name | % | SAS | SBA | LG | MC | GB | SGB
---|---|---|---|---|---|---|---
LFR[94] | 1 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.4 | 0.2
10 | 0.2 | 0.2 | 0.2 | 0.4 | 0.6 | 0.4
25 | 0.6 | 0.2 | 0.4 | 0.2 | 0.2 | 0.4
IMDB[54] | 1 | 0.7 | 0.7 | 1.0 | 0.7 | 0.7 | 0.7
5 | 1.7 | 1.3 | 0.0 | 0.0 | 0.7 | 1.7
10 | -1.0 | 0.0 | 0.7 | 1.3 | 1.7 | 0.0
25 | -1.3 | -1.0 | -3.3 | 0.0 | 0.3 | 0.3
ENZ[17] | 1 | 5.6 | 2.2 | 5.6 | 5.6 | 6.1 | 5.6
5 | 2.2 | 2.8 | 0.0 | -2.2 | 1.1 | 1.1
10 | 2.8 | 3.9 | 1.7 | 3.3 | 2.8 | 2.8
25 | 2.8 | 0.0 | 1.1 | 1.1 | 2.2 | 1.1

In particular, graph2vec [16] has been used to embed the graphs in a vector
space (considering the nodes degree as features) and a Multi-Layer Perceptron
to perform classification on graphs. In the case of LFR, where the classes are
clearly separated and the classification accuracy of the model trained with no
augmentation is already high, it is possible to obtain a little improvement
adding synthetic graphs to the train set. This improvement increases when the
distinction between different classes is non-trivial, as for IMDB, or in a
multi-class problem as ENZYMES. In general, a 1% augmentation of the initial
dataset is enough to obtain a significant improvement in terms of performance.
It is important to remark that, in most cases, graphons estimated using the
Gromow-Wasserstein Barycenters lead to better performances.

##  5 Conclusion

Data from many systems across various areas can be explicitly denoted as
graphs such as social networks, transportation networks, protein–protein
interaction networks. Deep learning approaches have a substantial role in
analyzing networks; while many data augmentation techniques exist for
numerical, image or text data, this field is still under-explored on graphs.
Most of the existing methods works by addition and ablation operations on the
existing graphs, which can be limiting. A more effective approach consists in
learning a generative model of graphs.
The goal of this paper is to introduce a new data augmentation framework that
rely on graphons estimated using the Gromow-Wasserstein Barycenters. As shown
by the previously reported results, such approach can enhance the performance
of Machine Learning models. In addition, since graphs are non-Euclidean
object, using a non-Euclidean distance (i.e., the Gromow-Wasserstein distance)
to estimate the graphons generally brings to better performance. In this
context, graphs are seen as discrete probability distributions supported by
their nodes equipped with the nodes degree distribution.
Recently, other generative methods for graphs have been proposed mainly based
on diffusion models [6, 12]. These models could fit well in the proposed
framework but it is important to remark that the computational complexity of
training diffusion models is much higher than estimating graphons.

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# Symmetry decomposition of relative entropies in conformal field theory

Hui-Huang<EMAIL_ADDRESS>
College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China

We consider the symmetry resolution of relative entropies in the 1+1
dimensional free massless compact boson conformal field theory (CFT) which
presents an internal $U(1)$ symmetry. We calculate various symmetry resolved
Rényi relative entropies between one interval reduced density matrices of CFT
primary states using the replica method. By taking the replica limit, the
symmetry resolved relative entropy can be obtained. We also take the XX spin
chain model as a concrete lattice realization of this CFT to perform numerical
computation. The CFT predictions are tested against exact numerical
calculations finding perfect agreement.

###### Contents

  1. 1 Introduction
  2. 2 Relative entopy in CFT
  3. 3 Symmetry resolution of entanglement entropy and relative entropy
    1. 3.1 Entanglement entropy and relative entropy in charge sectors
    2. 3.2 Symmetry resolution of entanglement entropy in CFT
  4. 4 Symmetry resolution of relative entropy in CFT
    1. 4.1 Resolution of relative entropy between the ground state and the vertex operator
    2. 4.2 Resolution of relative entropy between the derivative operator and the ground state
    3. 4.3 Resolution of relative entropy between the derivative operators and the vertex operators
  5. 5 Numerical tests
  6. 6 Conclusion
  7. A Correlation matrices and RDMs in XX spin chain

##  1 Introduction

In recent years, concepts and methods coming from quantum information theory
are playing more and more important roles in both high-energy physics and
condensed matter theory. In a many-body system, entanglement is a powerful
tool to characterize quantum phase transition and by studying its universal
features one can acquire knowledge of the underlying conformal field theory
(CFT). For reviews, see [1, 2, 3, 4]. In high-energy physics, entanglement is
also the key concept to understand the information paradox of black holes [5,
6, 7] through gauge/gravity duality [8, 9].

So far, most studies are focus on subsystem entanglement features of a single
quantum state. For some applications, the entanglement entropy for a given
subsystem can not provide enough information. One may wonder, how can we gain
insight when giving two different quantum states. It’s also important for us
to distinguish between subsystems in different states. In this respect,
relative entropy is an important quantity [10]. Relative entropy attracts a
great deal of attention during the past few years and has been extensively
studied [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The reason is that relative
entropy is relatively simple to calculate and is free of divergence in quantum
field theory. There also exist other quantities that can be used to
distinguish reduced density matrices (RDMs). For example, the quantum fidelity
[21] and the trace distance [22, 23] is two very commonly used concepts.

For two given states with reduced density matrices (RDMs) $\rho$ and $\sigma$,
the so-called relative entropy is defined as [24, 25]

$S(\rho\|\sigma)=\textrm{tr}(\rho\log\rho)-\textrm{tr}(\rho\log\sigma),$ (1.1)

which can be viewed as a measure of “distance” between the two quantum states.
In quantum field theory, the relative entropy can be obtained by using the
replica trick [12, 13]

$S(\rho\|\sigma)=\lim_{n\rightarrow 1}S_{n}(\rho\|\sigma)=\lim_{n\rightarrow
1}\frac{1}{1-n}\log{\frac{\textrm{tr}(\rho\sigma^{n-1})}{\textrm{tr}(\rho^{n})}},$
(1.2)

where we have defined the Rényi relative entropies as

$S_{n}(\rho\|\sigma)=\frac{1}{1-n}\log{\frac{\textrm{tr}(\rho\sigma^{n-1})}{\textrm{tr}(\rho^{n})}}.$
(1.3)

For a quantum many-body systems with global symmetry, one can decompose
entanglement into different symmetry sectors. In this respect, the authors of
reference [26] introduced a more refined notion of entanglement, the symmetry
resolved entanglement entropy. After this pioneering work, people have studied
a lot about symmetry resolution of entanglement properties for both pure
states[27, 28, 29, 30, 31, 32, 33] and mixed states [34, 35]. Moreover,
similar quantities have also been introduced in quantum field theories and in
the holographic settings [36, 37, 38, 39, 40, 41, 42, 43].

In this paper, we will mainly focus on the symmetry resolution of relative
entropies in CFT. More explicitly, we will consider the $U(1)$ symmetry
decomposition of relative entropy in free massless compact boson CFT using the
twist operator method. We will also check our universal CFT predictions
numerically in the XX spin chain.

The remaining part of this paper is organized as follows. In section 2, we
briefly review the CFT approach to the Rényi relative entropies between the
RDMs of two primary excited states. In section 3, we discuss how relative
entropies are distributed in different charge sectors. In this section, we
define all needed concepts concerning symmetry resolved relative entropy and
summarise the known results of the symmetry resolved entanglement entropy
which will be useful in the following sections. In section 4, we calculate
various symmetry resolved relative entropies between primary states in free
compact boson CFT. The CFT results are tested in section 5 against exact
numerical computations in the XX chain. Finally, we conclude in section 6 and
some technical details for numerical calculation are given in appendix A.

##  2 Relative entopy in CFT

In this section, let’s briefly review the replica trick to compute the
relative entropies of two reduced density matrices of excited states in 1+1
dimensional CFT. Consider a system with one spatial dimension and a
bipartition into two complementary regions $A$ and $\bar{A}$. We take
subsystem $A$ given by the interval $[u,v]$ with length $l=v-u$ and $\bar{A}$
is its complement with length $L-l$. Here $L$ is the total length of our
periodic 1D system. Given two (pure) states
$\ket{\Psi},\ket{\Phi}\in\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$,
the reduced density matrices of subsystem $A$ is defined by tracing over the
points not in $A$.

$\rho_{A,\Psi}=\textrm{tr}_{\bar{A}}\ket{\Psi}\bra{\Psi},\quad\rho_{A,\Phi}=\textrm{tr}_{\bar{A}}\ket{\Phi}\bra{\Phi}.$
(2.1)

The world sheet of the 1+1 dimensional CFT is an infinite cylinder with
circumference $L$ which can be parameterized by introducing the complex
coordinate $w=x+i\tau$. In this paper, we are only interested in the excited
states in CFT that correspond to local primary operators

$\ket{\Psi}=\Psi(-i\infty)\ket{0},$ (2.2)

where $\ket{0}$ is the CFT vacuum state and corresponds to the identity
operator $I$. Let us omit the index $A$ and denote the reduced density matrix
of a state $\ket{\Psi}$ to the subsystem $A$ by $\rho_{\Psi}$. Following the
standard procedure [44, 2], $\textrm{tr}(\rho_{I}^{n})$ can be obtained by
sewing cyclically $n$ copies of the above cylinders along with the interval
$[u,v]$. In contrast to the ground state case, the corresponding path-integral
representation of the density matrix $\rho=\ket{\Psi}\bra{\Psi}$ presents two
additional insertions of $\Psi(-i\infty)$ and $\Psi^{\dagger}(i\infty)$. In
this way, we end up with a $n$-sheeted Riemann surface $\mathcal{R}_{n}$ and
$\textrm{tr}(\rho_{\Psi}^{n})$ is given by a $2n$-point function on
$\mathcal{R}_{n}$ [45]

$\textrm{tr}(\rho_{\Psi}^{n})=\frac{Z_{n}}{Z_{1}^{n}}\frac{\langle\prod_{k=1}^{n}\Psi(w^{-}_{k})\Psi^{\dagger}(w^{+}_{k})\rangle_{\mathcal{R}_{n}}}{\langle\Psi(w^{-}_{1})\Psi^{\dagger}(w_{1}^{+})\rangle^{n}_{\mathcal{R}_{1}}},$
(2.3)

where $Z_{n}=\langle I\rangle_{\mathcal{R}_{n}}$ is the $n$-th moment of the
reduced density matrix of the ground state and
$w_{k}^{-}=-i\infty,w_{k}^{+}=i\infty$ are points where the operators are
inserted in the $k$-th copy.

In order to obtain the Rényi relative entropies between $\rho_{\Psi}$ and
$\rho_{\Phi}$, we further need to compute
$\textrm{tr}(\rho_{\Psi}\rho_{\Phi}^{n-1})$. Quite similar to the previous
case, and taking the normalization factor into account, we find

$\textrm{tr}(\rho_{\Psi}\rho_{\Phi}^{n-1})=\frac{Z_{n}}{Z_{1}^{n}}\frac{\langle\Psi(w_{1}^{-})\Psi^{\dagger}(w^{+}_{1})\prod_{k=2}^{n}\Phi(w^{-}_{k})\Phi^{\dagger}(w^{+}_{k})\rangle_{\mathcal{R}_{n}}}{\langle\Psi(w_{1}^{-})\Psi^{\dagger}(w_{1}^{+})\rangle_{\mathcal{R}_{1}}\langle\Phi(w_{1}^{-})\Phi^{\dagger}(w_{1}^{+})\rangle^{n-1}_{\mathcal{R}_{1}}},$
(2.4)

and the universal ratio

$G_{n}(\rho_{\Psi}\|\rho_{\Phi})\equiv\frac{\textrm{tr}(\rho_{\Psi}\rho_{\Phi}^{n-1})}{\textrm{tr}(\rho_{\Psi}^{n})}=\frac{\langle\Psi(w^{-}_{1})\Psi^{\dagger}(w^{+}_{1})\prod_{k=2}^{n}\Phi(w^{-}_{k})\Phi^{\dagger}(w^{+}_{k})\rangle_{\mathcal{R}_{n}}\langle\Psi(w_{1}^{-})\Psi^{\dagger}(w_{1}^{+})\rangle^{n-1}_{\mathcal{R}_{1}}}{\langle\prod_{k=1}^{n}\Psi(w^{-}_{k})\Psi^{\dagger}(w^{+}_{k})\rangle_{\mathcal{R}_{n}}\langle\Phi(w^{-}_{1})\Phi^{\dagger}(w_{1}^{+})\rangle^{n-1}_{\mathcal{R}_{1}}}.$
(2.5)

Knowing $G_{n}$, the Rényi relative entropy is simply given by

$S_{n}(\rho_{\Psi}\|\rho_{\Phi})=\frac{1}{1-n}\log
G_{n}(\rho_{\Psi}\|\rho_{\Phi}).$ (2.6)

We can apply the following sequence of conformal maps

$w\rightarrow
z=\left(\frac{\sin\frac{\pi(w-u)}{L}}{\sin\frac{\pi(w-u)}{L}}\right)^{\frac{1}{n}}\rightarrow
t=-i\log z$ (2.7)

to transform the $n$-sheet Riemann surface $\mathcal{R}_{n}$ into a single
cylinder. The transformation law of a primary field $\mathcal{O}$ is very
simple

$\mathcal{O}(z,\bar{z})=\left(\frac{dz}{dw}\right)^{-h_{\mathcal{O}}}\left(\frac{d\bar{z}}{d\bar{w}}\right)^{-\bar{h}_{\mathcal{O}}}\mathcal{O}(w,\bar{w}),$
(2.8)

with $(h_{\mathcal{O}},\bar{h}_{\mathcal{O}})$ the conformal weights of
$\mathcal{O}$. Applying the conformal maps in eq. (2.7), one can easily
express $G_{n}(\rho_{\Psi}\|\rho_{\Phi})$ in terms of correlation functions on
the cylinder

$G_{n}(\rho_{\Psi}\|\rho_{\Phi})=n^{2(n-1)(h_{\Psi}+\bar{h}_{\Psi}-h_{\Phi}-\bar{h}_{\Phi})}\frac{\langle\Psi(t^{-}_{1})\Psi^{\dagger}(t^{+}_{1})\prod_{k=2}^{n}\Phi(t^{-}_{k})\Phi^{\dagger}(t^{+}_{k})\rangle_{cy}\langle\Psi(t_{1}^{-})\Psi^{\dagger}(t_{1}^{+})\rangle^{n-1}_{cy}}{\langle\prod_{k=1}^{n}\Psi(t^{-}_{k})\Psi^{\dagger}(t^{+}_{k})\rangle_{cy}\langle\Phi(t^{-}_{1})\Phi^{\dagger}(t_{1}^{+})\rangle^{n-1}_{cy}},$
(2.9)

where $t_{k}^{\pm}$ are the points corresponding to $w_{k}^{\pm}$ through the
map $t(w)$

$t_{k}^{-}=\frac{\pi}{n}(x+2(k-1)),\qquad
t_{k}^{+}=\frac{\pi}{n}(-x+2(k-1)),\qquad k=1,2,\cdots,n.\quad
x=\frac{v-u}{L}=\frac{l}{L}.$ (2.10)

In the following, we will mainly focus on the theory of free massless compact
bosonic field $\varphi(z,\bar{z})$, with Euclidean action

$\mathcal{A}[\varphi]=\frac{1}{8\pi}\int
dzd\bar{z}\partial_{z}\varphi\partial_{\bar{z}}\varphi.$ (2.11)

This is a CFT with central charge $c=1$ and has two types of primary fields.
The first type is the vertex operators

$V_{\alpha,\bar{\alpha}}=:e^{i(\alpha\phi+\bar{\alpha}\bar{\phi})}:$ (2.12)

where $\phi,\bar{\phi}$ are chiral and anti-chiral parts of the bosonic field:
$\varphi(z,\bar{z})=\phi(z)+\bar{\phi}(\bar{z})$. The conformal weight of the
vertex operator is
$(h,\bar{h})=(\frac{\alpha^{2}}{2},\frac{\bar{\alpha}^{2}}{2})$. For
simplicity, we will consider holomorphic field $\bar{\alpha}=0$ only. The
$n$-point function of vertex operators on the complex plane is ($z_{i,j}\equiv
z_{i}-z_{j}$)[46]

$\langle\prod_{k}V_{\alpha_{k}}(z_{k})\rangle=\prod_{i<j}(z_{i,j})^{\alpha_{i}\alpha_{j}}.$
(2.13)

After the conformal map $t=-i\log z$ to the cylinder, this correlator becomes

$\langle\prod_{k}V_{\alpha_{k}}(t_{k})\rangle_{cy}=\prod_{i<j}(2\sin\frac{t_{i,j}}{2})^{\alpha_{i}\alpha_{j}}.$
(2.14)

The other type of primary field in this theory is the derivative operator
$i\partial\phi$ with conformal dimension $(h,\bar{h})=(1,0)$. The $2n$-point
function on the complex plane is given by [46]

$\langle\prod_{k=1}^{2n}i\partial\phi(z_{k})\rangle=\text{Hf}\left[\frac{1}{z_{i,j}^{2}}\right]_{1\leq
i,j\leq 2n},$ (2.15)

where $\text{Hf}(A)$ is the Haffnian of the $2n\times 2n$ matrix $A$

$\text{Hf}(A)=\frac{1}{2^{n}n!}\sum_{\sigma\in
S_{2n}}\prod_{i=1}^{n}A_{\sigma(2i-1),\sigma(2i)}.$ (2.16)

The Haffian in eq. (2.15) can be written as a determinant

$\text{Hf}\left[\frac{1}{z_{i,j}^{2}}\right]_{1\leq i,j\leq
2n}=\det\left[\frac{1}{z_{i,j}}\right]_{1\leq i,j\leq 2n}.$ (2.17)

In a cylinder parametrized by $t=-i\log z$, the correlator becomes

$\langle\prod_{k=1}^{2n}i\partial\phi(t_{k})\rangle_{cy}=\frac{1}{4^{n}}\det\left[\frac{1}{\sin(t_{i,j}/2)}\right]_{1\leq
i,j\leq 2n}.$ (2.18)

For this $2n$-point correlator evaluated at the $2n$-point list in eq. (2.10),
the analytic continuation has been obtained in [47, 48] and is given by

$\langle\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}=\frac{\Gamma^{2}(\frac{1+n+n\csc\pi
x}{2})}{\Gamma^{2}(\frac{1-n+n\csc\pi x}{2})}.$ (2.19)

Several relative entropies have been obtained in [13, 14], here we just report
the results. Firstly, the Rényi relative entropies between the ground state
and the vertex operator are given by

$S_{n}(\rho_{I}\|\rho_{V_{\alpha}})=S_{n}(\rho_{V_{\alpha}}\|\rho_{I})=\frac{\alpha^{2}}{1-n}\log\frac{\sin\pi
x}{n\sin\frac{\pi x}{n}}.$ (2.20)

By taking the replica limit $n\rightarrow 1$, the relative entropy is obtained
as

$S(\rho_{I}\|\rho_{V_{\alpha}})=S(\rho_{V_{\alpha}}\|\rho_{I})=\alpha^{2}(1-\pi
x\cot(\pi x)).$ (2.21)

The relative entropy between two vertex operators is given by

$S(\rho_{V_{\alpha}}\|\rho_{V_{\beta}})=S(\rho_{V_{\beta}}\|\rho_{V_{\alpha}})=(\alpha-\beta)^{2}(1-\pi
x\cot(\pi x)).$ (2.22)

The relative entropy between the derivative operator and the ground state is

$S(\rho_{i\partial\phi}\|\rho_{I})=2\log(2\sin(\pi x))+2-2\pi x\cot(\pi
x)+2\psi\left(\frac{\csc(\pi x)}{2}\right)+2\sin(\pi x).$ (2.23)

Finally, the relative entropy between the derivative operator and the vertex
operator is

$S(\rho_{i\partial\phi}\|V_{\alpha})=S(\rho_{i\partial\phi}\|\rho_{I})+S(\rho_{I}\|\rho_{V_{\alpha}}).$
(2.24)

##  3 Symmetry resolution of entanglement entropy and relative entropy

###  3.1 Entanglement entropy and relative entropy in charge sectors

Now assume that the system has an internal $U(1)$ symmetry with conserved
charge $Q$. We also take a bipartition of our system into two subsystems, $A$
and its complement $\bar{A}$ as before. When the conserved charge $Q$ is
local, it splits as $Q=Q_{A}+Q_{\bar{A}}$. We further assume that both $\rho$
and $\sigma$ are eigenstate of $Q$, which imply $[\rho,Q]=0,[\sigma,Q]=0$.
Tracing out the degree of freedom in $\bar{A}$, one obtains
$[\rho_{A},Q_{A}]=0,[\sigma_{A},Q_{A}]=0$. Then the density matrix $\rho_{A}$
and $\sigma_{A}$ can be written as block diagonal forms, in which each block
corresponds to a different charge sector with eigenvalue $q$ of $Q_{A}$

$\rho_{A}=\oplus_{q}\Pi_{q}\rho_{A}=\oplus_{q}p^{\rho}(q)\rho_{A}(q),\quad\sigma=\oplus_{q}\Pi_{q}\sigma_{A}=\oplus_{q}p^{\sigma}(q)\sigma_{A}(q),$
(3.1)

where $\Pi_{q}$ is the projector onto the eigenspace of $Q_{A}$ with fixed
eigenvalue $q$. We have

$\rho_{A}(q)=\frac{\Pi_{q}\rho_{A}}{\textrm{tr}(\Pi_{q}\rho_{A})},\quad\sigma_{A}(q)=\frac{\Pi_{q}\sigma_{A}}{\textrm{tr}(\Pi_{q}\sigma_{A})}.$
(3.2)

The denominators in the above equations are introduced to keep the
normalization $\textrm{tr}\rho_{A}(q)=1,\textrm{tr}\sigma_{A}(q)=1$, which
imply

$\textrm{tr}(\Pi_{q}\rho_{A})=p^{\rho}(q),\quad\textrm{tr}(\Pi_{q}\sigma_{A})=p^{\sigma}(q).$
(3.3)

Here $p^{\rho}(q)$ (or $p^{\sigma}(q)$, respectively) is the probability of
finding $q$ as the outcome of a measurement of $Q_{A}$ in state $\rho_{A}$
(resp. $\sigma_{A}$).

Our goal is to understand how the relative entropy is distributed in different
charged sectors. Let’s start with the resolution of von Neumann entanglement
entropy. The equation (3.1) implies the following decomposition of
entanglement entropy

$S(\rho_{A})=\sum_{q}p^{\rho}(q)S(\rho_{A}(q))-\sum_{q}p^{\rho}(q)\log
p^{\rho}(q)\equiv S^{c}+S^{f},$ (3.4)

where

$S(\rho_{A}(q))=-\textrm{tr}[\rho_{A}(q)\log\rho_{A}(q)].$ (3.5)

is the symmetry resolved entanglement entropy associated to $\rho_{A}(q)$. In
eq. (3.4), we have divided $S(\rho_{A})$ into two parts, $S^{c}$ and $S^{f}$,
which are called the configurational entanglement entropy and the fluctuation
entanglement entropy respectively. The configurational entanglement entropy
$S^{c}=\sum_{q}p^{\rho}(q)S(\rho_{A}(q))$, measuring the total entropy of all
the charged sectors. The fluctuation entanglement entropy
$S^{f}=-\sum_{q}p^{\rho}(q)\log p^{\rho}(q)$ takes into account the entropy
due to fluctuations of the eigenvalues of the charge.

In a similar way, we define the symmetry resolved Rényi relative entropies as

$S_{n}(\rho_{A}(q)\|\sigma_{A}(q))=\frac{1}{1-n}\log{\frac{\textrm{tr}(\rho_{A}(q)\sigma_{A}(q)^{n-1})}{\textrm{tr}(\rho_{A}(q)^{n})}}.$
(3.6)

After substituting the expression of $\rho_{A}(q)$ and $\sigma_{A}(q)$ given
in eq. (3.2) into the above equation, we obtain

$\displaystyle
S_{n}(\rho_{A}(q)\|\sigma_{A}(q))=\frac{1}{1-n}\log\frac{p^{\rho}(q)^{n}}{p^{\sigma}(q)^{n-1}p^{\rho}(q)}\frac{\textrm{tr}(\rho_{A}\sigma_{A}^{n-1}\Pi_{q})}{\textrm{tr}(\rho_{A}^{n}\Pi_{q})}$
(3.7)
$\displaystyle=-\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}+\frac{1}{1-n}\left[\log\textrm{tr}(\rho_{A}\sigma_{A}^{n-1}\Pi_{q})-\log\textrm{tr}(\rho_{A}^{n}\Pi_{q})\right].$
(3.8)

Taking the limit $n\rightarrow 1$, we find

$S(\rho_{A}(q)\|\sigma_{A}(q))=-\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}-\frac{1}{p^{\rho}(q)}\textrm{tr}(\Pi_{q}\rho_{A}\log\sigma_{A})+\frac{1}{p^{\rho}(q)}\textrm{tr}(\Pi_{q}\rho_{A}\log\rho_{A}).$
(3.9)

Multiplying both sides of the above equation with $p^{\rho}(q)$ and summing
over $q$, we get

$S(\rho_{A}\|\sigma_{A})=\langle
S(\rho_{A}(q)\|\sigma_{A}(q))\rangle_{p^{\rho}}+\left\langle\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}\right\rangle_{p^{\rho}},$
(3.10)

where

$\langle
S(\rho_{A}(q)\|\sigma_{A}(q))\rangle_{p^{\rho}}=\sum_{q}S(\rho_{A}(q)\|\sigma_{A}(q))p^{\rho}(q),$
(3.11)

is the averaged symmetry resolved relative entropy under the probability
distribution $p^{\rho}(q)$, and

$\left\langle\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}\right\rangle_{p^{\rho}}=\sum_{q}p^{\rho}(q)\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}$
(3.12)

is the classical relative entropy or Kullback-Leibler divergence of
probability distribution $p^{\rho}(q)$ and $p^{\sigma}(q)$. Here, for the
relative entropy, we find the equation (3.10) looks quite similar to eq.
(3.4). You can call the two terms on the right-hand side of eq. (3.10) the
configuration relative entropy and the fluctuation relative entropy
respectively if you will.

Let’s define the following generalized probability distributions

$p_{n}^{\rho|\sigma}(q)=\frac{\textrm{tr}(\rho_{A}\sigma_{A}^{n-1}\Pi_{q})}{\textrm{tr}(\rho_{A}\sigma_{A}^{n-1})},\quad
p_{n}^{\rho}(q)=\frac{\textrm{tr}(\rho_{A}^{n}\Pi_{q})}{\textrm{tr}(\rho_{A}^{n})},$
(3.13)

which are normalized as
$\sum_{q}p_{n}^{\rho|\sigma}(q)=\sum_{q}p_{n}^{\rho}(q)=1$. For $n=1$, since
$p_{1}^{\rho|\sigma}(q)=p_{1}^{\rho}(q)=p^{\rho}(q)$, these generalized
distributions are just the physical probability distribution of the subsystem
charge $Q_{A}$ in the state $\rho_{A}$. Using these generalized probabilities,
we can rewrite the symmetry resolved Rényi relative entropy as

$S_{n}(\rho_{A}(q)\|\sigma_{A}(q))=-\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}+\frac{1}{1-n}\log\frac{p_{n}^{\rho|\sigma}(q)}{p_{n}^{\rho}(q)}+S_{n}(\rho_{A}\|\sigma_{A}).$
(3.14)

Taking the limit $n\rightarrow 1$ of the above equation, we obtain the
symmetry resolved relative entropy

$S(\rho_{A}(q)\|\sigma_{A}(q))=-\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}-\frac{1}{p^{\rho}(q)}\partial_{n}(p_{n}^{\rho|\sigma}(q)-p_{n}^{\rho}(q))\big{|}_{n=1}+S(\rho_{A}\|\sigma_{A}).$
(3.15)

The average of the above equation over $p^{\rho}(q)$ also gives the equation
(3.10) using the fact that
$\sum_{q}\partial_{n}(p_{n}^{\rho|\sigma}(q)-p_{n}^{\rho}(q))\big{|}_{n=1}=0$.
It’s also useful to average equation (3.14) over $p^{\rho}(q)$ to give another
expression of the decomposition of the Rényi relative entropy

$S_{n}(\rho_{A}\|\sigma_{A})=\langle
S_{n}(\rho_{A}(q)\|\sigma_{A}(q))\rangle_{p^{\rho}}+\left\langle\log\frac{p^{\rho}(q)}{p^{\sigma}(q)}\right\rangle_{p^{\rho}}+\frac{1}{1-n}\sum_{q}p^{\rho}(q)\log\frac{p_{n}^{\rho|\sigma}(q)}{p_{n}^{\rho}(q)}.$
(3.16)

From eq. (3.14) and eq. (3.15), we see that to obtain the symmetry resolved
Rényi relative entropy and relative entropy, one needs to compute the
generalized probabilities $p_{n}^{\rho|\sigma}(q)$ and $p_{n}^{\rho}(q)$.
However, this is very hard in general due to the non-local feature of the
projector $\Pi_{q}$. Similar to the case of computing symmetry resolved
entanglement entropy, we can bypass this difficulty by defining the following
quantities

$p_{n}^{\rho|\sigma}(\mu)=\frac{\textrm{tr}(\rho_{A}\sigma_{A}^{n-1}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{A}\sigma_{A}^{n-1})},\quad
p_{n}^{\rho}(\mu)=\frac{\textrm{tr}(\rho_{A}^{n}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{A}^{n})},$ (3.17)

which turns out to be much easier to compute. From eq. (3.13), it’s easy to
see that $p_{n}^{\rho|\sigma}(q)$ and $p_{n}^{\rho}(q)$ can be obtained by
Fourier transformations222Here we have assumed that the eigenvalues of $Q_{A}$
are continuous. If the eigenvalues are integers, one needs to change the range
of the integral to $[-\pi,\pi]$.

$p_{n}^{\rho|\sigma}(q)=\int_{-\infty}^{\infty}\frac{d\mu}{2\pi}e^{-i\mu
q}p_{n}^{\rho|\sigma}(\mu),\quad
p_{n}^{\rho}(q)=\int_{-\infty}^{\infty}\frac{d\mu}{2\pi}e^{-i\mu
q}p_{n}^{\rho}(\mu).$ (3.18)

Here we have used the same notation but with a different argument to denote
the Fourier transform of the generalized probabilities
$p_{n}^{\rho|\sigma}(q)$ and $p_{n}^{\rho}(q)$.

For future use, it’s also useful to define the following ratio

$F^{\rho,\sigma}_{n}(\mu,A)\equiv\frac{p_{n}^{\rho|\sigma}(\mu)}{p_{n}^{\rho}(\mu)}=\frac{\textrm{tr}(\rho_{A}\sigma_{A}^{n-1}e^{i\mu
Q_{A}})\textrm{tr}(\rho_{A}^{n})}{\textrm{tr}(\rho_{A}^{n}e^{i\mu
Q_{A}})\textrm{tr}(\rho_{A}\sigma_{A}^{n-1})}.$ (3.19)

###  3.2 Symmetry resolution of entanglement entropy in CFT

From the analysis in the last subsection, we see that to compute the symmetry
resolved relative entropy, the first step is the calculation of Fourier
transformed generalized probabilities $p_{n}^{\rho|\sigma}(\mu)$ and
$p_{n}^{\rho}(\mu)$. In the free compact boson CFT, the later
($p_{n}^{\rho}(\mu)$, cf. (3.17)) has already been studied in the context of
symmetry resolution of entanglement entropy [49], where people called it the
(normalized) charged moments. In this subsection, we briefly review the
results and fix some notations that will be useful in the following sections.

Let’s first consider the ground state case. When $\rho$ is the ground state of
a CFT, $\textrm{tr}(\rho_{A}^{n}e^{i\mu Q_{A}})$ can be seen as a partition
function in the $n$-sheet Riemann surface $\mathcal{R}_{n}$ with an inserted
Aharonov-Bohm flux $\mu$. In two-dimensional CFT, the insertion of a flux
corresponds to a twisted boundary condition, which can be implemented by some
local fields acting on the boundary of subsystem $A$. This operator can be
seen as the composition of the branch point twist field $\mathcal{T}_{n}$ and
the $U(1)$ twist field $\mathcal{V}_{\mu}$ and we denote it by
$\mathcal{T}_{n,\mu}$ [26, 50]. The form factors and vacuum expectation values
(VEVs) of the composite twist field in integrable field theories have been
obtained in [51, 52] recently. If subsystem $A$ is an interval $[u,v]$, then
one can identify

$e^{i\mu Q_{A}}=\mathcal{V}_{\mu}(u,0)\mathcal{V}_{-\mu}(v,0).$ (3.20)

More precisely, we have the following relations

$\textrm{tr}(\rho_{I,A}^{n}e^{i\mu Q_{A}})=\langle e^{i\mu
Q_{A}}\rangle_{\mathcal{R}_{n}}=\langle\mathcal{V}_{\mu}(u,0)\mathcal{V}_{-\mu}(v,0)\rangle_{\mathcal{R}_{n}}=\langle\mathcal{T}_{n,\mu}(u,0)\tilde{\mathcal{T}}_{n,\mu}(v,0)\rangle_{\mathcal{R}_{1}},$
(3.21)

where $\tilde{\mathcal{T}}_{n,\mu}$ is the anti-twist field. The conformal
weight of $\mathcal{T}_{n,\mu}$ and $\tilde{\mathcal{T}}_{n,\mu}$ are the same
and are given by

$h_{n,\mu}=h_{n}+\frac{h_{\mu}}{n},\qquad h_{n}=\frac{c}{24}(n-\frac{1}{n}),$
(3.22)

where $c$ is the central charge of the CFT. Then, on the cylinder of
circumference $L$, the two-point function of $\mathcal{T}_{n,\mu}$ in eq.
(3.21) is immediately obtained

$\textrm{tr}(\rho_{I,A}^{n}e^{i\mu
Q_{A}})=s_{n,\mu}\left(\frac{L}{\pi}\sin\frac{\pi
l}{L}\right)^{-\frac{c}{6}(n-\frac{1}{n})-2\frac{h_{\mu}+\bar{h}_{\mu}}{n}},$
(3.23)

where $s_{n,\mu}$ is the unknown non-universal normalization of the composite
twist field. The charged moments of the ground state are

$p^{I}_{n}(\mu)=\frac{\textrm{tr}(\rho_{I,A}^{n}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{I,A}^{n})}=\frac{\langle\mathcal{T}_{n,\mu}(u,0)\tilde{\mathcal{T}}_{n,\mu}(v,0)\rangle_{\mathcal{R}_{1}}}{\langle\mathcal{T}_{n}(u,0)\tilde{\mathcal{T}}_{n}(v,0)\rangle_{\mathcal{R}_{1}}}=\frac{s_{n,\mu}}{s_{n,0}}\left(\frac{L}{\pi}\sin\frac{\pi
l}{L}\right)^{-\frac{2}{n}(h_{\mu}+\bar{h}_{\mu})}.$ (3.24)

In the free compact bosonic field theory defined by the action eq. (2.11), the
$U(1)$ twist field $\mathcal{V}_{\mu}$ can be implemented by the vertex
operator

$\mathcal{V}_{\mu}=V_{\frac{\mu}{2\pi}}=e^{\frac{i\mu}{2\pi}\phi}$ (3.25)

with conformal weight
$(h_{\mu},\bar{h}_{\mu})=(\frac{1}{2}(\frac{\mu}{2\pi})^{2},\frac{1}{2}(\frac{\mu}{2\pi})^{2})$.
Then the charged moments of the ground state or the Fourier transformed
generalized probabilities are

$p^{I}_{n}(\mu)\sim\left(\frac{L}{\pi}\sin\frac{\pi
l}{L}\right)^{-\frac{\mu^{2}}{2n\pi^{2}}},$ (3.26)

which are Gaussian distributions and as a consequence, $p^{I}_{n}(q)$ are also
Gaussian distributions. In terms of its variance, we can write $p_{n}^{I}(q)$
as

$p_{n}^{I}(q)=\frac{1}{\sqrt{2\pi\langle\Delta
q^{2}\rangle_{n}^{I}}}\exp(-\frac{\Delta q^{2}}{2\langle\Delta
q^{2}\rangle_{n}^{I}}),$ (3.27)

where $\Delta q^{2}=(q-\bar{q})^{2}$ and at large $L$ the variance scales as

$\langle\Delta q^{2}\rangle_{n}^{I}=\int_{-\infty}^{\infty}dq\Delta
q^{2}p^{I}_{n}(q)=\frac{1}{n\pi^{2}}\log\left(\frac{L}{\pi}\sin\pi
x\right)+\gamma_{n}+o(1),$ (3.28)

where $\gamma_{n}$ is a non-universal constant related to $s_{n,\mu}$ and
$\bar{q}$ is the mean value of $q$ under the probability distribution
$p_{n}^{I}(q)$ also cannot be fixed by CFT.333In the XX spin chain, the exact
values of $\gamma_{n}$ and $\bar{q}$ have been derived in [27]. Since
$p^{I}_{n}(\mu)$ is Gaussian, we can rewrite it as

$p^{I}_{n}(\mu)=\exp\\{-\frac{1}{2}\langle\Delta
q^{2}\rangle_{n}^{I}\mu^{2}+i\mu\bar{q}\\}.$ (3.29)

For a excited state $\ket{\Psi}$, it’s useful to define the ratio

$f_{n}^{\Psi}(\mu,x)\equiv\frac{p^{\Psi}_{n}(\mu)}{p^{I}_{n}(\mu)}=\frac{\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\prod_{k=1}^{n}\Psi(t_{k}^{-})\Psi(t_{k}^{+})\rangle_{cy}}{\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\rangle_{cy}\langle\prod_{k=1}^{n}\Psi(t_{k}^{-})\Psi(t_{k}^{+})\rangle_{cy}}.$
(3.30)

When the excitation is induced by the vertex operator
$\Psi=V_{\alpha}=e^{i\alpha\phi}$, the computation of eq. (3.30) is
straightforward and the final result is rather simple [49]

$f_{n}^{V_{\alpha}}(\mu,x)=e^{i\mu\alpha x}.$ (3.31)

Then the charged moments of the vertex operator are

$p^{V_{\alpha}}_{n}(\mu)=f_{n}^{V_{\alpha}}(\mu,x)p_{n}^{I}(\mu)=e^{i\mu\alpha
x}p_{n}^{I}(\mu).$ (3.32)

The generalized probability distributions $p^{V_{\alpha}}_{n}(q)$ are obtained
by Fourier transformation

$p^{V_{\alpha}}_{n}(q)=p_{n}^{I}(q-\alpha x).$ (3.33)

From the above equation, we conclude that the generalized probability
distribution of vertex operators $p^{V_{\alpha}}_{n}(q)$ is also Gaussian and
having the same variance with the ground state distribution $p^{I}_{n}(q)$.

However, when the excitation is induced by the derivative operators
$\Psi=i\partial\phi$, the computation of eq. (3.30) is more complicated and we
briefly mention the details here. In this calculation, the most involved
correlator is

$\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}.$
(3.34)

In the paper [49], the authors conjectured a formula for this correlator

$\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}=\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\rangle_{cy}\mathcal{P}_{M}(\frac{i\mu}{2\pi}),$
(3.35)

where $\mathcal{P}_{M}(\lambda)=\det(M-\lambda)$ is the characteristic
polynomial of the $2n\times 2n$ matrix $M$

$M=\frac{1}{2}\begin{pmatrix}A&B\\\ -B^{T}&A\end{pmatrix},$ (3.36)

where the matrix elements are ($i,j=1,2,\cdots,n$)

$A_{ij}=\begin{cases}0&\text{if}\;i=j\\\
\frac{1}{\sin[\pi(j-i)/n]}&\text{else}\end{cases},\qquad
B_{ij}=\frac{1}{\sin[\pi(j-i-x)/n]}.$ (3.37)

The analytic continuation of $\mathcal{P}_{M}(\lambda)$ is given by

$\mathcal{P}_{M}(\lambda)=\frac{\Gamma(a_{n}(\lambda)+n+1)\Gamma(\bar{a}_{n}(\lambda)+n+1)}{\Gamma(a_{n}(\lambda)+1)\Gamma(\bar{a}_{n}(\lambda)+1)},$
(3.38)

where

$a_{n}(\lambda)=\frac{1}{2}(\frac{n}{\sin\pi x}-n-1)+i\lambda.$ (3.39)

Then after pluging eq. (3.35), eq. (3.38) and eq. (2.19) into eq. (3.30), we
obtain

$f_{n}^{i\partial\phi}(\mu,x)=\frac{\Gamma(a_{n}(\frac{i\mu}{2\pi})+n+1)\Gamma(\bar{a}_{n}(\frac{i\mu}{2\pi})+n+1)}{\Gamma(a_{n}(\frac{i\mu}{2\pi})+1)\Gamma(\bar{a}_{n}(\frac{i\mu}{2\pi})+1)}\frac{\Gamma^{2}(a_{n}(0)+1)}{\Gamma^{2}(a_{n}(0)+n+1)}.$
(3.40)

We can expand $f_{n}^{i\partial\phi}(\mu,x)$ in $\mu$

$\begin{split}f_{n}^{i\partial\phi}(\mu,x)&=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}b_{n,2k}\mu^{2k}\end{split}$
(3.41)

where $b_{n,2k}$ is the coefficient of $\mu^{2k}$ and the first two values are

$\begin{split}b_{n,0}=1,\quad
b_{n,2}=\frac{1}{2\pi^{2}}\left[\psi^{(1)}(a_{n}(0)+1)-\psi^{(1)}(a_{n}(0)+n+1)\right].\end{split}$
(3.42)

Here $\psi^{(n)}(z)$ is the polygamma function. For integer $n$, the infinite
series in eq. (3.41) terminate at $k=n$ and as $n\rightarrow 1$, all
coefficients except $b_{n,0},b_{n,2}$ vanish. Then we can write

$\begin{split}p_{n}^{i\partial\phi}(q)&=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}b_{n,2k}\int_{\infty}^{\infty}\frac{d\mu}{2\pi}e^{-i\mu\Delta
q}\exp\\{-\frac{1}{2}\langle\Delta q^{2}\rangle_{n}^{I}\mu^{2}\\}\mu^{2k}\\\
&=p_{n}^{I}(q)\sum_{k=0}^{\infty}\frac{b_{n,2k}}{(2k)!}H_{2k}\left(\frac{\Delta
q}{\sqrt{2\langle\Delta q^{2}\rangle_{n}^{I}}}\right)[2\langle\Delta
q^{2}\rangle_{n}^{I}]^{-k},\end{split}$ (3.43)

where $H_{2k}(z)$ is the $2k$-th Hermite polynomial with argument $z$.

For $n=1$, we have a very simple result

$p^{i\partial\phi}(q)=\left(1-\frac{c_{1}(q)}{\pi^{2}}\sin^{2}\pi
x\right)p^{I}(q)=\left(1-\frac{c_{1}(q)}{\pi^{2}}\sin^{2}\pi
x\right)\frac{1}{\sqrt{2\pi\langle\Delta q^{2}\rangle^{I}}}\exp(-\frac{\Delta
q^{2}}{2\langle\Delta q^{2}\rangle^{I}}),$ (3.44)

where $\langle\Delta q^{2}\rangle^{I}\equiv\langle\Delta q^{2}\rangle^{I}_{1}$
and we have defined

$c_{n}(q)\equiv\frac{\langle\Delta q^{2}\rangle_{n}^{I}-\Delta
q^{2}}{[\langle\Delta q^{2}\rangle_{n}^{I}]^{2}}.$ (3.45)

Clearly, $p^{i\partial\phi}(q)$ is non-Gaussian.

##  4 Symmetry resolution of relative entropy in CFT

In this section, we will focus on the computation of the symmetry resolved
relative entropies between RDMs of excited states induced by primary operators
in the free massless compact boson CFT. According to the analysis in section
3, in order to calculate the symmetry resolved relative entropy
$S(\rho_{\Psi}(q)\|\rho_{\Phi}(q))$ (see eq. (3.15)), we have to compute the
generalized probability distribution $p_{n}^{\Psi}(q)$ and
$p_{n}^{\Psi|\Phi}(q)$. Let’s first compute the Fourier transform of
$p_{n}^{\Psi}(q)$ and $p_{n}^{\Psi|\Phi}(q)$

$p_{n}^{\Psi}(\mu)=\frac{\textrm{tr}(\rho_{\Psi}^{n}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{\Psi}^{n})}=\frac{\langle\mathcal{V}_{\mu}(u_{1})\mathcal{V}_{-\mu}(v_{1})\prod_{k=1}^{n}\Psi(w_{k}^{-})\Psi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}}{\langle\prod_{k=1}^{n}\Psi(w_{k}^{-})\Psi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}},$
(4.1)

and

$p_{n}^{\Psi|\Phi}(\mu)=\frac{\textrm{tr}(\rho_{\Psi}\rho_{\Phi}^{n-1}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{\Psi}\rho_{\Phi}^{n-1})}=\frac{\langle\mathcal{V}_{\mu}(u_{1})\mathcal{V}_{-\mu}(v_{1})\Psi(w_{1}^{-})\Psi^{\dagger}(w_{1}^{+})\prod_{k=2}^{n}\Phi(w_{k}^{-})\Phi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}}{\langle\Psi(w_{1}^{-})\Psi^{\dagger}(w_{1}^{+})\prod_{k=2}^{n}\Phi(w_{k}^{-})\Phi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}}.$
(4.2)

Thus their ratio $F^{\Psi,\Phi}_{n}(\mu,x)$ is given by

$F^{\Psi,\Phi}_{n}(\mu,x)=\frac{\langle\mathcal{V}_{\mu}(u_{1})\mathcal{V}_{-\mu}(v_{1})\Psi(w_{1}^{-})\Psi^{\dagger}(w_{1}^{+})\prod_{k=2}^{n}\Phi(w_{k}^{-})\Phi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}\langle\prod_{k=1}^{n}\Psi(w_{k}^{-})\Psi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}}{\langle\Psi(w_{1}^{-})\Psi^{\dagger}(w_{1}^{+})\prod_{k=2}^{n}\Phi(w_{k}^{-})\Phi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}\langle\mathcal{V}_{\mu}(u_{1})\mathcal{V}_{-\mu}(v_{1})\prod_{k=1}^{n}\Psi(w_{k}^{-})\Psi^{\dagger}(w_{k}^{+})\rangle_{\mathcal{R}_{n}}}.$
(4.3)

One could apply the conformal transformation defined in eq. (2.7) to map the
correlators in eq. (4.3) onto the cylinder. In this mapping, all factors
proportional to $\frac{dt}{dw}$ coming from the transformation law of primary
fields canceled out, leaving us with

$F^{\Psi,\Phi}_{n}(\mu,x)=\frac{\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\Psi(t_{1}^{-})\Psi^{\dagger}(t_{1}^{+})\prod_{k=2}^{n}\Phi(t_{k}^{-})\Phi^{\dagger}(t_{k}^{+})\rangle_{cy}\langle\prod_{k=1}^{n}\Psi(t_{k}^{-})\Psi^{\dagger}(t_{k}^{+})\rangle_{cy}}{\langle\Psi(t_{1}^{-})\Psi^{\dagger}(t_{1}^{+})\prod_{k=2}^{n}\Phi(t_{k}^{-})\Phi^{\dagger}(t_{k}^{+})\rangle_{cy}\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\prod_{k=1}^{n}\Psi(t_{k}^{-})\Psi^{\dagger}(t_{k}^{+})\rangle_{cy}}.$
(4.4)

If one of the two states is the ground state $\ket{0}$, the above generic
formula simplifies to

$F^{I,\Phi}_{n}(\mu,x)=\frac{\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\prod_{k=2}^{n}\Phi(t_{k}^{-})\Phi^{\dagger}(t_{k}^{+})\rangle_{cy}}{\langle\prod_{k=2}^{n}\Phi(t_{k}^{-})\Phi^{\dagger}(t_{k}^{+})\rangle_{cy}\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\rangle_{cy}},$
(4.5)

and

$F^{\Psi,I}_{n}(\mu,x)=\frac{\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\Psi(t_{1}^{-})\Psi^{\dagger}(t_{1}^{+})\rangle_{cy}\langle\prod_{k=1}^{n}\Psi(t_{k}^{-})\Psi^{\dagger}(t_{k}^{+})\rangle_{cy}}{\langle\Psi(t_{1}^{-})\Psi^{\dagger}(t_{1}^{+})\rangle_{cy}\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\prod_{k=1}^{n}\Psi(t_{k}^{-})\Psi^{\dagger}(t_{k}^{+})\rangle_{cy}}.$
(4.6)

###  4.1 Resolution of relative entropy between the ground state and the
vertex operator

The first case we will consider is the symmetry decomposition of relative
entropy between the ground state and the excited state generated by a vertex
operator. Firstly, let’s compute

$\begin{split}F^{I,V_{\alpha}}_{n}(\mu,x)&=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}{\langle\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{\frac{\mu}{2\pi}}(-i\infty)\rangle_{cy}}\\\
&=\prod_{k=2}^{n}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{\alpha}(t_{k}^{-})\rangle_{cy}\prod_{k=2}^{n}\langle
V_{-\frac{\mu}{2\pi}}(-i\infty)V_{\alpha}(t_{k}^{-})\rangle_{cy}\times\\\
&\times\prod_{k=2}^{n}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\alpha}(t_{k}^{+})\rangle_{cy}\prod_{k=2}^{n}\langle
V_{-\frac{\mu}{2\pi}}(-i\infty)V_{-\alpha}(t_{k}^{+})\rangle_{cy}\\\
&=\lim_{\Lambda\rightarrow\infty}\prod_{k=2}^{n}\left(\frac{\sin\frac{1}{2}(\frac{\pi
x+2(k-1)}{n})-i\Lambda}{\sin\frac{1}{2}(\frac{\pi
x+2(k-1)}{n})+i\Lambda}\right)^{\frac{\alpha\mu}{2\pi}}\prod_{k=2}^{n}\left(\frac{\sin\frac{1}{2}(\frac{-\pi
x+2(k-1)}{n})-i\Lambda}{\sin\frac{1}{2}(\frac{-\pi
x+2(k-1)}{n})+i\Lambda}\right)^{-\frac{\alpha\mu}{2\pi}}\\\ &=e^{i\mu\alpha
x(1-\frac{1}{n})}.\end{split}$ (4.7)

Then we find

$p^{I|V_{\alpha}}_{n}(\mu)=F^{I,V_{\alpha}}_{n}(\mu,x)p^{I}_{n}(\mu)=e^{i\mu\alpha
x(1-\frac{1}{n})}p^{I}_{n}(\mu).$ (4.8)

Since $p^{I}_{n}(\mu)$ is a Gaussian distribution, we conclude
$p^{I|V_{\alpha}}_{n}(q)$ is also a Gaussian with the same variance of
$p_{n}^{I}(q)$. After Fourier transformation, we find

$p^{I|V_{\alpha}}_{n}(q)=p_{n}^{I}(q-\alpha x(1-1/n)).$ (4.9)

Then the symmetry resolved Rényi relative entropies can be easily derived

$\begin{split}S_{n}(\rho_{I}(q)\|\rho_{V_{\alpha}}(q))=-\log\frac{p^{I}(q)}{p^{V_{\alpha}}(q)}+\frac{1}{1-n}\log\frac{p_{n}^{I|V_{\alpha}}(q)}{p^{I}_{n}(q)}+S_{n}(\rho_{I}\|\rho_{V_{\alpha}})\\\
=-\log\frac{p^{I}(q)}{p^{I}(q-\alpha
x)}+\frac{1}{1-n}\log\frac{p_{n}^{I}(q-(1-1/n)\alpha
x)}{p^{I}_{n}(q)}+S_{n}(\rho_{I}\|\rho_{V_{\alpha}}).\end{split}$ (4.10)

After substituting the expression of $p_{n}^{I}(q)$ in eq. (3.27) and Rényi
relative entropy in eq. (2.21) into the above equation, we obtain

$S_{n}(\rho_{I}(q)\|\rho_{V_{\alpha}}(q))=\frac{2\alpha x\Delta
q-\alpha^{2}x^{2}}{2\langle\Delta q^{2}\rangle^{I}}-\frac{2n\alpha x\Delta
q+(1-n)\alpha^{2}x^{2}}{2n^{2}\langle\Delta
q^{2}\rangle_{n}^{I}}+\frac{\alpha^{2}}{1-n}\log\frac{\sin\pi
x}{n\sin\frac{\pi x}{n}}.$ (4.11)

The symmetry resolved relative entropy is obtained by taking the limit
$n\rightarrow 1$

$S(\rho_{I}(q)\|\rho_{V_{\alpha}}(q))=\alpha^{2}(1-\pi x\cot(\pi
x))-\frac{\alpha^{2}x^{2}}{2\langle\Delta q^{2}\rangle^{I}},$ (4.12)

which is $q$ independent up to order $(\log L)^{-1}$.

After a very similar calculation with eq. (4.7), we find

$\frac{\langle\prod_{k=1}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{\frac{\mu}{2\pi}}(-i\infty)\rangle_{cy}}{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=1}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}=e^{-i\mu\alpha
x}.$ (4.13)

Then we have

$\begin{split}F^{V_{\alpha},I}_{n}(\mu,x)&=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)V_{\alpha}(t_{1}^{-})V_{-\alpha}(t_{1}^{+})\rangle_{cy}\langle\prod_{k=1}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}{\langle
V_{\alpha}(t_{1}^{-})V_{-\alpha}(t_{1}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=1}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}\\\
&=e^{-i\mu\alpha x}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{\alpha}(t_{1}^{-})\rangle_{cy}\langle
V_{-\frac{\mu}{2\pi}}(-i\infty)V_{-\alpha}(t_{1}^{+})\rangle_{cy}\times\\\
&\times\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\alpha}(t_{1}^{+})\rangle_{cy}\langle
V_{-\frac{\mu}{2\pi}}(-i\infty)V_{\alpha}(t_{1}^{-})\rangle_{cy}\\\
&=e^{-i\mu\alpha
x}\lim_{\Lambda\rightarrow\infty}\left(-\frac{\sin\frac{1}{2}(\frac{\pi
x}{n}-i\Lambda)}{\sin\frac{1}{2}(\frac{\pi
x}{n}+i\Lambda)}\right)^{\frac{\mu\alpha}{2\pi}}\left(-\frac{\sin\frac{1}{2}(\frac{\pi
x}{n}-i\Lambda)}{\sin\frac{1}{2}(\frac{\pi
x}{n}+i\Lambda)}\right)^{\frac{\mu\alpha}{2\pi}}\\\ &=e^{-i\mu\alpha
x(1-\frac{1}{n})}.\end{split}$ (4.14)

Thus the corresponding Fourier transformed generalized probability
distributions can be found as

$p^{V_{\alpha}|I}_{n}(\mu)=F^{V_{\alpha},I}_{n}(\mu,x)p^{V_{\alpha}}_{n}(\mu)=F^{V_{\alpha},I}_{n}(\mu,x)f^{V_{\alpha}}_{n}(\mu,x)p^{I}_{n}(\mu)=e^{\frac{i\alpha\mu
x}{n}}p^{I}_{n}(\mu).$ (4.15)

Then we get

$p^{V_{\alpha}|I}_{n}(q)=p^{I}_{n}(q-\alpha x/n).$ (4.16)

Now the symmetry resolved Rényi relative entropies are easily obtained

$\begin{split}S_{n}(\rho_{V_{\alpha}}(q)\|\rho_{I}(q))=-\log\frac{p^{V_{\alpha}}(q)}{p^{I}(q)}+\frac{1}{1-n}\log\frac{p_{n}^{V_{\alpha}|I}(q)}{p^{V_{\alpha}}_{n}(q)}+S_{n}(\rho_{V_{\alpha}}\|\rho_{I})\\\
=\frac{\alpha^{2}x^{2}-2\alpha x\Delta q}{2\langle\Delta
q^{2}\rangle^{I}}+\frac{2n\alpha x\Delta
q-(1+n)\alpha^{2}x^{2}}{2n^{2}\langle\Delta
q^{2}\rangle_{n}^{I}}+\frac{\alpha^{2}}{1-n}\log\frac{\sin\pi
x}{n\sin\frac{\pi x}{n}}.\end{split}$ (4.17)

Taking the limit $n\rightarrow 1$, the symmetry resolved relative entropy is
given by

$S(\rho_{V_{\alpha}}(q)\|\rho_{I}(q))=\alpha^{2}(1-\pi x\cot(\pi
x))-\frac{\alpha^{2}x^{2}}{2\langle\Delta
q^{2}\rangle^{I}}=S(\rho_{I}(q)\|\rho_{V_{\alpha}}(q)).$ (4.18)

Finally, let’s compute the symmetry resolved relative entropy between two
vertex operators. As before, we begin with

$\begin{split}F^{V_{\alpha},V_{\beta}}_{n}(\mu,x)&=\frac{\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)V_{\alpha}(t_{1}^{-})V_{-\alpha}(t_{1}^{+})\prod_{k=2}^{n}V_{\beta}(t_{k}^{-})V_{-\beta}(t_{k}^{+})}{\langle
V_{\alpha}(t_{1}^{-})V_{-\alpha}(t_{1}^{+})\prod_{k=2}^{n}V_{\beta}(t_{k}^{-})V_{-\beta}(t_{k}^{+})\rangle_{cy}\langle\mathcal{V}_{\mu}(i\infty)\mathcal{V}_{-\mu}(-i\infty)\rangle_{cy}}[f_{n}^{V_{\alpha}}(\mu,x)]^{-1}\\\
&=e^{i\mu\alpha
x}(\mu,x)F^{V_{\alpha},I}_{n}(\mu,x)F^{I,V_{\beta}}_{n}(\mu,x)[f_{n}^{V_{\alpha}}(\mu,x)]^{-1}\\\
&=e^{i\mu(\alpha-\beta)(\frac{1}{n}-1)x},\end{split}$ (4.19)

which implies

$p^{V_{\alpha}|V_{\beta}}_{n}(q)=p_{n}^{I}(q-(\alpha-\beta)(1-1/n)x-\alpha
x).$ (4.20)

The symmetry resolved relative entropy can be derived in a similar way, and
the final result is

$S(\rho_{V_{\alpha}}(q)\|\rho_{V_{\beta}}(q))=(\alpha-\beta)^{2}(1-\pi
x\cot(\pi x))-\frac{(\alpha-\beta)^{2}x^{2}}{2\langle\Delta
q^{2}\rangle^{I}},$ (4.21)

which is also $q$ independent up to order $1/\log L$.

In this subsection, we find that all the symmetry resolved relative entropies
did not depend on the charge eigenvalue up to order $(\log L)^{-1}$, which
means the relative entropies are the same in the different charge sectors. We
must mention that the equipartition of relative entropy may be broken at
higher order in $1/\log L$.

###  4.2 Resolution of relative entropy between the derivative operator and
the ground state

In this subsection, we will consider a more complicated case, which is the
symmetry resolution of the relative entropy between the excited state
generated by a derivative operator and the ground state. Let’s start with

$\begin{split}F^{i\partial\phi,I}_{n}(\mu,x)&=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\rangle_{cy}\langle\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}}{\langle
i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}}\\\
&=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\rangle_{cy}}{\langle
i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\rangle_{cy}}[f_{n}^{i\partial\phi}(\mu,x)]^{-1}\\\
&=(1-\frac{\mu^{2}}{\pi^{2}}\sin^{2}\frac{\pi
x}{n})[f_{n}^{i\partial\phi}(\mu,x)]^{-1}.\end{split}$ (4.22)

Then we obtain

$\begin{split}p^{i\partial\phi|I}_{n}(\mu)&=F^{i\partial\phi,I}_{n}(\mu,x)p^{i\partial\phi}_{n}(\mu)=F^{i\partial\phi,I}_{n}(\mu,x)f_{n}^{i\partial\phi}(\mu,x)p_{n}^{I}(\mu)\\\
&=(1-\frac{\mu^{2}}{\pi^{2}}\sin^{2}\frac{\pi
x}{n})p_{n}^{I}(\mu).\end{split}$ (4.23)

After Fourier transformation, we find

$p^{i\partial\phi|I}_{n}(q)=\left(1-\frac{c_{n}(q)}{\pi^{2}}\sin^{2}\frac{\pi
x}{n}\right)p^{I}_{n}(q).$ (4.24)

The symmetry resolved Rényi relative entropies can be derived
straightforwardly from the following equation

$\begin{split}S_{n}(\rho_{i\partial\phi}(q)\|\rho_{I}(q))=-\log\frac{p^{i\partial\phi}(q)}{p^{I}(q)}+\frac{1}{1-n}\log\frac{p_{n}^{{i\partial\phi}|I}(q)}{p^{i\partial\phi}_{n}(q)}+S_{n}(\rho_{i\partial\phi}\|\rho_{I}).\\\
\end{split}$ (4.25)

In this case, to see whether equipartition of relative entropy hold at order
$1/\log L$, we can keep only the first order $\mu^{2}$ in the expansion of
$f_{n}^{i\partial\phi}(\mu,x)$, i.e. to approximate $p_{n}^{i\partial\phi}(q)$
as

$p_{n}^{i\partial\phi}(q)\simeq(1-\frac{1}{2}b_{n,2}c_{n}(q))p_{n}^{I}(q).$
(4.26)

Then in the physical regime $\Delta q$ of order 1, we have

$S_{n}(\rho_{i\partial\phi}(q)\|\rho_{I}(q))=S_{n}(\rho_{i\partial\phi}\|\rho_{I})+\frac{c_{1}(q)}{\pi^{2}}\sin^{2}\pi
x+\frac{c_{n}(q)}{1-n}(\frac{1}{2}b_{n,2}-\frac{1}{\pi^{2}}\sin^{2}\frac{\pi
x}{n})+\mathcal{O}((\log L)^{-2}).$ (4.27)

The symmetry resolved relative entropy is obtained after taking the limit
$n\rightarrow 1$

$\begin{split}S(\rho_{i\partial\phi}(q)\|\rho_{I}(q))=2\log(2\sin(\pi
x))+2-2\pi x\cot(\pi x)+2\psi\left(\frac{\csc(\pi x)}{2}\right)+2\sin(\pi
x)\\\ +\frac{c_{1}(q)}{4\pi^{2}}\left(\psi^{(2)}\left(\frac{\csc(\pi
x)}{2}\right)+12\sin^{2}(\pi x)+8\sin^{3}(\pi x)-4\pi x\sin(2\pi
x)\right)+\mathcal{O}((\log L)^{-2}).\end{split}$ (4.28)

From the above equation, in contrast to the previous case, we find that the
equipartition of relative entropy breaks down at order $1/\log L$.

###  4.3 Resolution of relative entropy between the derivative operators and
the vertex operators

In this subsection, we finally study the symmetry resolved relative entropy
between two excited states, generated by a derivative operator and a vertex
operator respectively. As usual, we first compute

$\begin{split}F^{i\partial\phi,V_{\alpha}}_{n}(\mu,x)&=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\langle\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}}{\langle
i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=1}^{n}i\partial\phi(t_{k}^{-})i\partial\phi(t_{k}^{+})\rangle_{cy}}\\\
&=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}{\langle
i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\rangle_{cy}}[f^{i\partial\phi}_{n}(\mu,x)]^{-1}.\end{split}$
(4.29)

To calculate the complicated correlators in the above equation, it’s useful to
introduce

$\mathcal{F}_{n}(\mu,x)=\frac{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}{\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\rangle_{cy}\langle\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}}.$
(4.30)

Then one can rewrite $F^{i\partial\phi,V_{\alpha}}_{n}(\mu,x)$ as

$F^{i\partial\phi,V_{\alpha}}_{n}(\mu,x)=\frac{\mathcal{F}_{n}(\mu,x)}{\mathcal{F}_{n}(0,x)}[f^{i\partial\phi}_{n}(\mu,x)]^{-1}$
(4.31)

Noticing that

$i\partial\phi(t)=\frac{1}{\epsilon}\frac{\partial}{\partial
t}V_{\epsilon}(t)\Big{|}_{\epsilon=0},$ (4.32)

we have [15, 53]

$\begin{split}\langle
V_{\frac{\mu}{2\pi}}(i\infty)&V_{-\frac{\mu}{2\pi}}(-i\infty)i\partial\phi(t_{1}^{-})i\partial\phi(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\\\
&=-\frac{1}{\epsilon^{2}}\frac{\partial}{\partial
t_{1}^{+}}\frac{\partial}{\partial t_{1}^{-}}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)V_{\epsilon}(t_{1}^{-})V_{-\epsilon}(t_{1}^{+})\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\Big{|}_{\epsilon=0}\\\
&=\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\frac{\mu}{2\pi}}(-i\infty)\prod_{k=2}^{n}V_{\alpha}(t_{k}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}C_{\epsilon=0,\alpha}(n,x).\end{split}$
(4.33)

Then we can write

$\mathcal{F}(\mu,x)=F^{I,V_{\alpha}}(\mu,x)C_{\epsilon=0,\alpha}(n,x)=e^{i\mu\alpha
x(1-\frac{1}{n})}C_{\epsilon=0,\alpha}(n,x)$ (4.34)

where

$C_{\epsilon,\alpha}(n,x)=-\frac{1}{\epsilon^{2}}\partial_{t_{1}^{+}}\partial_{t_{1}^{-}}\tilde{C}_{\epsilon,\alpha}(n,x),$
(4.35)

and

$\begin{split}&\tilde{C}_{\epsilon,\alpha}(n,x)=\langle
V_{\epsilon}(t_{1}^{-})V_{-\epsilon}(t_{1}^{+})\rangle_{cy}\\\ &\times\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{\epsilon}(t_{1}^{-})\rangle_{cy}\langle
V_{\frac{\mu}{2\pi}}(i\infty)V_{-\epsilon}(t_{1}^{+})\rangle_{cy}\langle
V_{-\frac{\mu}{2\pi}}(-i\infty)V_{\epsilon}(t_{1}^{-})\rangle_{cy}\langle
V_{-\frac{\mu}{2\pi}}(-i\infty)V_{-\epsilon}(t_{1}^{+})\rangle_{cy}\\\
&\times\prod_{k=2}^{n}\langle
V_{\epsilon}(t_{1}^{-})V_{\alpha}(t_{k}^{-})\rangle_{cy}\langle
V_{\epsilon}(t_{1}^{-})V_{-\alpha}(t_{k}^{+})\rangle_{cy}\langle
V_{-\epsilon}(t_{1}^{+})V_{\alpha}(t_{k}^{-})\rangle_{cy}\langle
V_{-\epsilon}(t_{1}^{+})V_{-\alpha}(t_{k}^{+})\rangle_{cy}.\end{split}$ (4.36)

After taking derivatives of $\tilde{C}_{\epsilon,\alpha}(n,x)$ and taking
$\epsilon=0$, one find

$\mathcal{F}_{n}(\mu,x)=\frac{e^{i\mu\alpha
x(1-\frac{1}{n})}}{4}\left(\csc^{2}\frac{\pi x}{n}+\alpha^{2}(n\cot\pi
x-\cot\frac{\pi x}{n})^{2}-\frac{2i\alpha\mu}{\pi}(n\cot\pi x-\cot\frac{\pi
x}{n})-\frac{\mu^{2}}{\pi^{2}}\right).$ (4.37)

It follows that

$\begin{split}&p_{n}^{i\partial\phi|V_{\alpha}}(\mu)=F^{i\partial\phi,V_{\alpha}}_{n}(\mu,x)p_{n}^{i\partial\phi}(\mu)=\frac{\mathcal{F}_{n}(\mu,x)}{\mathcal{F}_{n}(0,x)}p^{I}_{n}(\mu)\\\
&=\left(1-\frac{2i\mu d_{n}+\mu^{2}}{\pi^{2}\csc^{2}\frac{\pi
x}{n}+d_{n}^{2}}\right)e^{i\mu\alpha
x(1-\frac{1}{n})}p_{n}^{I}(\mu),\end{split}$ (4.38)

where we have defined

$d_{n}\equiv\pi\alpha(n\cot\pi x-\cot\frac{\pi x}{n}).$ (4.39)

After Fourier transformation, we obtain

$p_{n}^{i\partial\phi|V_{\alpha}}(q)=\left(1-\frac{2d_{n}\Delta\tilde{q}+c_{n}(\tilde{q})\langle\Delta
q^{2}\rangle^{I}_{n}}{(\pi^{2}\csc^{2}\frac{\pi x}{n}+d_{n}^{2})\langle\Delta
q^{2}\rangle^{I}_{n}}\right)p^{I}_{n}(\tilde{q}),\qquad\tilde{q}\equiv
q-\alpha x(1-\frac{1}{n}).$ (4.40)

The symmetry resolved Rényi relative entropies are given by

$\begin{split}S_{n}(\rho_{i\partial\phi}(q)\|\rho_{V_{\alpha}}(q))&=-\log\frac{p^{i\partial\phi}(q)}{p^{V_{\alpha}}(q)}+\frac{1}{1-n}\log\frac{p_{n}^{{i\partial\phi}|V_{\alpha}}(q)}{p^{i\partial\phi}_{n}(q)}+S_{n}(\rho_{i\partial\phi}\|\rho_{V_{\alpha}}).\end{split}$
(4.41)

Similar to the previous calculation, in the physical regime $\Delta q$ of
order 1, we have

$\begin{split}&S_{n}(\rho_{i\partial\phi}(q)\|\rho_{V_{\alpha}}(q))=S_{n}(\rho_{i\partial\phi}\|\rho_{V_{\alpha}})+\frac{2\alpha
x\Delta q-\alpha^{2}x^{2}}{2\langle\Delta q^{2}\rangle^{I}}-\frac{2n\alpha
x\Delta q+(1-n)\alpha^{2}x^{2}}{2n^{2}\langle\Delta q^{2}\rangle_{n}^{I}}\\\
&+\frac{c_{1}(q)}{\pi^{2}}\sin^{2}\pi
x+\frac{1}{1-n}\left(\frac{c_{n}(q)}{2}b_{n,2}-\frac{2d_{n}\Delta\tilde{q}+c_{n}(\tilde{q})\langle\Delta
q^{2}\rangle^{I}_{n}}{(\pi^{2}\csc^{2}\frac{\pi x}{n}+d_{n}^{2})\langle\Delta
q^{2}\rangle^{I}_{n}}\right)+\mathcal{O}((\log L)^{-2}).\end{split}$ (4.42)

The symmetry resolved relative entropy is obtained after taking the limit
$n\rightarrow 1$

$\begin{split}&S(\rho_{i\partial\phi}(q)\|\rho_{V_{\alpha}}(q))=(2+\alpha^{2})(1-\pi
x\cot(\pi x))+2\log(2\sin(\pi x))+2\psi\left(\frac{\csc(\pi
x)}{2}\right)+2\sin(\pi x)\\\
&+\frac{c_{1}(q)}{4\pi^{2}}\left(\psi^{(2)}\left(\frac{\csc(\pi
x)}{2}\right)+12\sin^{2}(\pi x)+8\sin^{3}(\pi x)-4\pi x\sin(2\pi
x)\right)-\frac{\alpha^{2}x^{2}}{2\langle\Delta q^{2}\rangle^{I}}\\\
&+\frac{\alpha\Delta q(2x\sin^{2}\pi x+(\pi\sin 2\pi x-2\pi^{2}x)\langle\Delta
q^{2}\rangle^{I})}{\pi^{2}[\langle\Delta
q^{2}\rangle^{I}]^{2}}+\mathcal{O}((\log L)^{-2}).\end{split}$ (4.43)

From the above equation we know that the equipartition of relative entropy
also breaks down at order $1/\log L$.

##  5 Numerical tests

(a)

Figure 1: Numerical data of $|p_{n}^{V_{1}|I}(\mu)|$ in the XX spin chain. The
full lines are the CFT predictions, eq. (4.15) with eq. (3.29). Here we
consider $n=1,2,3$ and $x=1/18,1/6,1/2$ with $L=1800$. The agreement is very
well for small $\mu$, but it worsens as $\mu$ gets closer to $\pm\pi$ and as
$n$ get larger.

We now make some numerical tests of the universal CFT results obtained in the
previous section. In this section, we will use the XX spin chain model as a
concrete lattice model to check our CFT predictions. The Hamiltonian of the XX
spin chain is given by

$H=-\frac{1}{4}\sum_{j=1}^{L}(\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{y}_{j}\sigma^{y}_{j+1}-h\sigma^{z}_{j}),$
(5.1)

where $\sigma^{x,y,z}_{j}$ are the Pauli matrices acting on the $j$-th site
and we impose periodic boundary conditions. For simplicity, we assume that
$h=0$ and the length of chain $L$ multiples of 4. We are only interested in
the spatial bipartition of the system where subsystem $A$ is given by $l$
continuous lattice sites.

We did not manage to compute symmetry resolved relative entropies numerically.
Instead, we numerically calculate the Fourier transformed generalized
probabilities $p^{\rho|\sigma}_{n}(\mu)$ for integer $n$, which are the key
integrant in the computation of symmetry resolved Rényi relative entropies.
They are defined as

$p^{\rho|\sigma}_{n}(\mu)=\frac{\textrm{tr}(\rho_{A}\sigma^{n-1}_{A}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{A}\sigma^{n-1}_{A})}$ (5.2)

In the limit $L\rightarrow\infty$ with $x=l/L$ kept fixed, our numerical
results should converge to the CFT computations for $p^{\rho|\sigma}_{n}(\mu)$
calculated from eq. (4.2). The technical details of the numerical computation
are discussed in appendix A.

The numerical results for the function $|p_{n}^{V_{1}|I}(\mu)|$ are reported
in Fig. 1 for different $n$ and different subsystem sizes $l$, where we have
used the exact results for the ground state variance given in [27]. As shown
in the figure, the agreement between numerical data and CFT prediction is
excellent for small $\mu$, while it gets worse for larger values of $\mu$ and
$n$.

In Fig. 2, we report the numerical data for the quantities
$|p_{n}^{i\partial\phi|V_{1}}(\mu)|$ for various $n$ and $l$. From eq. (4.38),
it’s clear that $|p_{n}^{i\partial\phi|V_{1}}(\mu)|$ is non-Gaussian although
it’s not easy to see this from the figure. In this case, the numerical results
and the CFT predictions also match very well for small values of $\mu$ and
$n$.

(a)

Figure 2: Numerical data of $|p_{n}^{i\partial\phi|V_{1}}(\mu)|$ in the XX
spin chain. The full lines are the CFT predictions, eq. (4.38) with eq.
(3.29). Here we consider $n=1,2,3$ and $x=1/20,1/10,1/5$ with $L=1000$. The
agreement is very well for small $\mu$, but it worsens as $\mu$ gets closer to
$\pm\pi$ and as $n$ get larger.

In Fig. 3, we show the numerical results of the ratio
$p_{n}^{i\partial\phi|I}(\mu)/p_{n}^{I}(\mu)$ for different $n$ and $l$. From
eq. (4.23), it’s easy to see that $p_{n}^{i\partial\phi|I}(\mu)$ is also non-
Gaussian. This figure shows the non-Gaussian feature clearly and the agreement
between the numerical data and CFT results is perfect for small values of
$\mu$ and $n$.

(a)

Figure 3: Numerical data of $p_{n}^{i\partial\phi|I}(\mu)/p_{n}^{I}(\mu)$ in
the XX spin chain. The full lines are the CFT predictions, eq. (4.23). Here we
consider $n=1,2,3$ and $x=1/10,1/4,1/2$ with $L=2000$. Again, the agreement is
very well for small $\mu$, but it worsens as $\mu$ gets closer to $\pm\pi$ and
as $n$ get larger.

##  6 Conclusion

In this paper, we study the $U(1)$ symmetry resolution of relative entropies
between primary states in the free massless compact boson CFT and its concrete
lattice realization, the XX spin chain. We obtain various exact results from
the CFT calculation using the replica method. We also carefully test our CFT
predictions with the exact lattice calculations in the XX spin chain and find
perfect agreements.

We must mention that the symmetry resolved relative entropies cannot be
obtained directly by our numerical method. Instead, we just compute the
Fourier transformed generalized probabilities numerically. It would be very
interesting to further numerically confirm our CFT results of symmetry
resolved relative entropies by other methods.

Several generalizations of this paper would be worth investigating. For
example, one can consider the $U(1)$ symmetry resolved relative entropies in
other CFTs (such as Ising and other minimal models) and the corresponding
lattice models or consider the extension to Wess-Zuminon-Witten models which
have non-abelian symmetries. One can also work out the symmetry resolution of
other entanglement-related quantities. For example, a natural extension could
be to consider the mutual information or the trace distance.

## Acknowledgments

The work was supported by the National Natural Science Foundation of China,
Grant No. 12005081.

##  Appendix A Correlation matrices and RDMs in XX spin chain

The Hamiltonian of the XX spin chain is given by

$H=-\frac{1}{4}\sum_{j=1}^{L}(\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{y}_{j}\sigma^{y}_{j+1}-h\sigma^{z}_{j}),$
(A.1)

where $\sigma^{x,y,z}_{j}$ are the Pauli matrices acting on the $j$-th site
and we impose periodic boundary conditions. After a Jordan-Wigner
transformation

$c_{j}=\left(\prod_{k=1}^{j-1}\sigma_{k}^{z}\right)\frac{\sigma_{j}^{x}-i\sigma_{j}^{y}}{2},\qquad
c_{j}^{\dagger}=\left(\prod_{k=1}^{j-1}\sigma_{k}^{z}\right)\frac{\sigma_{j}^{x}+i\sigma_{j}^{y}}{2},$
(A.2)

the spin chain Hamiltonian is mapped into a free fermion Hamiltonian on the
lattice

$H=-\frac{1}{2}\sum_{j=1}^{L}\left[c_{j}^{\dagger}c_{j+1}+c_{j+1}^{\dagger}-2h(c_{j}^{\dagger}c_{j}-\frac{1}{2})\right],$
(A.3)

where $c_{j},c^{\dagger}_{j}$ are fermionic annihilation and creation
operators, satisfying $\\{c_{i},c^{\dagger}_{j}\\}=\delta_{ij}$. We will
impose anti-periodic boundary conditions to the fermions
$c_{L+1}=-c_{1},c^{\dagger}_{L+1}=-c^{\dagger}_{1}$. For simplicity, we assume
that $h=0$ and the length of chain $L$ multiples of 4. The Hamiltonian eq.
(A.3) can be diagonalized by Fourier transformation

$b_{k}=\frac{1}{\sqrt{L}}\sum_{l=1}^{L}c_{l}e^{i\phi_{k}l},\quad
b_{k}^{\dagger}=\frac{1}{\sqrt{L}}\sum_{l=1}^{L}c_{l}^{\dagger}e^{-i\phi_{k}l},\quad\phi_{k}=\frac{2\pi
k}{L}$ (A.4)

Then

$H=\sum_{k\in\Omega}\epsilon_{k}(b_{k}^{\dagger}b_{k}-\frac{1}{2}),$ (A.5)

where $\epsilon_{k}=-\cos k$ and the corresponding $\Omega$ is

$\Omega=\\{\pm\frac{1}{2},\pm\frac{3}{2},\cdots\pm\frac{L-1}{2}\\}.$ (A.6)

The eigenstates of the Hamiltonian can be characterized by a set of momenta
$K$, $\ket{K}=\prod_{k\in K}b_{k}^{\dagger}\ket{0}$. The ground state is a
Fermi sea with Fermi momentum $k_{F}=\pi/2$ and is half-filling with fermion
number $n_{F}=L/2$, characterized by the set of momenta:
$\\{\pm\frac{1}{2},\pm\frac{3}{2},\cdots\pm\frac{n_{F}-1}{2}\\}$. Low-lying
excited states are obtained by removing/adding particles in momentum space
close to the Fermi surface. The correspondence between low-lying excitations
in XX chain and primary excited states in CFT is described in detail in [45].
This model has a $U(1)$ symmetry with the conserved charge
$Q=\sum_{j=1}^{L}c_{j}^{\dagger}c_{j}$.

We are interested in the spatial bipartition of the system where subsystem $A$
is given by $l$ contiguous lattice sites. The reduced density matrix of the
pure state $\ket{K}$ can be written as

$\rho_{A}=\det{C_{A}^{K}}\exp\left(\sum_{i,j}\log[((C_{A}^{K})^{-1}-1)]_{ij}c_{i}^{\dagger}c_{j}\right).$
(A.7)

where the $l\times l$ matrix
$[C_{A}^{K}]_{m,n}=\bra{K}c_{m}^{\dagger}c_{n}\ket{K}$ is the correlation
matrix restricted in $A$. The element of $C_{A}^{K}$ is given by

$[C_{A}^{K}]_{mn}=\frac{1}{L}\sum_{k\in K}e^{i\phi_{k}(m-n)}.$ (A.8)

It useful to introduce $2L$ Majorana modes

$a_{2m-1}=c_{m}+c_{m}^{\dagger},\qquad a_{2m}=i(c^{\dagger}_{m}-c_{m}).$ (A.9)

For an interval with $l$ sites of the spin chain in a state $\ket{K}$, one
defines the Majorana correlation matrix

$\langle a_{r}a_{s}\rangle_{K}=\delta_{rs}+\Gamma^{K}_{rs}$ (A.10)

with $\Gamma\in\mathcal{M}_{2N}(\mathbb{C})$:

$\Gamma^{K}=\begin{pmatrix}T^{K}_{0}&T^{K}_{1}&\cdots&T^{K}_{l-1}\\\
T^{K}_{-1}&T^{K}_{0}&\cdots&T^{K}_{l-2}\\\ \vdots&\vdots&\ddots&\vdots\\\
T^{K}_{1-l}&T^{K}_{2-l}&\cdots&T^{K}_{0}\end{pmatrix},\qquad
T^{K}_{m-n}=\begin{pmatrix}f^{K}_{m-n}&g^{K}_{m-n}\\\
-g^{K}_{n-m}&f^{K}_{m-n}\end{pmatrix},$ (A.11)

and

$\begin{split}&f^{K}_{m-n}=[C^{K}_{A}]_{mn}-[C^{K}_{A}]_{nm}=\frac{1}{L}\sum_{k\in
K}e^{i\phi_{k}(m-n)}-\frac{1}{L}\sum_{k\in K}e^{-i\phi_{k}(m-n)},\\\
&g_{m-n}^{K}=-i[C^{K}_{A}]_{mn}-i[C^{K}_{A}]_{nm}+\delta_{mn}=-\frac{i}{L}\sum_{k\in
K}e^{-i\phi_{k}(m-n)}+\frac{i}{L}\sum_{k\notin
K}e^{i\phi_{k}(m-n)}.\end{split}$ (A.12)

The $2^{l}\times 2^{l}$ RDM are completely determined by the correlation
matrix $\Gamma^{K}$.

$\rho_{\Gamma_{1}}\rho_{\Gamma_{2}}=\textrm{tr}(\rho_{\Gamma_{1}}\rho_{\Gamma_{2}})\rho_{\Gamma_{1}\times\Gamma_{2}},$
(A.13)

where

$\textrm{tr}(\rho_{\Gamma_{1}}\rho_{\Gamma_{2}})=\sqrt{\left|\frac{1+\Gamma_{1}\Gamma_{2}}{2}\right|},$
(A.14)

and one defines the product rule [54]

$\Gamma_{1}\times\Gamma_{2}=1-(1-\Gamma_{2})(1+\Gamma_{1}\Gamma_{2})^{-1}(1-\Gamma_{1}).$
(A.15)

Now, by associativity, one can obtain the trace of the product of arbitrary
number of RDMs

$\textrm{tr}(\rho_{\Gamma_{1}}\rho_{\Gamma_{2}}\cdots)=\textrm{tr}(\rho_{\Gamma_{1}}\rho_{\Gamma_{2}})\textrm{tr}(\rho_{\Gamma_{1}\times\Gamma_{2}}\cdots).$
(A.16)

Then the above formula is rather useful to calculate the Rényi relative
entropy. However, when we want to know the symmetry resolution of relative
entropy, we need to calculate

$p^{\rho|\sigma}_{n}(\mu)=\frac{\textrm{tr}(\rho_{A}\sigma^{n-1}_{A}e^{i\mu
Q_{A}})}{\textrm{tr}(\rho_{A}\sigma^{n-1}_{A})}.$ (A.17)

We can also view $e^{i\mu Q_{A}}$ as some RDM with Majorana correlation matrix
$\Gamma^{\mu}$ since

$\begin{split}&e^{i\mu
Q_{A}}=e^{i\mu\sum_{j=1}^{l}c_{j}^{\dagger}c_{j}}=\prod_{j=1}^{l}(e^{i\mu}c_{j}^{\dagger}c_{j}+c_{j}c_{j}^{\dagger})\\\
&=(1+e^{i\mu})^{l}\prod_{j=1}^{l}[p_{j}c_{j}c_{j}^{\dagger}+(1-p_{j})c_{j}^{\dagger}c_{j}],\end{split}$
(A.18)

where $p_{j}=\frac{1}{1+e^{i\mu}}$. The Majorana correlation matrix
$\Gamma^{\mu}$ have the same structure with $\Gamma^{K}$ (cf. (A.11)) but with
different block matrices

$f^{\mu}_{m-n}=0,\quad
g^{\mu}_{m-n}=\frac{i(1-e^{i\mu})}{e^{i\mu}+1}\delta_{mn}.$ (A.19)

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