Aspects of $N$-partite information
in conformal field theories
César A. Agón${}^{\text{\Hades}}$, Pablo Bueno${}^{\text{\Zeus,\Cupido}}$,
Oscar Lasso Andino${}^{\text{\Poseidon}}$ and Alejandro Vilar
López${}^{\text{\Apollon}}$

${}^{\text{\Hades}}$ Instituto Balseiro, Centro Atómico Bariloche 8400-S.C. de
Bariloche, Río Negro, Argentina

${}^{\text{\Zeus}}$ CERN, Theoretical Physics Department, CH-1211 Geneva 23,
Switzerland

${}^{\text{\Cupido}}$ Departament de Física Quàntica i Astrofísica, Institut
de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1, E-08028
Barcelona, Spain

${}^{\text{\Poseidon}}$Escuela de Ciencias Físicas y Matemáticas, Universidad
de Las Américas,  Redondel del Ciclista, Antigua Vía a Nayón, C.P.170124,
Quito, Ecuador

${}^{\text{\Apollon}}$ Physique Théorique et Mathématique and International
Solvay Institutes, Université Libre de Bruxelles, C.P. 231, B-1050 Brussels,
Belgium

We present several new results for the $N$-partite information, $I_{N}$, of
spatial regions in the ground state of $d$-dimensional conformal field
theories. First, we show that $I_{N}$ can be written in terms of a single
$N$-point function of twist operators. Using this, we argue that in the limit
in which all mutual separations are much greater than the regions sizes, the
$N$-partite information scales as $I_{N}\sim r^{-2N\Delta}$, where $r$ is the
typical distance between pairs of regions and $\Delta$ is the lowest primary
scaling dimension. In the case of spherical entangling surfaces, we obtain a
completely explicit formula for the $I_{4}$ in terms of 2-, 3- and 4-point
functions of the lowest-dimensional primary. Then, we consider a three-
dimensional scalar field in the lattice. We verify the predicted long-distance
scaling and provide strong evidence that $I_{N}$ is always positive for
general regions and arbitrary $N$ for that theory. For the $I_{4}$, we find
excellent numerical agreement between our general formula and the lattice
result for disk regions. We also perform lattice calculations of the mutual
information for more general regions and general separations both for a free
scalar and a free fermion, and conjecture that, normalized by the
corresponding disk entanglement entropy coefficients, the scalar result is
always greater than the fermion one. Finally, we verify explicitly the
equality between the $N$-partite information of bulk and boundary fields in
holographic theories for spherical entangling surfaces in general dimensions.

###### Contents

  1. 1 Introduction
  2. 2 $N$-partite information for general CFTs
    1. 2.1 General formula in terms of twist operators
    2. 2.2 Long-distance behavior of $I_{N}$
      1. 2.2.1 The $C_{ij}$ coefficients
      2. 2.2.2 Explicit formulas for $I_{2}$
      3. 2.2.3 Explicit formulas for $I_{3}$
      4. 2.2.4 Explicit formulas for $I_{4}$
  3. 3 Free scalar CFT
    1. 3.1 Mutual information
    2. 3.2 Tripartite information
    3. 3.3 Four partite information
  4. 4 Lattice calculations in $d=3$
    1. 4.1 Lattice setup and models
    2. 4.2 Long-distance $I_{N}$ for a free scalar
    3. 4.3 $I_{2}$ at arbitrary separations for free scalars and fermions
  5. 5 $N$-partite information in holography
  6. 6 Final comments
  7. A Coefficients for the $N$-partite information
    1. A.1 Computation of $I_{2}$ coefficients
    2. A.2 Computation of $I_{3}$ coefficients
    3. A.3 Computation of $I_{4}$ coefficients
      1. A.3.1 Computation of $c_{4:3}^{(1,1,2)}$
      2. A.3.2 Computation of $c_{4:3}^{(2,2)}$
      3. A.3.3 Computation of $c_{4:4}^{(1,1,1,1)}$
      4. A.3.4 Computation of $c_{4:4}^{(2,2)}$
  8. B Some useful integrals
  9. C Identifying the leading scaling in the lattice
  10. D Analytic mutual-information toy models
    1. D.1 EMI model
    2. D.2 MEMI model

##  1 Introduction

In quantum field theory (QFT), vacuum expectation values are statistical
measures of the vacuum state fluctuations in the quantum fields, and a
complete knowledge of such expectation values can be used to fully
characterize the theory [1]. An alternative formulation of QFT can be
performed in terms of algebras associated to spacetime regions [2]
—prototypically, causal diamonds. In this, the rôle played by the vacuum
expectation values of the quantum fields in the usual formulation should be
played by statistical measures of the vacuum state in the corresponding
algebras. The natural quantities that one can associate to such algebras are
entanglement measures such as entanglement entropy (EE).

As a matter of fact, the EE of any pair of regions is ill-defined in the
continuum, although one can often construct well-defined regularized versions
of it which capture universal information about the continuum theory.111A few
examples in the context of conformal field theories (CFTs) are: the trace-
anomaly coefficients for even dimensional theories [3, 4, 5, 6, 7, 8], the
Euclidean sphere partition function in odd dimensions [9, 10], the stress-
tensor two-point function charge $C_{T}$ in general dimensions [11, 12, 13],
the thermal entropy coefficient $C_{S}$ in general dimensions [14, 15, 16],
and others [17, 18, 19, 20, 21, 22]. On the other hand, an entanglement-based
axiomatic approach to QFT would naturally require the use of measures which
can be defined properly in the continuum theory. A natural choice is the
mutual information (MI) which, given a pair of disjoint regions $A_{1},A_{2}$
can be defined in terms of the EE as

$I_{2}(A_{1},A_{2})\equiv S(A_{1})+S(A_{2})-S(A_{1}A_{2})\,.$ (1)

Strictly speaking, the RHS of this formula is only defined in the continuum as
a limit of regulated EEs, but there exists an alternative equivalent
definition which makes sense in the continuum theory in terms of relative
entropy —see e.g., [23]. The mutual information is positive semi-definite,
$I_{2}\geq 0$, monotonous under inclusion, and symmetric in its arguments,
among other properties —see e.g., [24]. It has been suggested that the long-
distance expansion of the mutual information of pairs of regions could be used
to systematically extract the operatorial content of the theory [24, 25].
Indeed, the exponents of the inverse distance powers appearing in such
expansion are linear combinations of the conformal dimensions of the theory
[26] and, in the case of spherical regions, the expansion can be organized as
a sum of conformal blocks associated to each primary field [27, 28, 29]. In
that case, the leading term is given by [30]

$I_{2}=c(2\Delta)\frac{R_{1}^{2\Delta}R_{2}^{2\Delta}}{r_{12}^{4\Delta}}\,,\quad\text{where}\quad
c(N\Delta)\equiv\frac{\sqrt{\pi}}{4}\frac{\Gamma[N\Delta+1]}{\Gamma[N\Delta+\frac{3}{2}]}\,,\quad(R_{1,2}\ll
r_{12})$ (2)

In this expression, which is valid in general dimensions, $R_{1},R_{2}$ are
the radii of the spheres, $r_{12}$ is the distance between them, and $\Delta$
is the lowest scaling dimension of the theory, which is assumed to correspond
to a scalar field.222Analogous formulas for the case in which the lowest
primary has an arbitrary spin have also been obtained [29, 31]. In the case of
a free fermion, the subleading term has been computed in [24]. For more
general regions, $c(2\Delta)$ is replaced by a complicated function of the
spacetime dimension, the geometry of the regions, and $\Delta$.

Subleading terms in the mutual information long-distance expansion are
expected to include information also about the Operator Product Expansion
(OPE) coefficients. In fact, one such coefficient appears already in the
analogous leading long-distance term in the case of the tripartite
information, $I_{3}$. Given three entangling regions $A_{1},A_{2},A_{3}$, this
is defined as

$I_{3}(A_{1},A_{2},A_{3})\equiv
S(A_{1})+S(A_{2})+S(A_{3})-S(A_{1}A_{2})-S(A_{1}A_{3})-S(A_{2}A_{3})+S(A_{1}A_{2}A_{3})\,.$
(3)

As opposed to the mutual information, the tripartite information can be both
positive and negative for different theories [32, 33]. When a given theory
satisfies $I_{3}\leq 0$ for arbitrary regions, the mutual information is said
to be “monogamous”, which is the case, for instance, of holographic theories
[34, 35] —see also [36, 37, 38]. On the other hand, examples of theories which
exhibit non-monogamous mutual informations include free fields [39]. Finally,
as argued in [25], a CFT with $I_{3}=0$ for general regions is only possible
in $d=2$ —such theory corresponding to a free fermion [39]. In the long-
distance regime, it has been shown that the tripartite information of three
spherical regions is given, for a general $d$-dimensional CFT, by333Observe
also that, in the case of a lowest-dimensional fermionic primary of dimension
$\Delta_{f}$, the leading power in $I_{3}$ is no longer
$\sim(R/r)^{6\Delta_{f}}$, but rather $\sim(R/r)^{6\tilde{\Delta}_{f}}$, where
$\Delta_{f}<\tilde{\Delta}_{f}\leq\Delta_{f}+\frac{1}{6}$, the saturation of
the inequality occurring at least for a free fermion [25]. [25]

$I_{3}=\left[\frac{2^{6\Delta}\Gamma(\Delta+\frac{1}{2})^{3}}{2\pi\Gamma(3\Delta+\frac{3}{2})}-c(3\Delta)(C_{\mathcal{O}\mathcal{O}\mathcal{O}})^{2}\right]\frac{R_{1}^{2\Delta}R_{2}^{2\Delta}R_{3}^{2\Delta}}{r_{12}^{2\Delta}r_{13}^{2\Delta}r_{23}^{2\Delta}}\,,\quad(R_{1,2,3}\ll
r_{12},r_{23},r_{13})$ (4)

where $\Delta$ is again the scaling dimension of the lowest scaling-dimension
primary of the theory, and $C_{\mathcal{O}\mathcal{O}\mathcal{O}}$ its three-
point structure constant.

In view of the above results, it is conceivable to imagine a procedure which
would bootstrap the CFT data from long-distance expansions of this kind. In
order to do this, one could either go to higher orders in the corresponding
long-distance expansion or, complementarily, consider the leading terms in
higher-$N$ generalizations of the mutual information. Given $N$ disjoint
regions $A_{1},\dots,A_{N}\equiv\\{A_{i}\\}$, a somewhat canonical choice is
the $N$-partite information, which can be defined as

$\displaystyle
I_{N}(A_{1},\dots,A_{N})\equiv-\sum_{\sigma}(-)^{|\sigma|}S(\sigma)\,,$ (5)

where the sum is over all the subsets $\sigma\subset\\{A_{1},\dots,A_{N}\\}$
and $|\sigma|$ is the number of elements in the subset. $I_{N}$ can be
alternatively written in terms of lower-partite informations, e.g., as

$I_{N}(\boldsymbol{\cdot},A_{N-1},A_{N})=I_{N-1}(\boldsymbol{\cdot},A_{N-1})+I_{N-1}(\boldsymbol{\cdot},A_{N})-I_{N-1}(\boldsymbol{\cdot},A_{N-1}A_{N})\,,$
(6)

where $\boldsymbol{\cdot}\equiv A_{1},\dots,A_{N-2}$, which makes manifest
both the fact that $I_{N}$ is a well-defined quantity in QFT and also that it
can be interpreted as a measure of the non-extensivity of $I_{N-1}$. Not much
is known about $I_{N}$ for general QFTs. In fact, studies of $I_{N}$ for
$N\geq 4$ in QFT have been mostly limited to holographic theories. In that
context, it has been argued that $I_{N}$ does not have a definite sign beyond
$N=3$ [34, 40]. The $N$-partite information also plays an important rôle in
various systematic approaches aimed at a full characterization of various
multipartite entanglement measures and their properties for holography
theories —see e.g., [41, 42, 43, 44, 45, 46]. Additional holographic
explorations involving the $N$-partite information include e.g., [47, 48]. In
the context of two-dimensional CFTs, the $N$-partite information has also been
studied using replica trick techniques in [49, 50].

In this paper we present several new results involving the $N$-partite
information of spatial regions in CFTs. First of all, we show that for general
CFTs in arbitrary dimensions, the $N$-partite information can be written as a
single $N$-point correlator of the form

$\displaystyle I_{N}(\\{A_{i}\\})=\lim_{n\to
1}\frac{\left(-\right)^{N+1}}{1-n}\langle\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\cdots\tilde{\Sigma}_{A_{N}}^{(n)}\rangle\,,$
(7)

where $\tilde{\Sigma}_{A_{i}}$ is the “twist operator” with support on $A_{i}$
which implements the identification between copies when computing the Rényi
entropy $S^{(n)}(A_{i})$ using the replica trick. Using this, we show that the
leading term in the long-distance expansion of the $N$-partite information
behaves as

$I_{N}(\\{A_{i}\\})\propto\left[\frac{R}{r}\right]^{2N\Delta}\,,\quad(R\ll r)$
(8)

for general theories and dimensions. Here, $\Delta$ is the lowest scaling
dimension of the theory and, for simplicity, we assumed all region sizes to be
equal to $R$ and the same for the relative distances (up to $\mathcal{O}(1)$
factors), which we denoted $r$. Generalizing the $N=2,3$ cases, we observe
that the leading term in $I_{N}$ generically contains $N$-,$(N-1)$-,$\dots$
and 2-point correlators of the smallest-dimension primary operator. In
particular, we obtain the following general formula for the long-distance
limit of the four-partite information in the case of spherical regions

$\displaystyle\frac{I_{4}}{R^{8\Delta}}=$
$\displaystyle+\left[\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle-\frac{1}{r_{12}^{2\Delta}r_{34}^{2\Delta}}-\frac{1}{r_{13}^{2\Delta}r_{24}^{2\Delta}}-\frac{1}{r_{14}^{2\Delta}r_{23}^{2\Delta}}\right]^{2}c(4\Delta)$
(9)
$\displaystyle-\left[\frac{1}{r_{13}^{\Delta}r_{14}^{\Delta}r_{34}^{2\Delta}r_{12}^{2\Delta}r_{23}^{\Delta}r_{24}^{\Delta}}+\frac{1}{r_{12}^{\Delta}r_{14}^{\Delta}r_{24}^{2\Delta}r_{13}^{2\Delta}r_{23}^{\Delta}r_{34}^{\Delta}}+\frac{1}{r_{12}^{\Delta}r_{13}^{\Delta}r_{23}^{2\Delta}r_{14}^{2\Delta}r_{24}^{\Delta}r_{34}^{\Delta}}\right]\frac{9\sqrt{\pi}\Gamma\left(3\Delta\right)^{2}(C_{\mathcal{O}\mathcal{O}\mathcal{O}})^{2}}{\Gamma\left(\Delta\right)^{2}\Gamma\left(4\Delta+\frac{3}{2}\right)}$
$\displaystyle+\left[\frac{1}{r_{12}^{2\Delta}r_{23}^{2\Delta}r_{34}^{2\Delta}r_{14}^{2\Delta}}+\frac{1}{r_{13}^{2\Delta}r_{23}^{2\Delta}r_{24}^{2\Delta}r_{14}^{2\Delta}}+\frac{1}{r_{12}^{2\Delta}r_{13}^{2\Delta}r_{24}^{2\Delta}r_{34}^{2\Delta}}\right]\left[\,8c_{4:4}^{(1,1,1,1)}(\Delta)-c(4\Delta)\right]\,.$

In this formula, $R\ll r_{ij}$ $\forall i\neq j$ and for simplicity we
considered equal-radii spheres, $c(4\Delta)$ was defined in eq. (2) and
$c_{4:4}^{(1,1,1,1)}(\Delta)$ is a complicated integral which appears
explicitly in eq. (214) and which we have not been able to integrate
analytically, with the exception of the $\Delta=1/2$ case, for which we
find444Additional numerical values are presented in Table 1.

$c_{4:4}^{(1,1,1,1)}(\Delta=1/2)=\frac{4}{45}+\frac{2}{3\pi^{2}}\,.$ (10)

Observe that the above formula for the $I_{4}$ contains a four-point function
of the leading primary which will be a theory-dependent function of the two
invariant cross-ratios.

We perform a highly non-trivial check of eq. (9) by computing $I_{4}$ in the
case of four identical disk regions of diameter $R$ centered at the corners of
a square of side length $\sqrt{2}r$ for a three-dimensional free scalar in the
lattice. The results read, respectively,

$\left.I_{4}\right|^{\rm eq.\,(\ref{I4-final0})}_{d=3\,{\rm
free\,scalar}}=\left[\frac{1}{180}+\frac{1}{6\pi^{2}}\right]\frac{R^{4}}{r^{4}}\simeq
0.0224\frac{R^{4}}{r^{4}}\,,\quad\left.I_{4}\right|^{\rm lattice}_{d=3\,{\rm
free\,scalar}}\simeq 0.0207\frac{R^{4}}{r^{4}}\,,\quad(R\ll r)$ (11)

which, given the limitations of the lattice, is an excellent agreement. We
carry out additional lattice calculations for the free scalar for
$N=2,3,4,5,6$. We show that the scaling given by eq. (8) is respected in all
cases for different shapes of the entangling regions. Our results suggest that
the $N$-partite information of a three-dimensional free scalar is positive for
general regions and arbitrary $N$, i.e., we provide evidence that

$\left.I_{N}(\\{A_{i}\\})\right|_{d=3\,{\rm
free\,scalar}}\overset{\rm(?)}{\geq}0\quad\forall\,\\{A_{i}\\},N\,,$ (12)

a result which most likely holds in general dimensions.

In the particular case of the mutual information, we perform additional
lattice calculations for different pairs of regions at arbitrary distances.
The dependence of mutual information on the shape of the entangling regions
has been previously studied e.g., in [51, 52, 53, 54, 55] for holographic
theories and e.g., in [56, 57, 58, 24] for free fields in the lattice and in
the so-called “Extensive Mutual Information” model. Here we consider both the
free scalar as well as a free fermion in the lattice and compute the mutual
information for pairs of identical regions with different shapes as a function
of the separation. We observe that, conveniently normalized by the
corresponding disk EE universal coefficient, $F_{0}$, the scalar results are
systematically larger than the fermion ones, which leads us to conjecture that

$\left.\frac{I_{2}(A_{1},A_{2})}{F_{0}}\right|_{d=3\,{\rm
free\,scalar}}\overset{(?)}{>}\left.\frac{I_{2}(A_{1},A_{2})}{F_{0}}\right|_{d=3\,{\rm
free\,fermion}}\quad\forall\,A_{1},A_{2}\,,$ (13)

holds for arbitrary pairs of regions.

Finally, we close with a discussion on holographic conformal field theories.
We first argue that at long distances the $N$-partite information of a
holographic theory equals the $N$-partite information of the dual bulk theory
in the gravitational background dual to the ground state of the CFT [59]. Then
we prove that, at leading order in the long distance expansion, both boundary
and bulk twist operators equal each other. This is done by explicit
computation using the universality of the modular flow for boundary spheres
and bulk hemispheres, as well as the extrapolate dictionary. The whole result
implies the equality between boundary and bulk $N$-partite information for the
cases in which each individual boundary region has spherical boundary.

The remainder of the paper is organized as follows. In Section 2 we present a
derivation of the $N$-partite information formula (7). We then study the long-
distance expansion of the twist operator and keep only the leading long-
distance term. A new method for determining the OPE coefficients of such term
is also presented and, in the particular case of a spherical boundary region,
this is evaluated explicitly in the $n\to 1$ limit, leading to a new explicit
formula —see eq. (37) below. We then present a diagrammatic method to organize
the computation of the leading contribution to the $N$-partite information at
long distances. The method is illustrated via examples, through the explicit
analysis of the $I_{2}$, $I_{3}$ and $I_{4}$ cases. We present closed formulas
for each of these quantities in the aforementioned regime. In Section 3 we
apply the general formulas derived in the previous section to the particular
case of a free scalar in arbitrary dimensions. We then consider particular
geometric arrangements of the regions in three spacetime dimensions and
determine the associated coefficients. In Section 4 we compute the $N$-partite
information (for $N=2,3,4,5,6$) for various entangling regions for a three-
dimensional free scalar in the lattice. We verify the leading scaling
predicted by our general formula and perform a highly non-trivial check of eq.
(9) in the case of disk regions, finding excellent agreement. In all cases, we
find $I_{N}$ to be positive for the free scalar. Additionally, also in the
lattice, we compute the mutual information for pairs of regions both for the
scalar and for a free fermion for arbitrary separations, verifying the
expected scalings at short and long separations. In the case of disk regions,
we provide analytic approximations valid for most separations. Based on the
results obtained, we conjecture the free scalar mutual information to be
greater than the free fermion one for general configurations. In Section 5, we
reinterpret our previous results from the perspective of holography. In that
context, we argue that, at leading order in the long-distance expansion, the
CFT twist operator associated to a spherical region equals the bulk twist
operator associated to the dual hemispherical region, establishing the
equality between boundary and bulk $N$-partite informations, as expected from
[59]. Appendix A presents an explicit computation of the various coefficients
appearing in the long-distance terms of $I_{2}$, $I_{3}$ and $I_{4}$. Those
coefficients are given in terms of sums of products of the twist OPE
coefficients $C_{ij}$. In order to carry out the various required computations
of this Appendix we present a collection of formulas (and in some cases its
derivations) in Appendix B. Appendix C contains a table where we compare
various possible fits to the lattice data obtained in the long-distance regime
for the free scalar and verify that the scaling predicted by eq. (8) is always
preferred by the data. Appendix D contains analytic results for the mutual
information of the pairs of regions considered in Section 4 for two toy models
which possess, respectively, the same scaling as the free scalar and the free
fermion for long separations.

##  2 $N$-partite information for general CFTs

In this Section we provide new general formulas for the $N$-partite
information of spatial regions in general CFTs and arbitrary dimensions. We
start by proving that $I_{N}$ can be written in terms of a single $N$-point
correlator of twist operators. Then, we obtain the general scaling of $I_{N}$
for long separations and obtain a new explicit formula for the OPE
coefficients of the leading long-distance term in the case of spherical
entangling surfaces.

###  2.1 General formula in terms of twist operators

As explained in the introduction, given $N$ non-empty disjoint regions we can
define the $N$-partite information as in eq. (5) or eq. (6). Explicitly, eq.
(5) takes the form

$\displaystyle I_{N}(\\{A_{i}\\})$ $\displaystyle=$
$\displaystyle\sum_{i}S(A_{i})-\sum_{i<j}S(A_{i}A_{j})+\sum_{i<j<k}S(A_{i}A_{j}A_{k})-\cdots+(-)^{N+1}S(A_{1}A_{2}\cdots
A_{N})\,.$

This formula has a natural separation in terms of the number of regions
involved in each of the entropies. We can thus use the following compact
notation to write the $N$-partite information

$\displaystyle I_{N}(\\{A_{i}\\})$ $\displaystyle=$
$\displaystyle-\sum_{\alpha=1}^{N}(-)^{\alpha}\sum_{i_{1}<\cdots<i_{\alpha}}S(A_{i_{1}}\cdots
A_{i_{\alpha}})\,.$ (14)

We will use this expression later on. In order to proceed, let us consider the
replica trick, which entails the evaluation of the Rényi entropies for integer
$n$ in the form

$\displaystyle S^{(n)}(A)=\frac{1}{1-n}\log\left[\frac{Z({\cal
C}^{(n)}_{A})}{Z^{n}}\right]\,,$ (15)

so that analytically continuing for real $n$ and taking the $n\to 1$ limit
yields the EE of region $A$,

$\displaystyle S(A)=\lim_{n\rightarrow 1}S^{(n)}(A)\,.$ (16)

In the above formula, ${\cal C}^{(n)}_{A}$ represents the replica manifold for
the $n$ copies of the original spacetime geometry, after suitably identifying
the region $A$ of the $i$-th copy with the $(i+1)$-th one, and
$n+1\leftrightarrow 1$. $Z(X)$ is the partition function of the theory defined
on the manifold $X$ (for simplicity we use $Z$ when the manifold is a single
copy of the original spacetime). Observe that replacing all EEs in the above
definitions of the $N$-partite information by Rényi entropies, it is
straightforward to define the $n$-th Rényi $N$-partite information,
$I_{N}^{(n)}$, which reduces to $I_{N}$ for $n\rightarrow 1$.

The identification process required to define the manifold ${\cal
C}^{(n)}_{A}$ can be implemented by introducing a twist operator
$\Sigma^{(n)}_{A}$ with support on the entangling region [3, 60, 61]. This
operator identifies the operators of different replicas in the region $A$,
namely, $\phi_{i}\to\phi_{i+1}$ when $x\in A$. $\Sigma^{(n)}_{A}$ can be
normalized such that we extract the contribution from the identity operator
and write555Notice that this definition implies
$\langle\tilde{\Sigma}_{A}^{(n)}\rangle=0$.

$\displaystyle\Sigma_{A}^{(n)}=\langle\Sigma_{A}^{(n)}\rangle(1+\tilde{\Sigma}_{A}^{(n)})\,\quad{\rm
with}\quad\langle\Sigma_{A}^{(n)}\rangle=\frac{Z({\cal
C}^{(n)}_{A})}{Z^{n}}\,.$ (17)

We would like to write down a closed formula for $I^{(n)}_{N}$ in terms of
correlators of $\tilde{\Sigma}_{A}^{(n)}$. Let us start by translating
equation (14) in the language of twist operators. More concretely, we take the
terms of order $\alpha$ which have the form

$\displaystyle\sum_{i_{1}<\cdots<i_{\alpha}}S(A_{i_{1}}\cdots
A_{i_{\alpha}})=\lim_{n\to
1}\sum_{i_{1}<\cdots<i_{\alpha}}\frac{1}{1-n}\log\langle\Sigma_{A_{i_{1}}}^{(n)}\cdots\Sigma_{A_{i_{\alpha}}}^{(n)}\rangle\,.$
(18)

Now, let us apply relation (17) to further massage the order-$\alpha$
correlators,

$\displaystyle\log\langle\Sigma_{A_{i_{1}}}^{(n)}\cdots\Sigma_{A_{i_{\alpha}}}^{(n)}\rangle$
$\displaystyle=$
$\displaystyle\log\left[\langle\Sigma_{A_{i_{1}}}^{(n)}\rangle\cdots\langle\Sigma_{A_{i_{\alpha}}}^{(n)}\rangle\right]+\log\Big{[}\Big{\langle}\left(1+\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\right)\cdots\left(1+\tilde{\Sigma}_{A_{i_{\alpha}}}^{(n)}\right)\Big{\rangle}\Big{]}\,,$
(19)

where we separated the expectation values of the individual twist operators
from their tilded parts. The part that contains the products of tilded twist
operators can be expanded and it becomes

$\displaystyle\log\Big{[}\Big{\langle}\left(1+\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\right)\cdots\left(1+\tilde{\Sigma}_{A_{i_{\alpha}}}^{(n)}\right)\Big{\rangle}\Big{]}=$
(20)
$\displaystyle\log\Big{[}1+\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\tilde{\Sigma}_{A_{i_{2}}}^{(n)}\rangle+\underbrace{\dots}_{\rm
other\,\,pairings}+\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\tilde{\Sigma}_{A_{i_{2}}}^{(n)}\tilde{\Sigma}_{A_{i_{3}}}^{(n)}\rangle+\underbrace{\dots}_{\rm
other\,\,triplets}+\cdots+\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\tilde{\Sigma}_{A_{i_{2}}}^{(n)}\cdots\tilde{\Sigma}_{A_{i_{\alpha}}}^{(n)}\rangle\Big{]}\,,$

where, as previously noted, there are not terms proportinal to
$\langle\tilde{\Sigma}_{A_{i}}^{(n)}\rangle$ as those are zero.

We start by analyzing the contribution of the twist normalization factors to
the $N$-partite information. This requires studying the quantity

$\displaystyle\sum_{\alpha=1}^{N}\left(-\right)^{\alpha}\left[\sum_{i_{1}<\cdots<i_{\alpha}}\frac{1}{1-n}\left[\log\langle\Sigma_{A_{i_{1}}}^{(n)}\rangle+\cdots+\log\langle\Sigma_{A_{i_{\alpha}}}^{(n)}\rangle\right]\right]\,.$
(21)

Notice that we have separated the logarithm of the product of expectations
values of twist operators appearing in (19) into the sum of logarithms of each
individual expectation value. For each order $\alpha$, the number of ordered
$\alpha$ indices is simply given by the number of combinations of $\alpha$
elements one can take from a set of $N$ objects, i.e., by the binomial
coefficient $\binom{N}{\alpha}$. For each choice of ordered indices we have
the sum of $\alpha$ different terms, therefore, the total number of terms of
the form $\log\langle\Sigma^{(n)}_{A_{j}}\rangle$ is
$\alpha\cdot\binom{N}{\alpha}$ for each $\alpha$. On the other hand, among all
these terms the logarithm of the twist operator of a particular region should
appear the same number of times regardless of the chosen region, since the
formula treats all regions on an equal footing. Thus, we conclude that for
each $\alpha$ we have

$\displaystyle\sum_{i_{1}<\cdots<i_{\alpha}}\frac{1}{1-n}\left[\log\langle\Sigma_{A_{i_{1}}}^{(n)}\rangle+\cdots\log\langle\Sigma_{A_{i_{\alpha}}}^{(n)}\rangle\right]=\frac{\alpha}{N}\binom{N}{\alpha}\sum_{i=1}^{N}\frac{1}{1-n}\log\langle\Sigma_{A_{i}}^{(n)}\rangle\,.$
(22)

Summing the above equation over $\alpha$ with the appropriate signs one gets
that eq. (21) equals

$\displaystyle\left(\sum_{\alpha=1}^{N}\left(-\right)^{\alpha}\binom{N-1}{\alpha-1}\right)\sum_{i=1}^{N}\frac{1}{1-n}\log\langle\Sigma_{A_{i}}^{(n)}\rangle\,.$
(23)

Shifting the summation index $\alpha$ by $1$, the term in parenthesis can be
identified with the binomial expansion of $(1-1)^{N-1}$, and therefore,
vanishes identically for $N>1$. This shows that $I_{N}$ for $N>1$ does not
depend on the expectation value of the individual twist operators. Notice that
for $N=1$, $I_{1}$ reduces to the usual entanglement entropy and in that case
the normalization factor is the answer. This result is true even for the Renyi
$I^{(n)}_{N}$, as we did not need to take the $n\to 1$ limit to arrive at this
conclusion.

Now, we analyze the contribution of the second term in the right hand side of
(19) to $I_{N}$. The first thing to notice is that to simplify those
contributions, it is convenient to take the $n\to 1$ at this stage. The
correlator of an arbitrary number of products of $\tilde{\Sigma}^{(n)}$’s
vanishes as $(n-1)$ in the $n\to 1$ limit, therefore, the linear approximation
of the logarithm becomes exact in such a limit. This means that the
contribution to $I_{N}$ we are interested in is

$\displaystyle I_{N}\supset\lim_{n\to
1}\frac{1}{1-n}\sum_{\alpha=1}^{N}\left(-\right)^{\alpha}\\!\\!\\!\sum_{i_{1}<\cdots<i_{\alpha}}\Big{[}$
$\displaystyle+\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\tilde{\Sigma}_{A_{i_{2}}}^{(n)}\rangle+\underbrace{\dots}_{\rm
other\,\,pairings}+\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\tilde{\Sigma}_{A_{i_{2}}}^{(n)}\tilde{\Sigma}_{A_{i_{3}}}^{(n)}\rangle+\underbrace{\dots}_{\rm
other\,\,triplets}$ (24)
$\displaystyle+\cdots+\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\tilde{\Sigma}_{A_{i_{2}}}^{(n)}\cdots\tilde{\Sigma}_{A_{i_{\alpha}}}^{(n)}\rangle\Big{]}\,.$
(25)

For each $\alpha$ there would be contributions coming from correlators of each
order $K$ with $\alpha\geq K$. Thus we will study the contribution to $I_{N}$
from all correlators of order $K$ for each $K$. These correlators will only
appear for $\alpha\geq K$. Now, for each $\alpha\geq K$ the number of ordered
$\alpha$ indices appearing in eq. (24) is given by $\binom{N}{\alpha}$ and
each such term will contain $\binom{\alpha}{K}$ correlators of $K$ tilded
twist operators, since that is the number of groups of $K$ elements one can
made out of $\alpha$ operators. Finally, the number of times a given
correlator of $K$ operators appears in such term will be independent of the
choice of correlator and therefore we have that the contribution to $I_{N}$
from the order $K$ correlators is given by

$\displaystyle\lim_{n\to
1}\left(\sum_{\alpha=K}^{N}(-)^{\alpha}\frac{\binom{N}{\alpha}\binom{\alpha}{K}}{\binom{N}{K}}\right)\sum_{i_{1}<\cdots<i_{K}}\langle\tilde{\Sigma}_{A_{i_{1}}}^{(n)}\cdots\tilde{\Sigma}_{A_{i_{K}}}^{(n)}\rangle$
(26)

where the factor of $\binom{N}{K}$ in the denominator is the total number of
distinct $K$ correlators we can form out of $N$ twist operators, and thus, it
coincides with the number of terms appearing in the above sum over $K$
indices. Using the identity

$\displaystyle\frac{\binom{N}{\alpha}\binom{\alpha}{K}}{\binom{N}{K}}=\binom{N-K}{\alpha-K}\,,$
(27)

and shifting the summation index in eq. (26), we find that the term in
parenthesis is nothing but the binomial expansion of $(1-1)^{N-K}$, and thus,
it identically vanishes for $N>K$. This means that the only correlator of
twist operators that can contribute to the $N$-partite information is
precisely the correlator of $N$ twist operators. Hence, we find that the
$N$-partite information is given by

$\displaystyle I_{N}(\\{A_{i}\\})=\lim_{n\to
1}\frac{\left(-\right)^{N+1}}{1-n}\langle\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\cdots\tilde{\Sigma}_{A_{N}}^{(n)}\rangle\,.$
(28)

This result can be alternatively proved by induction. This method requires
guessing eq. (28), but such equation is the natural generalization of the
$I_{2}$ and $I_{3}$ cases worked out in detailed for example in [25].
Therefore, we start assuming that eq. (28) holds for $N=K-1$. Then, we use the
recursive definition of $I_{N}$ presented in eq. (6) to write down a formula
for $I_{K}$, namely,

$\displaystyle I_{K}(\\{A_{i}\\})=\lim_{n\to
1}\frac{\left(-\right)^{K}}{1-n}\Big{\langle}\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\cdots\tilde{\Sigma}_{A_{K-2}}^{(n)}\left(\tilde{\Sigma}_{A_{K-1}}^{(n)}+\tilde{\Sigma}_{A_{K}}^{(n)}-\tilde{\Sigma}_{A_{K-1}A_{K}}^{(n)}\right)\Big{\rangle}\,.$
(29)

Using eq. (17) for the operator $\tilde{\Sigma}_{A_{K-1}A_{K}}^{(n)}$ we have

$\displaystyle\tilde{\Sigma}_{A_{K-1}A_{K}}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{A_{K-1}}^{(n)}\Sigma_{A_{K}}^{(n)}}{\langle\Sigma_{A_{K-1}}^{(n)}\Sigma_{A_{K}}^{(n)}\rangle}-1=\frac{1+\tilde{\Sigma}_{A_{K-1}}^{(n)}+\tilde{\Sigma}_{A_{K}}^{(n)}+\tilde{\Sigma}_{A_{K-1}}^{(n)}\tilde{\Sigma}_{A_{K}}^{(n)}}{1+\langle\tilde{\Sigma}_{A_{K-1}}^{(n)}\tilde{\Sigma}_{A_{K}}^{(n)}\rangle}-1\,,$
(30)

where in the first equality we used
$\Sigma_{A_{K-1}A_{K}}^{(n)}=\Sigma_{A_{K-1}}^{(n)}\Sigma_{A_{K}}^{(n)}$ while
in the second we used eq. (17). Taking the $n\to 1$ limit, the above equation
reduces to

$\displaystyle\tilde{\Sigma}_{A_{K-1}A_{K}}^{(n)}$ $\displaystyle\approx$
$\displaystyle\tilde{\Sigma}_{A_{K-1}}^{(n)}+\tilde{\Sigma}_{A_{K}}^{(n)}+\tilde{\Sigma}_{A_{K-1}}^{(n)}\tilde{\Sigma}_{A_{K}}^{(n)}\,.$
(31)

Notice that we have ignored the extra term in the denominator. This is
possible because in the $n\to 1$ limit, the correlator of an arbitrary number
of twist operators goes as $(n-1)$ and therefore, any extra $(n-1)$ factor in
eq. (30) would not contribute to the $N$-partite information. Plugging eq.
(31) into eq. (29) leads to eq. (28) for $N=K$, and therefore it completes the
proof.

###  2.2 Long-distance behavior of $I_{N}$

With eq. (28) at hand, let us now study the long-distance behavior of the
$N$-partite information. In this regime, we can approximate each twist
operator by the product of local operators located at a convenient location
inside the region $X$ as [62, 63]

$\displaystyle\tilde{\Sigma}_{X}^{(n)}=\sum_{\\{k_{j}\\}\neq\mathbb{I}}C_{\\{k_{j}\\}}^{X}\prod_{j=0}^{n-1}\Phi^{(j)}_{k_{j}}(x_{X})\,.$
(32)

For the $j$-th copy, the $k_{j}$ label a complete set of operators (with the
identity removed). The leading contribution at long distances will come from
the lowest scaling dimension operator of the theory. Assuming for simplicity
that such operator is a scalar of conformal dimension $\Delta$, which we
denote by $\mathcal{O}$, then we can simply keep the term

$\displaystyle\tilde{\Sigma}^{(n)}_{X}\approx\sum_{i<j}C^{X}_{ij}\mathcal{O}^{i}(x_{X})\mathcal{O}^{j}(x_{X})\,.$
(33)

Observe that in principle there is also a contribution of the form
$\sum_{i}C^{X}_{i}\,{\cal O}^{i}(x_{X})$, but such single-sheet operators do
not contribute in the $n\to 1$ limit.666The argument is the following: the
coefficient $C_{i}$ has support on a single sheet and by the replica symmetry
it cannot depend on $i$, so it should be a constant. On the other hand, since
$\tilde{\Sigma}^{(n)}_{X}\to 1$ as $n\to 1$, then, one concludes that
$C_{i}\sim\mathcal{O}(n-1)$. In the computation of the $I_{N>1}$ we would
always have products of more than one of those coefficients. Therefore, they
cannot contribute to $I_{N}$ in the $n\to 1$ limit. We abbreviate
$\mathcal{O}^{i}(x_{X})$ with $\mathcal{O}^{i}_{X}$. We apply this formalism
to the particular case when $X$ is a ball of radius $R_{X}$. In that
situation, it is convenient to use the new coefficients $C_{ij}\equiv
C^{X}_{ij}R_{X}^{-2\Delta}$ which are scale independent and thus are the same
for all the ball-like regions involved.

The coefficients $C_{ij}$ can be computed by studying the effect of a twist
operator in the correlator of test operators far away from the region $X$, and
located on different sheets. This is, one considers

$\displaystyle\langle\tilde{\Sigma}^{(n)}_{X}\mathcal{O}^{i}(x)\mathcal{O}^{j}(x)\rangle\,,$
(34)

with $|x-x_{X}|\gg R_{X}$. Using the above equation, eq. (33) and the two-
point function

$\braket{\mathcal{O}^{i}(x)\mathcal{O}^{j}(x_{X})}=\frac{\delta^{ij}}{|x-x_{X}|^{2\Delta}}\,,$
(35)

one finds the following formula for $C_{ij}$

$\displaystyle
C_{ij}=\lim_{x\to\infty}\frac{|x-x_{X}|^{4\Delta}}{R_{X}^{2\Delta}}\langle\tilde{\Sigma}^{(n)}_{X}\mathcal{O}^{i}(x)\mathcal{O}^{j}(x)\rangle\,.$
(36)

The $C_{ij}$ coefficients are determined by the two-point function in replica
space. Earlier works on the type of computations presented in this paper did
not obtain explicitly the form of those coefficients but instead they managed
to compute sums over these coefficients in the $n\to 1$ limit. We devote
subsection 2.2.1 to study the $C_{ij}$ coefficients and argue that

$\displaystyle
C_{ij}=\frac{1}{\sin^{2\Delta}\left[\frac{\pi(i-j)}{n}\right]}\,,$ (37)

in the $n\to 1$ limit. Namely, the above expression is valid for the purposes
of computing entanglement related quantities, so after adding over all the
indices we need to keep only the piece proportional to $(n-1)$ in replica
computations.

Now, let us write down the correlator in eq. (28) by keeping only the leading
long distance piece appearing in eq. (33),

$\displaystyle\langle\prod_{\alpha=1}^{N}\tilde{\Sigma}_{A_{\alpha}}^{(n)}\rangle\,=\prod_{\alpha=1}^{N}R^{2\Delta}_{A_{\alpha}}\sum_{i_{\alpha}<j_{\alpha}}C_{i_{\alpha}j_{\alpha}}\langle\prod_{\beta=1}^{N}\mathcal{O}^{i_{\beta}}_{A_{\beta}}\mathcal{O}^{j_{\beta}}_{A_{\beta}}\rangle\,.$
(38)

The case in which all the indices appearing in the above correlator are
different reduces to a product of the expectation values of the individual
operators, and since $\langle\mathcal{O}^{i}_{A_{J}}\rangle=0$ such
configuration does not contribute. On the other hand, the configurations that
use the minimum number of sheets simply put all the twist operators on the
same pair of sheets. Following this philosophy, we will separate the possible
configurations in the correlator (38) in terms of the number of sheets $\cal
N$ with non-trivial operator insertions. Below, we will represent those
configurations using matrices where the columns represent the different sheets
while rows represent the different regions.

Before moving on, observe that eq. (38) makes evident the leading scaling of
the $N$-partite information at long distances. Indeed, if we assume for
simplicity that all ball regions are characterized by the same radius $R$ and
that each pair is separated a distance $r$, up to order-$1$ factors, we have

$I_{N}(\\{A_{i}\\})\propto\left[\frac{R}{r}\right]^{2N\Delta}\,,$ (39)

where the $r$-dependent piece comes from the $N$-point correlator of the
operators
$\mathcal{O}^{i_{\beta}}_{A_{\beta}}\mathcal{O}^{j_{\beta}}_{A_{\beta}}$
which, as we show explicitly below for the $N=2,3,4$ cases, gives rise to
various products of correlators of same-sheet different-region operators
involving a total of $2N$ operators each. Schematically, for each pair of
operators we get a contribution $\sim r^{-2\Delta}$, and so
$I_{N}(\\{A_{i}\\})\propto r^{-2N\Delta}$. While this result is derived
assuming the entangling surfaces to be spheres, the scaling in eq. (39) holds
for arbitrary regions as long as all relevant scales characterizing them are
much smaller than the mutual separations.

For the analysis of the correlator appearing in eq. (38) it is useful to
introduce the following graph representation of the twist operator,

$\tilde{\Sigma}_{A}\approx\sum_{i<j}C_{ij}\mathcal{O}^{i}_{A}\mathcal{O}^{j}_{A}\approx$$i$$j$$A$

where vertices represent the different sheets on which local operators are
supported, and the lines represent the spatial region on which the operator is
located. We also introduce some notation for the various coefficients
appearing in the $I_{N}$. We will denote them by
$c_{N:K}^{(i_{1},\cdots,i_{\alpha})}$, where the subindex $N$ stands for the
fact we are considering the $N$-partite information while the index $K$ means
that such coefficient is given in terms of a sum involving $K$ different
sheets. The super index $(i_{1},\cdots,i_{\alpha})$ characterizes the various
powers of $C_{ij}$ appearing in the expression for that coefficient. Notice
that the $c_{N:K}^{(i_{1},\cdots,i_{\alpha})}$ are symmetric under
permutations of the indices $\\{i_{1},\cdots,i_{\alpha}\\}$ and thus, we can
chose to denote them in increasing order.

####  2.2.1 The $C_{ij}$ coefficients

Let us now return to the problem of determining the functional form of the
$C_{ij}$ coefficients. In order to evaluate eq. (36), one needs a formula for
the correlator (34). Recently, one such formula was derived in [31]. For the
special case in which the operators in question are scalars, this takes the
form

$\displaystyle\langle\Omega|\Sigma^{(n)}_{A}{\cal O}^{l}(x){\cal
O}^{k}(x)|\Omega\rangle$ $\displaystyle=$
$\displaystyle\frac{\operatorname{tr}\left\\{{\cal
O}(x)\rho_{A}^{n-(l-k)}\,{\cal
O}(x)\rho_{A}^{(l-k)}\right\\}}{\operatorname{tr}\rho^{n}_{A}}\,.$ (40)

The above result can be put in a more familiar form in terms of the density
operator $\tilde{\rho}_{A}(n):=\rho_{A}^{n}/\operatorname{tr}{\rho_{A}^{n}}$.
Since $\rho_{A}$ is a density operator, and therefore Hermitian, completely
positive and satisfying $\operatorname{tr}{\rho_{A}}=1$; the same holds for
$\tilde{\rho}_{A}(n)$ for $n>0$. Thus eq. (40) becomes

$\displaystyle\langle\Omega|\Sigma^{(n)}_{A}{\cal O}^{l}(x){\cal
O}^{k}(x)|\Omega\rangle$ $\displaystyle=$
$\displaystyle\operatorname{tr}\left\\{\tilde{\rho}_{A}(n)\,{\cal
O}^{(n)}_{A}\left[x,i\frac{\tau_{kl}}{\pi}\right]{\cal O}(x)\right\\}\,,$ (41)

with $\tau_{kl}\equiv\pi(k-l)/n$. Notice that the operators on the RHS are
defined in the original CFT (a single sheet of the replica space), and

$\displaystyle{\cal
O}^{(n)}_{A}\left[x,s\right]\equiv\tilde{\rho}_{A}(n)^{-is}{\cal
O}(x)\tilde{\rho}_{A}(n)^{is}$ (42)

is the operator transformed by the modular flow induced by
$\tilde{\rho}_{A}(n)$. On the RHS of eq. (41) we have the correlation function
of a field with a modular-evolved field. Such correlator is analytic in the
strip with Im$\,s\in\left(0,1\right)$, and obeys the KMS periodicity between
the boundaries of the strip [64]. This implies that the correlator is well
defined on $0\leq\tau_{kl}\leq 1$. It follows from its definition that
$1/n<\tau_{kl}<1-1/n$ provided we ordered the indices such that $k>l$. We will
do so here.

The correlator on the RHS of eq. (41) can be studied for any density operator
$\rho$. This observation suggests to approximate the operator
$\tilde{\rho}_{A}(n)$ by $\rho_{A}$ in the $n\to 1$ limit. Any correction to
this approximation will be of order $(n-1)$ and thus it will not contribute to
the entropies. Taking the $n\to 1$ limit on the state while keeping the $n$
dependence on the modular parameters is the only consistent approximation of
that correlator. Thus, in this regime we have

$\displaystyle\langle\Omega|\Sigma^{(n)}_{A}{\cal O}^{l}(x){\cal
O}^{k}(x)|\Omega\rangle$ $\displaystyle\approx$
$\displaystyle\operatorname{tr}\left\\{\rho_{A}\,{\cal
O}_{A}\left[x,i\frac{\tau_{kl}}{\pi}\right]{\cal
O}(x)\right\\}=\Big{\langle}{\cal
O}_{A}\left[x,i\frac{\tau_{kl}}{\pi}\right]{\cal O}(x)\Big{\rangle}\,.$ (43)

This relation coincides with the so called $1/n$ prescription derived in [29].
Such prescription has been sucessfully applied in similar entanglement
computations in the literature, see for example [29, 28] and references
therein.

Now, we are ready to compute the $C_{ij}$ coefficients starting from eq. (36)
and the large $r$ limit of eq. (43). For a sphere of radius $R$, the modular
flow acts geometrically on local operators via

$\displaystyle r^{\pm}[s]=R\frac{\left(R+r^{\pm}\right)-e^{\mp 2\pi
s}\left(R-r^{\pm}\right)}{\left(R+r^{\pm}\right)+e^{\mp 2\pi
s}\left(R-r^{\pm}\right)}\,,$ (44)

where $r^{\pm}=r\pm t$ are standard null coordinates [9]. This flow can be
interpreted as a conformal transformation, which helps us derive the form in
which the unitary acts on the operator, this is

$\displaystyle{\cal O}_{A}[x,s]=U_{A}(s){\cal
O}(x)U_{A}^{\dagger}(s)=\Omega^{\Delta}[x,s]{\cal O}(x[s])\,,$ (45)

where $U_{A}(s)$ is the unitary that implements the modular evolution, while
$\Omega[x,s]$ is the conformal factor associated to the coordinate
transformation. This can be derived from eq. (44) and the relation
$dx[s]^{2}=\Omega^{2}[x,s]dx^{2}$ . When $r\to\infty$ we find

$\displaystyle\langle{\cal O}_{A}\left[x,s\right]{\cal
O}(x)\rangle=\frac{1}{\left(-\sinh^{2}[\pi
s]\right)^{\Delta}}\frac{R^{2\Delta}}{r^{4\Delta}}\,.$ (46)

Replacing $s\to i\tau_{kl}/\pi$ in the above formula, and plugging the result
into eq. (36) leads to our final formula for the $C_{kl}$, which reads

$\displaystyle C_{kl}=\frac{1}{\sin^{2\Delta}\tau_{kl}}\quad{\rm
with}\quad\tau_{kl}=\frac{\pi(k-l)}{n}\,.$ (47)

In Appendix A we use the above expression for the $C_{kl}$ to compute the
relevant coefficients appearing in the mutual and tripartite information at
long distances, this is done in sections A.1 and A.2 respectively. We
reproduce results for these quantities previously appeared in the literature,
which provides strong support for the validity of the above formula. In the
same appendix we also compute the coefficients appearing in the four-partite
information at long distances (which are all new in the literature).

####  2.2.2 Explicit formulas for $I_{2}$

Completely explicit formulas for $I_{N}$ in the long-distance regime can in
principle be obtained from eq. (28) on a case-by-case basis for arbitrary
values of $N$. In the following subsections we present such formulas for
$N=2,3,4$ in the case of spherical entangling surfaces. Starting with the
mutual information case, we have

$\displaystyle I_{2}(A_{1},A_{2})=\lim_{n\to
1}\frac{1}{n-1}\langle\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\rangle\,.$
(48)

As we have seen, in the long-distance regime the correlator can be written as

$\displaystyle\langle\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\rangle\,=\left(R_{1}R_{2}\right)^{2\Delta}\sum_{i<j}\sum_{k<l}C_{ij}C_{kl}\langle\mathcal{O}^{i}_{A_{1}}\mathcal{O}^{j}_{A_{1}}\mathcal{O}^{k}_{A_{2}}\mathcal{O}^{l}_{A_{2}}\rangle\,,$
(49)

where $R_{i}$ are the radii of the spheres. The various contributions can be
organized in terms of the number of sheets with non-trivial operator
insertions,
$\text{${\cal
N}=2,$}\quad\displaystyle\begin{pmatrix}1&\cdots\mathcal{O}^{i}_{A_{1}}&\cdots\mathcal{O}^{j}_{A_{1}}&\cdots
1\\\ 1&\cdots\mathcal{O}^{i}_{A_{2}}&\cdots\mathcal{O}^{j}_{A_{2}}&\cdots 1\\\
\end{pmatrix}\,,$ $i$$j$

as previously described. We define the following coefficient

$\displaystyle c_{2:2}^{(2)}\equiv\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j}C^{2}_{ij}\,,$ (50)

which we compute explicitly in eq. (178) below. Plugging the result into the
mutual information we get

$\displaystyle
I_{2}(A_{1},A_{2})=\frac{\sqrt{\pi}}{4}\frac{\Gamma\left(2\Delta+1\right)}{\Gamma\left(2\Delta+\frac{3}{2}\right)}\langle\mathcal{O}_{A_{1}}\mathcal{O}_{A_{2}}\rangle^{2}R_{1}^{2\Delta}R_{2}^{2\Delta}\,.$
(51)

In Fig. 1 we provide a graph representation of the long distance mutual
information in terms of standard spatial correlators.

$A_{1}$$A_{2}$ Figure 1: Graph representation of the mutual information. The
result involves the square of the correlator of the primary operator with the
lowest conformal dimension of the theory. In each correlator, the primary is
defined at two points lying at the (centers of each of the) spheres.

The above formula can be written more explicitly using the expression for the
correlator of the leading primary operator,

$\langle\mathcal{O}(x_{A_{1}})\mathcal{O}(x_{A_{2}})\rangle=\frac{1}{r_{12}^{2\Delta}}\,,$
(52)

where $r_{12}\equiv|x_{A_{1}}-x_{A_{2}}|$ is the distance between the spheres
centers. Then we get

$\displaystyle
I_{2}(A_{1},A_{2})=\frac{\sqrt{\pi}}{4}\frac{\Gamma\left(2\Delta+1\right)}{\Gamma\left(2\Delta+\frac{3}{2}\right)}\frac{R_{1}^{2\Delta}R_{2}^{2\Delta}}{r_{12}^{4\Delta}}\,,$
(53)

in agreement with previous results [26, 30, 29, 31].

####  2.2.3 Explicit formulas for $I_{3}$

Let us now consider the tripartite information. In that case, we have

$\displaystyle I_{3}(A_{1},A_{2},A_{3})=-\lim_{n\to
1}\frac{1}{n-1}\langle\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\tilde{\Sigma}_{A_{3}}^{(n)}\rangle\,.$
(54)

The leading contribution to $I_{3}$ comes from replacing the twist operators
by a product of two operators of lowest scaling dimension as in eq. (33),
which leads to the following expression for the correlator in eq. (54),

$\displaystyle\langle\tilde{\Sigma}_{A_{1}}^{(n)}\tilde{\Sigma}_{A_{2}}^{(n)}\tilde{\Sigma}_{A_{3}}^{(n)}\rangle\,=\left(R_{1}R_{2}R_{3}\right)^{2\Delta}\sum_{i<j}\sum_{k<l}\sum_{m<n}C_{ij}C_{kl}C_{mn}\langle\mathcal{O}^{i}_{A_{1}}\mathcal{O}^{j}_{A_{1}}\mathcal{O}^{k}_{A_{2}}\mathcal{O}^{l}_{A_{2}}\mathcal{O}^{m}_{A_{3}}\mathcal{O}^{n}_{A_{3}}\rangle\,.$
(55)

For ${\cal N}=2$ non-trivial sheets there is one only possible configuration
which has the following structure
$\text{${\cal
N}=2,$}\quad\displaystyle\begin{pmatrix}1&\cdots\mathcal{O}^{i}_{A_{1}}&\cdots\mathcal{O}^{j}_{A_{1}}&\cdots
1\\\ 1&\cdots\mathcal{O}^{i}_{A_{2}}&\cdots\mathcal{O}^{j}_{A_{2}}&\cdots 1\\\
1&\cdots\mathcal{O}^{i}_{A_{3}}&\cdots\mathcal{O}^{j}_{A_{3}}&\cdots 1\\\
\end{pmatrix}\,,$ $i$$j$

as previously described. The contribution of such term to eq. (55) is

$\displaystyle\sum_{i<j}C^{3}_{ij}\langle\mathcal{O}^{i}_{A_{1}}\mathcal{O}^{i}_{A_{2}}\mathcal{O}^{i}_{A_{3}}\rangle\langle\mathcal{O}^{j}_{A_{1}}\mathcal{O}^{j}_{A_{2}}\mathcal{O}^{j}_{A_{3}}\rangle\,.$
(56)

Notice in the above graph that each vertex represents a correlator. The number
of legs attached to it will correspond to the number of operators, thus, we
have a product of two three point functions, one at sheet $i$ and the other
one at sheet $j$.

The other non-zero configurations will necesarilly have two operators on each
row (sheet), one such example will be
$\text{${\cal
N}=3,$}\quad\displaystyle\begin{pmatrix}1&\cdots\mathcal{O}^{i}_{A_{1}}&\cdots\mathcal{O}^{j}_{A_{1}}&\cdots
1&\cdots 1\\\ 1&\cdots
1&\cdots\mathcal{O}^{j}_{A_{2}}&\cdots\mathcal{O}^{k}_{A_{2}}&\cdots 1\\\
1&\cdots\mathcal{O}^{i}_{A_{3}}&\cdots 1&\cdots\mathcal{O}^{k}_{A_{3}}&\cdots
1\\\ \end{pmatrix}\,,$ $i$$j$$k$

This graph is obtained by taking three vertices (corresponding to three
different sheets), and three lines (corresponding to the three twist
operators) and constructing a graph with the condition that on each vertex
there must be at least two lines connected to it. This leads to the single
graph above. The terms contributing to the correlator with such a graph are

$\displaystyle\sum_{i<j<k}C_{ij}C_{ik}C_{jk}\left(\langle\mathcal{O}^{i}_{A_{1}}\mathcal{O}^{i}_{A_{3}}\rangle\langle\mathcal{O}^{j}_{A_{1}}\mathcal{O}^{j}_{A_{2}}\rangle\langle\mathcal{O}^{k}_{A_{2}}\mathcal{O}^{k}_{A_{3}}\rangle+{\rm
permutations\,\,of}\,\,\\{A_{1},A_{2},A_{3}\\}\right)\,.$ (57)

The number of terms in the parenthesis with the exact same contribution is
$3!$, which equals the number of different ways we can label the sides of the
triangle with labels $A_{1},A_{2},$ and $A_{3}$. Thus, we conclude that the
$I_{3}$, is given by

$\displaystyle I_{3}=-\lim_{n\to
1}\frac{\left(R_{1}R_{2}R_{3}\right)^{2\Delta}}{n-1}\Big{[}$
$\displaystyle+\sum_{i<j}C^{3}_{ij}\langle\mathcal{O}^{i}_{A_{1}}\mathcal{O}^{i}_{A_{2}}\mathcal{O}^{i}_{A_{3}}\rangle\langle\mathcal{O}^{j}_{A_{1}}\mathcal{O}^{j}_{A_{2}}\mathcal{O}^{j}_{A_{3}}\rangle$
$\displaystyle+6\sum_{i<j<k}C_{ij}C_{jk}C_{ki}\langle\mathcal{O}^{i}_{A_{1}}\mathcal{O}^{i}_{A_{3}}\rangle\langle\mathcal{O}^{j}_{A_{1}}\mathcal{O}^{j}_{A_{2}}\rangle\langle\mathcal{O}^{k}_{A_{2}}\mathcal{O}^{k}_{A_{3}}\rangle\Big{]}\,.$
(58)

Let us now define the following coefficients

$\displaystyle c_{3:2}^{(3)}\equiv\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j}C^{3}_{ij}\qquad{\rm and}\qquad
c_{3:3}^{(1,1,1)}\equiv\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j<k}C_{ij}C_{jk}C_{ki}\,,$ (59)

which we explicitly compute in detail in Appendix A.2. The results appear in
eq. (180) and eq. (192), respectively. In terms of these expressions the
tripartite information adopts the final form [25]

$\displaystyle
I_{3}=\Bigg{[}\frac{2^{6\Delta}\Gamma\left(\Delta+\frac{1}{2}\right)^{3}}{2\pi\Gamma\left(3\Delta+\frac{3}{2}\right)}\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle-\frac{2^{6\Delta}\left(\Gamma\left(3\Delta+1\right)\right)^{2}}{\Gamma\left(6\Delta+2\right)}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle^{2}\,\,\Bigg{]}\left(R_{1}R_{2}R_{3}\right)^{2\Delta}\,,$
(60)

where we used the notation $\mathcal{O}_{i}\equiv\mathcal{O}_{A_{i}}$. Notice
that in cases in which $\langle\mathcal{O}\mathcal{O}\mathcal{O}\rangle=0$, we
have $I_{3}>0$. For those, the mutual information is non-monogamous at long
distances.

The above formula can also be illustrated graphically in the usual space
representation of correlators as shown in Fig. 2.

$A_{1}$$A_{3}$$A_{2}$

$A_{1}$$A_{3}$$A_{2}$

Figure 2: Graph representation of the two types of terms contributing to the
$I_{3}$ at long distances. Both contributions are represented via totally
connected diagrams, as expected from clustering.

There, simple lines represent correlators, while lines joint by a triangle
vertex represents three-point functions. Thus, diagrammatically, the
tripartite information is determined by two contributions: one made out of
products of two-point functions connecting the three regions and another one
which is the square of a three-point function.

The final formula for the tripartite information can be written in terms of
the relative separations of the spheres using eq. (52) and the three-point
function formula

$\langle\mathcal{O}(x_{A_{1}})\mathcal{O}(x_{A_{2}})\mathcal{O}(x_{A_{3}})\rangle=\frac{C_{\mathcal{O}\mathcal{O}\mathcal{O}}}{r_{12}^{\Delta}r_{23}^{\Delta}r_{13}^{\Delta}}\,,$
(61)

as in eq. (4).

####  2.2.4 Explicit formulas for $I_{4}$

Let us now move to the $N=4$ case. We have

$\displaystyle I_{4}(A_{1},A_{2},A_{3},A_{4})=\lim_{n\to
1}\frac{1}{n-1}\langle\tilde{\Sigma}_{{A_{1}}}^{(n)}\tilde{\Sigma}_{{A_{2}}}^{(n)}\tilde{\Sigma}_{{A_{3}}}^{(n)}\tilde{\Sigma}_{{A_{4}}}^{(n)}\rangle\,.$
(62)

Here again the leading contribution to $I_{4}$ comes from replacing the twist
operators by a product of two operators of lowest scaling dimension as in
(33), which leads to the following expression for the correlator in (62)

$\displaystyle\langle\tilde{\Sigma}_{1}^{(n)}\tilde{\Sigma}_{2}^{(n)}\tilde{\Sigma}_{3}^{(n)}\tilde{\Sigma}_{4}^{(n)}\rangle\,=\left(R_{1}R_{2}R_{3}R_{4}\right)^{2\Delta}\sum_{i<j}\sum_{k<l}\sum_{m<n}\sum_{p<q}C_{ij}C_{kl}C_{mn}C_{pq}\langle\mathcal{O}^{i}_{1}\mathcal{O}^{j}_{1}\mathcal{O}^{k}_{2}\mathcal{O}^{l}_{2}\mathcal{O}^{m}_{3}\mathcal{O}^{n}_{3}\mathcal{O}^{p}_{4}\mathcal{O}^{q}_{4}\rangle\,,$
(63)

where throughout the subsection we use the notation
$\tilde{\Sigma}_{{A_{i}}}\equiv\tilde{\Sigma}_{{i}}$ and
$\mathcal{O}_{i}\equiv\mathcal{O}_{A_{i}}$. Organizing the contributions to
eq. (63) in terms of the number ${\cal N}$ of sheets with non-trivial operator
insertions, one finds that for ${\cal N}=2$ there is one only possible
configuration, which has the following structure

$\displaystyle\text{${\cal
N}=2,$}\quad\left(\begin{array}[]{ccccc}1&\cdots\mathcal{O}^{i}_{1}&\cdots\mathcal{O}^{j}_{1}&\cdots
1\\\ 1&\cdots\mathcal{O}^{i}_{2}&\cdots\mathcal{O}^{j}_{2}&\cdots 1\\\
1&\cdots\mathcal{O}^{i}_{3}&\cdots\mathcal{O}^{j}_{3}&\cdots 1\\\
1&\cdots\mathcal{O}^{i}_{4}&\cdots\mathcal{O}^{j}_{4}&\cdots 1\\\
\end{array}\right)\,.$ (68)

The contribution of such term to eq. (63) is

$\displaystyle\sum_{i<j}C^{4}_{ij}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}\quad{\rm
thus}\quad I_{4}\supset
c_{4:2}^{(4)}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}R^{8\Delta}\,,$
(69)

where

$\displaystyle c_{4:2}^{(4)}\equiv\lim_{n\to
1}\frac{1}{n-1}\sum_{j<k}C^{4}_{ij}\,,$ (70)

and we can associate to it the graph presented in Fig. 3.

$i$$j$ Figure 3: Graph representation of the contribution of the correlator of
four twist operators to $I_{4}$ when the twist operators have non-trivial
support on only two sheets. In the above graph vertices represent sheets while
lines represents operators on particular regions. Observe that the number of
lines ending at a vertex gives the order of the associated correlator.

The next configurations we study are the ones that are defined on three
different sheets, i.e., ${\cal N}=3$. Let us illustrate one of such
configurations in matrix form,

$\displaystyle\text{${\cal
N}=3,$}\quad\left(\begin{array}[]{cccccc}1&\cdots\mathcal{O}^{i}_{1}&\cdots
1&\cdots\mathcal{O}^{k}_{1}&\cdots 1\\\ 1&\cdots
1&\cdots\mathcal{O}^{j}_{2}&\cdots\mathcal{O}^{k}_{2}&\cdots 1\\\
1&\cdots\mathcal{O}^{i}_{3}&\cdots\mathcal{O}^{j}_{3}&\cdots 1&\cdots 1\\\
1&\cdots\mathcal{O}^{i}_{4}&\cdots\mathcal{O}^{j}_{4}&\cdots 1&\cdots 1\\\
\end{array}\right)\,,$ (75)

which corresponds to the left graph in Fig. 4. The other two graphs in that
figure have similar matrix representations obtained from the above by
permutation of columns. In the graphs of Fig. 4, the indices $\\{i,j,k\\}$
have been ordered and, therefore, they represent different contributions to
$I_{4}$.

$i$$j$$k$

$i$$j$$k$

$i$$j$$k$

Figure 4: Graph representation of one class of contributions of the correlator
of four twist operators to $I_{4}$ when the twist operators have non-trivial
support on three sheets. In the above graph, vertices represent sheets while
lines represents operators on particular regions.

However, all of them will be proportional to the product of the following
correlators

$\displaystyle\left(\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+{\rm
permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim equivalence}\right)\,,$ (76)

where we include only the permutations that lead to independent
configurations. For instance, in the matrix representation of eq. (75) the
interchange $3\leftrightarrow 4$ leads to the same configuration and thus, it
should not be included into the permutations of eq. (76). Thus, the number of
terms in eq. (76) equals the different orderings of $4$ regions where two of
these are indistinguishable. This corresponds to ${4!}/{2!}=12$
configurations. The coefficient can be obtained by summing over the
contributions from all the graphs presented in Fig. 4, which can be grouped
together into the single coefficient

$\displaystyle c_{4:3}^{(1,1,2)}\equiv\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j<k}\left[C^{2}_{ij}C_{jk}C_{ki}+C_{ij}C^{2}_{jk}C_{ki}+C_{ij}C_{jk}C^{2}_{ki}\right]\,.$
(77)

In the above formula each term represents the coefficient associated to the
contribution of each graph from left to right. Thus, the contribution to
$I_{4}$ coming from those graphs is given by

$\displaystyle I_{4}\supset
C_{4:3}^{(1,1,2)}\left(\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+{\rm
perms.\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim equivalence}\right)\,R^{8\Delta}\,.$
(78)

For ${\cal N}=3$, there is another set of configurations contributing to
$I_{4}$ which are represented graphically in Fig. 5.

$i$$j$$k$

$i$$j$$k$

$i$$j$$k$

Figure 5: Graph representation of another class of contributions of the
correlator of four twist operators to $I_{4}$ when the twist operators have
non-trivial support on three sheets.

The contributions of each of these graphs are proportional to the following
correlators

$\displaystyle\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,\,{\rm+\,\,permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim
equivalence}\Big{)}\,,$ (79)

where the number of terms is given by the number of different ways of
labelling each of these graphs with four letters taking into account that
lines connecting the same pair of sheets are indistinguishable. Thus we have
${4!}/{(2!2!)}=6$ configurations. The coefficients associated to each graph
can be grouped together into the following coefficient

$\displaystyle c_{4:3}^{(2,2)}\equiv\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j<k}\left[C^{2}_{ij}C^{2}_{ik}+C^{2}_{ij}C^{2}_{jk}+C^{2}_{ik}C^{2}_{jk}\right]\,,$
(80)

where each term corresponds with the contribution of one of the graphs in Fig.
5, from left to right. Thus, we find the total contribution to $I_{4}$ of
these terms is given by

$\displaystyle I_{4}\supset
C_{4:3}^{(2,2)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,\,{\rm+\,\,permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim
equivalence}\Big{)}R^{8\Delta}\,.$ (81)

For ${\cal N}=4$, there are two different configurations. The first one is of
the form

$\displaystyle\text{${\cal N}=4$
}\quad\left(\begin{array}[]{cccccc}1&\cdots\mathcal{O}^{i}_{1}&\cdots\mathcal{O}^{j}_{1}&\cdots
1&\cdots 1&\cdots 1\\\ 1&\cdots
1&\cdots\mathcal{O}^{j}_{2}&\cdots\mathcal{O}^{k}_{2}&\cdots 1&\cdots 1\\\
1&\cdots 1&\cdots 1&\cdots\mathcal{O}^{k}_{3}&\cdots\mathcal{O}^{l}_{3}&\cdots
1\\\ 1&\cdots\mathcal{O}^{i}_{4}&\cdots 1&\cdots
1&\cdots\mathcal{O}^{l}_{4}&\cdots 1\\\ \end{array}\right)\,,$ (86)

which corresponds to the left graph in Fig. 6. However, as in the previous
cases, for this given ordering of indices there would be a total of three
independent graphs as illustrated in that figure.

$i$$l$$k$$j$

$i$$l$$k$$j$

$i$$l$$k$$j$

Figure 6: Graph representation of one class of contributions of the correlator
of four twist operators to $I_{4}$ when the twist operators have non-trivial
support on four sheets.

All these graphs give contributions proportional to the following correlators

$\displaystyle\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\,\,+{\rm
permutations\,of}\,\,\\{1,2,3,4\\}\Big{)}\,,$ (87)

where in this case, every labelling of the lines for each graph gives rise to
a non-equivalent configuration and thus there is a total of $4!=24$
configurations. The associated coefficients can then be put together in the
following single coefficient

$\displaystyle c_{4:4}^{(1,1,1,1)}=\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j<k<l}\left[C_{ij}C_{jk}C_{kl}C_{li}+C_{ij}C_{jl}C_{lk}C_{ki}+C_{il}C_{lj}C_{jk}C_{ki}\right]\,,$
(88)

where each term in the right hand side comes from a graph in Fig. 6 from left
to right. Thus the total contribution to $I_{4}$ from the configurations
described by the above graphs is

$\displaystyle
I_{4}\supset\,C_{4:4}^{(1,1,1,1)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\,\,+{\rm
permutations\,of}\,\,\\{1,2,3,4\\}\Big{)}R^{8\Delta}\,.$ (89)

The second set of configurations for ${\cal N}=4$ are described in the graphs
of Fig. 7.

$i$$l$$k$$j$

$i$$l$$k$$j$

$i$$l$$k$$j$

Figure 7: Graph representation of another class of contributions of the
correlator of four twist operators to $I_{4}$ when the twist operators have
non-trivial support on four sheets.

Here each graph gives a contribution proportional to the following correlators

$\displaystyle\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle^{2}\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}\,\,{\rm+\,\,permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim
equivalence}\Big{)}\,,$ (90)

where the number of terms equal the number of non-equivalent ways of labelling
the four lines of the associated graph. This is a total of ${4!}/{(2!2!)}=6$
configurations. The coefficients to which each of the above graphs contributes
can be grouped together into

$\displaystyle c_{4:4}^{(2,2)}=\lim_{n\to
1}\frac{1}{n-1}\sum_{i<j<k<l}\left[C^{2}_{il}C^{2}_{jk}+C^{2}_{ij}C^{2}_{kl}+C^{2}_{ik}C^{2}_{jl}\right]\,.$
(91)

This gives a total contribution to $I_{4}$ of the form

$\displaystyle I_{4}\supset
c_{4:4}^{(2,2)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle^{2}\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}\,\,{\rm+\,\,permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim
equivalence}\Big{)}R^{8\Delta}\,.$ (92)

For ${\cal N}>4$ since there is a total of $8$ operators there would
necessarily be a sheet with only one operator insertion and therefore it would
vanish for all those configurations. Putting all the contributions from eqs.
(69), (78), (81), (89) and (92) together leads to the following formula for
$I_{4}$ in the long-distance regime

$\displaystyle\frac{I_{4}}{R^{8\Delta}}=$
$\displaystyle+c_{4:2}^{(4)}\,\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}$
$\displaystyle+\,c_{4:3}^{(1,1,2)}\,\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle+{\rm
permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim equivalence}\Big{)}$
$\displaystyle+\,c_{4:3}^{(2,2)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,\,{\rm+\,\,permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim
equivalence}\Big{)}$
$\displaystyle+\,c_{4:4}^{(1,1,1,1)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\,\,+{\rm
permutations\,of}\,\,\\{1,2,3,4\\}\Big{)}$
$\displaystyle+\,c_{4:4}^{(2,2)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle^{2}\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}\,\,{\rm+\,\,permutations\,\,of}\,\,\\{1,2,3,4\\}/{\rm\sim
equivalence}\Big{)}\,.$ (93)

After a close inspection one realizes that the terms appearing in the last
line should be absent from a four-partite information, since those will not
have the required scaling behaviour as one separates regions. In other words,
they represent disconnected contributions. However, similar disconnected terms
will also come from certain contributions to the four-point functions
appearing on the other terms. We should be able to write down the final answer
in terms of connected contributions only.777We thank Horacio Casini for
pointing this out to us. We do so next

$\displaystyle\frac{I_{4}}{R^{8\Delta}}=$
$\displaystyle+c_{4:2}^{(4)}\,\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}-\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle^{2}\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}-\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle^{2}\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle^{2}-\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle^{2}\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle^{2}\Big{)}$
$\displaystyle+2\,c_{4:3}^{(1,1,2)}\,\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$
$\displaystyle\qquad\qquad+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,$
$\displaystyle\qquad\qquad+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle\,\Big{)}$
$\displaystyle+\,2\,c_{4:3}^{(2,2)}\Big{[}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle-\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\Big{)}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,$
$\displaystyle\qquad\qquad+\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle-\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\Big{)}\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\,$
$\displaystyle\qquad\qquad+\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle-\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\Big{)}\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\Big{]}\,$
$\displaystyle+\,8\,c_{4:4}^{(1,1,1,1)}\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\,\,+\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\,$
$\displaystyle\qquad\qquad\quad+\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,\Big{)}$
$\displaystyle+2\left[\,c_{4:4}^{(2,2)}+c_{4:3}^{(2,2)}+\frac{1}{2}c^{(4)}_{4:2}\,\,\right]\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle^{2}\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle^{2}+\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle^{2}\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle^{2}\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle^{2}\Big{)}\,,$
(94)

where we have taken into account the number of terms associated to each
coefficient and written them explicitly. Thus, consistency with the long-
distance expected behavior (expectation from clustering principle) implies
that the last combination of coefficients should exactly cancel. That is
precisely what happens.

All the coefficients appearing in the above formula are computed in Appendix
A, where we find

$\displaystyle
c_{4:2}^{(4)}=\frac{2^{8\Delta}\left[\Gamma\left(4\Delta+1\right)\right]^{2}}{2\,\Gamma\left(8\Delta+2\right)}\,,\qquad
c_{4:3}^{(1,1,2)}=-\frac{9\sqrt{\pi}\Gamma\left(3\Delta\right)^{2}}{4\Gamma\left(\Delta\right)^{2}\Gamma\left(4\Delta+\frac{3}{2}\right)}\,,$
$\displaystyle
c_{4:3}^{(2,2)}=-\frac{2^{8\Delta}\left[\Gamma\left(4\Delta+1\right)\right]^{2}}{2\,\Gamma\left(8\Delta+2\right)}\,,\quad{\rm
and}\quad
c_{4:4}^{(2,2)}=\frac{2^{8\Delta}\left[\Gamma\left(4\Delta+1\right)\right]^{2}}{4\,\Gamma\left(8\Delta+2\right)}\,.$
(95)

from eqs. (195), (205), (210), and (247) respectively. Unfortunately, we were
not able to find a completely explicit formula for $c_{4:4}^{(1,1,1,1)}$. The
expression for this coefficient appears in eq. (214), whose numerical
evaluation for a few particular values of $\Delta$ (analytically for
$\Delta=1/2$ and possibly also for $\Delta=1$) we present in table 1. As
anticipated, we find that the last combination of coefficients in eq. (94)
vanishes exactly, which represents a non-trivial consistency check of our
calculations.

Additionally, the relation between $c_{4:2}^{(4)}$ and $c_{4:3}^{(2,2)}$ from
eq. (2.2.4) suggests that the $I_{4}$ can be written in a more compact form in
terms of the square of the connected four-point function, namely,

$\displaystyle\frac{I_{4}}{R^{8\Delta}}=$
$\displaystyle+\Big{[}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle-\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle-\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle-\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\Big{]}^{2}\left[\frac{2^{8\Delta}\left[\Gamma\left(4\Delta+1\right)\right]^{2}}{2\,\Gamma\left(8\Delta+2\right)}\right]\
$
$\displaystyle-\Big{[}\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle$
$\displaystyle\qquad\quad+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,$
$\displaystyle\qquad\quad+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{4}\rangle\,\Big{]}\left[\frac{9\sqrt{\pi}\Gamma\left(3\Delta\right)^{2}}{2\Gamma\left(\Delta\right)^{2}\Gamma\left(4\Delta+\frac{3}{2}\right)}\right]$
$\displaystyle+\Big{[}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\,\,\,$
$\displaystyle\qquad\quad+\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,\Big{]}\left[\,8c_{4:4}^{(1,1,1,1)}-\frac{2^{8\Delta}\left[\Gamma\left(4\Delta+1\right)\right]^{2}}{\Gamma\left(8\Delta+2\right)}\right]\,.$
(96)

Using the explicit form of the two- and three-point functions, the final
expression for the $I_{4}$ can be written as in eq. (9). Finally, observe that
this final formula can be expressed via a space representation of the
correlators contributing to it as shown in Fig. 8.

$A_{1}$$A_{4}$$A_{3}$$A_{2}$

$A_{1}$$A_{4}$$A_{3}$$A_{2}$

$A_{1}$$A_{4}$$A_{3}$$A_{2}$

Figure 8: We provide a graph representation of the various terms contributing
to $I_{4}$. Single lines represent two-point correlators, lines converging at
a triangle vertex represent three-point correlators while lines converging at
a square vertex represents connected four-point correlators. Observe that each
diagram gives a completely connected contribution to $I_{4}$.

In view of eq. (2.2.4) and the results previously obtained for $N=2,3$, it is
clear that the leading long-distance term of the $N$-partite information will
be given by some linear combination of products of $2$-, $3$-, $\dots$,
$(N-1)$-, $N$-point functions of the smallest-conformal-dimension primary of
the corresponding theory. While such information should also be available (at
least in principle) within the mutual information expansion, the results
presented here support the idea of using the $N$-partite information expansion
as a complementary approach. Given a CFT for which the input information are
all the $N$-partite informations corresponding to ball regions in the long-
distance regime, one could systematically identify the scaling dimension of
the leading primary as well as all its $N$-point correlators as follows:
first, $\Delta$ would be extracted from the mutual information expansion in
eq. (2); then, eq. (4) would be used to extract the three-point coefficient;
eq. (9) would be used to extract the four-point function; analogous formulas
would yield the higher $N$-point functions. Information about subleading
primaries will start appearing at the next order in the long-distance
expansions. Determining the precise way in which this will happen and how one
should systematically proceed in order to extract the CFT data involving such
operators requires further work. However, we know that an essential part in
such process will require to upgrade the expansion in terms of local replica
operators (32) to an expansion in terms of “OPE blocks” associated to each
replica primary [27] —remember that each “OPE block” includes the resumed
contribution of a given primary as well as their associated descendants [65].
Doing so would reduce the type of operators included in the $N$-partite
information expansion to the replica primaries. For a recent discussion on the
“OPE block” expansion applied to the tripartite information see [25].

##  3 Free scalar CFT

In this section we obtain even more explicit formulas for the $N$-partite
information (for $N=2,3,4$) in the particular case of a CFT which has a free
scalar as its lowest-dimensional operator. We particularize our formulas to
three dimensions, obtaining analytic results which in section 4 we compare
with lattice calculations. We close with some conjectures regarding the sign
of $I_{N}$ for free-scalar CFTs in general dimensions.

Let us consider then a CFT which has a free scalar as its lowest-dimensional
primary operator. This includes, obviously, free scalar theories, but also
theories which have a free sector. Various simplifications occur in that case.
First, the conformal dimension of a free scalar field is given by

$\displaystyle\Delta_{\rm free\,scalar}=\frac{d-2}{2}\,,$ (97)

and so it is a half-integer. Second, correlators including an odd number of
operators vanish. Finally, correlators involving an even number of fields are
completely determined by two-point correlators by virtue of Wick’s theorem.
For instance, the four-point function becomes

$\displaystyle\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}\rangle=\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle+\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\,,$
(98)

and similarly for higher-point correlators. As a consequence, the long-
distance $N$-partite information of a free theory must be given by a single
term of the form

$\displaystyle I^{\rm
free\,scalar}_{N}(\\{A_{i}\\})=C_{N}\left(\prod_{\alpha=1}^{N}\Big{\langle}\mathcal{O}_{A_{\alpha}}\mathcal{O}_{A_{\alpha+1}}\Big{\rangle}+\,{\rm
permutations\,\,of\,}\,\\{A_{1},\cdots,A_{N}\\}\right)R^{N(d-2)}\,,$ (99)

where $A_{N+1}\equiv A_{1}$ and $R$ is the radii of the spheres. This is due
to the fact that the only completely connected $2N$-point correlators on $N$
regions that can be constructed with two-point correlators are the ones
presented above.

###  3.1 Mutual information

In the case of the mutual information, the general formula at long distances
in the case of spherical entangling surfaces was presented in eq. (53). For a
free scalar such formula reduces to

$\displaystyle I^{\rm
free\,scalar}_{2}=\frac{2^{2(d-2)}\Gamma\left(d-1\right)^{2}}{2\,\Gamma\left(2(d-1)\right)}\frac{R^{4\Delta}}{r^{4\Delta}}\,,$
(100)

where $R$ is the radius of the balls which we assumed to be the same, and $r$
is the center-to-center distance between the regions.

For comparison with the lattice results we consider the case of $d=3$. Also,
we rewrite the result using the parameters $R_{\rm latt}\equiv 2R$, $r_{\rm
latt}\equiv r$. We have

$\displaystyle\left.I^{\rm
free\,scalar}_{2}\right|_{d=3}=\frac{1}{48}\frac{R_{\rm latt}^{2}}{r_{\rm
latt}^{2}}\approx 0.08333x^{-2}\,,$ (101)

where $x\equiv r_{\rm latt}/R_{\rm latt}$.

###  3.2 Tripartite information

The tripartite information at long distances is given by eq. (60). For a free
scalar this reduces to

$\displaystyle I^{\rm
free\,scalar}_{3}=\frac{2^{3(d-2)}\Gamma\left(\tfrac{d-1}{2}\right)^{3}}{2\pi\Gamma\left(\tfrac{3(d-1)}{2}\right)}\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\,R^{3(d-2)}\,.$
(102)

Now, let us consider a geometric configuration where the balls are located at
the vertices of an equilateral triangle. Then, we have

$\displaystyle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle=\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle=\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle=\frac{1}{r^{d-2}}\,,$
(103)

where $r$ is the length of the side of triangle. In that case, eq. (102)
reduces to

$\displaystyle I^{\rm
free\,scalar}_{3}=\frac{2^{3(d-2)}\Gamma\left(\frac{d-1}{2}\right)^{3}}{2\pi\Gamma\left(\frac{3(d-1)}{2}\right)}\frac{R^{3(d-2)}}{r^{3(d-2)}}\,,$
(104)

where $R$ is the radius of the balls, which we assumed to be the same for the
three.

Again, for comparison with the lattice results, we consider the case of $d=3$
and use instead the parameters $R_{\rm latt}$ and $r_{\rm latt}$ which are
related to the previous ones by $R_{\rm latt}=2R$ and $r_{\rm
latt}=r/\sqrt{3}$. $R_{\rm latt}$ is just the diameter of each disk, while
$r_{\rm latt}$ is the radius of the circumference containing the vertices of
the equilateral triangle. In terms of these parameters we have

$\displaystyle\left.I^{\rm
free\,scalar}_{3}\right|_{d=3}=\frac{1}{12\,\sqrt{3}\,\pi}\frac{R_{\rm
latt}^{3}}{r_{\rm latt}^{3}}\approx 0.01531x^{-3}$ (105)

where $x\equiv r_{\rm latt}/R_{\rm latt}$.

###  3.3 Four partite information

For the $I_{4}$, the long-distance expression for spherical entangling
surfaces was given in eq. (2.2.4). This expression simplifies now for two
reasons. First, the three-point function of the free scalar exactly vanishes,
which makes the whole second term vanish. Second, the four-point function
reduces to products of two-point functions as in eq. (98), which makes the
first term vanish as well. We are then left with

$\displaystyle I^{\rm free\,scalar}_{4}=$
$\displaystyle+\left[\,8c_{4:4}^{(1,1,1,1)}\left(\frac{d-2}{2}\right)-\frac{2^{4(d-2)}\left[\Gamma\left(2d-3\right)\right]^{2}}{\,\Gamma\left(4d-6\right)}\right]\Big{(}\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle+\,\,$
(106)
$\displaystyle+\,\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\,+\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle\,\Big{)}R^{4(d-2)}\,.$

Now, let us consider a configuration in which the spherical regions are
located at the corners of a square of side $r$. In that case the correlators
become

$\displaystyle\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle=\langle\mathcal{O}_{2}\mathcal{O}_{3}\rangle=\langle\mathcal{O}_{3}\mathcal{O}_{4}\rangle=\langle\mathcal{O}_{4}\mathcal{O}_{1}\rangle=\frac{1}{r}\,,$
(107)

and

$\displaystyle\langle\mathcal{O}_{1}\mathcal{O}_{3}\rangle=\langle\mathcal{O}_{2}\mathcal{O}_{4}\rangle=\frac{1}{\sqrt{2}r}\,.$
(108)

Plugging this into eq. (106) results in

$\displaystyle I^{\rm
free\,scalar}_{4}=16\left[\,c_{4:4}^{(1,1,1,1)}\left(\frac{d-2}{2}\right)-\frac{2^{4(d-2)}\left[\Gamma\left(2d-3\right)\right]^{2}}{8\,\Gamma\left(4d-6\right)}\right]\frac{R^{4(d-2)}}{r^{4(d-2)}}\,.$
(109)

Like in the previous cases, we restrict to the case of $d=3$ and use the
parameters $R_{\rm latt}=2R$ and $r_{\rm latt}=r/\sqrt{2}$ where $R_{\rm
latt}$ is the diameter of the disks while $r_{\rm latt}$ is the radius of the
circle that passes through the vertices of the square. Thus, in terms of these
parameters we have

$\displaystyle\left.I_{4}^{\rm
free\,scalar}\right|_{d=3}=\left[\frac{1}{180}+\frac{1}{6\pi^{2}}\right]\frac{R_{\rm
latt}^{4}}{r_{\rm latt}^{4}}\simeq 0.0224\frac{R_{\rm latt}^{4}}{r_{\rm
latt}^{4}}\,,$ (110)

where we used the analytic result obtained for
$c_{4:4}^{(1,1,1,1)}(\Delta=1/2)$ in eq. (241).

Evaluating $c_{4:4}^{(1,1,1,1)}$ for other values of $\Delta$ we can determine
the leading long-distance coefficient in the spheres $I_{4}$ for a free-scalar
CFT in higher dimensions. In particular, using the results presented in table
1, we find

$\displaystyle\left.I_{4}^{\rm free\,scalar}\right|_{d=4}$
$\displaystyle=\frac{256}{105}\frac{R^{8}}{r^{8}}\,,$ (111)
$\displaystyle\left.I_{4}^{\rm free\,scalar}\right|_{d=5}$
$\displaystyle\simeq 5.70678\frac{R^{12}}{r^{12}}\,,$ (112)
$\displaystyle\left.I_{4}^{\rm free\,scalar}\right|_{d=6}$
$\displaystyle\simeq 15.4904\frac{R^{16}}{r^{16}}\,.$ (113)

As we can see, the coefficient is always positive and grows with $d$. In view
of these results, it is tempting to conjecture that the $I_{N}$ for $N=2,3,4$
will be positive for free-scalar CFTs in arbitrary dimensions and for general
regions. In fact, lattice results for $d=3$ and for $N=1,2,\dots,6$ suggest
the positivity of $I_{N}$ for the free scalar CFT for arbitrary $N$ and
general regions. Thus, we conjecture that the $N$-partite information for
theories with a free scalar as their lowest-dimensional primary is always
positive for arbitrary $N$ and $d$. While such a claim in the $d\geq 3$ case
relies exclusively on our result for the leading long-distance term in the
case of spherical regions, it is natural to expect that moving the regions
closer should tend to monotonically increase the absolute value of the
$N$-partite information on general grounds (and hence its overall sign would
not change with respect to the long-distance situation). Similarly, it is
reasonable to expect that changing the shape of the regions will not modify
the sign of the $I_{N}$ either. This is related to the question of whether or
not the sign of $I_{N}$ is completely fixed for a given theory independently
of the shape of the entangling regions and their relative separations. As far
as we know, all available evidence supports a positive answer to this
question, but we are not aware of a general proof.

##  4 Lattice calculations in $d=3$

In this section we compute the $N$-partite information (for $N=2,3,4,5,6$) for
various entangling regions assembled forming regular $N$-gon configurations
for a three-dimensional free scalar in the lattice. First we focus on the
long-distance regime. We verify that the leading scaling obtained for $I_{N}$
in this regime in Section 2.2 is fulfilled in all cases and study the
dependence on the shape of the entangling regions. In the case of disk
regions, we obtain numerical approximations for the coefficient of the leading
terms, which can be tested against analytic calculations using the results of
Section 2.2 for $N=2,3,4$, finding excellent agreement. Our results suggest
that $I_{N}$ for a three-dimensional free scalar is positive for all $N$ and
for general regions and configurations. In subsection 4.3 we perform lattice
calculations of the mutual information for pairs of regions both for a free
scalar and a free fermion for general separations. In all cases, we also
present analytic results for two toy models which approximate the respective
scalings of each of the free models at long separations. In the case of disk
regions, we provide analytic approximations for the whole range of values both
for the scalar and the fermion. We determine various scalings at small and
large separations for the different regions and observe that, normalized by
the respective disk entanglement entropy coefficients, the free scalar mutual
information is always greater than the fermion one, a feature which we
conjecture to be true for arbitrary regions.

###  4.1 Lattice setup and models

In the following subsection we perform lattice calculations for a free scalar
CFT. In order to do that, we consider a set of scalar fields and conjugate
momenta $\phi_{i},\pi_{j}$, $i,j=1,\dots,N$ labelled by their positions at the
square lattice —each subindex $i$ makes reference to a position
$(x_{i},y_{i})$. The fields satisfy canonical commutation relations,
$[\phi_{i},\pi_{j}]=i\delta_{ij}$ and
$[\phi_{i},\phi_{j}].=[\pi_{i},\pi_{j}]=0$. Then, given some Gaussian state
$\rho$, the EE for some entangling region $A$ can be obtained from the
correlators $X_{ij}\equiv\operatorname{tr}(\rho\phi_{i}\phi_{j})$ and
$P_{ij}\equiv\operatorname{tr}(\rho\pi_{i}\pi_{j})\,$ as follows [66, 56],

$S(A)=\operatorname{tr}\left[(C_{A}+1/2)\log(C_{A}+1/2)-(C_{A}-1/2)\log(C_{A}-1/2)\right]\,,$
(114)

where $C_{A}\equiv\sqrt{X_{A}P_{A}}$ and $(X_{A})_{ij}\equiv X_{ij}$,
$(P_{A})_{ij}=P_{ij}$ (with $i,j\in A$) are the restrictions of the
correlators to the sites inside $A$. We compute the $N$-partite information
using eq. (114) along with eq. (5). Setting the lattice spacing to one, the
Hamiltonian is given by

$H=\frac{1}{2}\sum_{i,j=-\infty}^{\infty}\left[\pi^{2}_{i,j}+(\phi_{i+1,j}-\phi_{i,j})^{2}+(\phi_{i,j+1}-\phi_{i,j})^{2}\right]\,,$
(115)

and the correlators corresponding to the vacuum state are given by [56]

$\displaystyle X_{(0,0),(i,j)}$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{-\pi}^{\pi}dx\int_{-\pi}^{\pi}dy\frac{\cos(jy)\cos(ix)}{\sqrt{2(1-\cos
x)+2(1-\cos y)}}\,,$ (116) $\displaystyle P_{(0,0),(i,j)}$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{-\pi}^{\pi}dx\int_{-\pi}^{\pi}dy\cos(jy)\cos(ix)\sqrt{2(1-\cos
x)+2(1-\cos y)}\,.$ (117)

Figure 9: We show two examples of the $N$-gon arrangements of entangling
regions considered in the lattice. The one in the left figure corresponds to
$N=3$ for rectangles with dimensions $L\times 2L$. The one in the right figure
corresponds to $N=5$ for disk regions. In each case, $r$ is defined as the
radius of the circumference which shares center with the $N$-gon and passes
through the center of the entangling regions.

In the third subsection, where we compute mutual information of various
regions for arbitrary separations, we also consider a free Dirac fermion CFT.
For that, we take fields $\psi_{i}$, $i=1,\dots,N$ defined at the lattice
points, satisfying canonical anticommutation relations,
$\\{\psi_{i},\psi_{j}^{\dagger}\\}=\delta_{ij}$. For a given Gaussian state
$\rho$, we define the correlators matrix
$D_{ij}\equiv\operatorname{tr}\small(\rho\psi_{i}\psi_{j}^{\dagger}\small)$.
Then, similarly to the scalars case, the EE for some entangling region $A$ can
be obtained as [56]

$S(A)=-\operatorname{tr}\left[D_{A}\log
D_{A}+(1-D_{A})\log(1-D_{A})\right]\,,$ (118)

where $D_{A}$ is the restriction of $D_{ij}$ to the lattice sites lying within
$A$. Our mutual information calculations are performed using this formula for
the corresponding regions $A$, $B$ and $A\cup B$. The lattice Hamiltonian for
the free fermion is given by

$H=-\frac{i}{2}\sum_{n,m}\left[\left(\psi^{\dagger}_{m,n}\gamma^{0}\gamma^{1}(\psi_{m+1,n}-\psi_{m,n})+\psi^{\dagger}_{m,n}\gamma^{0}\gamma^{2}(\psi_{m,n+1}-\psi_{m,n})\right)-{\rm
h.c.}\right]\,,$ (119)

and the correlators in the vacuum state read [56]

$D_{(n,k),(j,l)}=\frac{1}{2}\delta_{n,j}\delta_{kl}-\int_{-\pi}^{\pi}dx\int_{-\pi}^{\pi}dy\frac{\sin(x)\gamma^{0}\gamma^{1}+\sin(y)\gamma^{0}\gamma^{2}}{8\pi^{2}\sqrt{\sin^{2}x+\sin^{2}y}}e^{i(x(n-j)+y(k-l))}\,.$
(120)

For our numerical calculations, we use a lattice of size 200, so we can
consider points separated at most 200 units. In the second subsection, we
analyze the behavior of the $N$-partite information at long distances for
$N=2,3,4,5,6$, for several (simple) entangling regions and arrangements —i.e.,
relative positions of the blocks with respect to one another. For a given $N$,
we consider two arrangements: one in which the blocks are located at the
vertices of a regular $N$-gon, and one in which they are all aligned. For
clarity reasons, we only present explicit results for the $N$-gon arrangement
—see Fig. 9— but we also mention the analogous linear-arrangement results when
something qualitatively different is obtained in that case. For each setup, we
consider five different elementary blocks: disks, squares, rectangles with
dimensions $L\times 2L$ (along the horizontal and vertical axes,
respectively), rectangles with dimensions $L\times 4L$, and rectangles with
dimensions $L\times 6L$. We study the $N$-partite information for sets of $N$
identical elementary blocks of those kinds. We do this as a function of the
ratio $x\equiv r_{\rm latt}/R_{\rm latt}$, where $R_{\rm latt}$ is the
characteristic size of the blocks and $r_{\rm latt}$ is the characteristic
separation among them. On the other hand, in the third subsection, we focus on
the mutual information $I_{2}$, but we compute it for general separations,
both for the scalar and the fermion. In that case, results are presented for:
disks, squares, rectangles with dimensions $L\times 2L$, rectangles with
dimensions $L\times 3L$, rectangles with dimensions $L\times 4L$ and
rectangles with dimensions $L\times 5L$.

In the continuum limit, the above lattice models yield results corresponding
to a free scalar field and a free fermion, respectively. In order to obtain
such values, we proceed as follows. Given a particular configuration and
keeping $x$ fixed, we compute $I_{N}$ for increasingly greater values of
$R_{\rm latt}$ and $r_{\rm latt}$. In every case, we find that such values
clearly tend to certain constant asymptotic values as $R_{\rm latt},r_{\rm
latt}\rightarrow\infty$. In order to obtain those, we perform fits of the data
points using functions $\\{1,1/x,1/x^{2},\dots\\}$. Reliable values are
obtained whenever the resulting constant values hardly depend on the order at
which we stop adding fitting functions. This is the situation encountered for
the present models in all cases considered and therefore the results presented
can be directly attributed to the continuum theories. In the case of the free
fermion, we need to take into account the “doubling” in the number of
fermionic degrees of freedom which occurs in the lattice. In the present
three-dimensional case, the Dirac fermion result is obtained by dividing the
final result by 4.

The quantities $R_{\rm latt}$ and $r_{\rm latt}$ (which for the remainder of
the Section we denote simply by $R$ and $r$, respectively) are precisely
defined as follows —see Fig. 9. For square and rectangle blocks, $R$ is the
number of lattice links of the largest side. For instance, this means that in
a square of $5\times 5$ lattice _points_ , $R=4$; while in a $L\times 4L$
rectangle of $4\times 13$ lattice points, $R=12$ ($L=3$ in this example). For
disk blocks, $R$ is the number of lattice links of the diameter. Thus, a
circle with $R=12$ has $13$ lattice points along its diameter. The definition
of a circle in a lattice is not precise, so we define the circle of size $R$
to be the set of lattice points at a distance less than or equal to $R/2$ to
the central point. Regarding the magnitude $r$, for $N=2$ we define it to be
the distance between the rightmost boundary point of the left region and the
leftmost boundary point of the right one, whereas for an $N\geq 3$-gon
arrangement, it corresponds to the radius of the circle in which we
circumscribe the polygon. This is not the precise distance between adjacent
sites except in a particular case (the hexagon), but the difference is always
an $\mathcal{O}(1)$ number which does not change the functional dependence we
want to identify. On the other hand, for the linear arrangement and $N\geq 3$,
$r$ is the separation between the centers of adjacent blocks, measured in
number of lattice links.

When presenting the mutual information results in subsection 4.3, we will
normalize the results by the universal coefficient appearing in the EE of a
disk region. For general CFTs, the EE of an arbitrary smooth region $A$ is
given by

$S(A)=c_{0}\frac{{\rm perimeter}(\partial
A)}{\delta}-F(A)+\mathcal{O}(\delta)\,,$ (121)

where $c_{0}$ is a non-universal constant, $\delta$ is a UV cutoff and $F(A)$
is a dimensionless and universal coefficient. In the case of a disk region, we
denote this quantity by $F_{0}$, namely, $F(\partial A=\mathbb{S}^{1})\equiv
F_{0}$.888$F_{0}$ and its properties have been further studied in different
contexts and for various theories —see e.g., [67, 68, 51, 69, 52, 70, 71] and
references therein. As it turns out, $F_{0}$ satisfies a number of interesting
properties. On the one hand, it coincides with the Euclidean free energy of
the corresponding CFT on a three-sphere [9]. In addition, an entropic
monotonicity theorem for general three-dimensional theories has been proven in
[72] and the c-function coincides with $F_{0}$ at the fixed points —see also
[73, 52, 74]. Finally, as shown in [58], for arbitrary CFTs it happens that

$F(A)/F_{0}\geq 1\quad\text{for all regions }A\quad\text{and }\quad
F(A)=F_{0}\Leftrightarrow A=\text{disk}\,,$ (122)

namely, disks globally minimize the EE universal term for general three-
dimensional CFTs. In view of this, in order to compare the mutual information
results of different theories, it is natural to normalise the results by
$F_{0}$ in each case, and that is what we do. For the free scalar and the free
Dirac fermion we have, respectively [75, 76]

$F_{0}^{\rm s}=\frac{1}{16}\left[2\log
2-\frac{3}{\pi^{2}}\zeta(3)\right]\,,\quad F_{0}^{\rm
f}=\frac{1}{8}\left[2\log 2+\frac{3}{\pi^{2}}\zeta(3)\right]\,.$ (123)

###  4.2 Long-distance $I_{N}$ for a free scalar

Based on the results obtained in Section 2.2, we expect the $N$-partite
information for a free scalar in general dimensions to scale as

$I_{N}^{\rm free\,scalar}\sim(1/x)^{(d-2)N}\quad\text{for}\quad x\equiv r/R\gg
1\,.$ (124)

Here, we used the result for the free-scalar conformal dimension, $\Delta_{\rm
free\,scalar}=(d-2)/2$. Hence, in $d=3$ we expect $I_{N}^{\rm
free\,scalar}\sim(1/x)^{N}$. In order to test this, for each set of entangling
regions we produce points at five distinct (large) values of $x$, and we fit
them to $(1/x)^{N-1}$, $(1/x)^{N}$, and $(1/x)^{N+1}$ curves. In Table 2,
which we present in Appendix C, we have compiled the resulting coefficients of
determination $\mathcal{R}^{2}$ —not to be confused with the size of the
regions— for each fit and highlight the one which corresponds to the best one.

The scaling predicted by eq. (124) is strongly favoured by all the sets of
regions and for all the values of $N$ considered. Considering even greater
values of $N$ becomes increasingly challenging due to technical reasons
associated to the lattice, but the pattern is clear. Repeating the analysis
using the linear arrangement instead, we find very similar results and the
same evidence in favor of eq. (124).999In that case, for $N=6$ the $1/x^{5}$
scaling is slightly favoured with respect to $1/x^{6}$ for a couple of sets of
regions. However, this is most likely due to the fact that in the linear
arrangement setup, studying the long-distance regime is more problematic in a
finite lattice because, as opposed to the $N$-gon case, all separations
between regions have to be large along the same direction.

Let us now present the explicit results found for the $N$-partite information
of the free scalar corresponding to the configurations and regions explained
above. The data points obtained appear plotted in Fig. 10.

Figure 10: For a free scalar in the lattice, we plot the $I_{N}$ (continuum
values) for $N=2,3,4,5,6$ for sets of: squares, disks, $L\times 2L$
rectangles, $L\times 4L$ rectangles and $L\times 6L$ rectangles (in each plot
moving from the uppermost curve to the bottom one) placed in a regular $N$-gon
configuration. The solid curves correspond to fits of the form eq. (125),
including two terms. The orange dashed lines in the $I_{2}$, $I_{3}$ and
$I_{4}$ plots correspond to curves $(1/12)/(x+1)^{2}$,
$1/(12\sqrt{3}\pi)\cdot(1/x^{3})$ and $(1/180+1/(6\pi^{2}))/x^{4}$
respectively, which are the exact leading behaviours. In the last two cases,
the subleading term (unknown) has a non-negligible effect, which explains the
discrepancy with the points and the numerical fits (the dashed gray lines
represent the leading terms in the numerical approximations which include two
terms, namely: $0.0832/(x+1)^{2}$, $0.0155/x^{3}$ and $0.0207/x^{4}$).

As explained in the previous subsection, for every data set we find that a
function of the form $I_{N}\propto(1/x)^{N}$ fits the points very well.
Naturally, we expect subleading corrections to this behavior, namely,

$I_{N}^{\rm
free\,scalar}=\alpha_{(N)}(1/x)^{N}+\beta_{(N)}(1/x)^{N+1}+\dots\quad(x\gg 1)$
(125)

Now, the values of $\alpha_{(2)}$ and $\alpha_{(3)}$ in the case of disk
regions have been obtained analytically in [30] and [25], respectively. They
read

$\displaystyle\left.\alpha_{(2)}\right|_{\rm disks}$
$\displaystyle=\frac{1}{12}\simeq 0.083333\,,$ (126)
$\displaystyle\left.\alpha_{(3)}\right|_{\rm disks}$
$\displaystyle=\frac{1}{12\sqrt{3}\pi}\simeq 0.0153147\,.$ (127)

In these two cases, performing a fit to the data points which includes the
subleading contribution —i.e., of the form (125) without the dots— results in
a numerical prediction for $\alpha_{(2)}$ and $\alpha_{(3)}$ which is slightly
better than the one obtained when only the leading term is considered when
compared to the analytic result. In particular, we find101010Observe that in
the case of the mutual information of the disk regions, defining $r$ as the
distance between the leftmost point of the right disk and the rightmost point
of the left disk, rather than the separation between centers, the leading
scaling is $\sim\alpha_{(2)}|_{\rm disks}/(x+1)^{2}$. For $x\gg 1$, this is
equivalent to $\alpha_{(2)}|_{\rm
disks}\cdot[1/x^{2}-4/x^{3}+\mathcal{O}(x^{-4})]$, so we could have chosen to
present the numerical results as an expansion on $1/x$ rather than $1/(x+1)$.
However, the definition of $x$ we use here is more convenient for the analysis
of the following subsection, and performing a meaningful comparison with the
leading analytic behavior of our lattice points represented as a function of
$x$ in Fig. 10 requires using $1/(x+1)$.

$\displaystyle\left.I_{2}^{\rm free\,scalar}\right|_{\rm disks}$
$\displaystyle\simeq{0.0832}\,/(x+1)^{2}+0.00629\,/(x+1)^{3}\,,\quad(x\gg 1)$
(128) $\displaystyle\left.I_{3}^{\rm free\,scalar}\right|_{\rm disks}$
$\displaystyle\simeq{0.0155}\,(1/x)^{3}-0.00861\,(1/x)^{4}\,,\quad(x\gg 1)$
(129)

as the best fits. Hence, the lattice calculations for the disks mutual and
tripartite informations reproduce the analytic results with remarkable
accuracy (they differ by $\sim 0.16\%$ and $\sim 1.21\%$, respectively). For
comparison, fits which only include the leading term yield $0.0836$ and
$0.0144$ respectively for the leading coefficients (discrepancies $\sim
0.32\%$ and $\sim 6.35\%$ respectively). In view of this, we proceed similarly
for the $N=4$ case, finding

$\displaystyle\left.I_{4}^{\rm free\,scalar}\right|_{\rm disks}$
$\displaystyle\simeq 0.0207\,(1/x)^{4}-0.0155\,(1/x)^{5}\,,\quad(x\gg 1)$
(130)

The leading coefficient can be compared with the exact formula result in eq.
(110) obtained in the previous section. As we can see, both results differ by
$\sim 8\%$, which is considerably more than for the $N=2,3$ cases, but still a
very good agreement in view of the limitations of the lattice.111111These
limitations include the fact that disks are difficult to approximate in a
square lattice, or the fact that increasing the number of regions makes it
more difficult to reliably obtain the continuum limit in a finite lattice.
This is a highly non-trivial check of our general formula eq. (9) for the
long-distance four-partite information of ball regions. On the other hand, for
the $N=5,6$ cases, we find

$\displaystyle\left.I_{5}^{\rm free\,scalar}\right|_{\rm disks}$
$\displaystyle\simeq 0.0446\,(1/x)^{5}-0.0699\,(1/x)^{6}\,,\quad(x\gg 1)$
(131) $\displaystyle\left.I_{6}^{\rm free\,scalar}\right|_{\rm disks}$
$\displaystyle\simeq 0.183\,(1/x)^{6}-0.489\,(1/x)^{7}\,,\quad(x\gg 1)$ (132)

The leading-term coefficients could be eventually compared with formulas
analogous to eq. (9).

Performing similar fits for the square and rectangular regions, we find for
the respective mutual information coefficients121212As mentioned above, in
this case we define the $\beta_{(2)}$ so that $I_{2}^{\rm
free\,scalar}\simeq\alpha_{(2)}\,/(x+1)^{2}+\beta_{(2)}\,/(x+1)^{3}+\dots$

$\displaystyle\left.\alpha_{(2)}\right|_{L\times L}$ $\displaystyle\simeq
0.124\,,\quad\left.\alpha_{(2)}\right|_{L\times 2L}\simeq
0.0682\,,\quad\left.\alpha_{(2)}\right|_{L\times 4L}\simeq
0.0412\,,\quad\left.\alpha_{(2)}\right|_{L\times 6L}\simeq 0.0325\,,$ (133)
$\displaystyle\left.\beta_{(2)}\right|_{L\times L}$ $\displaystyle\simeq
0.0150\,,\quad\left.\beta_{(2)}\right|_{L\times 2L}\simeq
0.00126\,,\quad\left.\beta_{(2)}\right|_{L\times
4L}\simeq-0.000502\,,\quad\left.\beta_{(2)}\right|_{L\times
6L}\simeq-0.000636\,.$

Observe that, as we make the rectangles thinner, the coefficients
$\alpha_{(2)}$ tend to decrease. It is easy to check that they are very
accurately approximated —see Fig. 11— by a function of the form

$\left.\alpha_{(2)}\right|_{L\times\xi L}\simeq
0.015215+0.102340(1/\xi)+0.0073286(1/\xi)^{2}\,.$ (134)

From this, we can extract the limiting value corresponding to arbitrarily thin
rectangles,

$\left.\alpha_{(2)}\right|_{L\times\xi L}\overset{\xi\gg
1}{\simeq}0.015215\,.$ (135)

Note that in order for this to hold one needs to be careful with the order of
limits (we are taking $x\gg 1$ and $\xi\gg 1$ at the same time). In
particular, eq. (135) is valid as long as we keep $r\gg\xi L$, i.e., the
expression is valid for pairs of very thin rectangles which are separated a
distance much greater than the length of their longest side. It would be
interesting to compute this coefficient analytically.131313An analogous
calculation of the leading coefficient as a function of $\xi$ was performed in
the case of a free fermion in [24]. In that case, the leading scaling is
$I_{2}^{\rm free\,fermion}\sim\mathcal{O}(x^{-4})$ instead.

Figure 11: We plot the values of the leading coefficient in the mutual
information of a free scalar field in the long-distance regime for pairs of
rectangle regions of dimensions $L\times\xi L$, as a function of $\xi$. The
solid curve is the fit presented in the main text, and the dashed line, the
asymptotic value corresponding to $\xi\rightarrow\infty$ (namely, very thin
rectangles).

On the other hand, we find for the $I_{3}$,

$\displaystyle\left.\alpha_{(3)}\right|_{L\times L}$ $\displaystyle\simeq
0.0280\,,\quad\left.\alpha_{(3)}\right|_{L\times 2L}\simeq
0.0112\,,\quad\left.\alpha_{(3)}\right|_{L\times 4L}\simeq
0.00512\,,\quad\left.\alpha_{(3)}\right|_{L\times 6L}\simeq 0.00355\,,$ (136)
$\displaystyle\left.\beta_{(3)}\right|_{L\times L}$
$\displaystyle\simeq-0.0168\,,\quad\left.\beta_{(3)}\right|_{L\times
2L}\simeq-0.00551\,,\quad\left.\beta_{(3)}\right|_{L\times
4L}\simeq-0.00211\,,\quad\left.\beta_{(3)}\right|_{L\times
6L}\simeq-0.00136\,,$

for the $I_{4}$,

$\displaystyle\left.\alpha_{(4)}\right|_{L\times L}$ $\displaystyle\simeq
0.0442\,,\quad\left.\alpha_{(4)}\right|_{L\times 2L}\simeq
0.0125\,,\quad\left.\alpha_{(4)}\right|_{L\times 4L}\simeq
0.00418\,,\quad\left.\alpha_{(4)}\right|_{L\times 6L}\simeq 0.00251\,,$ (137)
$\displaystyle\left.\beta_{(4)}\right|_{L\times L}$
$\displaystyle\simeq-0.0409\,,\quad\left.\beta_{(4)}\right|_{L\times
2L}\simeq-0.00626\,,\quad\left.\beta_{(4)}\right|_{L\times
4L}\simeq-0.000682\,,\quad\left.\beta_{(4)}\right|_{L\times
6L}\simeq-0.0000901\,,$

for the $I_{5}$,

$\displaystyle\left.\alpha_{(5)}\right|_{L\times L}$ $\displaystyle\simeq
0.152\,,\quad\left.\alpha_{(5)}\right|_{L\times 2L}\simeq
0.0226\,,\quad\left.\alpha_{(5)}\right|_{L\times 4L}\simeq
0.00543\,,\quad\left.\alpha_{(5)}\right|_{L\times 6L}\simeq 0.00279\,,$ (138)
$\displaystyle\left.\beta_{(5)}\right|_{L\times L}$
$\displaystyle\simeq-0.281\,,\quad\left.\beta_{(5)}\right|_{L\times
2L}\simeq-0.0294\,,\quad\left.\beta_{(5)}\right|_{L\times
4L}\simeq-0.00409\,,\quad\left.\beta_{(5)}\right|_{L\times
6L}\simeq-0.00144\,,$

and for the $I_{6}$,

$\displaystyle\left.\alpha_{(6)}\right|_{L\times L}$ $\displaystyle\simeq
1.10\,,\quad\left.\alpha_{(6)}\right|_{L\times 2L}\simeq
0.158\,,\quad\left.\alpha_{(6)}\right|_{L\times 4L}\simeq
0.0557\,,\quad\left.\alpha_{(5)}\right|_{L\times 6L}\simeq 0.0171\,,$ (139)
$\displaystyle\left.\beta_{(6)}\right|_{L\times L}$
$\displaystyle\simeq-4.02\,,\quad\left.\beta_{(6)}\right|_{L\times
2L}\simeq-0.752\,,\quad\left.\beta_{(6)}\right|_{L\times
4L}\simeq-0.0311\,,\quad\left.\beta_{(5)}\right|_{L\times 6L}\simeq-0.0591\,.$

The resulting curves for the various $I_{N}$ are shown in Fig. 10 and
approximate the data points very well in all cases. Let us mention that, as a
matter of fact, fits which only include the leading term in eq. (125) also do
a very good job at fitting the data —in most cases the curves are
indistinguishable from the ones presented in the figure. However, as mentioned
earlier, the numerical results for the coefficients $\alpha_{(2)}$ and
$\alpha_{(3)}$ in the case of disk regions are more precisely obtained
including also the subleading term, and so we choose to perform fits of that
kind for all the rest of configurations. For comparison, in Fig. 10 we have
included, in the case of square configurations, the best fits obtained from
ansatze of the form $I_{N}\propto(1/x)^{N-1}$ and $I_{N}\propto(1/x)^{N+1}$.
It is obvious from the figure that those are clearly wrong.

Observe that for every $N$, the values of $I_{N}^{\rm free\,scalar}$ for all
possible sets of regions are positive. It seems then reasonable to speculate
that the free scalar $N$-partite information is positive for all possible
regions and configurations. The results obtained replacing the $N$-gon
arrangement by the aligned one also support this claim. In view of the exact
results obtained in the previous section, it is in fact natural to conjecture
that $I_{N}^{\rm free\,scalar}\geq 0$ for all possible regions and in general
dimensions. It would be interesting to find additional support for this
conjecture.

###  4.3 $I_{2}$ at arbitrary separations for free scalars and fermions

In this subsection we compute the mutual information of pairs of entangling
regions of different shapes for three-dimensional free scalars and free
fermions in the lattice for general separations. We also include results
corresponding to two analytic toy models whose long-distance behaviours mimic
the free fermion and free scalar ones, respectively. The toy models, which are
introduced in detail in Appendix D, correspond to the so-called “Extensive
Mutual Information” (EMI) model [39] and a modified version of it, which we
call “Modified EMI” (MEMI) model. In the case of disk regions, we use the toy
models to produce very precise analytic approximations to the free scalar and
free fermion mutual informations for general separations. Based on the
results, we conjecture that the free scalar mutual information is greater than
the fermion one for arbitrary regions (both conveniently normalized by their
respective $F_{0}$).

Figure 12: We plot, in a $\log$-$\log$ scale, the mutual information
(normalized by the disk EE coefficient $F_{0}$) for pairs of identical regions
for free scalars (blue), free fermions (red) and the analytic toy models EMI
and MEMI⋆ (gray and purple, respectively) as a function of the dimensionless
separation $x\equiv r/R$, ($R\equiv\xi L$). The upper plots correspond to
square (left) and disk (right) regions, respectively, whereas the middle and
lower plots correspond to rectangle regions of sides $L\times(2,3,4,5)L$. The
thick blue and red lines represent the analytic approximations explained in
the main text. The orange dotted lines correspond to the leading long-distance
approximations obtained for the free fermion in [24].

We present our results for the mutual information in a $\log$-$\log$ scale.
This allows for a better appreciation of the short- and long-distance
behaviours. On the one hand, observe that for rectangles of sides $L\times\xi
L$ separated a distance $r\equiv x\cdot\xi L$ along the $Y$ axis we have, for
general CFTs,

$I_{2}(x)/F_{0}\overset{x\rightarrow 0}{=}\frac{k/F_{0}}{x}+\dots$ (140)

where $k$ is the universal coefficient characterizing the EE of a thin strip
region. Hence, in a doubly logarithmic scale, for any CFT we should have a
linear behaviour with unit slope for short separations, since
$\log(I_{2}/F_{0})=\log(k/F_{0})-\log x+\dots$ in that case. Such straight
lines should cut the $Y$ axis at a value $\log(k/F_{0})$. The values of such
constants are known for free scalars and fermions. In particular, we have [56]

$k^{\rm s}\simeq 0.0397\,,\quad k^{\rm f}\simeq 0.0722\,.$ (141)

For two disk regions, the limit $r\rightarrow 0$ corresponds to the situation
in which the disks touch at a point.141414See e.g., [57] for examples of both
finite and divergent mutual informations for pairs of regions with a single
contact point. In that case, the results for the analytic toy models we
consider in Appendix D suggest that the mutual information diverges, for
general CFTs, as

$I_{2}(x)/F_{0}\overset{x\rightarrow
0}{=}\frac{\gamma/F_{0}}{\sqrt{x}}+\mathcal{O}(x^{0})\,,$ (142)

where $\gamma$ is a new theory-dependent coefficient. On the other hand, for
long separations and general regions, we know that free scalars and fermions
behave, respectively, as

$\left.I_{2}(x)/F_{0}\right|_{{\rm free\,scalar}}\overset{x\rightarrow
0}{=}\alpha_{2}^{\rm
s}\cdot(1/x)^{2}+\mathcal{O}(x^{-3})\,,\quad\left.I_{2}(x)/F_{0}\right|_{\rm
free\,fermion}\overset{x\rightarrow 0}{=}\alpha_{2}^{\rm
f}\cdot(1/x)^{4}+\mathcal{O}(x^{-5})\,,$ (143)

and so, for long separations we expect the log-log curves to approach straight
lines of slope $-2$ and $-4$, respectively.

We present the results in Fig. 12. The different plots correspond to the
mutual information of pairs of identical regions separated along the $X$ axis
corresponding to: squares, disks and increasingly thinner rectangles with
lengths: $L\times 2L$, $L\times 3L$, $L\times 4L$ and $L\times 5L$,
respectively. In all cases, we have included solid lines representing the
short- and long-distance behaviors appearing in eq. (140) and eq. (143). In
the latter case, the curves represent fits of the form (143) —including three
subleading terms in each case— to the data points corresponding to separations
$x=3,4,5,6,7$. The dotted orange lines correspond to approximations of the
form $\left.I_{2}(x)/F_{0}\right|_{\rm free\,fermion}\simeq\alpha_{2}^{\rm
f}\cdot(1/x)^{4}$ with the coefficients $\alpha_{2}^{\rm f}$ obtained in [24].
In all cases, the numerical fits tend to the corresponding curves. In the case
of the squares and the rectangles, the data points seem to tend, in all cases,
to the short-distance approximations given by eq. (140), which is a good check
of the numerics. In the same plot we include the exact analytic results
obtained in Appendix D for each set of regions for the EMI and MEMI models
—see eqs. (282), (289), (294) and (D.2). For the MEMI model, the value of
$F_{0}$ turns out to vanish identically, and so we normalize the corresponding
curves in a way such that they tend to the free scalar results for
$x\rightarrow 0$. As we can see, the EMI curves are quite similar to the
fermion ones, and so are the MEMI ones to the free scalar ones. This is
particularly so in the case of the disks. In that case, we can actually
approximate the free scalar and free fermion results by the MEMI and EMI ones
for rather general values of $x$. In particular, normalizing the MEMI and EMI
formulas by the long-distance coefficients of the free scalar and free
fermion, respectively, we find the approximations

$\displaystyle\left.I_{2}(x)/F_{0}\right|^{\rm disks}_{{\rm free\,scalar}}$
$\displaystyle\simeq\frac{1.29802}{(1+x)\sqrt{x(2+x)}}\,,$
$\displaystyle(x\gtrsim 0.17)$ (144)
$\displaystyle\left.I_{2}(x)/F_{0}\right|^{\rm disks}_{{\rm free\,fermion}}$
$\displaystyle\simeq\frac{2.43574\left[\sqrt{x(2+x)}-2x(2+x)(1+x-\sqrt{x(2+x)})\right]}{x(1+x)(2+x)}\,,$
$\displaystyle(x\gtrsim 0.2)$ (145)

These appear plotted as the thick blue and red lines respectively in the disks
plot in Fig. 12 and, as we can see, they approximate the data points very well
for arbitrary values of $x$ greater than (roughly) $0.2$. This is no
coincidence. Indeed, as shown in [24], in the case of spherical entangling
surfaces in arbitrary dimensions, the EMI model result is exactly equal to the
contribution to the free-fermion mutual information produced by the conformal
block corresponding to the leading replica primary operator —namely, a
conserved current $\bar{\psi}_{i}\gamma_{\mu}\psi_{j}$, where $i,j$ are
replica indices. The EMI model does not contain any additional terms which do
contribute to the free fermion mutual information, but such leading primary
block contribution is large enough to reproduce most of the full free-fermion
result for most values of $x$. As a matter of fact, something analogous
happens with the free scalar and the MEMI model, namely, the conformal block
making the largest contribution to the free-scalar mutual information in the
case of spheres corresponds exactly to the MEMI result in general dimensions
—in this case, the leading conformal block corresponds to a replica operator
of the form $\phi_{i}\phi_{j}$. Again, such contribution, given by eq. (144)
in the three-dimensional case, approximates the full free-scalar result very
precisely for sufficiently large values of $x$.

Near the touching points, we find the following fits for the scalar and the
fermion, respectively,

$\displaystyle\left.I_{2}(x)/F_{0}\right|_{{\rm free\,scalar}}^{\rm disks}$
$\displaystyle\overset{x\rightarrow
0}{\simeq}+1.3125/\sqrt{x}-1.3522+0.41962\sqrt{x}\,,$ (146)
$\displaystyle\left.I_{2}(x)/F_{0}\right|_{{\rm free\,fermion}}^{\rm disks}$
$\displaystyle\overset{x\rightarrow
0}{\simeq}+0.95599/\sqrt{x}-1.7427+0.83507\sqrt{x}\,.$ (147)

These appear represented as solid lines in the plot. Computing lattice data
points near $x\rightarrow 0$ is increasingly challenging and so the values of
$\gamma/F_{0}|_{\rm free\,scalar}$ and $\gamma/F_{0}|_{\rm free\,fermion}$
reported here should be taken with a grain of salt. They should be improved by
additional calculations near the touching region.

We observe that the curves for both models are monotonically decreasing as a
function of the separation for all sets of regions and that the scalar results
lie above the fermion ones for all regions and separations, a trend which
persists as we make the entangling regions thinner. It seems then reasonable
to conjecture that

$\left.I_{2}(A,B)/F_{0}\right|_{{\rm
free\,scalar}}>\left.I_{2}(A,B)/F_{0}\right|_{{\rm free\,fermion}}\,,$ (148)

for arbitrary pairs of regions. Similar hierarchies —with the scalar
systematically producing greater results— for EE universal coefficients in
three dimensions have been previously observed e.g., in [13, 77, 58].

##  5 $N$-partite information in holography

In this section, we would like to reinterpret our analytic results of Section
2 from the holographic perspective. According to the proposal presented in
[59], the entanglement entropy of an arbitrary region $A$ in a holographic
theory dual to Einstein gravity can be obtained from the formula

$\displaystyle{}S(A)=\underset{m\sim A}{\rm min}{\frac{{\rm
area}(m(A))}{4G_{N}}}+S_{b}(A^{b})\,,$ (149)

which includes the usual Ryu-Takayanagi term [78] plus the leading quantum
correction. In this expression, $m(A)$ is the minimal area surface homologous
to $A$, $A^{b}$ is the bulk homology region with boundary $\partial
A^{b}=A\cup m(A)$ and $S_{b}(A^{b})$ represents its associated bulk von
Neumann entropy.

For a boundary region made of disconnected subregions, $A=\cup_{i}^{K}A_{i}$,
it is known that, for sufficiently large mutual separations, the corresponding
minimal-area surface becomes disconnected and equal to the union of the
individual extremal surfaces associated to the subregions

$\displaystyle m(\cup_{i=1}^{K}A_{i})=\cup_{i=1}^{K}m(A_{i})\,.$ (150)

Thus, at long distances the minimal-area contribution to the holographic
$N$-partite information cancels out from its definition and should be given
entirely by the $N$-partite information of the bulk dual theory, namely,

$\displaystyle I_{N}(\\{A_{i}\\}_{N})=I^{b}_{N}(\\{A^{b}_{i}\\}_{N})\,.$ (151)

In this section we will prove the above equality for largely separated
boundary spherical regions and its associated bulk hemispheres. We will
achieve this by employing the same techniques developed in Section 2 to the
bulk theory in the AdS background. Our result generalizes the analogous one
obtained in [30] for the case of the holographic mutual information.

Our main observations are the following. First, following the arguments
presented in [30], we conclude that, at large separations, the twist operator
associated to a bulk region can be approximated as

$\displaystyle\tilde{\Sigma}^{(n)}_{A^{b}}\approx\sum_{i<j}C^{b}_{ij}\phi^{i}(r^{b}_{A^{b}})\phi^{j}(r^{b}_{A^{b}})\,,$
(152)

in the multicopy bulk theory, where the bulk OPE coefficients can be computed
via

$\displaystyle
C^{b}_{ij}=\lim_{|x-x_{A}|\to\infty}G(r^{b},r^{b}_{A^{b}})^{-2}\langle\tilde{\Sigma}^{(n)}_{A^{b}}\phi^{i}(r^{b})\phi^{j}(r^{b})\rangle\,,$
(153)

where $r^{b}$ are bulk Poincaré AdS coordinates so that $r^{b}=(x,z)$, $x$
being the flat space boundary coordinates. $G(r,r_{A^{b}})$ is the two-point
function of the bulk field $\phi(r^{b})$ dual to the operator ${\cal O}(x)$.
In particular, for large separations, namely, for $|x-x_{A}|\gg R_{A}$, we
have

$\displaystyle
G(r^{b},r^{b}_{A^{b}})\approx\alpha^{2}_{\Delta}\frac{z^{\Delta}z_{A^{b}}^{\Delta}}{|x-x_{A}|^{2\Delta}}\,,$
(154)

where the coefficient $\alpha_{\Delta}$ comes from a normalization relation
between bulk and boundary two-point functions [79]. Indeed, we could arrive at
the above formula from the bulk two-point function and the extrapolate
dictionary [80], this is

$\displaystyle\phi(x,z)\approx\alpha_{\Delta}z^{\Delta}{\cal O}(x),$ (155)

which is the leading term in the near boundary expansion of the dual bulk
operator.

The above statement generalizes to arbitrary $N$-point correlation functions.
Namely, the bulk $N$-point correlation function can be obtained using (155)
for each bulk operator

$\displaystyle\langle\phi(x_{1},z_{1})\cdots\phi(x_{N},z_{N})\rangle\approx\alpha^{N}_{\Delta}z_{1}^{\Delta}\cdots
z_{N}^{\Delta}\langle{\cal O}(x_{1})\cdots{\cal O}(x_{N})\rangle\,,$ (156)

provided all distances $|x_{i}-x_{j}|\gg|z_{k}|$ for all $\\{i,j,k\\}$. This
is precisely the regime we are considering when computing correlation
functions of bulk twist operators. The relation (156) holds trivially for bulk
replica operators as well with the obvious generalization.

In our derivation of the operator formula for $I_{N}(\\{A_{i}\\}_{N})$ in
terms of twist operators we did not make any assumption about the theory being
conformal or being defined on flat space. Hence, the same formula should apply
to $I^{b}_{N}(\\{A^{b}_{i}\\}_{N})$, this is

$\displaystyle I^{b}_{N}(\\{A^{b}_{i}\\}_{N})=\lim_{n\to
1}\frac{\left(-\right)^{N+1}}{1-n}\langle\tilde{\Sigma}_{A^{b}_{1}}^{(n)}\tilde{\Sigma}_{A^{b}_{2}}^{(n)}\cdots\tilde{\Sigma}_{A^{b}_{N}}^{(n)}\rangle\,.$
(157)

The same happens with our derivation of the twist OPE coefficient $C_{ij}$ of
Section 2.2.1, more precisely, Eq. (43). Thus, for the bulk twist operator we
can similarly write

$\displaystyle\langle\Omega|\Sigma^{(n)}_{A^{b}}{\phi}^{l}(r^{b}){\phi}^{k}(r^{b})|\Omega\rangle\approx\Big{\langle}{\phi}_{A^{b}}\left[r^{b},i\frac{\tau_{kl}}{\pi}\right]{\phi}(r^{b})\Big{\rangle}\,,$
(158)

where ${\phi}_{A^{b}}\left[r^{b},s\right]$ is the bulk field operator evolved
by the bulk modular flow associated to region $A^{b}$, this is

$\displaystyle{\phi}_{A^{b}}\left[r^{b},s\right]=\rho^{is}_{A^{b}}\phi(r^{b})\rho^{-is}_{A_{b}}\,,\quad{\rm
and}\quad\tau_{kl}=\frac{\pi(k-l)}{n}\,.$ (159)

It turns out that the bulk modular flow for the region dual to a spherical
region is also explicitly known. The CFT modular flow for a spherical region
is generated by the conformal Killing vector which preserves the associated
causal cone, and it is given by (44). Such Killing vector has a natural
extension into the bulk where it becomes the AdS Killing vector field which
preserves the associated dual bulk hemispherical causal cone. This flow is
given by [9]

$\displaystyle r_{b}^{\pm}[s]=R\frac{\left(R+r_{b}^{\pm}\right)-e^{\mp 2\pi
s}\left(R-r^{\pm}\right)}{\left(R+r^{\pm}\right)+e^{\mp 2\pi
s}\left(R-r_{b}^{\pm}\right)}\,,$ (160)

where $r_{b}^{\pm}=r^{b}\pm t$ are standard null coordinates. Notice that this
flow reduces to the boundary flow for $z=0$. The map generated by the bulk
modular flow acts as a diffeomorphism of the AdS metric and thus the action on
the bulk field is trivial, this is

$\displaystyle{\phi}_{A^{b}}\left[r^{b},i\frac{\tau_{kl}}{\pi}\right]=\phi\left(r^{b}\left[i\frac{\tau_{kl}}{\pi}\right]\right)\,,$
(161)

since $\phi(r^{b})$ transforms as a bulk scalar field. Using the above
relation into (158), together with the extrapolate dictionary (155) and the
explicit modular flow (160), we find

$\displaystyle\langle\Omega|\Sigma^{(n)}_{A^{b}}{\phi}^{l}(r^{b}){\phi}^{k}(r^{b})|\Omega\rangle\approx\alpha_{\Delta}^{2}\frac{\Omega^{\Delta}(x,i\frac{\tau_{kl}}{\pi})z^{2\Delta}}{|x\left[i\frac{\tau_{kl}}{\pi}\right]-x|^{2\Delta}}\,,$
(162)

for large $|x-x_{A}|$. Plugging this formula back into (153) and using (154)
we arrive at our final formula for the bulk twist OPE coefficients

$\displaystyle
C^{b}_{lk}=\frac{1}{\alpha_{\Delta}^{2}z_{A^{b}}^{2}}\lim_{|x-x_{A}|\to\infty}\frac{\Omega^{\Delta}(x,i\frac{\tau_{kl}}{\pi})}{|x\left[i\frac{\tau_{kl}}{\pi}\right]-x|^{2\Delta}}=\frac{C_{lk}}{\alpha_{\Delta}^{2}z_{A^{b}}^{2}}\,.$
(163)

This is the main result of this section. It gives a simple relation between
the OPE coefficients of the twist bulk operator and the ones of the boundary
theory. Together with the extrapolate dictionary (155) it allows us to
conclude the equality between boundary and bulk twist operators

$\displaystyle\tilde{\Sigma}^{(n)}_{A^{b}}\approx\tilde{\Sigma}^{(n)}_{A}\,,$
(164)

which holds inside correlation functions with operator insertions far away
from the locations of the twist operator. Therefore, at large separations we
conclude that

$\displaystyle I^{b}_{N}(\\{A^{b}_{i}\\}_{N})=I_{N}(\\{A_{i}\\}_{N})\,.$ (165)

##  6 Final comments

In this paper we have presented several new results involving the $N$-partite
information of spacetime regions in conformal field theories. Our main results
appear summarized in the introduction. Here, we close with a few comments
regarding possible extensions.

As we have seen, our final formula for $I_{4}$ includes a coefficient which we
left expressed in terms of a triple integral which we were not able to
evaluate explicitly for general values of $\Delta$ —see eq. (214). It would
certainly be interesting to see it done.

On the other hand, it should be possible (although increasingly challenging)
to obtain explicit formulas for the long-distance leading term of the $I_{N}$
for $N\geq 5$ in the case of spherical entangling regions. In each case, the
formulas should involve a linear combination of products of $2$-,$3$-,$\dots$,
$N$-point functions of the leading primary. In relation with the CFT
reconstruction program mentioned above, obtaining information about subleading
primaries requires going beyond leading order in the long-distance expansion.
As a concrete question in this direction, it would be interesting to determine
precisely how much information about a CFT one can reconstruct from the