††institutetext: Department of Theoretical Physics
Tata Institute for Fundamental Research, Mumbai 400005

# Towards a classification of holographic multi-partite entanglement measures

Abhijit Gadde, Vineeth Krishna, Trakshu Sharma<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we systematically study the measures of multi-partite
entanglement with the aim of constructing those measures that can be computed
in probe approximation in the holographic dual. We classify and count general
measures as invariants of local unitary transformations. After formulating
these measures in terms of permutation group elements, we derive conditions
that a probe measure should satisfy and find a large class of solutions. These
solutions are generalizations of the multi-entropy introduced in Gadde:2022cqi
. We derive their holographic dual with the assumption that the replica
symmetry is unbroken in the bulk and check our prescription with explicit
computations in $2d$ CFTs. Analogous to the multi-entropy, the holographic
dual of these measures is given by the weighted area of the minimal brane-web
but with branes having differing tensions. We discuss the replica symmetry
assumption and also how the already known entanglement measures, such as
entanglement negativity and reflected entropy fit in our framework.

##  1 Introduction

One of the central problems of modern theoretical physics is understanding the
quantum nature of gravity. This problem manifests itself most prominently as
the black hole information paradox Hawking:1975vcx ; Hawking:1976ra . One tool
to tackle this problem is the AdS/CFT correspondence Maldacena:1997re ;
Witten:1998qj ; Gubser:1998bc . The correspondence provides us a definition of
the quantum theory of gravity in Anti de-Sitter (AdS) space as a conformal
field theory (CFT) living on the boundary. For the dual gravitational theory
to be local, the CFT must have certain properties such as a large central
charge and a large gap in the spectrum of higher spin operators
Heemskerk:2009pn . Coupled with quantum information theoretic ideas such as
quantum entanglement and quantum error correction, the AdS/CFT correspondence
has been effective in advancing our understanding of the black hole
information paradox Almheiri:2019hni ; Penington:2019kki . Quantum information
theory ideas have been useful in enriching the correspondence itself via the
so-called subregion duality Dong:2016eik ; Harlow:2016vwg ; Almheiri:2019hni .

The ingredient that has been crucial to making progress in this direction is
the Ryu-Takayanagi (RT) formula Ryu:2006bv ; Hubeny:2007xt  and its quantum
generalization Engelhardt:2014gca . The RT formula expresses the entanglement
entropy of a state in Hilbert spaces associated with two regions of space in
terms of a simple geometrical quantity in the dual gravitational theory viz.
the area of the minimal surface anchored on the boundaries of the regions in
question. Its quantum generalization - the so-called quantum extremal surface
formula Engelhardt:2014gca  \- takes into account the contribution of quantum
fluctuations of the bulk theory.

One thing to note is that the progress in understanding the black hole
information paradox and the local nature of AdS/CFT correspondence discussed
above has stemmed almost exclusively from the ideas related to _bi_ -partite
entanglement i.e. to entanglement between _two_ parties. It is natural to ask
how understanding the _multi_ -partite entanglement will further this
progress. With this motivation, in Gadde:2022cqi , we introduced a measure of
multi-partite entanglement with a “probe” property (described shortly) that is
desirable from the holographic point of view. In this paper, we continue to
pursue the same line and make progress toward the classification of such probe
multi-partite entanglement measures. In addition, we address several points
that were left for future work in Gadde:2022cqi . The somewhat mysterious RT
formula for entanglement was explained by Lewkowycz and Maldacena
Lewkowycz:2013nqa  using the replica trick and a certain clever analytic
continuation of the bulk geometry. Their work guided us in defining and
evaluating the multi-partite measure called multi-entropy in Gadde:2022cqi .
It continues to do so even in this paper.

The entanglement in a bi-partite pure state can be measured111In this paper,
we use the word “measure” rather liberally without requiring attributes such
as positivity and monotonicity under local quantum operations and classical
communications etc. that would be demanded by a quantum information theorist.
by Renyi entropies $S_{n}(\rho)$, defined as

$\displaystyle S_{n}(\rho)=\frac{1}{1-n}{\log}\Big{(}{\rm
Tr}\,\rho^{n}\Big{)}$ (1)

where $\rho$ is the density matrix obtained by tracing one of the two parties.
The Renyi entropy is symmetric in both parties even though the definition of
Renyi entropy obscures this symmetry. The von-Neumann entropy, also called the
entanglement entropy $S$ is defined as $-{\rm Tr}\rho\log\rho$. In many cases,
it turns out to be convenient to think of $S$ as the $n\to 1$ limit of
$S_{n}$, analytically continued in $n$. This way of computing the entanglement
entropy is known as the replica trick. Once ${\rm Tr}\rho^{n}$ is analytically
continued in $n$, then $S$ can also be written as,

$\displaystyle S=-\partial_{n}\,{\log}\Big{(}{\rm
Tr}\,\rho^{n}\Big{)}|_{n=1}=-\partial_{n}\,\Big{(}{\rm
Tr}\,\rho^{n}\Big{)}|_{n=1}.$ (2)

In the last equality, we use the fact that the state is normalized i.e. ${\rm
Tr}\,\rho=1$. From a holographic point of view, it is easier to characterize
the entanglement entropy compared to the Renyi entropies. This is because the
entanglement entropy is given simply by the area of a certain minimal area
surface in the dual geometry viz. the RT formula. The Renyi entropies, or
rather a certain combination of their derivatives with respect $n$
Dong:2016fnf , can also be computed holographically but unlike entanglement
entropy, they are not probes of the original dual geometry. Rather their
evaluation requires solving the gravitational equation of motion with the same
infra-red boundary conditions but with a conical singularity along a certain
co-dimension 2 locus. We call the quantities that are computed _in the dual
geometry_ as opposed to those that are computed in a new solution, the probe
quantities. The probe measure of entanglement is the one that is computed by a
probe quantity. With this definition of the probe measure, we can say that the
only probe measure in the family of Renyi entropies is the entanglement
entropy. In this work, we first identify the multi-partite entanglement
measures that admit a holographic dual with the same infra-red boundary
conditions. These are organized into families where the members are indexed by
an integer $n$. We then use a similar sort of $n\to 1$ limit within each
family to construct the measures that are probes of the original geometry.

###  1.1 Overview of the holographic computation

In this section, we will give a quick overview of the probe multi-partite
measure, named multi-entropy, that was introduced in Gadde:2022cqi . Its
construction will form the basis of the present work. During this overview, we
will also need to refer to Lewkowycz:2013nqa , so we will review its relevant
aspects as well.

####  1.1.1 Bi-partite

Let us start with the bi-partite entanglement, say between party
${\mathtt{a}}$ and party ${\mathtt{b}}$. As stated earlier, the pure state of
two parties is represented by a density matrix
$(\rho_{\mathtt{a}})_{\alpha}^{\beta}$, obtained after tracing out, say party
${\mathtt{b}}$. So $\rho_{\mathtt{a}}$ is an operator acting on ${\cal
H}_{\mathtt{a}}$. We denote it graphically as in figure 1. With this
presentation of $\rho_{\mathtt{a}}$, the ${\cal E}_{n}\equiv{\rm
Tr}\rho_{\mathtt{a}}^{n}$ that enters the definition of Renyi entropy is
denoted as a necklace with $n$ beads as shown in figure 1.

Figure 1: Graphical notation for the density matrix $\rho_{\mathtt{a}}$ and
${\cal E}_{n}\equiv{\rm Tr}\rho_{\mathtt{a}}^{n}$. Here we have defined the
case of $n=4$. The thread with $\rho$’s is periodically identified to make a
necklace.

This necklace ${\cal E}_{n}$ has ${\mathbb{Z}}_{n}$ rotational symmetry. We
call the symmetry of the graph of $\rho$, the replica symmetry.

The path integral that computes ${\cal E}_{n}$ in quantum field theory ${\cal
Q}$ is constructed by cutting and pasting procedure as shown in figure 2.

Figure 2: The path integral that computes ${\cal E}_{n}$ in two-dimensional
quantum field theory

For simplicity of drawing, we have taken the example of a $2d$ theory. Even
though, in figure 2 we have taken the party ${\mathtt{a}}$ to consist of a
single interval, it could also comprise of a union of intervals and similarly
for party ${\mathtt{b}}$. The state $\psi$ is obtained by a path integral on
the southern hemisphere and its conjugate is obtained by a path integral on
the northern hemisphere. These two hemispheres are glued along ${\mathtt{b}}$
which corresponds to tracing out the party ${\mathtt{b}}$ the resulting space
is a sphere with slits along the ${\mathtt{a}}$ region, which corresponds to
the density matrix $\rho_{\mathtt{a}}$. The upper lip of the slit corresponds
to the ket index and the lower lip corresponds to the bra index of the density
matrix. To construct ${\rm Tr}\rho_{\mathtt{a}}^{n}$, we take $n$ copies of
$\rho_{a}$ in a row and glue the upper lip of the $i$-th copy to the lower lip
of the $i+1$ copy with periodic identification. The resulting space is a
sphere _branched_ or _ramified_ $n$ times at the boundary points of all
intervals. The ${\mathbb{Z}}_{n}$ replica symmetry of ${\rm
Tr}\rho_{\mathtt{a}}^{n}$ shows up as the symmetry of this space.
Alternatively, the space obtained by this type of gluing can be thought of as
being obtained with the insertion of appropriate twist operators in the direct
product of $n$ quantum field theories ${\cal Q}^{\otimes n}$. A general twist
operator in ${\cal Q}^{\otimes n}$ is labeled by an element $\sigma$ of the
permutation group $S_{n}$. If the region on the left (right) of the operator
in the replica set of kets is glued with permutation $\sigma_{l}$
($\sigma_{r}$) to the corresponding region in the replica set of bras then the
twist operator type is $\sigma_{l}^{-1}\sigma_{r}$.

The quantity ${\rm Tr}\rho_{\mathtt{a}}^{n}$ in CFTs with gravity duals can be
computed holographically by computing the action of the dominant bulk solution
that fills in the branched cover of the boundary. In Lewkowycz:2013nqa  this
computation was performed with the assumption that the dominant bulk solution
preserves the replica symmetry. Orbifolding by the replica symmetry, the
authors obtained a geometry with a co-dimension $2$ conical singularity of
conical angle $2\pi/n$. As a cone, the geometry admits analytic continuation
away from integer $n$ by considering general conical angles not restricted to
those of the form $2\pi/n$ with integer $n$. For von-Neumann entropy, we are
interested in computing the variation of the gravitational action at $n=1$. In
this limit, the conical singularity flattens out and the variation of the
gravity action is given by the area of the minimal surface (a geodesic, in the
case of $2d$ CFT). See Lewkowycz:2013nqa  for more details of this argument.

####  1.1.2 Multi-partite

Here we will only review Gadde:2022cqi  in the context of the tri-partite
multi-entropy. For a more detailed discussion, we refer the reader to
Gadde:2022cqi  and to section 2. For a tri-partite pure state on parties
${\mathtt{a}},{\mathtt{b}}$ and ${\mathtt{c}}$, Gadde:2022cqi  first
constructs the two party density matrix $\rho_{{\mathtt{a}}{\mathtt{b}}}$ by
tracing out party ${\mathtt{c}}$. This two-party density matrix $\rho$ with
the graphical notation is presented in figure 9. The measure ${\cal
E}_{n}^{({\mathtt{q}})}$ is now defined to be the value of the $n\times n$
square lattice shown in figure 9 obtained after contracting indices of
$\rho_{{\mathtt{a}}{\mathtt{b}}}$ as shown ($n$ is taken to be $3$ in the
figure and parties are taken to be ${1},{\mathtt{2}}$ and ${\mathtt{3}}$ with
party ${1}$ being traced out.). In this case, the replica symmetry is
${\mathbb{Z}}_{n}\otimes{\mathbb{Z}}_{n}$ where one ${\mathbb{Z}}_{n}$
cyclically permutes the rows of the lattice and the other cyclically permutes
the columns. As in the bi-partite case, the calculation of ${\cal
E}_{n}^{({\mathtt{q}})}$ is mapped to the calculation of the partition
function on the appropriately branched sphere or equivalently as the
correlation function of appropriate twist operators.

A method similar to that of Lewkowycz:2013nqa  was employed to compute this
partition function holographically as the gravitation action of the dominant
bulk solution that fills in the branched cover of the boundary sphere.
Assuming that the dominant bulk solution preserves replica symmetry, its
orbifold with respect to ${\mathbb{Z}}_{n}\otimes{\mathbb{Z}}_{n}$ was
performed. The replica symmetry assumption is certainly true222This was shown
in three-party and three-interval case in Penington:2022dhr . The case of four
parties and four intervals is discussed in section 6.3. The case of arbitrary
number intervals and parties is treated in GaddeWIP . for $n=2$ and the
resulting geometry admits analytic continuation for real values of $n\leq 2$.
This geometry has three co-dimension $2$ conical singularities anchored at the
three endpoints of the intervals on the boundary that meet up in the bulk at a
co-dimension $3$ locus. For $2d$ CFT, this junction is a point in the $3d$
bulk. As in the bi-partite case, in the $n\to 1$ limit, the conical
singularity flattens out and the derivative of the gravitational action is
given by the area of the minimum area “soap-film” that is anchored on the edge
of the regions on the boundary. For $2d$ CFTs, this soap film is a trivalent
tree of geodesics of minimum total length. Please see Gadde:2022cqi  for the
details of this argument. The holographic computation for a more general class
of multi-partite entanglement measures is detailed in section 4.

###  1.2 Structure of the paper

The paper is structured as follows. In section 2, we consider general
entanglement measures as invariants of local unitary transformations. We count
them in the limit of large Hilbert space dimension and formulate them in terms
of permutation group elements. In section 3, we specialize to the entanglement
measures in holographic conformal field theories and determine the conditions
that a probe measure should satisfy and find a large class of solutions to
these constraints. These probe conditions are based on the analysis of
holographic entanglement entropy by Lewkowycz and Maldacena Lewkowycz:2013nqa
. In section 4, following Gadde:2022cqi , we derive the probe holographic dual
of this class of measures with the assumption that the replica symmetry is
unbroken in the bulk. We also conjecture the quantum extension of the
prescription. We discuss the marriage of our multi-partite measures with
holographic purification in section 5. In section 6, we check our proposal for
two-dimensional conformal field theories with large central charge and discuss
the issue of bulk replica symmetry. We discuss a potential payoff from multi-
partite measures as a more refined bulk reconstruction using a form of multi-
partite error correction in section 7. In appendix A, we give details of the
counting of multi-partite measures for large dimensional Hilbert spaces. In
appendix B, we discuss the analytic continuation involved in the definition of
the probe measure. In particular, we present a numerical approach for the
analytic continuation and discuss its validity. This appendix is based on the
papers regge-viano ; osti_4065624 . Appendix C, gives details of four-point
and five-point conformal block computation using the so-called “monodromy
method” in $2d$ CFTs with large central charge.

##  2 Classification of multi-partite entanglement measures

In quantum information literature, entanglement measures are usually defined
for mixed states. They obey several physically motivated properties such as
positivity on the “inseparable” states and monotonicity under local operations
and classical communications (LOCC). See Horodecki:2009zz  for a detailed
review and relevant references. In this paper, we will be interested in the
entanglement measures for pure states and we will define them in what is
perhaps the most general fashion. Consider a normalized quantum state
$|\Psi\rangle\in\otimes_{{\mathtt{a}}=1}^{\mathtt{q}}{\cal H}_{\mathtt{a}}$ of
a ${\mathtt{q}}$-party system. Let $d_{\mathtt{a}}$ be the dimension of the
Hilbert space ${\mathcal{H}}_{\mathtt{a}}$ and
$|\alpha_{\mathtt{a}}\rangle,\,\alpha_{\mathtt{a}}=1,\ldots,d_{\mathtt{a}}$ be
its arbitrarily chosen orthonormal basis. In $|\alpha_{\mathtt{a}}\rangle$
basis, the state $|\Psi\rangle$ is given as

$\displaystyle|\Psi\rangle=\sum_{\alpha_{1}=1}^{d_{1}}\ldots\sum_{\alpha_{\mathtt{q}}=1}^{d_{\mathtt{q}}}\,\,\psi_{\alpha_{1}\ldots\alpha_{\mathtt{q}}}\,\,|\alpha_{1}\rangle\otimes\ldots\otimes|\alpha_{\mathtt{q}}\rangle.$
(3)

We call $\psi_{\alpha_{1}\ldots\alpha_{\mathtt{q}}}$, the wavefunction.

  * •

A q-party entanglement measure $\cal E$ is a function of $\psi$ (and its
complex conjugate) that is invariant under “local unitary transformations”.

These transformations take the form
$\otimes_{{\mathtt{a}}=1}^{\mathtt{q}}{\cal U}_{\mathtt{a}}\in
U(d_{\mathtt{a}})$ where ${\cal U}_{\mathtt{a}}$ is the unitary transformation
that acts on ${{\mathcal{H}}_{\mathtt{a}}}$. If we think of the local unitary
group $\prod_{\mathtt{a}}U(d_{\mathtt{a}})$ as the gauge symmetry and $\psi$’s
and $\bar{\psi}$’s as ${\mathtt{q}}$-fundamental and ${\mathtt{q}}$-anti-
fundamental “operators” then the measures are simply the gauge invariant
operators. We construct the local unitary invariants by taking multiple copies
of $\psi$’s and $\bar{\psi}$’s and contracting the fundamental indices
$\alpha_{\mathtt{a}}$ of $\psi$ with anti-fundamental indices333There may be
invariants of the unitary group that are not of this type. For example, one
can consider $|\Psi\rangle\in{\cal H}_{1}\otimes{\cal H}_{2}$ such that ${\rm
dim}\,{\cal H}_{i}=d$. In this case, a local unitary invariant can be
constructed by taking $d$ copies of only $\psi$ and contracting its party
${1}$ as well as party ${\mathtt{2}}$ indices with the two separate $d$-index
$\epsilon$ tensors. Because we are interested in Hilbert spaces of arbitrary
dimensions, in particular of infinite dimension, we do not consider such
baryon-type unitary invariants. There are even more exotic local unitary
invariants such as the so-called tensor eigenvalues lim2006singular ;
10.5555/3240740 . We do not know whether it is possible to express those in
terms of simple products of $\psi$’s and $\bar{\psi}$’s considered here.
Restricting only to the class of invariants considered here is sufficient for
the purposes of this paper. $\alpha_{\mathtt{a}}$ of $\bar{\psi}$. As a
result, the number of $\psi$’s and the number of $\bar{\psi}$’s is the same in
any entanglement measure. We call this number the replica number $n$. We index
the replicas by the superscript $(i)$. The Hilbert space
${\mathcal{H}}_{\mathtt{a}}$ in the $i$-th replica is denoted as
${\mathcal{H}}_{\mathtt{a}}^{(i)}$ and its basis as
$|\alpha_{\mathtt{a}}^{(i)}\rangle$. The wavefunction of the $i$-th replica is
then $\psi_{\alpha_{1}^{(i)}\ldots\alpha_{\mathtt{q}}^{(i)}}$ and its
conjugate is $\bar{\psi}^{\alpha_{1}^{(i)}\ldots\alpha_{\mathtt{q}}^{(i)}}$.
The analogy with gauge invariance can be pushed further and be made into an
equivalence. Consider a $0$-dimensional gauge theory with gauge group
$\prod_{\mathtt{a}}U(d_{\mathtt{a}})$ coupled with a complex bosonic field
$\psi$ that transforms in the fundamental representation with respect to each
of the $U(d_{\mathtt{a}})$ factors at weak coupling. The set of local unitary
invariants that we are after is precisely the set of gauge invariant operators
of this quantum mechanical theory. The Hilbert space of the theory is spanned
by these gauge invariant operators.

To classify such measures, it is convenient to introduce a graphical notation
in which the wavefunction $\psi$ is denoted as a ${\mathtt{q}}$-valent black
vertex as in figure 3.

Figure 3: Graphical notation for the wavefunction $\psi$ and its conjugate
$\bar{\psi}$.

The vertex has outgoing colored edges as they stand for fundamental indices
$\alpha_{\mathtt{a}}$ of distinct Hilbert spaces ${\mathcal{H}}_{\mathtt{a}}$.
Similarly $\bar{\psi}$ is denoted as a ${\mathtt{q}}$-valent white vertex with
incoming colored edges. The index contraction in ${\mathcal{H}}_{\mathtt{a}}$
is denoted as joining a white vertex with a black vertex with the
appropriately colored edge. An entanglement measure $\cal E$ is then a bi-
partite graph made out of these vertices with no dangling edges. Two examples
of such measures are given in figure 4.

Figure 4: Examples of multi-partite measures.

The first of the figures is simply the squared norm of the state and the
second is a non-trivial measure.

The set of all entanglement measures forms a commutative ring. Recall that a
ring is a set with two binary operations, addition and multiplication with
multiplication being distributive over addition. The addition is commutative
and has an inverse. Multiplication need not have an inverse but is
associative. In case multiplication is also commutative, the ring is known as
a commutative ring. It is convenient to characterize the ring in terms of its
generators. The generating set of a ring is a set of elements that generates
the whole ring through inverse and the two binary operations. The simplest
example of a commutative ring is the ring of integers. In this case, the
generating set can be taken to be the set consisting of a single element $1$.
The generating set of the ring of polynomials $P(x)$ with integer coefficients
can be taken to be the monomial $x$. When there are no relations between
generators then the ring is said to be generated freely. Both the rings
mentioned above are freely generated.

The set of invariants described above consists of connected as well as
disconnected graphs. The disconnected graphs are obtained by multiplying
connected graphs together so it is useful to focus our attention only on the
connected graphs. In the quantum mechanical gauge theory, the connected and
disconnected graphs correspond to “single-trace” and “multi-trace” operators
respectively. The single traces generate the entire set of invariants by
multiplication and hence provide a natural generating set for the ring. For
the finite-dimensional case i.e. when all $d_{\mathtt{a}}$ are finite, not all
elements of this generating set are independent. This is because the total
number of components of $\psi$ is finite while the total number of single
traces i.e. connected graphs is infinite. It is interesting to understand the
independent single trace generators and characterize the relations among them
if there are any.

Let us take the example of a bi-partite state. Tracing over the second party
we get the density matrix $\rho$ which transforms in the adjoint
representation with respect to $U(d_{1})$. The connected graphs correspond
actually to single traces ${\rm Tr}\rho^{i}$. Thanks to Caley-Hamilton
theorem444Caley-Hamilton theorem states that for any $n\times n$ matrix $A$,
$p_{A}(A)=0$ where $p_{A}(\lambda)\equiv{\rm det}(\lambda{\mathbb{I}}-A)$ is
the characteristic polynomial of $A$. It expresses $A^{n}$ in terms of lower
matrix powers of $A$. If the rank of the matrix is not maximal then one can
construct a lower-order polynomial by removing the factors of $\lambda$. Then
$A^{r}$ is expressed in terms of lower powers of $A$., we know that not all
such invariants are independent. More precisely, ${\rm Tr}\rho^{i}$ for $i<r$
are all independent where $r$ is the rank of the matrix and the rest can be
expressed as their linear combinations. We will not be able to characterize
the independent generators for a general number of parties but we will be able
to count them, along with the relations between them if there are any.

###  2.1 Counting

The main tool that we will use to count the number of local unitary invariants
is the so-called plethystic program. It first made an appearance in physics in
the context of counting gauge invariant operators Feng:2007ur . Since then it
has been used extensively for computing the superconformal indices of gauge
theories. See Gadde:2020yah  for a pedagogical introduction to the
superconformal index and its computation. The counting problem is defined as
follows. The number operator $n$ of $\psi$’s is an operator acting on the
vector space $V$ of local invariant operators. As remarked earlier, it is the
same as the replica number. The partition function on the measures is defined
as

$\displaystyle\hat{Z}^{({\mathtt{q}})}(x)\equiv{\rm
Tr}_{V}x^{n}=\sum_{n=0}^{\infty}\,P_{n}^{({\mathtt{q}})}\,x^{n}.$ (4)

When expanded in the powers of $x$, the coefficient of $x^{n}$ is the number
of local unitary invariants with replica number $n$. While we are at it, let
us also define the partition function over vector space $V_{C}$ of _connected_
graphs as $Z(x)\equiv{\rm Tr}_{V_{C}}x^{n}$. It is known that these two
partition functions are related to each other via the so-called plethystic
exponential (PE) and plethystic logarithm (PL) respectively.

$\displaystyle\hat{Z}(x)$
$\displaystyle={\mathtt{PE}}[Z(x)]=\exp\Big{(}\sum_{k=1}^{\infty}\frac{1}{k}Z(x^{k})\Big{)}$
(5) $\displaystyle Z(x)$
$\displaystyle={\mathtt{PL}}[\hat{Z}(x)]=\sum_{k=1}^{\infty}\frac{\mu(k)}{k}\log\Big{(}\hat{Z}(x^{k})\Big{)}.$
(6)

Here the function $\mu(x)$ is the Möbius function. It is defined as

$\mu_{k}=\begin{cases}0&\text{if $k$ has repeated prime factors}\\\ 1&\text{if
$k=1$}\\\ (-1)^{n}&\text{if $k$ is a product of $n$ distinct
primes.}\end{cases}$ (7)

The partition function over bosonic multi-particle states is always the PE of
the single-particle states. Although the formula looks daunting, it can be
understood in terms of a simple example. If there is a single particle then
its partition function is $Z(x)=x$. Then ${\mathtt{PE}}[Z(x)]=1/(1-x)$. This
is simply the partition function over multi-particle states of identical
bosonic particles. So the operation of PE can be thought of as “multi-
particling” the single particle partition function. It essentially implements
$x+y+\ldots\to(1-x)^{-1}(1-y)^{-1}\ldots$.

The partition function over disconnected graphs $\hat{Z}(x)$ is computed by a
matrix integral over unitary groups. In the quantum mechanical gauge theory,
we first consider the “single letter” partition function over the wavefunction
“field” $\psi$ and its complex conjugate $\bar{\psi}$. Recall that $\psi$
transforms in the fundamental representation with respect to all the unitary
factors $\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}U(d_{\mathtt{a}})$.

$\displaystyle
z(x,{\mathcal{U}}_{{\mathtt{a}}})=x\,\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}{\rm
Tr}({\mathcal{U}}_{\mathtt{a}})+\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}{\rm
Tr}({\mathcal{U}}_{\mathtt{a}}^{\dagger}).$ (8)

Here $x$ denotes the contribution from a single replica $\psi$ and the factor
$\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}{\rm Tr}({\mathcal{U}}_{\mathtt{a}})$ is
the character of the fundamental representation of the local unitary group
$\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}U(d_{\mathtt{a}})$. To compute
$\hat{Z}^{({\mathtt{q}})}(x)$ we first compute the multi-letter partition
function by taking the PE of $z(x,{\mathcal{U}}_{\mathtt{a}})$ and then
projecting onto gauge invariants by integrating over the unitary group
$\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}U(d_{\mathtt{a}})$ with the Haar
measure.

$\displaystyle\hat{Z}^{({\mathtt{q}})}(x)$
$\displaystyle=\int\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}d{\mathcal{U}}_{\mathtt{a}}\,{\mathtt{PE}}\,\Big{[}z(x,{\mathcal{U}}_{\mathtt{a}})\Big{]}=\int\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}d{\mathcal{U}}_{\mathtt{a}}\,\exp\Big{(}\sum_{k=1}^{\infty}\frac{1}{k}\,z(x^{k},{\mathcal{U}}_{\mathtt{a}}^{k})\Big{)}.$
(9)

This is the partition function over all _disconnected_ graphs of
${\mathtt{q}}$-party entanglement measures. To compute the partition function
$Z^{({\mathtt{q}})}(x)$ over _connected_ graphs we take the PL of
$\hat{Z}^{({\mathtt{q}})}(x)$.

For ${\mathtt{q}}=2$, with $d_{1}=d_{\mathtt{2}}=d$,

$\displaystyle Z^{({\mathtt{2}})}(x)=x+x^{2}+x^{3}+\ldots+x^{d-1}.$ (10)

The term $x^{i}$ is contributed by ${\rm Tr}\rho^{i}$. The series truncates
because the traces of higher powers of $\rho$ are not independent and are
obtained via linear combinations of lower powers.

For higher values of ${\mathtt{q}}$ it is difficult to evaluate
$Z^{({\mathtt{q}})}(x)$ for general values of $d_{\mathtt{a}}$. Even after
fixing $d_{\mathtt{a}}$, we can only compute this function as expansion in $x$
up to a few terms. Below we give $Z^{({\mathtt{q}})}(x)$ for
${\mathtt{q}}=3,4$ for the case of qubits i.e. for $d_{\mathtt{a}}=2$.

$\displaystyle Z^{({\mathtt{3}})}(x)$
$\displaystyle=x+3x^{2}+x^{3}+x^{4}+x^{6}-x^{12}.$ $\displaystyle
Z^{({\mathtt{4}})}(x)$
$\displaystyle=x+7x^{2}+12x^{3}+50x^{4}+111x^{5}+323x^{6}+568x^{7}+{\cal
O}(x^{8}).$ (11)

We have obtained this result using ${\mathtt{mathematica}}$. For
$Z^{({\mathtt{4}})}(x)$, we have obtained the terms up to $x^{60}$ and unlike
$Z^{({\mathtt{2}})}(x)$ and $Z^{({\mathtt{3}})}(x)$, this series does not
terminate. Let us pause to explain the terms with negative coefficients. As
explained earlier, $Z^{({\mathtt{q}})}(x)$ is a partition function over
connected graphs of invariants. When the ring of all invariants is not freely
generated by a subset of connected graphs then the PL counts not only
generators of the ring but also relations among them with a negative sign and
relations of relations (if they exist) with a positive sign and so on. That is
why the partition function $Z^{({\mathtt{q}})}(x)$ can have terms with both
positive and negative signs.

The ring of invariants is freely generated by connected graphs when all the
dimensions $d_{\mathtt{a}}$ are taken to be infinity. This is the case we will
focus on in the rest of the paper because this limit is relevant for counting
entanglement measures for quantum field theories where parties are chosen to
be distinct spatial subregions. In this case, the matrix integral (9) can be
performed using saddle point approximation and a compact expression for
$\hat{Z}^{({\mathtt{q}})}(x)$ can be obtained for any ${\mathtt{q}}$. We have
done this computation in appendix A and the result is produced below.

$\displaystyle\hat{Z}^{({\mathtt{q}})}(x)=\prod_{r=1}^{\infty}\left(\sum_{k=0}^{\infty}\Big{(}x^{r}r^{{\mathtt{q}}-2}\Big{)}^{k}\Big{(}k!\Big{)}^{{\mathtt{q}}-2}\right).$
(12)

As a quick sanity check, if we set ${\mathtt{q}}=1$, we get
$Z^{(\mathtt{1})}(x)\equiv{\mathtt{PL}}[\hat{Z}^{({1})}(x)]=x$. This is as
expected because the norm is the only local unitary invariant for a single-
party system. Also, when we set ${\mathtt{q}}=2$, we get the plethystic
logarithm
$Z^{(\mathtt{2})}(x)\equiv{\mathtt{PL}}[\hat{Z}^{({\mathtt{2}})}(x)]=x/(1-x)$.
This is nothing but the partition function over infinitely many Renyi
entropies, except for the first term $x$ which corresponds to the norm of the
state. For ${\mathtt{q}}={\mathtt{3}}$ we get

$\displaystyle Z^{({\mathtt{3}})}(x)=x+3x^{2}+7x^{3}+26x^{4}+97x^{5}+\ldots.$
(13)

The first $3$ non-trivial invariants that are quadratic in $\psi$ are the ones
shown in figure 5.

Figure 5: Three measures that are quadratic in $\psi$. The notation $3\times$
stands for the three measures obtained by permutation of parties i.e. colors.

The next $7$ invariants that are cubic in $\psi$ are shown in figure 6.

Figure 6: Seven measures that are cubic in $\psi$. The notation $3\times$
stands for the three measures obtained by permutation of parties i.e. colors.

One lesson from the expression (12) is that, unlike in the ${\mathtt{q}}=2$
case, the number of ${\mathtt{q}}\geq 3$ partite measures increases very
rapidly with the replica number. It is then a non-trivial task to look for
holographically probe measures in this vast zoo. Shortly we will come up with
a set of conditions that a measure must obey to have the probe property.

###  2.2 Graphs, permutations and replica symmetry

In this subsection, we will formulate the entanglement measures in terms of
permutations of replicas. A general measure $\cal E$ can be written in terms
of contractions of fundamental indices of $n$-replicas of $\psi$ with anti-
fundamental indices of $n$-replicas of $\bar{\psi}$ as follows

$\displaystyle{\cal E}$
$\displaystyle=\Big{(}\psi_{\alpha_{1}^{(1)}\ldots\alpha_{\mathtt{q}}^{(1)}}\ldots\psi_{\alpha_{1}^{(n)}\ldots\alpha_{\mathtt{q}}^{(n)}}\Big{)}\Big{(}\bar{\psi}^{\beta_{1}^{(1)}\ldots\beta_{\mathtt{q}}^{(1)}}\ldots\bar{\psi}^{\beta_{1}^{(n)}\ldots\beta_{\mathtt{q}}^{(n)}}\Big{)}\delta^{\vec{\alpha}_{1}}_{\sigma_{1}\cdot\vec{\beta}_{1}}\ldots\delta^{\vec{\alpha}_{\mathtt{q}}}_{\sigma_{\mathtt{q}}\cdot\vec{\beta}_{\mathtt{q}}}$
$\displaystyle{\rm
where}\quad\delta^{\vec{\alpha}_{\mathtt{a}}}_{\sigma_{\mathtt{a}}\cdot\vec{\beta}_{\mathtt{a}}}$
$\displaystyle\equiv\delta^{\alpha_{\mathtt{a}}^{(1)}}_{\beta_{\mathtt{a}}^{(\sigma_{\mathtt{a}}\cdot
1)}}\ldots\delta^{\alpha_{\mathtt{a}}^{(n)}}_{\beta_{\mathtt{a}}^{(\sigma_{\mathtt{a}}\cdot
n)}}$ (14)

If we call the pair of $\psi$ and $\bar{\psi}$ as a replica then the measure
${\cal E}$ is labeled by ${\mathtt{q}}$ permutation elements
$\sigma_{\mathtt{a}}$ of the permutation group $S_{n}\equiv G$ acting on the
replica set. The element $\sigma_{\mathtt{a}}$ indicates how the $n$
fundamental indices of party ${\mathtt{a}}$ are contracted with $n$ anti-
fundamental indices. Graphical presentation of a measure with
$n=4,{\mathtt{q}}=3$ is given in figure 7.

Figure 7: Describing a multi-partite measure in terms of permutations. The
permutations associated with red, blue and green parties, in the cycle
notation, are $(1234),(12)(34)$ and $(1)(23)(4)$ respectively

The red-colored edges of all the replicas of $\psi$ are connected to those of
replicas of $\bar{\psi}$ with a permutation element $\sigma_{1}$ that is the
one cycle $(1234)$. Similarly, the permutation elements for party $2$ (blue
colored edges) and $3$ (green colored edges) respectively are $(12)(34)$ and
$(1)(23)(4)$. From this discussion, it is also clear that given a
${\mathtt{q}}$-colored regular bi-partite graph on $n$ white and $n$ black
nodes, one can always read off a set of ${\mathtt{q}}$ elements of the
permutation group $\\{\sigma_{1},\ldots,\sigma_{\mathtt{q}}\\}$ such that the
graph is a presentation of the measure ${\cal
E}_{\sigma_{1}\ldots\sigma_{\mathtt{q}}}$.

Let us denote the subgroup of $G$ generated by $\sigma_{\mathtt{a}}$ as $K$.
In special cases, the measure $\cal E$ enjoys a certain symmetry that we
suggestively call the replica symmetry. This is the symmetry of the bipartite
graph corresponding to $\cal E$. It is generated by the elements of $G$ that
commute with all $\sigma_{\mathtt{a}}$’s. In other words, it is the commutant
of $K$ in $G$, usually denoted as $c_{G}(K)$.

##  3 Towards holographic probe measures

We are interested in computing multi-partite entanglement measures for
holographic CFT states i.e. for CFT states that admit semiclassical dual
geometry. In this section, we will motivate successively stringent conditions
on general multi-partite entanglement measures to obtain a measure that for
holographic states is computed by a probe of the dual geometry. In deriving
these conditions, we will closely follow Lewkowycz and Maldacena’s derivation
Lewkowycz:2013nqa  of the Ryu-Takayanagi formula for entanglement entropy.

###  3.1 Maximally symmetric measures

The CFT path integral that computes any multi-partite measure ${\cal
E}(\sigma_{1},\ldots,\sigma_{\mathtt{q}})$ is formulated straightforwardly by
considering replica number worth of copies of the CFT path integral on ${\cal
M}$ and “cutting and pasting” along the ${\mathtt{q}}$ regions as specified by
the permutation elements $\sigma_{\mathtt{a}}$. The replicated manifold ${\cal
M}_{n}$ constructed thus has special loci at the boundaries of ${\mathtt{q}}$
regions with excess conical angles. Moreover, the background fields enjoy
symmetry under the replica symmetry group defined above.

For the bipartite Renyi-entropy $S^{(2)}_{n}$, Lewkowycz and Maldacena have
computed the CFT path integral using the dual gravitational theory
Lewkowycz:2013nqa . A crucial part of their derivation is the assumption that
the dominant bulk solution ${\cal B}_{n}$ also preserves the replica symmetry.
To derive the Ryu-Takayanagi formula, it is then convenient to orbifold the
bulk solution by the replica symmetry group. After orbifolding, the boundary
goes back to being the “un-replicated” original boundary ${\cal M}$. This is
because the replica symmetry group acts freely and transitively on the replica
set. So, if we wish to characterize the multi-partite entanglement measures
that are computed by dual geometries with the un-replicated original boundary
then the replica symmetry group must satisfy the same condition. We formalize
this condition as:

  * •

Maximal symmetry condition: The replica symmetry group $c_{G}(K)$ contains an
abelian subgroup $H$ that acts freely and transitively on the replica set.

Strictly speaking, the replica symmetry subgroup $H$ need not be abelian but
we consider the abelian case in this paper to make the analysis tractable. It
would be interesting to consider non-abelian generalizations in the future.
The measures whose replica symmetry satisfies the maximal symmetry condition
are called maximally symmetric measures. In the rest of the subsection, we
will characterize all the maximally symmetric measures.

Recall that transitive means that for every $x,y$ of the replica set there
exists $h\in H$ such that $h\cdot x=y$ and free means that no element of $H$
has a fixed point. Transitive abelian subgroups of $G$ have the following
properties. See scott64 , section 10.3:

  * •

They are freely acting. So, strictly speaking, the condition of free action in
the maximal symmetry condition is not necessary.

  * •

They are maximal i.e. they are commutants of themselves. So we have
$c_{G}(H)=H$.

The second property will be important for characterizing all maximally
symmetric measures.

When a group $H$ acts on a set $X$ then, according to Burnside’s lemma,

$\displaystyle|{X}/{H}|=\frac{1}{|H|}\sum_{h\in H}\,|{X}_{h}|.$ (15)

Here the notation $|X|$ stands for the cardinality of the set $X$ and
${X}_{h}$ is the set of points in ${X}$ left fixed by the action of $h\in{H}$.
Then the transitive-ness of the action implies $|{X}/{H}|=1$ and free-ness
implies $|{X}_{g}|=0$ for $g\neq{\rm id}$. So, in particular,
$|{H}|=|{X}|\equiv n$ because ${X}_{\rm id}={X}$. So the subgroup $H$ must
contain as many elements as the number of replicas. In particular, using the
fundamental theorem of algebra, $H$ must take the form
${\mathbb{Z}}_{n_{1}}\otimes\ldots\otimes{\mathbb{Z}}_{n_{k}}$ with
$\prod_{k}n_{k}=n$. There is a simple way of understanding the action of $H$
on $X$. We take the replica set $X$ as $H$ itself. Then the free and
transitive action in question is simply the left (or right) multiplicative
action on $H$ onto itself.

Now we characterize the measures that have $H$ as their replica symmetry
group. Using the group theoretic property of commutants, if $S_{1}$ and
$S_{2}$ are two subgroups of $G$ satisfying $S_{1}\subseteq S_{2}$ then
$c_{G}(S_{2})\subseteq c_{G}(S_{1})$. Also for any subgroup $S$ of $G$,
$S\subseteq c_{G}(c_{G}(S))$. As the subgroup $H\subseteq c_{G}(K)$, we get
$c_{G}(c_{G}(K))\subseteq c_{G}(H)$. Using $c_{G}(H)=H$ and $K\subseteq
c_{G}(c_{G}(K))$ we get, $K\subseteq H$. As $K$ is the group generated by
$\sigma_{\mathtt{a}}$, maximally symmetric measures are now completely
characterized by explicitly specifying the associated permutation elements
$\sigma_{\mathtt{a}},\,{\mathtt{a}}=1,\ldots,{\mathtt{q}}$. We simply pick
them arbitrarily from _some_ abelian group $H$! The chosen abelian group $H$
is the desired subgroup of the replica symmetry group $c_{G}(K)$ that acts
freely and transitively on the replicas.

#### Examples

The von Neumann entropy is a maximally symmetric measure. It corresponds to
the following choice of permutations for the two parties

$\displaystyle\sigma_{{1}}={\rm id},\qquad\sigma_{{\mathtt{2}}}=g,$ (16)

where $g$ is the generator of a ${\mathbb{Z}}_{n}$ subgroup. These two
permutation elements commute with each other.

Entanglement negativity is also a maximally symmetric measure. It is defined
for a tripartite pure state with the following choice of permutation elements.

$\displaystyle\sigma_{{1}}={\rm
id},\qquad\sigma_{{\mathtt{2}}}=g,\qquad\sigma_{\mathtt{3}}=g^{-1}.$ (17)

These elements are mutually commuting.

The other popular measure of tri-partite entanglement, the reflected entropy,
is not maximally symmetric because the permutation elements
$\sigma_{\mathtt{a}}$ are not mutually commuting Dutta:2019gen . We will
discuss other aspects of the reflected entropy further in section 5.

####  3.1.1 Formulation in terms of the density matrix

We formulated bi-partite graphs in terms of permutation elements
$\sigma_{\mathtt{a}}$. However, the map from the graph to permutation elements
is not unique. There are multiple collections of permutation elements
$\\{\sigma_{1},\ldots,\sigma_{\mathtt{q}}\\}$ which give the same graph. This
is due to the freedom of relabeling the vertices. Relabeling of the $\psi$
vertices ($\bar{\psi}$ vertices) gives rise to the left (right) multiplication
freedom. However, it is more convenient to think of this redundancy as left-
multiplication and conjugation as follows.

  * •

Left-multiplication: we can relabel only the black vertices keeping the labels
of the white vertices fixed. This gives the equivalence of $\cal E$ under
left-multiplication by an arbitrary group element $h$ i.e.

$\displaystyle{\cal E}(\sigma_{1},\ldots,\sigma_{\mathtt{q}})={\cal
E}(h\cdot\sigma_{1},\ldots,h\cdot\sigma_{\mathtt{q}})\qquad\forall h\in G.$
(18)

  * •

Conjugation: In addition to this, we can relabel both black and white vertices
simultaneously. This results in the equivalence under conjugation by an
arbitrary element $g$.

$\displaystyle{\cal E}(\sigma_{1},\ldots,\sigma_{\mathtt{q}})={\cal
E}(g^{-1}\cdot\sigma_{1}\cdot g,\ldots,g^{-1}\cdot\sigma_{\mathtt{q}}\cdot
g)\qquad\forall g\in G.$ (19)

It is convenient to gauge fix the left multiplication freedom by fixing
$\sigma_{1}={\rm id}$. Interestingly, it gives a convenient formulation of the
measure in terms of the density matrix obtained by tracing out party ${1}$.

Choosing $\sigma_{1}={\rm id}$ provides a canonical correspondence between
$\psi$’s and $\bar{\psi}$’s. We now take the set of pairs connected by party
${1}$ index as the replica set. Because contraction of the party ${1}$ index
is also the density matrix on the rest of the parties, we effectively treat
the set of density matrices as the set of replicas. We have explained this in
figure 8.

Figure 8: Measure corresponding to choosing $\sigma_{1}$ and $\sigma_{2}$ to
be two independent generators of ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$. It
has three identical connected components. Each of the component corresponds to
${\rm Tr}\rho^{3}$.

Consider $H={\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ and take
$\sigma_{1}=g_{1}\otimes{\rm id}$ and $\sigma_{2}={\rm id}\otimes g_{2}$ where
$g_{i}$’s are generators of ${\mathbb{Z}}_{3}$. The resulting bi-partite graph
is shown in figure 8. As we can see, it consists of three identical
disconnected components. If we gauge fix $\sigma_{1}={\rm id}$, this
identifies the $\psi$ and $\bar{\psi}$ that are connected by the red arrows as
a density matrix and the blue arrows give the contractions of the indices
corresponding to the second party. Each of the three connected components
gives the familiar ${\rm Tr}\rho_{2}^{3}$. Sometimes it is convenient to
describe the measure in this way in terms of the density matrices.

Figure 9: A single connected component of the measure corresponding to
choosing $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$ to be three independent
generators of ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$.

In figure 9, we have shown a single connected component, out of the three
identical ones, of the measure obtained with
$H={\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ with
$\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ being their independent generators
respectively.

If we gauge fix some $\sigma_{\mathtt{b}}={\rm id}$ instead, the measure might
look quite different as a graph constructed out of the density matrices
(obtained after tracing out party ${\mathtt{b}}$) but it is the same measure
as obtained by gauge fixing $\sigma_{1}={\rm id}$. This is simply a
consequence of the left-multiplication gauge invariance. To see this
explicitly, we simply need to observe that the group generated by
$\sigma_{1}^{-1}\sigma_{\mathtt{a}}$ is the same as the one generated by
$\sigma_{\mathtt{b}}^{-1}\sigma_{\mathtt{a}}$ for any ${\mathtt{b}}$.

###  3.2 Holographic probe family

Let us analyze the conditions that are further required to get a holographic
probe measure of entanglement like the Ryu-Takayanagi formula.

In the case of bi-partite Renyi measure $S^{(2)}_{n}$, the replica symmetric
bulk solution ${\cal B}_{n}$ has certain special co-dimension $2$ loci ${\cal
L}_{n}$ that are fixed under the replica symmetry group ${\mathbb{Z}}_{n}$.
These loci emanate from points of conical excess on the replicated boundary
${\cal M}_{n}$. Orbifolding ${\cal B}_{n}$ by the replica symmetry group takes
the boundary back to ${\cal M}$ and yields a bulk geometry ${\widetilde{\cal
B}}_{n}$ that has conical defects ${\widetilde{\cal L}}_{n}$. These are
essentially images of ${\cal L}_{n}$ under orbifolding. The conical opening
angle at ${\widetilde{\cal L}}_{n}$ is $2\pi/n$. The path integral for the
Renyi-entropy thus gets reduced to the computation of bulk action in the
presence of a conical defect. The advantage of this reformulation is that the
bulk description admits a natural analytic continuation in $n$. To compute the
Renyi measure for a real $n$, we simply consider the singularity with conical
opening angle $2\pi/n$ and to obtain von Neumann entropy we take $n\to 1$
limit. In this limit, the conical opening angle approaches $2\pi$ and the
defect flattens out. The action for this almost smooth geometry can then be
computed in the probe fashion on the bulk solution ${\cal B}$ dual to the CFT
state on the un-replicated boundary manifold ${\cal M}$. So the crucial
property of the bulk solution is that it is labeled by a single integer $n$
and admits analytic continuation in $n$ such that in the $n\to 1$ limit, the
opening angles of all conical defects approach $2\pi$.

Motivated by Lewkowycz and Maldacena’s construction, we wish to define a
family of multi-partite maximally symmetric measures ${\cal
E}_{n}(\sigma_{\mathtt{a}})$ labeled by a single integer $n$. We will consider
the condition,

  * •

Probe family condition: The permutation elements $\sigma_{\mathtt{a}}(n)$ are
such that the cycle lengths of all the cycles in
$\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}$ go to $1$ as $n\to 1$ for all
parties ${\mathtt{a}},{\mathtt{b}}$.

We will call the family of maximally symmetric measures obeying the above
condition a holographic probe family or simply, a probe family. We will show
later that conical singularities in the orbifolded geometry associated with
the probe family flatten out in the $n\to 1$ limit. As a result, the measure
corresponding to the limit $n\to 1$ can be computed by a probe on the original
solution ${\cal B}$. We call the $n=1$ element of the family (assuming it
exists, see section 3.4), a probe measure.

#### Examples

The family of Renyi entropies is a probe family and the associated probe
measure is the von Neumann entropy.

For entanglement negativity, the permutation elements
$\sigma_{1}^{-1}\sigma_{2},\sigma_{1}^{-1}\sigma_{3}$ and
$\sigma_{3}^{-1}\sigma_{2}$ are $g,g^{-1}$ and $g^{2}$ respectively. Here $g$
is the generator of a cyclic group. When $|g|$ is even, say $n=2k$ then
$g^{2}$ consists of two cycles of length $k=n/2$ each. So even though
$|g|=|g^{-1}|\to 1$ as $n\to 1$, cycle lengths of $g^{2}$ go to $1/2$. So
clearly it does not form a probe measure family. However, if $|g|$ is odd, say
$n=2k+1$ then $g^{2}$ consists of a single cycle of the same length. The cycle
lengths of $g,g^{-1},g^{2}$ all go to $1$ as $n\to 1$ i.e. $k\to 0$. So
entanglement negativity, for odd replica number, does form a probe measure
family.

It seems difficult to solve for the probe family condition completely because
lengths of cycles in $\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}$ depend on
the details of $\sigma_{\mathtt{a}}$ and $\sigma_{\mathtt{b}}$ in general and
not just on their equivalence classes. However, we will soon find a large
class of solutions where the equivalence class of
$\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}$ can be computed just from the
equivalence class of $\sigma_{\mathtt{a}}$ and $\sigma_{\mathtt{b}}$.

###  3.3 Special symmetric measures and special probes

We call the maximally symmetric measures whose permutation elements
$\sigma_{\mathtt{a}}$ satisfy the following condition,

  * •

Special symmetry condition:

$\displaystyle\langle\sigma_{\mathtt{a}}\rangle\cap\langle\sigma_{\mathtt{b}}\rangle={\rm
id}\qquad{\rm for}\qquad{\mathtt{a}}\neq{\mathtt{b}}$ (20)

a special symmetric measure. Here $\langle g\rangle$ is the cyclic group
generated by $g$. If the order of $g$ is $|g|$ i.e. if $|g|$ is the smallest
integer such that $g^{|g|}=1$ then $\langle g\rangle$ is isomorphic to
${\mathbb{Z}}_{|g|}$. The condition
$\langle\sigma_{\mathtt{a}}\rangle\cap\langle\sigma_{\mathtt{b}}\rangle={\rm
id}$ for ${\mathtt{a}}\neq{\mathtt{b}}$ morally means that the generators
$\sigma_{\mathtt{a}}$ are linearly independent. In particular, this means
$H=K=\otimes_{{\mathtt{a}}=1}^{{\mathtt{q}}}\langle\sigma_{\mathtt{a}}\rangle=\otimes_{\mathtt{a}}{\mathbb{Z}}_{|\sigma_{\mathtt{a}}|}$.
So essentially, a special symmetric measure is labeled by a choice of
${\mathtt{q}}$ integers $m_{\mathtt{a}}\equiv|\sigma_{\mathtt{a}}|$. We will
denote it as ${\cal E}^{\\{m_{1},m_{\mathtt{2}},\ldots\\}}$.

An important property of special symmetric measures is that the order of the
element $\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}$ can be characterized in
terms of the order of $\sigma_{\mathtt{a}}$ and $\sigma_{\mathtt{b}}$,

$\displaystyle|\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}|={\rm
lcm}(|\sigma_{\mathtt{a}}|,|\sigma_{\mathtt{b}}|).$ (21)

This property lets us straightforwardly construct a holographic probe family.
The ${\mathtt{q}}$-partite measure constructed in Gadde:2022cqi , corresponds
to choosing $|\sigma_{\mathtt{a}}|=n$ for all ${\mathtt{a}}$. Then
$|\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}|$ is also $n$. The fact that
$|\sigma_{\mathtt{a}}|\to 1$ in the $n\to 1$ limit implies that the length of
all the cycles goes to $1$ in the $n\to 1$ limit as required by the
holographic probe condition. Explicitly, this probe family is defined
as555This probe family was called Renyi multi-entropy in Gadde:2022cqi  and
was defined without the factor $1/n^{{\mathtt{q}}-1}$.

$\displaystyle
S_{n}^{({\mathtt{q}})}\equiv\frac{1}{1-n}\frac{1}{n^{{\mathtt{q}}-1}}\log\Big{(}{\cal
E}^{\\{n,n,\ldots\\}}\Big{)}.$ (22)

The associated probe measure obtained was termed multi-entropy in
Gadde:2022cqi .

Because we organized the multi-partite measures in this systematic manner, we
can quickly recognize that a significantly larger class of holographic probe
measures can be constructed. One obtains a holographic probe measure starting
from any special symmetric measure. If $|\sigma_{\mathtt{a}}|=m_{\mathtt{a}}$
for a special symmetric measure then the family of special symmetric measures
with $|\sigma_{\mathtt{a}}|=m_{\mathtt{a}}^{n-1}$ is a probe measure family.
This is because

$\displaystyle|\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}|=({\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}}))^{n-1}$ (23)

and it goes to $1$ as $n\to 1$ for arbitrary integers $m_{\mathtt{a}}$. We
will call the probe family obtained from special symmetric measures ${\cal
E}^{\\{m_{1},m_{\mathtt{2}},\ldots\\}}$ as a special probe family. Explicitly,
we define it to be

$\displaystyle
S_{n}^{\\{m_{1},m_{\mathtt{2}},\ldots\\}}\equiv\frac{1}{1-n}\frac{1}{(\prod_{\mathtt{a}}m_{\mathtt{a}})^{n-1}}\log\Big{(}{\cal
E}^{\\{m_{1}^{n-1},m_{\mathtt{2}}^{n-1},\ldots\\}}\Big{)}.$ (24)

The special probes are then ${\rm lim}_{n\to
1}S_{n}^{\\{m_{1},m_{\mathtt{2}},\ldots\\}}$. We will denote them as
$S^{\\{m_{1},m_{\mathtt{2}},\ldots\\}}$.

####  3.3.1 Number of connected components

How many connected components does the special symmetric measure have? This
simple-sounding question has an interesting combinatorial answer. The number
of replicas in a single connected component is equal to the order of the group
$\tilde{K}$ generated by
$\sigma_{1}^{-1}\sigma_{\mathtt{a}}\equiv\tilde{\sigma}_{\mathtt{a}}$. Note
that the group generated by $\sigma_{\mathtt{b}}^{-1}\sigma_{\mathtt{a}}$ for
all ${\mathtt{a}}$ and any fixed ${\mathtt{b}}$ is $\tilde{K}$. The number of
connected components of the special symmetric measure is $|H|/|\tilde{K}|$.
Let us discuss the computation of $|\tilde{K}|$ in terms of in terms of
$|\sigma_{\mathtt{a}}|=m_{\mathtt{a}}$.

For ${\mathtt{q}}=2$, the group $\tilde{K}$ is generated by
$\sigma_{1}^{-1}\sigma_{\mathtt{2}}\equiv\tilde{\sigma}_{\mathtt{2}}$. It is
well known that for independent elements $\sigma_{1}$ and
$\sigma_{\mathtt{2}}$, $|\tilde{\sigma}_{\mathtt{2}}|={\rm
lcm}(m_{1},m_{\mathtt{2}})$.

For ${\mathtt{q}}=3$, the group $\tilde{K}$ is generated by
$\sigma_{1}^{-1}\sigma_{\mathtt{2}}\equiv\tilde{\sigma}_{2},\sigma_{1}^{-1}\sigma_{\mathtt{3}}\equiv\tilde{\sigma}_{\mathtt{3}}$.
Their orders are ${\rm lcm}(m_{1},m_{\mathtt{2}})\equiv\tilde{m}_{2}$ and
${\rm lcm}(m_{1},m_{\mathtt{3}})\equiv\tilde{m}_{\mathtt{3}}$ respectively. If
$\tilde{\sigma}_{1}$ and $\tilde{\sigma}_{\mathtt{2}}$ were independent,
$|\tilde{K}|$ would be product of their orders $\tilde{m}_{1}\tilde{m}_{2}$.
However, they are not independent since they obey the relation
$(\tilde{\sigma}_{\mathtt{2}}^{-1}\tilde{\sigma}_{\mathtt{3}})^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}=(\sigma_{\mathtt{2}}^{-1}\sigma_{\mathtt{3}})^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}={\rm id}$. In order to compute
$|\tilde{K}|$, we divide $\tilde{m}_{1}\tilde{m}_{\mathtt{2}}$ by the order of
the group generated by the relation
$(\tilde{\sigma}_{\mathtt{2}}^{-1}\tilde{\sigma}_{\mathtt{3}})^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}$ assuming $\tilde{\sigma}_{\mathtt{2}}$
and $\tilde{\sigma}_{\mathtt{3}}$ to be independent. For that, we will compute
the order of $\tilde{\sigma}_{\mathtt{2}}^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}$ and $\tilde{\sigma}_{\mathtt{3}}^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}$ and the order of the group generated by
the relation is the ${\rm lcm}$ of these two orders.

$\displaystyle|\tilde{\sigma}_{\mathtt{2}}^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}|$ $\displaystyle=\frac{{\rm
lcm}(\tilde{m}_{\mathtt{2}},{\rm lcm}(m_{\mathtt{2}},m_{\mathtt{3}}))}{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}=\frac{{\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})}{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}$
$\displaystyle|\tilde{\sigma}_{\mathtt{3}}^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}|$ $\displaystyle=\frac{{\rm
lcm}(\tilde{m}_{\mathtt{3}},{\rm lcm}(m_{\mathtt{2}},m_{\mathtt{3}}))}{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}=\frac{{\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})}{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}$
$\displaystyle|(\tilde{\sigma}_{\mathtt{2}}^{-1}\tilde{\sigma}_{\mathtt{3}})^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}|$ $\displaystyle={\rm
lcm}(|\tilde{\sigma}_{\mathtt{2}}^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}|,|\tilde{\sigma}_{\mathtt{3}}^{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}|)=\frac{{\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})}{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}.$ (25)

This gives,

$\displaystyle|\tilde{K}|=\frac{{\rm lcm}(m_{1},m_{\mathtt{2}})\cdot{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})\cdot{\rm lcm}(m_{\mathtt{3}},m_{1})}{{\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})}.$ (26)

This is symmetric between $m_{1},m_{\mathtt{2}},m_{\mathtt{3}}$ as expected.

For higher values of ${\mathtt{q}}$, the problem is slightly more involved. It
turns out to be useful to think of $|\tilde{K}|$ as a group generated by
$\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}$ for _all_
${\mathtt{a}},{\mathtt{b}}$ rather than only by
$\sigma_{1}^{-1}\sigma_{\mathtt{b}}$.666We thank Arvind Nair for discussion on
this point. To characterize the group, we characterize the relations (and
relations among relations, if any and so on) among them. The answer for
$|\tilde{K}|$ will be manifestly symmetric in all the parties with this
approach.

Let us again take the case of ${\mathtt{q}}=3$. The generators of $\tilde{K}$
are $\sigma_{1}^{-1}\sigma_{\mathtt{2}}\equiv
h_{{1}{\mathtt{2}}},\sigma_{\mathtt{2}}^{-1}\sigma_{\mathtt{3}}\equiv
h_{{\mathtt{2}}{\mathtt{3}}}$ and $\sigma_{\mathtt{3}}^{-1}\sigma_{1}\equiv
h_{{\mathtt{3}}{1}}$. If these elements were independent, then the group would
have been ${\mathbb{Z}}_{{\rm
lcm}(m_{1},m_{\mathtt{2}})}\otimes{\mathbb{Z}}_{{\rm
lcm}(m_{\mathtt{2}},m_{\mathtt{3}})}\otimes{\mathbb{Z}}_{{\rm
lcm}(m_{\mathtt{3}},m_{1})}$. But they are not independent and obey the
relation
$h_{{1}{\mathtt{2}}}h_{{\mathtt{2}}{\mathtt{3}}}h_{{\mathtt{3}}{1}}={\rm id}$.
The group generated by the relation has the order ${\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})$ because it is the ${\rm lcm}$ of
the order of $h_{{1}{\mathtt{2}}},h_{{\mathtt{2}}{\mathtt{3}}}$ and
$h_{{\mathtt{3}}{1}}$. The size of $\tilde{K}$ is then

$\displaystyle|\tilde{K}|=\frac{{\rm lcm}(m_{1},m_{\mathtt{2}})\cdot{\rm
lcm}(m_{1},m_{\mathtt{2}})\cdot{\rm lcm}(m_{1},m_{\mathtt{2}})}{{\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})}.$ (27)

This analysis generalizes readily to higher values of ${\mathtt{q}}$. Let us
associate a vertex for every group element $\sigma_{\mathtt{a}}$ and denote
the generator $\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}\equiv
h_{{\mathtt{a}}{\mathtt{b}}}$ as an edge from vertex ${\mathtt{a}}$ to
${\mathtt{b}}$. We will have $\,{}^{q}C_{2}$ such edges777Here
${}^{n}C_{m}=n!/(m!(n-m)!)$ is the binomial coefficient. It counts the number
of ways $m$ objects can be chosen from the set of $n$ distinct objects.
forming a skeleton of a ${\mathtt{q}}$-simplex888A ${\mathtt{q}}$-simplex is a
generalization of notion of tetrahedron to ${\mathtt{q}}$ dimensions. Examples
of a few low-dimensional simplices are the following. A $0$-simplex is a
point. $1$-simplex is obtained by connecting a point to the $0$-simplex giving
us an interval. A $2$-simplex is obtained by connecting a point to the $1$
simplex with two edges and also filling it in with a face giving us a
triangle. A $3$-simplex is obtained by connecting a point to the $2$-simplex
by three edges, three triangular faces and a three-dimensional tetrahedral
“face”. In general, a ${\mathtt{q}}$-simplex is a ${\mathtt{q}}$-dimensional
polytope which is the convex hull of its ${\mathtt{q}}+1$ vertices, no three
of which are collinear.. The relation among these corresponds to a closed loop
and hence a face. For example, in the ${\mathtt{q}}=3$ case, the simplex is a
triangle and the relation
$h_{{1}{\mathtt{2}}}h_{{\mathtt{2}}{\mathtt{3}}}h_{{\mathtt{3}}{1}}$ clearly
corresponds to the closed loop.

For ${\mathtt{q}}=4$, the simplex is the tetrahedron. There are four relations
among the $\,{}^{4}C_{2}$ edges corresponding to the $4$ faces. The order of
these relations are ${\rm lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})$, ${\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{4}})$, ${\rm
lcm}(m_{1},m_{\mathtt{4}},m_{\mathtt{3}})$ and ${\rm
lcm}(m_{\mathtt{4}},m_{\mathtt{2}},m_{\mathtt{3}})$. However, these relations
are not independent and have relations among them. These relations of
relations correspond to closed two-dimensional surfaces formed out of faces or
a 3-simplex. In this case, there is only one. The order of the corresponding
element is ${\rm lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}},m_{\mathtt{4}})$
because this is the ${\rm lcm}$ of the above four numbers. All in all, this
gives the size of $|\tilde{K}|$ to be

$\displaystyle|\tilde{K}|=\frac{{\rm
lcm}(m_{1},m_{\mathtt{2}},m_{\mathtt{3}},m_{\mathtt{4}})\prod_{{\mathtt{a}}<{\mathtt{b}}}{\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}})}{\prod_{{\mathtt{a}}<{\mathtt{b}}<{\mathtt{c}}}{\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}},m_{\mathtt{c}})}.$ (28)

One can write down the formula for $|\tilde{K}|$ for higher values of
${\mathtt{q}}$ using this “inclusion-exclusion” type principle.

###  3.4 About analytic continuation

The definition of the special probes as well as multi-entropy involves an
analytic continuation of the probe family $S^{\\{m_{\mathtt{a}}\\}}_{n}$, that
is defined for integer $n>1$ to complex $n$ and then taking the $n\to 1$
limit. As we will discuss in section 4, the analytic continuation in $n$ does
exist for holographic states if we assume that the dominant bulk saddle
preserves replica symmetry. However, it is _a priori_ not obvious that it
exists for general quantum states. The problem of analytic continuation can be
formulated as a question in complex analysis:

  * •

Does there exist an analytic function $f(z)$, that is unique in some sense,
that takes given values $f(n)=f_{n}$ (input data) on positive integers?

First of all, the question asked above is ambiguous. We have to specify the
sense in which we expect the answer to be unique. One solution to this problem
is provided by Carlson’s theorem. It states the following:

  * •

A function $g(z)$ that is analytic for ${\rm Re}(z)>0$ and satisfies

$\displaystyle|g(z)|\leq Ae^{c|z|},\quad g(iy)\leq Ae^{By},\quad B<\pi$ (29)

for some real constants $A$ and $c$ for all $z\in{\mathbb{C}}$ and
$y\in{\mathbb{R}}$ and vanishes for all non-negative integer must be
identically zero.

We will call the analyticity and boundedness conditions on $g(z)$ that are
required by the theorem as Carlson’s conditions. The boundedness condition in
the imaginary direction is crucial otherwise $\sin(\pi z)$ forms an immediate
counterexample. In particular, this also means that to disallow such a
possibility we need to take the bounding exponent $B<\pi$.

The idea is to require that the putative analytic continuation $f(z)$ obeys
Carlson’s conditions. Thanks to Carlson’s theorem this analytic continuation
would then be unique. For the analytic continuation of $f_{n}={\rm
Tr}\,\rho^{n}$ imposing Carlson’s condition is perfectly suited. This is
because $f_{n}=\sum_{i}\lambda_{i}^{n}$ with999$\lambda$ can equal $1$ in case
of a rank $1$ density matrix. In that case, $f_{n}={\rm Tr}\rho^{n}=1$ which
analytically continues to $f(z)=1$. $0\leq\lambda_{i}<1$ where $\lambda_{i}$
are (possibly infinitely many) eigenvalues of $\rho$. With this form, it is
clear that $f(z)$ defined as $f(z)=\sum_{i}\lambda_{i}^{z}$ obeys Carlson’s
analyticity and boundedness conditions. This is why Carlson’s condition is
often used to analytically continue the Renyi entropy $S_{n}$ away from
integer values of $n$.

For multi-partite case, _a priori_ it is not clear if the quantity ${\cal
E}^{\\{m_{1}^{n-1},m_{2}^{n-1},\ldots\\}}$ or even ${\cal
E}^{\\{n,n,\ldots\\}}$ is compatible with Carlson’s conditions. Let us
explain. Let us define the generalized ${\mathtt{q}}$-partite GHZ state as

$\displaystyle|\Psi\rangle_{\rm
GHZ}\equiv\sum_{i}\lambda_{i}|e^{1}_{i}\rangle\otimes|e^{2}_{i}\rangle\otimes\ldots|e^{{\mathtt{q}}}_{i}\rangle$
(30)

where all the parties ${\cal H}_{\mathtt{a}}$ are take to have the same
dimension and $|e_{i}^{{\mathtt{a}}}\rangle$ is a set of orthonormal basis
vectors. The measure ${\cal E}^{\\{n,n,\ldots\\}}$ for this state is given by
Gadde:2022cqi ,

$\displaystyle{\cal
E}^{\\{n,n,\ldots\\}}=(\sum_{i}|\lambda_{i}|^{2n^{{\mathtt{q}}-1}})^{n}.$ (31)

If we demand that the analytic continuation of ${\cal E}^{\\{n,n,\ldots\\}}$
is obtained by replacing $n$ by a complex number $z$ then it is clear that the
resulting function does not obey Carlson’s conditions. Even if we only
analytically continue the quantity inside the bracket in this way, this
problem persists. It might be that the simple lift of $n$ to $z$ in the
formula (31) is not the correct analytic continuation for the generalized GHZ
state and that the correct analytic continuation still obeys Carlson’s
conditions. Here, by correct analytic continuation we mean two things, first
is that it should be unique subject to some conditions and second, it should
match the analytic continuation for holographic states performed in section 4.
In what follows, we will outline a numerical check of whether the analytic
continuation that obeys Carlson’s conditions exists.

This question has been discussed in regge-viano  in the context of analytic
continuation in angular momentum and a necessary condition on the input data
$f_{n}$ has been derived in osti_4065624 101010We thank Geoff Penington for
pointing out this paper to us.. Below we will review these issues. It is not
difficult to convince oneself that if we wish to uniquely analytically
continue a function that is defined on say, even positive integers, we need to
demand Carlson’s conditions but with $B<\pi/2$. A generalized version of
Carlson’s theorem, due to 10.2307/1992882 , implies that if we want uniquely
analytically continue a function that is defined on positive points with
density $D$ per unit then we should require $B<\pi D$.

From this discussion, we can see that the input data $f_{n}$ can not be
completely independent to have a unique analytic continuation satisfying
Carlson’s conditions. The argument is as follows. Let us assume a function
$f(z)$ satisfying all Carlson’s conditions exists. In particular, $f(iy)\leq
e^{C\pi y},y\in{\mathbb{R}}$ for some $C<1$. As discussed earlier, we only
need points with density $C$ per unit to determine such a function but because
the input data is specified for all positive integers i.e. on the set with
density $1$, it must not be completely independent. It is not clear what
conditions on the input data are necessary and sufficient for the existence of
analytic continuation in the above sense; however in osti_4065624 , a set of
infinitely many necessary conditions is formulated. These conditions express
$f_{l}$ in terms of all the rest of the $f_{n}$ for all $l$. We have
reproduced the proposed analytic continuation and the consistency conditions
it needs to satisfy, in appendix B for the reader’s convenience. These
conditions of osti_4065624  are amenable to numerical analysis. If the probe
family measures fail these conditions, we need to look for other appropriate
conditions that guarantee uniqueness and also match with the analytic
continuation performed for the holographic states in section 4. We will not
undertake this problem here but continue with the analysis of these measures
for holographic states.

##  4 Holographic dual of the special probes

In this section, we will derive a holographic prescription to compute the
special probe measure in CFTs that admit a semiclassical dual. This discussion
closely parallels the derivation of the holographic prescription for multi-
entropy discussed in Gadde:2022cqi  which in turn is inspired by the
derivation of the Ryu-Takayanagi formula for the entanglement entropy by
Lewkowycz and Maldacena Lewkowycz:2013nqa .

Let the state $|\Psi\rangle$ be defined on a time-symmetric Cauchy slice
${\mathcal{R}}$ of a $D$-manifold ${\mathcal{M}}$. It is given by a Euclidean
path integral on the half-space ${\mathcal{M}}_{\Psi}$ such that
$\partial{\mathcal{M}}_{\Psi}={\mathcal{R}}$. The dual bra $\langle\Psi|$ is
constructed by Euclidean path integral on the other half
${\mathcal{M}}_{\bar{\Psi}}$. It is obtained from ${\mathcal{M}}_{\Psi}$ by
reflecting about ${\mathcal{R}}$. The squared norm of $\Psi$ is the partition
function ${\mathcal{Z}}_{\mathcal{M}}$ on ${\mathcal{M}}$. We normalize the
state such that ${\mathcal{Z}}_{\mathcal{M}}=1$. Let us decompose
${\mathcal{R}}$ into ${\mathtt{q}}$ number of disjoint regions
${\mathcal{R}}_{\mathtt{a}}$, such that
$\cup_{\mathtt{a}}{\mathcal{R}}_{\mathtt{a}}={\mathcal{R}}$. Let the Hilbert
space on region ${\mathcal{R}}_{\mathtt{a}}$ be ${\mathcal{H}}_{\mathtt{a}}$.
We are interested in computing the special probe measures for the state
$|\Psi\rangle$ under the decomposition
$\otimes_{\mathtt{a}}{\mathcal{H}}_{\mathtt{a}}$. For theories that admit a
weakly coupled gravity dual, this problem can be addressed holographically.

The probe family measure ${S}_{n}^{\\{m_{\mathtt{a}}\\}}$ involves working
with $|H_{n}|=\prod_{{\mathtt{a}}=1}^{{\mathtt{q}}}m_{\mathtt{a}}^{n-1}$
number of replica copies of the theory. The tensor product theory admits co-
dimension two twist operators corresponding to the permutation group
${\mathbb{S}}_{|H_{n}|}$. For every pair of regions
$({\mathcal{R}}_{\mathtt{a}},{\mathcal{R}}_{\mathtt{b}})$ that share a
boundary, we insert the twist operator
${{\mathcal{O}}}_{\sigma_{\mathtt{a}}^{-1}\sigma_{{\mathtt{b}}}}$ on the
common boundary. The measure $S_{n}^{\\{m_{\mathtt{a}}\\}}$ is then given by
(appropriately normalized) logarithm of the correlation function of these
twist operators. Let us denote the resulting replicated manifold as ${\cal
M}_{n}$. Following Lewkowycz:2013nqa , we will proceed to compute this
partition function holographically. Let ${\mathcal{B}}_{n}$ be the dominant
gravity solution such that $\partial{\mathcal{B}}_{n}={\mathcal{M}}_{n}$. At
leading order in $G_{N}$,

$\displaystyle{\log}\,{\mathcal{Z}}_{{\mathcal{M}}_{n}}=-{\mathcal{S}}_{\rm
grav}({\mathcal{B}}_{n})+|H_{n}|{\mathcal{S}}_{\rm grav}({\mathcal{B}}_{1}).$
(32)

Here ${\mathcal{S}}_{\rm grav}({\cal X})$ is the gravitational action
evaluated on the solution ${\cal X}$. The second term on the right-hand side
of equation (32) serves to normalize the state.

The background fields on the manifold ${\mathcal{M}}_{n}$ enjoy a replica
symmetry $H_{n}$. This is the symmetry generated by the permutation elements
$\sigma_{\mathtt{a}}$ associated with all the twist operators. Following
Lewkowycz:2013nqa , we will make

  * •

Replica symmetry assumption: The dominant bulk solution ${\cal B}_{n}$ filling
in the replicated boundary manifold ${\cal M}_{n}$ corresponding to the probe
family measure preserves replica symmetry.

We will offer support for this assumption in section 6 where we compute
$S_{n}^{\\{m_{\mathtt{a}}\\}}$ in $2d$ CFTs with large central charge directly
using CFT methods. The solution ${\mathcal{B}}_{n}$ consists of co-dimension
$2$ loci that are invariant under certain subgroups of the replica symmetry
group. Some of these loci, called “external”, are anchored at the fixed points
on the boundary (these are locations of twist operator insertions on
${\mathcal{M}}$) while the rest are “internal”. Let us denote the loci that
are anchored at the fixed points corresponding to the twist operator
${\mathcal{O}}_{g}$ as ${\mathcal{L}}_{g}$. They are invariant under the
cyclic subgroup of $H_{n}$ generated by $g$. The quotient group $H_{n}/\langle
g\rangle$ acts on one such locus to generate a family of them that is
$|H_{n}/\langle g\rangle|$ in number. Generically, two ${\mathcal{L}}$s can
merge to form a different ${\mathcal{L}}$. Merging obeys the algebra
${\mathcal{L}}_{g_{1}}\cdot{\mathcal{L}}_{g_{2}}\to{\mathcal{L}}_{g_{1}g_{2}}$.
In fact, because the merging of ${\cal L}_{g_{1}}$ and ${\cal L}_{g_{2}}$ to
form ${\cal L}_{g_{1}g_{2}}$ happens on the locus that is invariant under both
$g_{1}$ and $g_{2}$, the members of ${\cal L}_{g_{1}}$ family that are related
to each other by ${g_{2}}$ action merge with members of ${\cal L}_{g_{2}}$
family that are related to each other by $g_{1}$ action. The branched
structure of the replicated manifold ${\cal M}_{n}$ along with the invariant
loci and their merging is depicted in figure 10. Figure 11 presents a
simplified view of the invariant loci and their merging when the boundary
manifold ${\cal M}_{n}$ is unwrapped. The internal loci come about because of
such merging. Below we will assume that

  1. 1.

Every fixed point locus is of the form
${\mathcal{L}}_{\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}}$.

Figure 10: Branching of the replicated manifold ${\cal M}_{n}$ and the fixed
points. We also show the invariant loci that emanate from the boundary fixed
points. They merge according to the rule
${\mathcal{L}}_{g_{1}}\cdot{\mathcal{L}}_{g_{2}}\to{\mathcal{L}}_{g_{1}g_{2}}$.
Figure 11: Families of invariant loci anchored on the boundary. They merge
according to the rule
${\mathcal{L}}_{g_{1}}\cdot{\mathcal{L}}_{g_{2}}\to{\mathcal{L}}_{g_{1}g_{2}}$.

We now make use of the replica symmetry in the bulk to construct the orbifold
$\widetilde{\mathcal{B}}_{n}\equiv{\mathcal{B}}_{n}/H_{n}$. Due to symmetry,
the classical gravitational actions on the two spaces are related as

$\displaystyle{\mathcal{S}}_{\rm
grav}({\mathcal{B}}_{n})=|H_{n}|\,{\mathcal{S}}_{\rm
grav}(\widetilde{\mathcal{B}}_{n}).$ (33)

The orbifold $\widetilde{\mathcal{B}}_{n}$ has a nice property that
$\partial\widetilde{\mathcal{B}}_{n}={\mathcal{M}}$. The family of
${\mathcal{L}}_{g}$ loci discussed above become a _single_ conical singularity
of opening angle $2\pi/|g|$ in the orbifold $\widetilde{\mathcal{B}}_{n}$.
This is because orbifolding by quotient group $H_{n}/\langle g\rangle$
identifies all the members of the family and orbifolding by the remaining
$\langle g\rangle$, which fixes the locus ${\mathcal{L}}_{g}$, gives rise to
the conical singularity. Let us denote this singularity as
$\widetilde{\mathcal{L}}_{g}$. Let us denote the web created by these
singularities as ${\mathcal{W}}$. Consider a co-dimension $1$ slice
${\mathcal{C}}\in\widetilde{\mathcal{B}}_{n}$ that contains ${\mathcal{W}}$
and with the property $\partial{\mathcal{C}}={\mathcal{R}}$. There are
multiple such slices but the exact choice doesn’t matter for the following
discussion. Every singularity becomes a co-dimension $1$ wall in
${\mathcal{C}}$ and the web ${\mathcal{W}}$ yields its chamber decomposition.
As we move from ${\mathcal{R}}_{\mathtt{a}}$ to ${\mathcal{R}}_{\mathtt{b}}$
with ${\mathtt{b}}\neq{\mathtt{a}}$, through ${\mathcal{C}}$, we must
encounter at least one wall because the permutation elements
$\sigma_{\mathtt{a}}$ and $\sigma_{\mathtt{b}}$ are different. At this stage,
we make an assumption about ${\mathcal{W}}$ that

  1. 2.

There is no chamber that lies completely in the interior of ${\mathcal{C}}$.

As a result, we get a one-to-one map between the chambers and boundary
regions. Let us denote the chamber adjacent to ${\mathcal{R}}_{\mathtt{a}}$ as
${\mathcal{C}}_{\mathtt{a}}$. It has the property
$\partial{\mathcal{C}}_{a}\cap{\mathcal{M}}={\mathcal{R}}_{\mathtt{a}}$. The
web ${\mathcal{W}}$ consists of only those walls that separate
${\mathcal{C}}_{\mathtt{a}}$ and ${\mathcal{C}}_{\mathtt{b}}$ for some
$({\mathtt{a}},{\mathtt{b}})$. Such a wall must be of the type
$\widetilde{\mathcal{L}}_{\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}}$. The
parent ${\mathcal{L}}\in{\mathcal{B}}_{n}$ must also be of the same type. This
justifies our assumption 1. The configuration is summarized in figure 12.

Figure 12: Here we have denoted the boundary ${\cal M}$ and the regions of
interest ${\cal R}_{\mathtt{a}}$, the bulk ${\cal B}$ and the chambers ${\cal
C}_{\mathtt{a}}$. The chambers ${\cal C}_{\mathtt{a}}$ and ${\cal
C}_{\mathtt{b}}$ are separated by walls of the type
$\widetilde{\mathcal{L}}_{\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}}$.

To compute the probe family measure using equations (32) and (33), we need to
evaluate the gravitational action on the orbifold solution
$\widetilde{{\mathcal{B}}}_{n}$. In what follows, we will specialize to the
case of Einstein gravity. The orbifold solution is a smooth solution of
Einstein’s equations with constant negative curvature except at
$\widetilde{{\mathcal{L}}}_{g}$’s where it has a conical singularity of
opening angle $2\pi/|g|$. The gravitational action for this solution is
computed as follows Lewkowycz:2013nqa ; Dong:2013qoa . To find the solution it
is convenient to engineer the conical singularities with the help of a cosmic
brane. It is known that a co-dimension two cosmic brane with the action

$\displaystyle{\mathcal{S}}_{\rm br}^{(k)}=\frac{k-1}{4kG_{N}}\int
dy^{D-1}\sqrt{h}=\frac{k-1}{4kG_{N}}A$ (34)

gives rise to a conical singularity with the opening angle $2\pi/k$ around its
support Unruh:1989hy ; Boisseau:1996bp . So we consider a brane with the
action $S_{\rm br}^{(|g|)}$ supported on every locus
${\widetilde{\mathcal{L}}}_{g}$. In this way, we get a cosmic brane-web
${\mathcal{W}}$ in which branes of differing tensions are joined together. The
solution is then computed by solving the equations coming from the action
${\cal S}_{\rm grav}+{\cal S}_{\rm br}(n)$. We then need to evaluate only the
action $S_{\rm grav}$ on the solution. This is because the cosmic brane is not
actually present at the singularity but merely used as a trick to model the
singularity.

A priori, there can be additional terms in the brane action supported only at
the meeting locus. To compute such terms we excise the tubular neighborhood of
${\widetilde{\mathcal{L}}}$’s of size $a$ and focus at their co-dimension
three junction. As multiple ${\widetilde{\mathcal{L}}}$’s meet, their
corresponding tubular neighborhoods also meet forming corners as shown in
figure 13.

Figure 13: Top view of the junction of tubular neighborhoods of three
${\widetilde{\mathcal{L}}}^{\prime}s$.

For the gravitational variational principle to be well-defined, we need to add
the so-called Hayward term at the corner just as the way we add the Gibbons-
Hawking-York (GHY) term on the boundary.

$\displaystyle{\cal S}_{\rm Hayward}=-\frac{1}{8\pi G_{N}}\int
d\xi^{D-2}\,(\theta-\pi)\sqrt{\gamma}.$ (35)

Here $\theta$ is the angle subtended by the two boundaries at the corner,
$\xi$ is the coordinate along the corner and $\gamma$ is the induced metric on
the corner. The contribution of the Hayward term is extensive along the co-
dimension $3$ meeting locus and the proportionality constant is computed by
integrating the Hayward term along the one-dimensional corner seam of the
tubular neighborhood as shown in figure 13. This integral is proportional to
$a$ and vanishes as we take $a\to 0$. This had to be the case on dimensional
grounds because the contribution at the meeting locus must take the form
$\sim\frac{1}{G_{N}}\ell^{(3)}_{\rm meeting}\ell_{\rm scale}$. Here
$\ell^{(3)}_{\rm meeting}$ is the length (co-dimension $3$) of the meeting
locus in ${\mathcal{W}}$ and $\ell_{\rm scale}$ is some length scale needed to
obtain a dimensionless answer. The only length scale that could serve this
purpose is111111Another length scale present in the problem is the AdS scale
$\ell_{\rm AdS}$, however, it is irrelevant to this completely local
computation. $a$ which we take to $0$. This argument also shows that there is
no extra contribution to the action even at higher co-dimension meeting loci.

Finding the solution to the theory ${\mathcal{S}}_{\rm
grav}+{\mathcal{S}}_{\rm br}(n)$ for general $n$ is still a daunting task (see
Headrick:2010zt ; Hung:2011nu ; Dong:2016fnf  for computation of bi-partite
Renyi entropy). But to compute the special probe, only the limit $n\to 1$ is
relevant. As the only $n$ dependence appears in the coefficients in
${\mathcal{S}}_{\rm br}(n)$ analytically through
$|\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}|={\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}})^{n-1}$, the solution can be analytically
continued away from $n$ integer. In the limit $n\to 1$, the tensions of all
the branes go to zero and the solution can be found in the probe limit as the
brane web configuration that extremizes ${\mathcal{S}}_{\rm br}$ in the fixed
background ${\mathcal{B}}_{1}$. This brane web consists of branes with
differing tensions. Every brane wall separating the chambers
${\mathcal{C}}_{\mathtt{a}}$ and ${\mathcal{C}}_{\mathtt{b}}$ has a
vanishingly small tension proportional to $\log({\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}}))$. We are interested in evaluating
$\partial_{n}{\cal S}_{\rm grav}|_{n=1}$. This can be done using the same
techniques as Lewkowycz:2013nqa ; Dong:2013qoa ; Dong:2016fnf . We use the
geometry that was just discussed near equation (35) where the tubular
neighborhood of the singular locus has been excised. The variation of
gravitational solution vanishes in the bulk for any $\delta_{g_{\mu\nu}}$.
Taking the variation with respect to $n$, we see that the contribution only
comes from the GHY term on the boundary. The Hayward term at the corner does
not contribute as discussed earlier. The contribution of the GHY term has been
evaluated in Lewkowycz:2013nqa ; Dong:2013qoa ; Dong:2016fnf . It produces the
area of the singular locus weighted by the tension. Explicitly,

$\displaystyle\partial_{n}{\mathcal{S}}_{\rm grav}|_{n=1}=\frac{1}{4G_{N}}{\rm
min}_{\cal
W}\left\\{\sum_{\widetilde{L}_{\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}}\in{\mathcal{W}}}\log({\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}}))A_{{\mathtt{a}},{\mathtt{b}}}\right\\}\equiv\frac{1}{4G_{N}}A_{\rm
min}^{\cal W}.$ (36)

Here $A_{{\mathtt{a}},{\mathtt{b}}}$ is the area of the brane wall that
separates the chamber ${\mathcal{C}}_{\mathtt{a}}$ and
${\mathcal{C}}_{\mathtt{b}}$ (if there exists such a wall). The sum is over
all the brane walls and minimization is over all brane webs ${\cal W}$. The
weighted sum of areas, after minimization, is denoted as $A^{\cal W}_{\rm
min}$. From our discussion it follows that the brane web ${\mathcal{W}}$ obeys
the topological conditions,

  1. 1.

${\mathcal{W}}$ is anchored at the boundaries of all the regions
${\mathcal{R}}_{\mathtt{a}}$’s.

  2. 2.

${\mathcal{W}}$ contains sub-webs that are homologous to all the regions
${\mathcal{R}}_{\mathtt{a}}$’s.

The second condition is the reformulation of the statement that between any
two chambers ${\mathcal{C}}_{\mathtt{a}}$ and ${\mathcal{C}}_{\mathtt{b}}$
there must be at least one wall. As the solution minimizes the area subject to
these conditions, it doesn’t allow any chamber that lies completely in the
interior of ${\mathcal{C}}$. This justifies our assumption 2.

This, along with equations (24) and (32), shows that the special probe measure
is given a simple formula

$\displaystyle S^{\\{m_{\mathtt{a}}\\}}=\frac{1}{4G_{N}}A_{\rm min}^{\cal W}.$
(37)

In other words, it is equal to the $1/4G_{N}$ times the minimal weighted area
of the brane web ${\mathcal{W}}$.

In the case of $2d$ CFTs, all the loci ${\widetilde{\cal L}_{g}}$ are one
dimensional. Solving the equations of motion for the cosmic brane web means
that the individual segments of the brane web are geodesics and that they
balance tension at the tri-valent junction. So, for the solution of the
tension balance to exist, it is needed that the magnitude of the three tension
vectors involved in the junction satisfy the triangle inequality. Happily, for
any three positive numbers $(m_{1},m_{\mathtt{2}},m_{\mathtt{3}})$, the three
quantities $\log({\rm lcm}(m_{1},m_{2})),\log({\rm lcm}(m_{2},m_{3}))$ and
$\log({\rm lcm}(m_{1},m_{3}))$ always satisfy the triangle inequality! Hence
the solution to the tension balance always exists for the special probes.

If we pick all the integers $m_{\mathtt{a}}=m$ then the special probe measure
is simply $\log(m)$ times the multi-entropy computed in Gadde:2022cqi . In
this case, the tensions of all the brane walls are equal and the formula
produces the soap film prescription of Gadde:2022cqi . In the general case,
the soap film prescription is extended to allow the differing tensions to
individual soap-films as discussed above.

Interestingly, the brane web corresponding to multi-entropy has appeared
already in the discussion of holographic multi-partite entanglement
Harper:2021uuq ; Harper:2022sky . In this paper, the author considers
configurations of the so-called hyperthreads that maximize a certain type of
flow. The solution to this linear program is provided precisely by the brane
webs of multi-entropy. It would be interesting to explore the connection
between multi-entropy and hyperthreads further.

###  4.1 Quantum correction

The discussion in Gadde:2022cqi  of quantum correction to multi-entropy
generalizes straightforwardly to the more general case of special probe
measures. The quantum corrections to the special probe measure are obtained
using the replica trick in the bulk. The web ${\cal W}$ gives a decomposition
of the bulk Cauchy slice into regions, called chambers above, ${\cal
C}_{\mathtt{a}}$’s. In the un-orbifolded geometry ${\cal B}_{n}$, the bulk
regions ${\cal C}_{\mathtt{a}}$’s get glued to each other precisely in the
same way boundary regions ${\cal R}_{\mathtt{a}}$’s were glued. This evaluates
nothing but the same special probe measure for the bulk quantum fields. If
this replica trick was performed in the bulk geometry ${\mathcal{B}}$ along
${\cal W}$, it would have given, in the $n\to 1$ limit, the bulk probe measure
$S^{\\{m_{\mathtt{a}}\\}}_{\rm bulk}({\mathcal{W}})$ corresponding to the
chamber decomposition ${\mathcal{C}}_{a}$ directly, however, because the
replica trick was performed on $\widetilde{\mathcal{B}}_{n}$, it is not
immediately clear that what we get is $S^{\\{m_{\mathtt{a}}\\}}_{\rm
bulk}({\mathcal{W}})$. This situation is similar to the bi-partite case
Faulkner:2013ana  (see also Barrella:2013wja ). There, the difference between
the two quantities is captured by changing the classical solution by
${\mathcal{O}}(G_{N})$ to account for the one-loop expectation value of the
stress tensor. This changes the area and hence the entanglement entropy by
${\mathcal{O}}(1)$. We expect a similar formula to give the sub-leading
correction to the special probe measure.

$\displaystyle
S^{\\{m_{\mathtt{a}}\\}}=\sum_{\widetilde{L}_{\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}}\in{\mathcal{W}}}\log({\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}}))\frac{\langle\hat{A}_{{\mathtt{a}},{\mathtt{b}}}\rangle}{4G_{N}}+S^{\\{m_{\mathtt{a}}\\}}_{\rm
bulk}({\mathcal{W}})+{\rm c.t.}.$ (38)

Here $\hat{A}_{{\mathtt{a}},{\mathtt{b}}}$ is the area operator of the
$({\mathtt{a}},{\mathtt{b}})$ wall of the minimal web ${\cal W}$ and ${\rm
c.t.}$ are the counter-terms that render $S_{\rm
bulk}^{({\mathtt{q}})}({\mathcal{W}})$ finite.

Following Engelhardt:2014gca , Dong:2017xht  we conjecture a formula that is
valid to all orders in $1/G_{N}$ perturbation theory: special probe measure
$S^{\\{m_{\mathtt{a}}\\}}$ is given by the above formula but ${\mathcal{W}}$
is not the ordinary area minimizing web but rather the “quantum extremal web”
i.e. the web that minimizes the combination

$\displaystyle\sum_{\widetilde{L}_{\sigma_{\mathtt{a}}^{-1}\sigma_{\mathtt{b}}}\in{\mathcal{W}}}\log({\rm
lcm}(m_{\mathtt{a}},m_{\mathtt{b}}))\frac{A_{{\mathtt{a}},{\mathtt{b}}}}{4G_{N}}+S^{\\{m_{\mathtt{a}}\\}}_{\rm
bulk}({\mathcal{W}}).$ (39)

For general gravitational theories, we expect that the special probe measure
is given by Wald-type corrections as computed in Dong:2013qoa ; Hung:2011xb
for the bi-partite case.

##  5 Marriage with holographic purification

Perhaps the most well-known multi-partite measure that has been studied in the
holographic context is the so-called reflected entropy. It is defined for a
tri-partite pure state $|\psi_{ABC}\rangle$ or equivalently a bi-partite mixed
state $\rho_{AB}$.

Before we review the definition of reflected entropy, let us review the notion
of canonical purification. Consider a density matrix $\rho_{A}\in{\rm
End}\,H_{A}$ on party $A$. It admits the Schmidt decomposition

$\displaystyle\rho_{A}=\sum_{i}\lambda_{i}|i\rangle\langle i|$ (40)

with positive $\lambda_{i}$’s and mutually orthonormal basis vectors
$|i\rangle$. The canonical purification of $\rho_{A}$ is the following pure
state “$|\sqrt{\rho_{A}}\rangle$” in $H_{A}\otimes H_{A^{\star}}$ where
$H_{A^{\star}}$ is the isomorphic “reflection” of $H_{A}$.

$\displaystyle|\sqrt{\rho_{A}}\rangle\equiv\sum_{i}\sqrt{\lambda_{i}}|i\rangle|i^{\star}\rangle.$
(41)

Here $|i^{\star}\rangle$ is the basis of $H_{A^{\star}}$. Now the reflected
entropy between $A$ and $B$ in the mixed state $\rho_{AB}$ can be defined
straightforwardly. It is simply the von Neumann entropy of $AA^{\star}$ (or
equivalently of $BB^{\star}$) in the canonical purification
$|\sqrt{\rho_{AB}}\rangle\in H_{A}\otimes H_{B}\otimes H_{A^{\star}}\otimes
H_{B^{\star}}$.

Following Engelhardt:2017aux ; Engelhardt:2018kcs ; Faulkner:2018faa , Dutta
and Faulkner Dutta:2019gen  gave a prescription for constructing the
holographic dual of the state $|\sqrt{\rho_{AB}}\rangle$ starting from the
holographic dual of the original pure state $|\psi_{ABC}\rangle$. For
simplicity, let us assume that all three parties have been defined on the
special section at the moment of time symmetry. Then the holographic dual of
$|\sqrt{\rho_{AB}}\rangle$ is obtained by cutting the dual slice of
$|\psi_{ABC}\rangle$ at the Ryu-Takayanagi surface of $C$ and gluing it to its
reflected copy along the cut. As the Ryu-Takayanagi surface is of vanishing
extrinsic curvature, this procedure does yield a solution to Einstein’s
equations. Once the dual geometry to $|\sqrt{\rho_{AB}}\rangle$ is obtained,
entanglement entropy of $AA^{\star}$ is computed using the Ryu-Takayanagi
prescription on the new geometry. This gives the answer that is two times the
minimal cut of the $AB$ entanglement wedge in the original geometry. This is
graphically denoted in figure 14.

Figure 14: The reflected geometry dual to the canonically purified state
$|\sqrt{\rho_{AB}}\rangle$ is denoted in the right figure. The entanglement
entropy of $AA^{\star}$ is twice the entanglement wedge cut as shown in the
left figure.

Even though the reflected entropy is defined for a tri-partite pure state, the
idea can be extended to define multi-partite entanglement measures. We think
of canonical purification ${\cal CP}$ as a map from a ${\mathtt{q}}$-partite
pure state with a distinguished party to a $2({\mathtt{q}}-1)$-partite pure
state.

$\displaystyle{\cal
CP}[|\psi_{A_{1},\ldots,A_{\mathtt{q}}},A_{i}\rangle]=|\sqrt{\rho_{A_{1},\ldots,A_{\mathtt{q}}}}\rangle$
(42)

Here $A_{i}$ is absent from the subscript set of $\rho$. The pure state on the
right-hand side is on $\otimes_{j\neq i}H_{A_{j}}\otimes H_{A_{j}^{\star}}$.
The map can be applied once again on
$|\sqrt{\rho_{A_{1},\ldots,A_{\mathtt{q}}}}\rangle$ with the choice of a
distinguished party and so on. Note that, while doing so,
$|\sqrt{\rho_{A_{1},\ldots,A_{\mathtt{q}}}}\rangle$ can be thought of as
either $2({\mathtt{q}}-1)$-partite state or we can re-partition the parties to
obtain a $k$-partite state with $k<2({\mathtt{q}}-1)$. One can obtain a rich
class of multi-partite measures of the original state
$|\psi_{A_{1},\ldots,A_{\mathtt{q}}}\rangle$ by evaluating a general multi-
partite measure ${\cal E}$ on the pure state obtained after successive
application of ${\cal CP}$ on $|\psi_{A_{1},\ldots,A_{\mathtt{q}}}\rangle$.
Even the entanglement entropy corresponding to any bipartition would be a
multi-partite entanglement measure of the original state. The reflected
entropy is just one such measure.

The canonical purification procedure of Dutta:2019gen  is precisely the map
${\cal CP}$. Remarkably it is closed on holographic states. Applying this
multiple times constructs a bulk solution with multiple boundary regions. Any
probe measure can then be evaluated in the new geometry. As argued above, this
measure is then a multi-partite measure also of the original holographic
state. In figure 15, we show two applications of ${\cal CP}$ on a holographic
state and tri-partite probe measures evaluated on the resulting bulk geometry.

Figure 15: The first figure shows the geometry dual to the pure state on
$ABCD$. The second figure shows the reflected geometry dual to the canonical
purification of the density matrix on $ABC$. The bottom right shows the
geometry dual to another purification, this time of the density matrix on
$AA^{\star}B$. We have also denoted the tri-partite probe measure
$S^{\\{m_{\mathtt{a}}\\}}(AA^{\star},{\widetilde{A^{\star}}},B\tilde{B}\tilde{A})$
in red. The resulting brane web is described in the original geometry through
reflections across certain minimal surfaces. For simplicity, all the minimal
length loci have been drawn as straight lines.

All in all, we have shown how the class of computable multi-partite measures
of a holographic state can be enlarged by marrying the holographic canonical
purification of Dutta:2019gen  with our probe measures defined in section 3.

The idea of canonically purifying a given holographic state multiple times has
previously appeared in Umemoto:2018jpc ; Bao:2018gck ; Bao:2019zqc ;
Chu:2019etd ; Bhattacharya:2020ymw ; Harper:2020wad .

##  6 Computations in $AdS_{3}/CFT_{2}$

In this section, we will check our holographic prescription in $2d$ conformal
field theories. To avoid clutter, we will focus on the case of special probe
measures with all the integers $m_{\mathtt{a}}=m$ taken equal121212The
discussion applies directly to the case of multi-entropy after removing all
factors of $\log m$.. For the case of three parties and three intervals, when
the results are relatively less complicated, we will also give results for the
case of unequal $m_{\mathtt{a}}$’s.

###  6.1 Using holographic prescription

In this subsection, we will compute the special probe measure between regions
of a particular Cauchy slice ${\mathcal{R}}$ of a $2d$ holographic CFT. We
will take ${\mathcal{R}}$ to be circular and at the moment of time symmetry.
We extend it into the bulk so that the bulk slice ${\mathcal{C}}$ also has the
moment of time symmetry. ${\mathcal{C}}$ is a two-dimensional hyperbolic disk
bounded by the circular conformal boundary ${\mathcal{R}}$. For our
computation, we only need to focus on this $2d$ bulk-$1d$ boundary system in
this time slice. The special probe measure is expected to be covariant under
the conformal symmetry of the boundary. For the calculation at hand, it is
sufficient to impose the subgroup of conformal symmetry that preserves
${\mathcal{R}}$. It is $SO(2,1)$, also the isometry of ${\mathcal{C}}$. To
impose conformal constraints, it is convenient to carry out the calculations
in the so-called embedding space. It is ${\mathbb{R}}^{2,1}$ on which
$SO(2,1)$ acts linearly. The boundary coordinates $P$ obey $P^{2}=0$ with the
projective identification $P\sim\lambda P$ and the bulk coordinates $X$ obey
$X^{2}=-1$ (we have set the AdS radius to $1$). In what follows, we will fix
the projective identification $P\sim\lambda P$ by picking the slice $P_{3}=1$.
In this gauge, the boundary is a circle of radius $1$.

### ${\mathtt{q}}=3$

Let us first compute the probe measure $S^{\\{m,m,m\\}}$ between the three
connected regions defined by points $P_{1},P_{2},P_{3}$ on ${\mathcal{R}}$.
According to our holographic prescription for the special probe measure,

$\displaystyle S^{\\{m,m,m\\}}=\frac{\log m}{4G_{N}}{\rm
min}_{X}\Big{(}\sum_{i}\ell(P_{i},X))\Big{)}.$ (43)

Here $X$ is some bulk point and $\ell(P,X)$ denotes the geodesic distance
between the points $P$ and $X$, ${\rm min}_{X}$ means that the expression is
to be minimized over the bulk point $X$. See figure 16.

Figure 16: Geodesic network to compute the tri-partite special probe measure.

In this dimension, the soap-film prescription is more usefully thought of as a
minimal geodesic network prescription. The geodesic distance between a bulk
point from the boundary is infinite. It is regularized as follows. We think of
the boundary point $P$ as a limit $\alpha\to\infty$ of the bulk point
$Z=\alpha P+Y$ with $P\cdot Y=0,Y^{2}=-1$.

$\displaystyle\ell(Z,X)$ $\displaystyle=\cosh^{-1}(|Z\cdot
X|)=\cosh^{-1}(|(\alpha P+Y)\cdot X|)$ $\displaystyle=\log(2|P\cdot
X|)+\log(\alpha).$ (44)

We will regularize the length $\ell(P,X)$ by subtracting the fixed infinity,
$\log(\alpha)$, for all boundary points $P$. Then

$\displaystyle S^{\\{m,m,m\\}}=\frac{\log m}{4G_{N}}{\rm
min}\Big{(}\sum_{i}\log(2|P_{i}\cdot X|))\Big{)}$ (45)

subject to the constraint $X^{2}=-1$. This is done by introducing a Lagrange
multiplier $\lambda$ that multiplies $(X^{2}+1)$ and solving for $P_{i}\cdot
X$. We give the solution for $X$ below.

$\displaystyle
X=\frac{1}{\sqrt{3}}\Big{(}\sqrt{\frac{P_{23}}{P_{12}P_{13}}}P_{1}+\sqrt{\frac{P_{13}}{P_{12}P_{23}}}P_{2}+\sqrt{\frac{P_{12}}{P_{13}P_{23}}}P_{3}\Big{)}.$
(46)

Here $P_{ij}\equiv-2P_{i}\cdot P_{j}$. This can be deduced purely from
conformal symmetry as $X$ is the only $SO(2,1)$ vector invariant under
$P_{i}\sim\lambda_{i}P_{i}$ with $X^{2}=-1$ that is symmetric in all
$P_{i}$’s. Substituting this in equation (45), we get

$\displaystyle S^{\\{m,m,m\\}}=\frac{\log
m}{4G_{N}}\Big{[}3\log\Big{(}\frac{2}{\sqrt{3}}\Big{)}+\frac{1}{2}\log\Big{(}P_{12}P_{23}P_{31}\Big{)}\Big{]}.$
(47)

Note that

$\displaystyle e^{-\frac{4G_{N}}{\log
m}S^{\\{m,m,m\\}}}=\Big{(}\frac{\sqrt{3}}{2}\Big{)}^{3}(P_{12}P_{23}P_{31})^{-1/2}.$
(48)

has the same conformal transformation properties as the three-point function
of local operators of dimension $1$ inserted at $P_{i}$.

Let us now consider the case where $m_{\mathtt{a}}$ are unequal. In this case,
the tension of the three segments $P_{1}\cdot X,P_{2}\cdot X$ and $P_{3}\cdot
X$ is unequal and is given by $\log({\rm lcm}(m_{1},m_{2})),\log({\rm
lcm}(m_{2},m_{3}))$ and $\log({\rm lcm}(m_{3},m_{1}))$. Let us label these
tensions $t_{3},t_{1}$ and $t_{2}$ respectively. The special probe measure

$\displaystyle S^{\\{m_{1},m_{2},m_{3}\\}}=\frac{1}{4G_{n}}{\rm
min}\Big{(}\sum_{i}t_{i}\log(2|P_{i}\cdot X|)\Big{)}$ (49)

The minimization over length is performed with respect $X$ satisfying
$X^{2}=-1$. This is achieved with the help of Lagrange multiplier $\lambda$.
We define

$\displaystyle g(\lambda,X)\equiv\sum_{i}t_{i}\log(2|P_{i}\cdot
X|)+\lambda(X^{2}+1)$ (50)

and vary with respect to both $\lambda$ and $X$. The solutions are,

$\displaystyle\lambda$ $\displaystyle=\frac{t_{1}+t_{2}+t_{3}}{2}$
$\displaystyle P_{1}\cdot X$
$\displaystyle=t_{1}\sqrt{\frac{(-t_{1}+t_{2}+t_{3})}{(t_{1}-t_{2}+t_{3})(t_{1}+t_{2}-t_{3})(t_{1}+t_{2}+t_{3})}}\sqrt{\frac{P_{12}P_{13}}{P_{23}}}$
$\displaystyle P_{2}\cdot X$
$\displaystyle=t_{2}\sqrt{\frac{(t_{1}-t_{2}+t_{3})}{(-t_{1}+t_{2}+t_{3})(t_{1}+t_{2}-t_{3})(t_{1}+t_{2}+t_{3})}}\sqrt{\frac{P_{12}P_{23}}{P_{13}}}$
$\displaystyle P_{3}\cdot X$
$\displaystyle=t_{3}\sqrt{\frac{(t_{1}+t_{2}-t_{3})}{(-t_{1}+t_{2}+t_{3})(t_{1}-t_{2}+t_{3})(t_{1}+t_{2}+t_{3})}}\sqrt{\frac{P_{13}P_{23}}{P_{12}}}$
(51)

Substituting in equation (49), we get

$\displaystyle 4G_{N}\,S^{\\{m_{1},m_{2},m_{3}\\}}$
$\displaystyle=\log\Big{(}\frac{\chi(2t_{1})\chi(2t_{2})\chi(2t_{3})}{\chi(-t_{1}+t_{2}+t_{3})\chi(t_{1}-t_{2}+t_{3})\chi(t_{1}+t_{2}-t_{3})\chi(t_{1}+t_{2}+t_{3})}\Big{)}$
$\displaystyle+\log\Big{(}P_{12}^{\frac{(t_{1}+t_{2}-t_{3})}{2}}P_{23}^{\frac{(-t_{1}+t_{2}+t_{3})}{2}}P_{13}^{\frac{(t_{1}-t_{2}+t_{3})}{2}}\Big{)},\qquad\qquad\chi(x)=x^{\frac{x}{2}}.$
(52)

### ${\mathtt{q}}=4$

Let us look at the special probe measure of four connected regions defined by
the points $P_{1},\ldots,P_{4}$. It is given by the minimum of the two
extremal geodesic networks given in figure 17.

Figure 17: Geodesic network to compute the 4-partite special probe measure.

We will refer to them suggestively as s-channel and t-channel geodesic
networks. From conformal symmetry, we expect it to take the form

$\displaystyle\frac{4G_{N}}{\log
m}S^{\\{m,m,m,m\\}}=\log(P_{12}P_{34})+f(z),\quad
z\equiv\Big{(}\frac{P_{12}P_{34}}{P_{13}P_{24}}\Big{)}^{\frac{1}{2}}.$ (53)

The coefficient of the first term $\log(P_{12}P_{23})$ is determined by
requiring $e^{-\frac{4G_{N}}{\log m}S{\\{m,m,m,m\\}}}$ transform as a
conformal correlation function of operators with dimension $1$. Here $z$ is
the usual cross-ratio of four points. It is clear that near $z=0$,
$S^{\\{m,m,m,m\\}}$ is given by the s-channel network and near $z=1$, it is
given by t-channel network, with phase transition occurring at an intermediate
value of $z$. For concreteness let us take $z$ to be finite but close to $0$.
We can compute $S^{\\{m,m,m,m\\}}$ without actually solving the extremization
problem but rather by using symmetry and ${\mathtt{q}}=3$ computation. To
compute $f(z)$, we use conformal transformation to arrange the point
symmetrically as shown in figure 18.

Figure 18: Geodesic network to compute the 4-partite special probe measure.

The parametrization of $P_{i}$ in terms of $\theta$ is

$\displaystyle P_{1}=(-\cos\theta,\sin\theta,1),\quad
P_{2}=(-\cos\theta,-\sin\theta,1)$ $\displaystyle
P_{3}=(\cos\theta,-\sin\theta,1),\quad P_{4}=(\cos\theta,\sin\theta,1)$ (54)

This gives $z=\sin^{2}\theta$. Now realize that the length of the minimal
four-point geodesic network is the sum of lengths of minimal three-point
networks $(P_{1},P_{2},Q)$ and $(P_{3},P_{4},Q^{\prime})$ minus the “diameter”
$(Q,Q^{\prime})$. We get,

$\displaystyle S^{\\{m,m,m,m\\}}=\frac{\log
m}{4G_{N}}\Big{[}6\log\Big{(}\frac{2}{\sqrt{3}}\Big{)}+\log\Big{(}\frac{1+\sqrt{1-z}}{2z}\Big{)}+\log(P_{12}P_{34})\Big{]}.$
(55)

When $z$ is closer to $1$, the t-channel network dominates and
$S^{\\{m,m,m,m\\}}$ is obtained from the above answer by permuting $P_{1}$ and
$P_{3}$.

### ${\mathtt{q}}=5$

To compute $S^{\\{m,m,m,m,m\\}}$ for five connected regions the same trick can
be used. The number of cross-ratios for five points in $1d$ is two. Unlike in
the ${\mathtt{q}}=4$ case, there is no canonical choice of cross-ratios, we
simply parametrize this space with two angles $\theta_{1},\theta_{2}$ labeling
the cyclically symmetric configuration of points shown in figure 19.

Figure 19: Symmetric configuration of the geodesic network to compute the
general 5-partite special probe measure.

The points are $P_{1}=(1,0,1)$ and

$\displaystyle P_{2,3}$
$\displaystyle=(\cos(\frac{2\pi}{3}\mp\theta_{1}),\sin(\frac{2\pi}{3}\mp\theta_{1}),1),$
$\displaystyle P_{4,5}$
$\displaystyle=(\cos(\frac{2\pi}{3}\mp\theta_{2}),\sin(\frac{2\pi}{3}\mp\theta_{2}),1).$
(56)

The length of the minimal geodesic network is just the sum of lengths of
three-point networks $(P_{2},P_{3},Q)$ and $(P_{4},P_{5},Q^{\prime})$ minus
half the diameter. This can be easily evaluated. We get

$\displaystyle S^{\\{m,m,m,m,m\\}}$ $\displaystyle=\frac{\log
m}{4G_{N}}\left[9\log\Big{(}\frac{2}{\sqrt{3}}\Big{)}+\frac{1}{2}\log(P_{12}P_{23}P_{34}P_{45}P_{51})\right.$
(57)
$\displaystyle\left.+\frac{1}{2}\log\left(\left(\frac{3}{4}\right)^{3}\frac{(1+\cos\theta_{1})^{2}(1+\cos\theta_{2})^{2}}{(1+\cos(\frac{\pi}{3}+\theta_{1}))(1+\cos(\frac{\pi}{3}+\theta_{1}))(1+\cos(\frac{\pi}{3}+\theta_{1}+\theta_{2}))}\right)\right].$

Here we have made a cyclically symmetric choice of the “prefactor”
$(P_{12}P_{23}P_{34}P_{45}P_{51})^{1/2}$ in $e^{-\frac{4G_{N}}{\log
m}S^{\\{m,m,m,m,m\\}}}$ to saturate the unit conformal dimensions of all the
operators.

###  6.2 CFT computation

Now we move on to compute special probe measures with $m_{\mathtt{a}}=m$ for
$3$, $4$ and $5$ for connected regions directly from CFT. We do this by first
computing the twist operator correlation function ${\cal
E}^{\\{m^{n-1},m^{n-1},\ldots\\}}$ using CFT techniques. The special probe
measure is then computed using the formula

$\displaystyle S^{\\{m,m,\ldots\\}}=\lim_{n\to 1}\frac{1}{1-n}\log\Big{(}{\cal
E}^{\\{m^{n-1},m^{n-1},\ldots\\}}\Big{)}.$ (58)

As a warm-up consider the bi-partite case ${\mathtt{q}}=2$.

$\displaystyle{\cal E}^{\\{m^{n-1},m^{n-1}\\}}$
$\displaystyle=\langle{{\mathcal{O}}}_{\sigma_{1}^{-1}\sigma_{2}}(P_{1}){{\mathcal{O}}}_{\sigma_{2}^{-1}\sigma_{1}}(P_{2})\rangle$
(59)

where $\sigma_{1}$ and $\sigma_{2}$ are two independent
${\mathbb{Z}}_{m^{n-1}}$ generators. This correlation function has been
computed in Lunin:2000yv  using the Liouville action for uniformization for
general conformal field theories. It only depends on the conjugacy class of
the permutation element associated with the twist operator.

$\displaystyle{\cal
E}^{\\{m^{n-1},m^{n-1}\\}}=(P_{12})^{-\Delta_{\sigma_{1}^{-1}\sigma_{2}}}.$
(60)

For a permutation element of the conjugacy class $\\{p_{k}\\}$ ($p_{k}$ is the
number of $k$ cycles), we have

$\displaystyle\Delta=\frac{c}{12}\sum_{k}p_{k}\Big{(}k-\frac{1}{k}\Big{)}.$
(61)

For twist operators of special probes for ${\mathtt{q}}$ parties, only
$p_{m^{n-1}}=(m^{n-1})^{{\mathtt{q}}-1}$ and the rest are $0$. so
$\Delta_{\hat{\sigma}_{1}^{-1}\hat{\sigma}_{2}}=\frac{c}{12}(m^{n-1})^{{\mathtt{q}}-2}(m^{2(n-1)}-1)\equiv\Delta^{({\mathtt{q}})}_{n}(m)$.
Using equation (58) it follows that

$\displaystyle S^{\\{m,m\\}}=\frac{c\,\log m}{6}\log(P_{12}).$ (62)

Using the holographic dictionary $cG_{N}=3/2$, we see that
$e^{-\frac{4G_{N}}{\log m}S^{\\{m,m\\}}}$ indeed transforms like a two-point
function of operators with dimension $1$ as expected.

### ${\mathtt{q}}=3$

In this case,

$\displaystyle{\cal E}^{\\{m^{n-1},m^{n-1},m^{n-1}\\}}$
$\displaystyle=\langle{{\mathcal{O}}}_{\sigma_{1}^{-1}\sigma_{2}}(P_{1}){{\mathcal{O}}}_{\sigma_{2}^{-1}\sigma_{3}}(P_{2}){{\mathcal{O}}}_{\sigma_{3}^{-1}\sigma_{1}}(P_{3})\rangle$
(63)
$\displaystyle=C_{n}(m)\,(P_{12}P_{23}P_{31})^{-\frac{1}{2}\Delta_{n}^{(3)}(m)}.$
(64)

All the twist operators in this correlation function have the same conformal
dimension $\Delta_{n}^{({\mathtt{q}})}(m)$. The OPE coefficient $C_{n}(m)$ can
not be fixed using conformal symmetry. In the cases where the replicated
manifold turns out to be a sphere the OPE coefficient can be computed using
uniformization, see Lunin:2000yv ; Lunin:2001pw . Unfortunately, this approach
seems difficult in our case since the genus of the replicated manifold
increases with $n$. Pressing on,

$\displaystyle S^{\\{m,m,m\\}}$ $\displaystyle=\frac{c\log
m}{6}\Big{[}\kappa(m)+\frac{1}{2}\log(P_{12}P_{23}P_{31})\Big{]}$
$\displaystyle{\rm where}\qquad\kappa(m)$ $\displaystyle=\frac{1}{\log
m}\log\Big{(}\partial_{n}\Big{(}\frac{C_{n}(m)}{C_{1}^{n}(m)}\Big{)}_{n=1}\Big{)}.$
(65)

We can’t compute $\kappa(m)$ from CFT as we don’t know $C_{n}(m)$. If our
holographic prescription is correct then comparing the above expression to the
one obtained using holographic prescription (47), we arrive at the conjecture

$\displaystyle\kappa(m)=3\log\Big{(}\frac{2}{\sqrt{3}}\Big{)}.$ (66)

### ${\mathtt{q}}=4$

In this case, we need the four-point function of twist operators.

$\displaystyle{\cal E}^{\\{m^{n-1},m^{n-1},m^{n-1},m^{n-1}\\}}$
$\displaystyle=\langle{{\mathcal{O}}}_{\sigma_{1}^{-1}\sigma_{2}}{{\mathcal{O}}}_{\sigma_{2}^{-1}\sigma_{3}}{{\mathcal{O}}}_{\sigma_{3}^{-1}\sigma_{4}}{{\mathcal{O}}}_{\sigma_{4}^{-1}\sigma_{1}}\rangle.$
(67)

The operators are inserted at $P_{1},\ldots,P_{4}$ respectively. When the
cross-ratio $z$ is in a finite neighborhood of $0$ (of $1$), it can be shown
that for a large $c$ conformal field theory, the correlator is dominated by
the Virasoro block of the lightest operator in the s-channel (t-channel). This
operator is the twist operator ${{\mathcal{O}}}_{\sigma_{1}^{-1}\sigma_{3}}$
(${{\mathcal{O}}}_{\sigma_{2}^{-1}\sigma_{4}}$).

$\displaystyle{\cal
E}^{\\{m^{n-1},m^{n-1},m^{n-1},m^{n-1}\\}}=\tilde{C}_{n}(m){\tilde{C}^{\prime}}_{n}(m)(P_{12}P_{34})^{-\Delta_{n}^{(4)}(m)}F_{\Delta^{(4)}_{n}(m)}(z,\bar{z}).$
(68)

Here $F_{\Delta}(z,\bar{z})$ is the Virasoro block of an operator of dimension
$\Delta$. The OPE coefficients $\tilde{C}_{n}(m)$ and
$\tilde{C}^{\prime}_{n}(m)$ are the three point function coefficients in
$\langle{{\mathcal{O}}}_{\hat{\sigma}_{1}^{-1}\hat{\sigma}_{2}}{{\mathcal{O}}}_{\hat{\sigma}_{2}^{-1}\hat{\sigma}_{3}}{{\mathcal{O}}}_{\hat{\sigma}_{3}^{-1}\hat{\sigma}_{1}}\rangle$
and
$\langle{{\mathcal{O}}}_{\hat{\sigma}_{1}^{-1}\hat{\sigma}_{3}}{{\mathcal{O}}}_{\hat{\sigma}_{3}^{-1}\hat{\sigma}_{4}}{{\mathcal{O}}}_{\hat{\sigma}_{4}^{-1}\hat{\sigma}_{1}}\rangle$
respectively. As our permutation elements are symmetric under the exchange of
parties, both these three-point functions are the same and are equal to
$C_{n}(m)$ in equation (63). This gives

$\displaystyle S^{\\{m,m,m,m\\}}=\frac{c\log
m}{6}\Big{[}2\kappa(m)+\log(P_{12}P_{34})-\partial_{n}\log
F_{\Delta_{n}^{(4)}(m)}|_{n=1}\Big{]}.$ (69)

The $\log F_{\Delta}$ can be computed for large $c$ CFT in a perturbative
expansion in $\Delta$ using the so-called monodromy method. This method is
reviewed in appendix C. Because $\Delta\to 0$ in the $n\to 1$ limit, the
special probe $S^{\\{m,m,m,m\\}}$ can be computed just from the first term in
the perturbative expansion. We have done this computation in appendix C.1 and
have found the result to agree with the holographic expectation (55). Note
that the conformal symmetry only guarantees that the holographic probe measure
depends on the cross-ratio, the fact that it is the same function of cross-
ratio that we get from the CFT (up to a constant) serves as a check of our
holographic prescription.

### ${\mathtt{q}}=5$

We will parametrize the five points in the same way as in section 6. For
simplicity, we will only consider the channel shown in figure 19. The
correlation function is dominated by the lightest Virasoro block in this
channel when $\theta_{1}$ and $\theta_{2}$ are small but finite. The OPE
coefficients appearing at all three vertices are the same, $C_{n}(m)$. So we
have,

$\displaystyle S^{\\{m,m,m,m,m\\}}$ $\displaystyle=\frac{c\log
m}{6}\Big{[}3\kappa(m)+\frac{1}{2}\log(P_{12}P_{23}P_{34}P_{45}P_{51})$
$\displaystyle-\partial_{n}\log
F_{\Delta_{n}^{(5)}(m),\Delta_{n}^{(5)}(m)}(\theta_{1},\theta_{2})|_{n=1}\Big{]}.$
(70)

Here $F_{\Delta_{1},\Delta_{2}}(\theta_{1},\theta_{2})$ is the conformal block
in the channel displayed in figure 19. We again use the monodromy method for
the case of five points and we compute the leading term in $\log
F_{\Delta_{n}^{(5)},\Delta_{n}^{(5)}}$ in appendix C.2. The resulting
$S^{\\{m,m,m,m,m\\}}$ agrees with the holographic prediction (57). Again, _a
priori_ this agreement is not guaranteed so this match does strengthen the
check of our holographic prescription.

####  6.2.1 General argument

A beautiful argument - using the $SL(2,C)$ Chern-Simons theory description of
gravity in AdS3 \- has been given in Hijano:2015rla  that shows in full
generality that the logarithm of a general point conformal block, at leading
order in $G_{N}$, is given by the length of the minimal geodesic network up to
an overall constant. The constant is fixed by taking the OPE limit. We will
reproduce this argument here for the reader’s convenience. For details, please
see Hijano:2015rla .

Consider the Fefferman-Graham expansion of the locally $AdS_{3}$ metric in the
presence of a massive particle

$\displaystyle
ds^{2}=d\rho^{2}+e^{2\rho}g^{(0)}_{\mu\nu}dx^{\mu}dx^{\nu}+g^{(2)}_{\mu\nu}dx^{\mu}dx^{\nu}+\ldots$
(71)

with $g^{(0)}_{\mu\nu}dx^{\mu}dx^{\nu}=dzd\bar{z}$. Let the world-line of the
massive particle, i.e. with the action $S_{\rm particle}=2M\int
d\lambda\sqrt{h}$ where $h$ is the induced metric on the world-line, pierce
the boundary at $z_{0}$. We wish to solve Einstein’s equations near this
point. Expanding the Einstein equations at leading order in large $\rho$, the
only non-vanishing equations are

$\displaystyle g_{z\bar{z}}^{(2)}$ $\displaystyle=2\pi M\delta^{(2)}(z-z_{0})$
$\displaystyle\partial_{\bar{z}}g_{zz}^{(2)}-\partial_{z}g_{z\bar{z}}^{(2)}$
$\displaystyle=-4\pi p_{z}\delta^{(2)}(z-z_{0})$
$\displaystyle\partial_{z}g_{\bar{z}\bar{z}}^{(2)}-\partial_{\bar{z}}g_{z\bar{z}}^{(2)}$
$\displaystyle=-4\pi p_{\bar{z}}\delta^{(2)}(z-z_{0}).$ (72)

Here $p_{\mu}$ is the momentum defined through,
$\frac{c}{6}p^{\mu}=2M\,dx^{\mu}/d\lambda$ with $c$ being the Brown-Henneaux
central charge $c=3/2G$. When it is large, the mass is related to the
conformal dimension as $M=\Delta\equiv\delta c/6$. Using the fact that
boundary stress tensor is $T_{\mu\nu}=g^{(2)}_{\mu\nu}$, singular terms in the
holomorphic stress tensor can be computed using the above expression.

$\displaystyle
T(z)=\frac{\delta}{(z-z_{0})^{2}}+\frac{2p_{z}}{z-z_{0}}+\ldots$ (73)

As the momentum $p_{z}$ is canonical conjugate to the coordinate $z$, we have
$p_{z}=dS_{\rm particle}/dz$. In terms of $T(z)$, the metric takes the form

$\displaystyle
ds^{2}=d\rho^{2}-Tdz^{2}-\bar{T}d{\bar{z}}^{2}+(e^{2\rho}+T\bar{T}e^{-2\rho})dzd\bar{z}.$
(74)

At this point, it is useful to pass to the Chern-Simons description of
$AdS_{3}$ gravity. The $SL(2)\times SL(2)$ connection that corresponds to the
above metric is

$\displaystyle A=\begin{pmatrix}\frac{1}{2}d\rho&e^{-\rho}Tdz\\\
-e^{\rho}dz&-\frac{1}{2}d\rho\end{pmatrix},\qquad\bar{A}=\begin{pmatrix}-\frac{1}{2}d\rho&e^{\rho}dz\\\
-e^{-\rho}\bar{T}dz&\frac{1}{2}d\rho\end{pmatrix}.$ (75)

In the absence of matter, $T(z)$ obeys the equation $\partial_{\bar{z}}T(z)=0$
and hence the holonomy of this connection around a closed contour would be
zero. However, when the contour encloses a massive particle, the holonomy of
$A$ is nontrivial due to the conical singularity. Computing this holonomy is
equivalent to computing the monodromy of a two-component vector function
$\psi(z)$ that satisfies the equation

$\displaystyle\frac{d\psi}{dz}=A\psi.$ (76)

The bottom component of $\psi$ then obeys,

$\displaystyle\psi_{2}^{\prime\prime}+T(z)\psi_{2}=0.$ (77)

This is exactly the differential equation that appears in the monodromy method
(102). Moreover, the function $T(z)$ given in equation (73) is exactly equal
to the one appearing in (103) with $p_{z}$ playing the role of the accessory
parameter $c_{\mathtt{a}}(z)$. When the $\psi_{2}$ is taken around a group of
external sources for the massive operators, its path is homologous to a closed
loop around the world-line of the massive particle appearing in this channel.
The holonomy of $A$ around such a loop is such that it produces the monodromy
of $\psi_{2}$ that matches the one required in the monodromy method of
appendix C.

This argument shows in full generality that computation of the action of a
world-line network of massive particles is equivalent to the computation of
the corresponding Virasoro conformal block in the CFT with large central
charge $c$ (at leading order in $c$). Note that, although the actual
computation of either is possible only in perturbation theory in $\delta$
where $\Delta=\delta c/6$ i.e. in the limit $n\to 1$, the equivalence of the
two is shown exactly in $n$.

The minimal geodesic network in a hyperbolic disc is also called a Steiner
tree. The connection between Virasoro conformal blocks and Steiner trees has
been analyzed in detail in Alkalaev:2018nik . Some of our analysis in section
6 overlaps with theirs.

####  6.2.2 Limitation of the check

As remarked earlier, using the CFT methods we are only able to compute the
dominant conformal block in the twist operator correlation function. To
compute the correlation function itself in the large $c$ limit, we also need
to compute the three-point function coefficient $C_{n}(m)$ of twist operators
at large $c$. This computation is difficult to do as it maps to the
computation of the CFT partition function on a higher genus Riemann surface
which could potentially be non-universal. This does not allow the
uniformization methods of Lunin:2000yv ; Lunin:2001pw  to be applicable. The
holographic prescription gives a conjecture

$\displaystyle\kappa(m)=3\log\Big{(}\frac{2}{\sqrt{3}}\Big{)}$ (78)

where $\kappa(m)$ is defined in the equation (6). It would be nice to check
the same from the CFT. One possible way to do so could be along the lines of
Cardy:2017qhl . In this paper, the authors use the modular invariance of genus
$2$ partition function of 2d CFT to compute the average of the three-point
coefficient squared for heavy operators. If one believes, that the twist
operators considered here are generic enough that their three-point function
should match with the average, then methods of Cardy:2017qhl  could be used to
test the conjecture (78). We will not have anything more to say about this.

Even after fixing this value of $\kappa(m)$, there is yet another check. The
prediction from the CFT is that ${\mathtt{q}}$-partite special probe measure
for ${\mathtt{q}}$ connected regions must take the form,

$\displaystyle S^{\\{m,m,\ldots\\}}=({\mathtt{q}}-2)\kappa(m)+\partial_{n}\log
F^{({\mathtt{q}})}_{n}|_{n=1}$ (79)

where $F^{({\mathtt{q}})}$ is the Virasoro conformal block in some channel.
This is because there are ${\mathtt{q}}-2$ trivalent vertices that appear in a
${\mathtt{q}}$-point conformal block. The argument outlined in this section
shows that $S^{\\{m,m,\ldots\\}}_{\rm holographic}-\partial_{n}\log
F^{({\mathtt{q}})}_{n}|_{n=1}$ is a constant, say $\kappa^{({\mathtt{q}})}$.
It is then nontrivial that the ${\mathtt{q}}$ dependance of
$\kappa^{({\mathtt{q}})}$ is such that
$\kappa^{({\mathtt{q}})}=({\mathtt{q}}-2)\kappa$. We have checked this for
${\mathtt{q}}=4,5$. It would be nice to see that this continues to hold for
higher points and hopefully produce an argument that works for all higher
values of ${\mathtt{q}}$.

###  6.3 Comment on bulk replica symmetry

In this section, we will discuss the status of the replica symmetry
assumption. We will discuss this assumption only for the case of the probe
family $S^{({\mathtt{q}})}_{n}$ that arises from the special symmetric measure
${\cal E}^{\\{n,n,\ldots\\}}$ in $2d$ CFTs. Just like any other measure, this
measure is computed from the path integral on the ramified manifold obtained
by cutting and gluing replicas along given spatial regions in the fashion
specified by the permutations. The genus of the ramified boundary is easily
computed using the Riemann-Hurwitz formula

$\displaystyle
g_{({\mathtt{q}},n)}=1+\frac{n^{{\mathtt{q}}-2}}{2}(n({\mathtt{q}}-2)-{\mathtt{q}}).$
(80)

As expected, for ${\mathtt{q}}=2$ we get $g=0$ for all $n$. This is simply the
statement that the replicated manifold corresponding to the bi-partite Renyi
entropy between an interval and its complement is topologically a sphere. For
$({\mathtt{q}},n)=(3,2)$ also we get $g=0$. This case has been studied in
Penington:2022dhr . For $({\mathtt{q}},n)=(3,3),(4,2)$, the genus is $1$ and
for every other case $g\geq 3$. The case of $({\mathtt{q}},n)=(3,3)$ was also
considered in Penington:2022dhr  and it was argued that the dominant bulk
solution does not preserve the replica symmetry
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$. The argument is as follows.131313We
thank Geoff Penington for the discussion on this issue.

Configuration of three points on a sphere does not admit any modulus so the
complex structure of the ramified manifold is fixed. It is fixed to the
${\mathbb{Z}}_{3}$ symmetric point $\tau=e^{i\pi/3}$. As a result, the torus
has three shortest cycles and they are transformed into each other by the
action of a subgroup of the replica symmetry. The dominant bulk solution must
correspond to the torus handle-body where one of the three shortest cycles
becomes contractible in the bulk. This bulk solution breaks the symmetry
between the three cycles and hence also a part of the replica symmetry. This
is an example where the assumption that the dominant bulk saddle is replica
symmetry preserving is invalid. Does this mean that the replica symmetry
assumption is incorrect even for other values of $({\mathtt{q}},n)$? We have
verified the correctness of our prescription, which followed from the replica
symmetry assumption, for ${\mathtt{q}}\geq 4$ using conformal field theory
methods. However, it is possible that even in the absence of full replica
symmetry of the dominant bulk solution, the correct prescription may coincide
with the one obtained from CFT methods. Therefore, it would be useful to have
a direct bulk argument that supports the replica symmetry assumption at least
for ${\mathtt{q}}\geq 4$. In what follows, we will give such an argument for
$({\mathtt{q}},n)=(4,2)$. We will explicitly construct the dominant bulk
solution and show that it is replica symmetric.

Consider the measure $S_{2}^{(4)}$ where the parties correspond to four
connected regions on the equator of the sphere as shown in figure 20.

Figure 20: Configuration of twist operators on the equator.

For simplicity, we have taken the regions $1$ and $3$ to be smaller than $2$
and $4$. The equatorial configuration of twist operators maps to a rectangular
torus. The gluing pattern of all the regions in eight replicas is given in the
first diagram of figure 21. This is the graph with the density matrix on
parties $1,2,3$ as vertices.

Figure 21: The first figure gives the gluing pattern of all the replicas used
to construct the measure. The second figure shows the ramified manifold viz.
torus tessellated by the replicas. In the second picture, we identify the two
homologically independent cycles. They are denoted by the red and green dotted
lines. These cycles are then pulled back to the figure.

In the second diagram in figure 21, we have shown how the individual replicas
glue together to form the torus. The two homologically independent cycles of
the torus are denoted by the red and green dotted lines. Their pull back to
the graph is also shown in the same way. The action of the replica group
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ is clear on the
graph. The independent cycles are not mixed under the replica group action.
The shortest cycle of the torus is the one shown by a red dotted line. The
dominant bulk saddle fills out this cycle to form a hyperbolic genus-$1$
handle-body. Because the cycles are invariant under the replica symmetry group
the dominant bulk saddle is also invariant.

This construction is possible because $g_{(4,2)}=1$. For $n=2$ and higher
values of ${\mathtt{q}}$, the genus is larger and hence the analysis becomes
cumbersome but not completely intractable. Currently, we are in the process of
computing replica symmetric solutions for the case of $({\mathtt{q}},2)$ for
general ${\mathtt{q}}$ GaddeWIP . We don’t have anything definitive to say
about the status of replica symmetric solutions for higher values of $n$.

##  7 Discussion

### Usefulness to Quantum information theory

The measures that we have constructed are genuinely multi-partite. Let us
elaborate. One can find pairs of multi-partite states, known as isospectral
states, with the property that every density matrix constructed out of them by
partial trace has the same spectrum and even then they are not equivalent to
each other up to local unitary transformations. As a result, any combination
of bi-partite Renyi entropies and Von Neumann entropies yields identical
results. A very simple example is furnished by the following pair Nielsen_2001
; Gadde:2022cqi .

$\displaystyle|\psi_{1}\rangle=\frac{1}{\sqrt{3}}|000\rangle+\frac{\sqrt{2}}{\sqrt{3}}|111\rangle,\quad|\psi_{2}\rangle=\frac{1}{\sqrt{3}}(|100\rangle+|010\rangle+|001\rangle).$
(81)

However, as emphasized in Gadde:2022cqi , the multi-partite measures
introduced in this paper, do distinguish them. This illustrates their
usefulness.

Another useful property that a multi-partite entanglement measure should
satisfy is that it should be monotonic under coarse-graining Hein_2004  i.e.
if ${\cal M}^{({\mathtt{q}})}$ is a ${\mathtt{q}}$-partite measure evaluated
on a ${\mathtt{q}}$-partite state $|\psi\rangle$ and ${\cal
M}^{{\mathtt{q}}-1}_{[{\mathtt{a}},{\mathtt{b}}]}$ is a
$({\mathtt{q}}-1)$-partite measure evaluated on a $({\mathtt{q}}-1)$-partite
state obtained from $|\psi\rangle$ by identifying a pair of parties
${\mathtt{a}},{\mathtt{b}}$ and thinking of their tensor product as a single
party then it is desirable that the measure (or more appropriately the family
of measures ${\cal M}^{({\mathtt{q}})}$) obey

$\displaystyle{\cal M}^{({\mathtt{q}})}-{\cal
M}^{({\mathtt{q}}-1)}_{[{\mathtt{a}},{\mathtt{b}}]}\geq 0.$ (82)

This is intuitive because by identifying a pair of parties, the measure allows
for a scrambling unitary operator that mixes the two parties. It is expected
that the entanglement should reduce under such scrambling. We are currently
exploring this inequality for the measures discussed in this paper. Another
viewpoint on the inequality (82) can be obtained by thinking of the left-hand
side as a sort of “multi-partite” mutual information ${\cal
I}^{({\mathtt{q}})}_{{\mathtt{a}},{\mathtt{b}}}$ between the parties
${\mathtt{a}}$ and ${\mathtt{b}}$ so that the inequality (82) is a statement
of its positivity. To be clear, this is not the usual mutual information but a
novel one that has to do with the entanglement of parties ${\mathtt{a}}$ and
${\mathtt{b}}$ with themselves directly as well as “through” other parties. It
is then useful study conditions that lead to the vanishing of this “multi-
partite” mutual information ${\cal
I}^{({\mathtt{q}})}_{{\mathtt{a}},{\mathtt{b}}}$. The advantage of this point
of view will become clear in this next discussion point.

### Application to bulk reconstruction

Arguably the most important application of information-theoretic ideas to
AdS/CFT correspondence has been the so-called entanglement wedge
reconstruction Dong:2016eik ; Harlow:2016vwg . This idea relies on formulating
the AdS/CFT correspondence as an error-correcting code Almheiri:2014lwa  and
on the observation that boundary modular flow is equal to the bulk modular
flow Jafferis:2015del . It was also shown that a version of the quantum-
corrected Ryu-Takayanagi formula holds for any error-correcting code
Harlow:2016vwg . Let’s review the statement of entanglement wedge
reconstruction in the simplest context of the vacuum state.

Let the AdS/CFT map be an isometry $W$ from the bulk Hilbert space $H_{\rm
bulk}$ into the boundary Hilbert space $H_{\rm boundary}$. Decomposition of
the boundary into regions $A$ and its complement $\bar{A}$ corresponds to the
tensor decomposition

$\displaystyle H_{\rm boundary}=H_{A}\otimes H_{\bar{A}}.$ (83)

The Ryu-Takayanagi surface for a given boundary region $A$ divides the bulk
into the entanglement wedge $a$ of $A$ and its complement. This gives the
tensor decomposition of the bulk Hilbert space

$\displaystyle H_{\rm bulk}=H_{a}\otimes H_{\bar{a}}.$ (84)

We can now ask refined questions about the isometry for the above tensor
product decomposition. The entanglement wedge reconstruction is the statement
that bulk operator $O_{\rm bulk}$ acting on $H_{a}$ can be reconstructed on
the boundary as an operator $O_{\rm boundary}$ that only acts on $H_{A}$. A
similar statement can also be made for the region $\bar{A}$ and its
entanglement wedge $\bar{a}$. Alternatively, the reconstruction can also be
phrased in the language of error correcting codes viz. the encoding of $H_{a}$
is robust against arbitrary errors acting on $H_{\bar{A}}$. In simpler
language, any quantum operation on $\bar{A}$ will not destroy “information”