††institutetext: Martin Fisher School of Physics, Brandeis University,
Waltham, Massachusetts 02453, USA

# Bit threads and holographic entanglement of purification

Jonathan Harper and Matthew Headrick<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Generalizing the bit thread formalism, we use convex duality to derive dual
flow programs to the bipartite and multipartite holographic entanglement of
purification proposals and then prove several inequalities using these
constructions. In the multipartite case we find the flows exhibit novel
behavior which allows for a constrained flux on the boundary of the homology
region. We show this flux can be made distinct from bipartite terms and
reflects the truly multipartite portion of the holographic entanglement of
purification.

††preprint: BRX-TH-6642

##  1 Introduction

The study of geometry has been key to our current understanding of quantum
gravity. This is most strongly evidenced by the Ryu-Takayanagi (RT) formula,
which allows one to calculate the entanglement entropy as the area of a bulk
minimal surface for systems with holographic duals (in static states or states
with time-reflection symmetry). The entanglement entropy is the canonical
measure of entanglement for pure states. Recent work has attempted to define a
similar measure of entanglement for a mixed state on a bipartite region $AB$
as the cross section $E_{w}(A:B)$ of the joint homology region $r(AB)$, the
bulk region bounded by $AB$ and its RT surface $m(AB)$ (see fig. 1).111For
this paper we will only be working with the static or time reflection-
symmetric spacetimes, where our slice is a moment of time symmetry. The
homology region $r(AB)$ is a slice of the entanglement wedge $E_{W}[AB]$. As
we will briefly discuss in sec. 4, the conjecture has a natural generalization
to the covariant case. More precisely, $E_{w}(A:B)$ is the area of the minimal
surface in $r(AB)$ homologous to $A$ relative to $m(AB)$; we will denote this
surface $b_{p}(A:B)$.

Figure 1:  Illustration of the homology-region cross section $E_{w}(A:B)$,
conjectured to be dual to the entanglement of purification (1). The $AB$
homology region $r(AB)$ is shown in blue. $E_{w}(A:B)$ is defined as the area
of the minimal surface $b_{p}(A:B)$ in $r(AB)$ homologous to $A$ relative to
$m(AB)$.

This geometric quantity is conjectured to be dual to a certain measure of
quantum and classical correlations in the boundary theory, namely the
entanglement of purification (EOP) Umemoto_2018 ; Nguyen2018 .222Several
competing conjectures have also appeared in the literature. In Dutta:aa  the
homology-region cross section was calculated to be equal to the reflected
entropy: half the entanglement entropy of a canonical purification whose dual
geometry is a wormhole formed by gluing two copies of the homology region. In
Kudler-Flam:2018aa  the logarithmic negativity was proposed as a dual to the
homology region cross section and in Tamaoka_2019  the odd entanglement
entropy. (For recent progress in the context of the surface/state
correspondence Miyaji:2015aa  see Bao:2018ac ; Guo:2019aa , and in terms of
bit threads see bao:aa .) The EOP is defined as follows Terhal:2002aa :

$E_{p}(\rho_{AB})=\min_{|{\psi}\rangle_{AA^{\prime}BB^{\prime}}:\atop\psi_{AB}=\rho_{AB}}S(\psi_{AA^{\prime}})\,;$
(1)

here the minimization is over purifications
$|{\psi}\rangle_{AA^{\prime}BB^{\prime}}$ of $\rho_{AB}$, where $A^{\prime}$,
$B^{\prime}$ are auxiliary systems. In general this is a very difficult
quantity to calculate, especially in QFTs (see e.g. Bhattacharyya2018 ;
Caputa_2019 ; Bhattacharyya:2019aa ; Nguyen2018 ). Despite this difficulty
several general bounds are know for the EOP Bagchi:2015aa :
$\frac{1}{2}I(A:B)\leq E_{p}(A:B)\leq\min[S(A),S(B)]$ $E_{p}(A:BC)\geq
E_{p}(A:B)$ $E_{p}(A:BC)\geq\frac{1}{2}I(A:B)+\frac{1}{2}I(A:C)$ If $ABC$ is
pure then $E_{p}(A:B)+E_{p}(A:C)\geq E_{p}(A:BC)$ It has been shown via
geometric proofs using surfaces that the minimal homology region cross section
$E_{w}(A:B)$ too satisfies these conditions lending credibility to the
conjecture

$E_{p}(A:B)=E_{w}(A:B)\,.$ (2)

Additionally, it was shown that the minimal homology region cross section
satisfies the superadditivity condition $E_{w}(A\tilde{A}:B\tilde{B})\geq
E_{w}(A:B)+E_{w}(\tilde{A}:\tilde{B})$ which $E_{p}$ does not always satisfy.
If the conjecture holds, then this inequality would be a necessary criterion
for a system to have a holographic dual; this is similar to how the
holographic entanglement entropy always satisfies monogamy of mutual
information (MMI), even though general quantum systems may violate MMI
Hayden:2011ag ; Headrick2014 .

In this paper we take a complementary approach to the holographic surface
proposals: using the full power of convex duality and the max flow-min cut
theorem (MFMC), we derive the dual convex flow program

$E_{w}(A:B)\coloneqq\max_{v^{\mu}}\int_{A}\sqrt{h}\,n_{\mu}v^{\mu}\quad\text{such
that}\quad\nabla_{\mu}v^{\mu}=0,\quad|v|\leq 1,\quad
n_{\mu}v^{\mu}|_{m(AB)}=0\,,$ (3)

which works by explicitly forcing any flow to remain inside the homology
region $r(AB)$. This result was anticipated by Nguyen2018 ; Kudler-Flam:2018aa
; Agon:2018aa ; Bao:2018ac ; Ghodrati:2019aa ; Kudler-Flam:2019aa . Using this
construction we reprove the above bounds and interpret the result in terms of
bit threads.

Figure 2:  The holographic multipartite EOP proposal of Umemoto2018 ,
illustrated for the case of three regions. The joint RT surface
$\mathcal{O}\coloneqq m(\mathcal{A})$ is shown in red. This is partitioned
into regions $A_{i}^{\prime}$, and the total area of the corresponding minimal
surfaces $m(A_{i}A_{i}^{\prime})$ is minimized over partitions. The minimal
surfaces $\Sigma(A_{i})$ are shown as colored dashed lines; their union is
$\Sigma$. Also shown as a dotted line is the minimal surface $b_{p}(A_{1})$
which computes the bipartite EOP $E_{w}(A_{1})$.

We then generalize our analysis to the case of multipartite EOP. Given a state
$\rho_{\mathcal{A}}$ on a set $\mathcal{A}=\\{A_{i}\\}$ of systems, this
quantity is defined as

$E_{p}(\mathcal{A})=\min_{|{\psi}\rangle_{\mathcal{A}\mathcal{A}^{\prime}}:\atop\psi_{\mathcal{A}}=\rho_{\mathcal{A}}}\sum_{i}S(\psi_{A_{i}A^{\prime}_{i}})\,,$
(4)

where $\mathcal{A}^{\prime}=\\{A^{\prime}_{i}\\}$ is a set of auxiliary
systems and the minimization is over purifications
$|{\psi}\rangle_{\mathcal{A}\mathcal{A}^{\prime}}$ of $\rho_{\mathcal{A}}$
Bao:2018aa ; Umemoto2018 . The holographic dual to (4) was conjectured by
Umemoto2018  to be given as the area of a certain closed surface $\Sigma$
defined as follows. Consider all possible ways of partitioning the joint RT
surface $\mathcal{O}\coloneqq m(\mathcal{A})$ into non-overlapping regions
$A_{i}^{\prime}$, and for each partition find the homologous minimal surface
$m(A_{i}A_{i}^{\prime})$; $\Sigma$ is their union, minimized over partitions
(see fig. 2):

$E_{w}(\mathcal{A})\coloneqq\min_{\\{A_{i}^{\prime}\\}=\atop\text{partition of
}\mathcal{O}}\sum_{i}\operatorname{area}(m(A_{i}A_{i}^{\prime}))=\sum_{i}\operatorname{area}(\Sigma(A_{i}))=\operatorname{area}(\Sigma)\,.$
(5)

To find the flow program dual to (5), we must first make it into a convex
program; this involves certain subtleties not present in the proof of the
usual max flow-min cut theorem (see appendix B). Modulo those issues, we find
that the dual flow program involves a set $\mathcal{V}:=\\{v_{i}\\}$ of flows,
one for each region $A_{i}$, along with a function $\alpha$ on $\mathcal{O}$
which defines a boundary condition for all of the flows:

$\begin{split}E_{w}(\mathcal{A})=\max_{\mathcal{V},\alpha}\left(\sum_{i}\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}+\int_{\mathcal{O}}\sqrt{h}\,\alpha\right)\\\
\text{such that}\quad\nabla_{\mu}v_{i}^{\mu}=0\,,\;|v_{i}|\leq
1\,,\;\left.n_{\mu}v^{\mu}_{i}\right|_{\mathcal{O}}=\alpha\,.\end{split}$ (6)

We can understand this program heuristically as follows. If we set $\alpha=0$,
then the flows are decoupled from each other; each $v_{i}$ is simply
maximizing the flux from the corresponding region $A_{i}$ to the other regions
$\cup_{j\neq i}A_{j}$. The optimal value is then just the sum of those
bipartite cross-sections:

$\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}=E_{w}(A_{i})$ (7)

where for brevity we write $E_{w}(A_{i})$ for $E_{w}(A_{i}:\cup_{j\neq
i}A_{j})$. Turning on $\alpha$ allows each flow $v_{i}$ to have additional
flux coming out of $\mathcal{O}$. However, the flows are now coupled to each
other via the boundary condition $n_{\mu}v_{j}^{\mu}|_{\mathcal{O}}=\alpha$.
This constraint forces the other flows $v_{j}$ ($j\neq i$) to also have this
additional flux, limiting the amount of flux they can send to different
regions. A maximal flow configuration is found by balancing the competition to
maximize the flux of each flow. The duality guarantees this occurs exactly
when each flow saturates on the corresponding portion of $\Sigma$.

It is clear from the definition (4) that the multipartite EOP cannot be less
than the sum of the respects bipartite EOPs:

$E_{p}(\mathcal{A})\geq\sum_{i}E_{p}(A_{i})\,,$ (8)

This suggests that the residue

$E_{p}(\mathcal{A})-\sum_{i}E_{p}(A_{i})\,,$ (9)

reflects the amount of entanglement that is “truly multipartite” among the
parties of $\mathcal{A}$. The conjectured holographic duals obey the same
inequality:

$E_{w}(\mathcal{A})\geq\sum_{i}E_{w}(A_{i})\,.$ (10)

This can be shown from the minimal surfaces Umemoto2018 , or from the flows
using the above analysis, since the right-hand side is the maximum value of
the objective subject to the additional constraint $\alpha=0$ (see Theorem 1
below).

We will show that, within the set of all solutions (or maximizing
configurations) of (6), there exists a subset of particularly nice solutions
in which (7) holds, i.e. the flux of $v_{i}$ emanating from $A_{i}$ saturates
on the bipartite surface $b_{p}(A_{i})$. Hence the last term in the objective,
the flux on $\mathcal{O}$, equals the residue of (10):

$\int_{\mathcal{O}}\sqrt{h}\,\alpha=E_{w}(\mathcal{A})-\sum_{i}E_{w}(A_{i})\,,$
(11)

suggesting again that this quantity reflect the truly multipartite
entanglement.

The picture in terms of bit threads is that we have a different “color” of
thread for each region $A_{i}$. These types of threads do not interact with
each other in the bulk. They are only coupled via the constraint that, at any
point on $\mathcal{O}$ where a thread of one color begins, a thread of each
other color must also begin. In the “nice” solutions mentioned in the last
paragraph, the $A_{i}$ threads connecting $A_{i}$ to $A_{j}$ ($j\neq i$)
represent bipartite entanglement, while the ones connecting $\mathcal{O}$ to
$A_{j}$ represent truly multipartite entanglement.

In section 2 we articulate how to express the minimal homology region cross
section in term of flows by utilizing relative homology and convex duality. We
prove a simple theorem which describes how the addition of constraints affects
convex optimization problems. Utilizing this theorem we reprove that all of
the bipartite inequalities which the EOP satisfies are also satisfied by the
minimal homology region cross section. In section 3 we generalize to the
multipartite case and provide a derivation using convex duality and proofs of
several bounds. We find that with some simple restrictions we can “gauge” to a
class of thread configurations which have distinct bipartite and multipartite
contributions. In 4 we comment on some future directions of potential work. In
appendix A we review the definition and basic properties of the multiflow as
well as give a flow-based proof of MMI. In appendix B we discuss a subtlety in
the convex relaxation of the holographic proposal (5). In appendix C we
dualize another interesting class of surfaces which allow for intersection in
the bulk. Comparing to the holographic multipartite EOP, we demonstrate a
common mechanism for how the manifold is able to provide the extra flux
necessary to overcome geometric obstacles. Finally, in appendix D we provide
evidence using an explicit bit thread construction of the multipartite
information that the multipartite squashed entanglement will holographically
always be saturated.

The methods of convex optimization, convex duality, and convex relaxation are
used throughout this paper. Those unfamiliar with these methods may find it
useful to review the relevant sections of Boyd_2004 ; Freedman2017 ;
Headrick_2018 .

While this paper was in preparation, the paper Du:2019aa  appeared, which has
some overlap with section 2 of this paper.

##  2 Bipartite entanglement of purification

###  2.1 Flows and relative homology

Starting with a Riemannian manifold $M$ which is dual to a state of a boundary
CFT on $\partial M$, the entanglement entropy of a boundary region
$A\subset\partial M$ is given by the RT formula Ryu_2006  as the minimal
surface homologous to $A$

$S(A)=\min_{m\sim A}\operatorname{area}(m).$ (12)

We define a flow $v^{\mu}$ as a norm-bounded divergenceless vector field on
$M$. That is $\nabla_{\mu}v^{\mu}=0$ and $|v^{\mu}|\leq 1$. Via an application
of convex duality the RT formula can be expressed in terms of flows as the
maximum flux through $A$ among all choices of flows Freedman2017

$S(A)=\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}.$ (13)

It was shown in Headrick_2018  that we can make the following generalization
of the standard max flow-min cut (MFMC) theorem for flows: If we define the
minimal surface with respect to the relative homology
$A\,\text{rel}\,\mathcal{O}$ of a boundary region $\mathcal{O}$ this has the
effect of allowing the minimal surface to begin or end on $\mathcal{O}$,
increasing the space of allowed surface. On the flow side this is equivalent
to imposing the constraint $n_{\mu}v^{\mu}|_{\mathcal{O}}=0$. This means no
threads can end on or pass through $\mathcal{O}$.

Figure 3: Shrinking $M$ to $M^{\prime}$.

In order to make use of this we can further generalize to the case when
$\mathcal{O}$ is instead an arbitrary codimension 1 surface whose boundary
lies entirely on $\partial M$. In other words we no longer require that
$\mathcal{O}$ be restricted to lie on the boundary $\partial M$. To prove this
result we apply convex dualization to the max flow program

$\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}\quad\text{s.t.}\quad
n_{\mu}v^{\mu}|_{\mathcal{O}}=0.$ (14)

First we construct the Lagrangian function on $M$

$L(v^{\mu},\psi,\phi,\gamma)=\int_{A}\sqrt{h}n_{\mu}v^{\mu}+\int_{M}\sqrt{g}[-\psi\nabla_{\mu}v^{\mu}+\phi(1-|v^{\mu}|)]+\gamma\int_{\mathcal{O}}\sqrt{h}n_{\mu}v^{\mu}.$
(15)

To proceed we make the following observation: since the flow cannot penetrate
the surface $\mathcal{O}$ and the minimal surface will always be located where
the threads achieve their maximum density, the minimal surface corresponding
to the dual solution can never be located in a region of the manifold
contained entirely in $\mathcal{O}$ or between the boundary and $\mathcal{O}$.
As a result without loss of generality we are free to instead consider the
manifold $M^{\prime}$ which is constructed by shrinking the bulk by removing
all regions the threads cannot reach (see fig. 3). This ensures $\mathcal{O}$
will be a part of the boundary $\partial M^{\prime}$.

Now that $\mathcal{O}$ is a part of the boundary we can make the constraint
$n_{\mu}v^{\mu}|_{\mathcal{O}}=0$ implicit and integrate by parts

$L(v^{\mu},\psi,\phi)=\int_{\partial
M^{\prime}\backslash\mathcal{O}}\sqrt{h}n_{\mu}v^{\mu}(\chi_{A}-\psi)+\int_{M^{\prime}}\sqrt{g}[v^{\mu}\partial_{\mu}\psi+\phi(1-|v^{\mu}|)].$
(16)

Integrating out $v^{\mu}$ and minimizing we get the dual program

$\min_{\psi,\phi}\int_{M^{\prime}}\sqrt{g}\phi\quad s.t.\quad\psi|_{\partial
M^{\prime}\backslash\mathcal{O}}=\chi_{A},\quad\phi\geq|\partial_{\mu}\psi|=\min_{m\sim
A\,\text{rel}\,\mathcal{O}}\operatorname{area}(m)\;\text{on}\,M^{\prime}.$
(17)

We may now view the surface as being located on $M$ instead of $M^{\prime}$
and arrive at the MFMC result

$\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}\quad\text{s.t.}\quad
n_{\mu}v^{\mu}|_{\mathcal{O}}=0\overset{\text{MFMC}}{\iff}\min_{m\sim
A\,\text{rel}\,\mathcal{O}}\operatorname{area}(m)\;\text{on}\,M.$ (18)

###  2.2 Flow formulation

Figure 4: A maximal flow of (19) whose flux calculates $E_{w}(A:B)$.

One can make use of (18) to define $E_{w}(A:B)$ in terms of flows: First,
determine the homology region $r(AB)$ and its boundary in the bulk. From this
construct the submanifold $M^{\prime}$ with boundary $\partial
M^{\prime}=\partial r(AB)=A\cup B\cup m(AB)$. By using relative homology to
force the thread configurations to remain in $M^{\prime}$ we can then by (18)
define $E_{w}(A:B)$ in terms of flows:

$E_{w}(A:B)\coloneqq\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}\quad
s.t.\quad n_{\mu}v^{\mu}|_{m(AB)}=0\iff\min_{b_{p}(A:B)\sim
A\,\text{rel}\,m(AB)}\operatorname{area}(b_{p}(A:B))$ (19)

where $v^{\mu}$ is a flow (see fig. 4). We stress that the explicit
construction of $M^{\prime}$ is not important since we can view the flow as
being either on $M$ or $M^{\prime}$. The existence of $M^{\prime}$ provides a
simple argument that all the standard results of convex optimization will
carry over to this submanifold.

###  2.3 Relaxation of the homology region

In this subsection we will reformulate our flow program for the minimal
homology region cross section by removing the need to refer to the boundary of
the homology region. To provide some intuition consider the following: as
before, split the boundary $\partial M$ into three regions $A,B,R:=(A\cup
B)^{c}$ and calculate the maximum flux from $A$ to $B$ such that
$n_{\mu}v^{\mu}|_{R}=0$. Such a maximum flow will not be contained to $r(AB)$.
What we need is a way to force the threads to remain in $r(AB)$ so that the
flux will correctly calculate $E_{w}(A:B)$.

Figure 5: Left: The flow $x^{\mu}$ acts to change the geometry $v^{\mu}$ can
probe. Right: On the reduced manifold the maximum flux of $v^{\mu}$ gives the
holographic EOP.

To do this consider an arbitrary partition of the manifold $M$ by a surface
homologous to $R$, call it $\mathcal{O}$. For such a choice of $\mathcal{O}$
we calculate a maximum flow from $R$ to $\mathcal{O}$ and then minimize the
maximum flux with respect to choices of partition. This has the effect of
forcing $\mathcal{O}$ to extend far enough into the bulk such that the region
between $R$ and $\mathcal{O}$ contains the minimal surface. Call this set of
partitions $\mathcal{X}$. By the divergencelessness condition all such
elements of $\mathcal{X}$ will allow the same maximal flux. Next we calculate
the maximum flow from $A$ to $B$ such that $n_{\mu}v^{\mu}|_{\mathcal{O}}=0$
where $\mathcal{O}\in\mathcal{X}$ . By construction the partition which gives
the maximum space for these threads will be the one where $\mathcal{O}$ is the
minimal surface homologous to $R$. $E_{w}(A:B)$ is the flux of this flow (see
fig. 5):

$\mathcal{X}=\\{\mathcal{O}\\}\text{ s.t.
}\min_{\mathcal{O}}\max_{x^{\mu}}\int_{R}x$ (20)
$E_{w}(A:B)=\max_{v^{\mu},\mathcal{O}\in\mathcal{X}}\int_{A}v\text{ s.t.
}n_{\mu}v^{\mu}|_{\mathcal{O}}=0.$ (21)

As a result of maximizing the flux on $\mathcal{O}$, the flows naturally find
the minimal surface homologous to $\mathcal{O}$ which forms the boundary of
the homology region. From this perspective the threads of $x^{\mu}$ act to
change the geometry of the manifold by reducing the region that the threads of
$v^{\mu}$ can occupy. This means that there are a class of partitions of the
manifold on which we will be able to reproduce $E_{w}(A:B)$. This could be
anticipated by noting that the maximal flow only saturates at the minimal
homology region cross section. There is freedom in how the threads are
arranged which gives rise to a number of allowed partitions. While we will not
in general make use of this additional freedom, it is appealing to see how the
threads can give rise to the change in geometry. Since the homology region can
always be canonically chosen using this procedure, we will in general make
this choice leaving the generation of the manifold partition via threads
implicit.

###  2.4 Adding constraints

The following bound will be used throughout the paper and is a key tool to
proving many results. Therefore we state it as a theorem:

######  Theorem 1.

Let $P$ be a convex maximization program:

$P:maximize\ f_{0}(y)\ subject\ to\ f_{i}(y)\leq 0\ \forall i\ and\
h_{j}(y)=0\ \forall j$ (22)

with solution $y^{*}\in\mathcal{F}$ where $\mathcal{F}$ is the set of feasible
points. Let $\tilde{P}$ be a second convex maximization program obtained by
imposing an additional set of constraints $g_{m}(y)\leq 0\ \forall m\ and\
l_{n}(y)=0\ \forall n$ on $P$:

$\tilde{P}:maximize\ f_{0}(y)\ subject\ to\ f_{i}(y)\leq 0\ \forall i\ ,\
h_{j}(y)=0\ \forall j,\ g_{m}(y)\leq 0\ \forall_{m}\ and\ l_{n}(y)=0\ \forall
n$ (23)

and similarly define $\tilde{y}^{*}\in\tilde{\mathcal{F}}$. Then
$f_{0}(y^{*})\geq f_{0}(\tilde{y}^{*})$.

###### Proof.

Suppose the opposite that is $f_{0}(y^{*})<f_{0}(\tilde{y}^{*})$. By
construction $\tilde{\mathcal{F}}\subset\mathcal{F}$ which means
$\tilde{y}^{*}\in\mathcal{F}$ This leads to a contradiction since then $y^{*}$
is not maximal in $\mathcal{F}$. ∎

Figure 6: The two possible cases: In each $f_{0}(y^{*})\geq
f_{0}(\tilde{y}^{*})$.

In other words there are two possible cases (see fig. 6): If
$y^{*}\in\tilde{\mathcal{F}}$ then necessarily $y^{*}=\tilde{y}^{*}$, but if
$y^{*}\in\mathcal{F}\backslash\tilde{\mathcal{F}}$ then $f_{0}(y^{*})$ must be
greater then every point in $\tilde{\mathcal{F}}$. If not then there would be
a point in $\tilde{\mathcal{F}}$ which is greater and we would be back to the
first case.

Theorem 1 states that whenever we have a convex maximization program and we
add additional constraints, the original solution will always be an upper
bound to the new one. Thus, by relating different programs in this way we can
easily derive inequalities between them. This method allows for many of the
inequalities involving EOP to be proven using our flow construction (19).

###  2.5 Conditional EOP

Before continuing we wish to mention a straightforward generalization of the
EOP. A proposal for holographic conditional EOP was defined in Bao2018 ;
Bao:2018aa  as the minimum cross section of the bulk region $r(ABC)\backslash
r(C)$. This is easily accounted for using bit threads by a modification of the
boundary condition.

$E_{w}(A:B|C)\coloneqq\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}\quad
s.t.\quad n_{\mu}v^{\mu}|_{\mathcal{O}[AB|C]}=0$ (24)

where $\mathcal{O}[AB|C]\coloneqq\partial(r(ABC)\backslash r(C))\backslash
A\cup B$. $E_{w}(A:B|C)$ satisfies a modified version of each of the
inequalities that $E_{w}(A:B)$ satisfies. The proofs remain essentially
unchanged.

###  2.6 Bounds

We now proceed by deriving the inequalities that the EOP satisfies using our
flow formulation for the holographic proposal. This is accomplished by noting
that restricting to a smaller submanifold (i.e. decreasing the volume of the
homology region) or adding topological obstructions, such as black holes, can
only reduce the space threads can occupy. These restrictions are implemented
by imposing additional constraints on the flow which increases the region of
relative homology. Because of Theorem 1, imposing a constraint can never
increase the maximal flow which means the unconstrained problem is an upper
bound.

In contrast lower bounds can be found by explicit construction. If a
particular thread configuration is always allowed when maximizing (in the
feasible set of the convex program) then the true optimal value of the program
will always be at least this value.

####  2.6.1  $\frac{1}{2}I(A:B)\leq E_{w}(A:B)\leq\min[S(A),S(B)]$

##### Upper bound

$S(A)$ and $E_{w}(A:B)$ only differ as flow optimization problems by the
addition of the constraint $n_{\mu}v^{\mu}|_{m(AB)}=0$. Furthermore, we note
that the flow which gives $E_{w}(A:B)$ is in the feasible set of flows of
$S(A)$. Thus by Theorem 1 $S(A)\geq E_{w}(A:B)$. The same argument is true for
$S(B)$ therefore $E_{w}(A:B)\leq\min[S(A),S(B)]$.

##### Lower bound

Figure 7: A flow which gives $\frac{1}{2}I(A:B)$ is suboptimal to the program
(19).

Following Freedman2017  , we can explicitly define the flow
$\frac{1}{2}(v(A,B)-v(B,A))$ where $v(A,B)$ is the maximum flow through $A$
and $AB$. By construction
$\frac{1}{2}I(A:B)=\int_{A}\frac{1}{2}(v(A,B)-v(B,A))$. This flow is non zero
only in the homology region because both of the individual flows $v(A,B)$ and
$v(B,A)$ saturated on $m(AB)$. The result of taking their difference is
therefore a flow which satisfies the condition $n_{\mu}v^{\mu}|_{m(AB)}=0$ and
is in the feasible set of $E_{w}(A:B)$. Since it is an allowed flow when
maximizing, $E_{w}(A:B)$ will always be at least this value therefore
$\frac{1}{2}I(A:B)\leq E_{w}(A:B)$ (see fig. 7).

Together we have:

$\frac{1}{2}I(A:B)\leq E_{w}(A:B)\leq\min[S(A),S(B)]$ (25)

##### Saturation

Figure 8: Examples of situations where AL is saturated.

The upper bound, $E_{w}(A:B)=\min[S(A),S(B)]$, will saturate if the Araki-Lieb
(AL) inequality $S(A)\leq S(AB)+S(B)$ is saturated Nguyen2018  (see fig. 8).
Without loss of generality let $S(B)\leq S(A)$. In the bulk AL will be
saturated if and only if $m(A)=m(AB)\cup m(B)$ and $r(A)=r(AB)\backslash r(B)$
Headrick2014 .

In this case it is always possible find a flow that simultaneously maximizes
the flux through $A,B,AB$ (which is an exceptional case of the nesting
property of Freedman2017 ). As a result, the part of this flow which goes to
$B$ is the maximal flow and gives $E_{w}(A:B)$. That is when we impose
$n_{\mu}v^{\mu}|_{m(AB)}=0$ none of the maximal flows which give $S(B)$ cross
$m(AB)$ and thus imposing the constraint to get $E_{w}(A:B)$ cannot decrease
the flow.

If we have a bipartite pure state both inequalities saturate as $S(AB)=0$ and
$m(A)=m(B)$. Trivially, $m(AB)=\emptyset$ and $E_{w}(A:B)=S(A)=S(B)$.
Therefore, $I(A:B)=2S(A)$ and $\frac{1}{2}I(A:B)=E_{w}(A:B)$.

####  2.6.2 $E_{w}(A:BC)\geq E_{w}(A:B)$

We can define the two convex programs as:

$\begin{split}E_{w}(A:BC)&=\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}\quad\text{s.t.}\quad
n_{\mu}v^{\mu}|_{m(ABC)}=0\\\
E_{w}(A:B)&=\max_{v^{\mu}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}\quad\text{s.t.}\quad
n_{\mu}v^{\mu}|_{m(ABC)}=0,\;n_{\mu}v^{\mu}|_{m(AB)}=0\end{split}$ (26)

Figure 9: The reduction to $r(ABC)$. Imposing a second constraint reduces the
manifold further to $r(AB)$.

Since we can get $E_{w}(A:B)$ by starting with $E_{w}(A:BC)$ and imposing an
additional constraint and the maximal flow of $E_{w}(A:B)$ is an allowed flow
of $E_{w}(A:BC)$, by Theorem 1 $E_{w}(A:B)$ can only be as big as
$E_{w}(A:BC)$ (see fig. 9)

$E_{w}(A:BC)\geq E_{w}(A:B).$ (27)

####  2.6.3 $E_{w}(A\tilde{A}:B\tilde{B})\geq
E_{w}(A:B)+E_{w}(\tilde{A}:\tilde{B})$

Figure 10: The green surfaces are the additional boundary constraints which
must be imposed to separate $r(AB)$ and $r(\tilde{A}\tilde{B})$.

This superadditivity inequality always holds in the bulk, but will not hold on
the boundary for a generic state. As a result it may be a criterion for
determining whether a state has a holographic description similar to MMI
Bhattacharyya2018 . The maximal flow which gives
$E_{w}(A:B)+E_{w}(\tilde{A}:\tilde{B})$ is an allowed flow of the convex
program of $E_{w}(A\tilde{A}:B\tilde{B})$, but
$E_{w}(A:B)+E_{w}(\tilde{A}:\tilde{B})$ has the additional constraint
$n_{\mu}v^{\mu}|_{\mathcal{O}^{\prime}}=0$ which acts to separate $r(AB)$ and
$r(\tilde{A}\tilde{B})$ (see fig. 10). As a result

$E_{w}(A\tilde{A}:B\tilde{B})\geq E_{w}(A:B)+E_{w}(\tilde{A}:\tilde{B})$ (28)

by Theorem 1.

##### Saturation

This inequality will saturate if and only if the boundary state is a product
$\rho_{(A\tilde{A})(B\tilde{B})}=\rho_{AB}\otimes\rho_{\tilde{A}\tilde{B}}$.
When this happens the mutual information $I(AB:\tilde{A}\tilde{B})=0$. In the
bulk this means the two homology regions $r(AB)$ and $r(\tilde{A}\tilde{B})$
are disconnected Headrick2014  and implies there will be no maximal flow which
has threads connecting $AB$ and $\tilde{A}\tilde{B}$. In this case no
additional constraints are needed, $\mathcal{O}^{\prime}=\emptyset$ and the
two programs are equivalent sharing the same maximum.

### Multiple Regions

In order to proceed to cases where we deal with more than two boundary regions
independently at a time we need to utilize some additional technology. Because
now the flow will be connecting multiple regions we need a way to keep track
of the threads extending between each region and ensure that the entire
combination is still a valid flow. To this end we make use of the multiflow
construction of Cui:2018aa  which accomplishes just this. Essentially, a
multiflow is a collection of flows such that the linear combination of any
flows in a region (referred to here as a subflow) is itself a flow. Those
interested can find definitions and a flow proof of MMI in A.

In what follows, we will use multiflows to prove some remaining inequalities
involving EOP. These inequalities involve more than two regions, but contain
only quantities which are bipartite. As a consequence their proof requires
only (25) and MMI (84). Since each of these inequalities has been proven using
bit threads we cite the direct proofs using these properties.

####  2.6.4 $E_{w}(A:BC)\geq\frac{1}{2}I(A:B)+\frac{1}{2}I(A:C)$

By directly applying MMI (84) to (25) we get

$E_{w}(A:BC)\geq\frac{1}{2}I(A:BC)\geq\frac{1}{2}I(A:B)+\frac{1}{2}I(A:C).$
(29)

####  2.6.5 If $ABC$ is pure $E_{w}(A:B)+E_{w}(A:C)\geq E_{w}(A:BC)$

Consider a maximal multiflow $\mathcal{V}_{m}$ for a pure state of $\partial
M=ABC$. By maximality we know the single interval subflows are saturated

$S(A)=E_{w}(A:BC)=\int_{A}v_{A}\quad S(B)=\int_{B}v_{B}\quad
S(C)=\int_{C}v_{C}.$ (30)

As a consequence of purity the two interval subflows are saturated as well and
can be related to the single interval subflows

$v_{AB}=-v_{C},\quad v_{BC}=-v_{A},\quad v_{AC}=-v_{B}.$ (31)

This implies the mutual information can be written simply as

$I(A:B)=S(A)+S(B)-S(AB)=2\int_{A}v_{AB}.$ (32)

Let $v^{\prime}_{AB}$ be the maximal flow on the manifold $r(AB)$ whose flux
gives $E_{w}(A:B)$ similarly define $v^{\prime\prime}_{AC}$ on $r(AC)$ for
$E_{w}(A:C)$. Using the definition of the flows and the lower bound of (25) we
can rederive the inequality

$\begin{split}E_{w}(A:B)+E_{w}(A:C)=\int_{A}(v^{\prime}_{AB}+v^{\prime\prime}_{AC})\geq\frac{1}{2}(I(A:B)+I(A:C))=\\\
\int_{A}(v_{AB}+v_{AC})=\int_{A}v_{A}=E_{w}(A:BC).\end{split}$ (33)

##  3 Multipartite entanglement of purification

We will now generalize to the case of more than two boundary regions. The
multipartite EOP was defined by Bao:2018aa ; Umemoto2018  in the boundary CFT
as

$E_{p}(\mathcal{A})=\min_{|{\psi}\rangle_{\mathcal{A}\mathcal{A}^{\prime}}}\sum_{i}S(A_{i}A^{\prime}_{i})$
(34)

where $\mathcal{A}=\\{A_{i}\\}$, $\mathcal{A}^{\prime}=\\{A^{\prime}_{i}\\}$,
and the minimization is over purifications of $\rho_{\mathcal{A}}$.

Figure 11: The holographic multipartite EOP proposal of Umemoto2018 .

The holographic dual to (34) was conjectured by Umemoto2018  to be given as
the minimal closed surface among all possible partitions of the boundary
region $\mathcal{O}$ into regions $A_{i}^{\prime}$ each belonging to
$A_{i}$333Historically, another surface proposal for the multipartite EOP was
considered in Bao:2018aa  which allows for bulk intersection. Being an
interesting class of of surfaces in the bulk we provide a dualization of these
in C. (see fig. 11)

$E_{w}(\mathcal{A})\coloneqq\min_{\mathcal{A}^{\prime}}\sum_{i}\operatorname{area}(m(A_{i}A_{i}^{\prime}))=\sum_{i}\operatorname{area}(\Sigma(A_{i}))=\operatorname{area}(\Sigma)$
(35)

where $\Sigma\coloneqq\cup_{i}\Sigma(A_{i})$.

Before dualizing there is another geometric quantity we wish to mention for
its importance in the interpretation of our construction. Consider a maximal
multiflow which remains in the homology region $r(\mathcal{A})$:444In making
this definition we have chosen to sum over all of the flows. This amounts to a
different normalization than that of the bipartite case. This has been done
for clarity and to remain notationally consistent with Bao:2018aa ;
Umemoto2018 .

$\operatorname{area}(b_{p}(\mathcal{A}))\coloneqq\max_{\mathcal{V}_{m}}\sum_{i,\;j}\int_{A_{i}}v_{ij}\text{
s.t. }n_{\mu}v_{ij}^{\mu}|_{m(\mathcal{A})}=0\,.$ (36)

Figure 12: The sum of the area of the bipartite homology region cross sections
for multiple regions is given by the flux of a maximal multiflow confined to
$r(\mathcal{A})$.  Figure 13: The surface $b_{p}(\mathcal{A})$ is a geometric
obstacle to the maximal flow. Additional flux is needed on $\mathcal{O}$ in
order to get flows which saturate on $\Sigma$.

As a result of Theorem 2 the flux of this flow calculates precisely the sum of
the bipartite homology region cross sections which is the minimal surface dual
to the maximal multiflow on $r(\mathcal{A})$

$\operatorname{area}(b_{p}(\mathcal{A}))=\sum_{i}E_{w}(A_{i}:\cup_{i\neq
j}A_{j}).$ (37)

This surface is the natural geometric obstruction to flows in the bulk, but by
definition is purely bipartite (see fig. 12). We will find that any
formulation of multipartite entanglement must inherently find a way to
circumvent this obstacle (see fig. 13).555The surface $b_{p}(A_{i})$ is the
minimal surface homologous to $A_{i}$ relative to $\mathcal{O}$. That is we
can define a part of $\mathcal{O}$ called $A^{\prime}_{i}$ which “belongs” to
$A_{i}$. Finding the surface $\Sigma(A_{i})$ is similar except for the
additional condition that after splitting $\mathcal{O}$ among all of
$\mathcal{A}$ there can be nothing left, everything must be assigned. This is
more restrictive in that the resulting surface must be closed

###  3.1 Derivation

####  3.1.1 Convex and integer relaxation

We will now derive our flow dual to (35) using convex dualization. For clarity
we will explicitly go through all the steps. In order to dualize we must first
write this optimization problem as a convex program. Starting with (35) we
perform a convex relaxation by introducing scalar fields $\\{\psi_{i}\\}$
which smear the surface to form level sets in the bulk. For each $\psi_{i}$ we
must impose boundary conditions based upon where the surface ends. We arrive
at the relaxed program

$\min_{\\{\psi_{i}\\}}\sum_{i}\int_{M}\sqrt{g}|\nabla_{\mu}\psi_{i}|\quad\text{s.t.}\quad\psi_{i}|_{A_{j}}=\delta_{ij},\quad\sum_{i}\psi_{i}|_{\mathcal{O}}=1\quad\text{On
$\mathcal{O}$: }\psi_{i}\in\\{0,1\\}.$ (38)

Here the constraints on $\mathcal{O}$ force a division: a piece of
$\mathcal{O}$ must belong to only one $A_{i}$. The problem with the program as
written is that it is an integer program and as a result is not convex. In
order to make use of the tools of convex optimization it is necessary to
perform an additional relaxation to the program. This is done by increasing
the domain of the $\psi_{i}$’s to be over the reals666Such a relaxation of
variables from being integer valued is very common in the field of integer
programming. Notably, if one has a linear objective and the constraints are
totally unimodular then this relaxation will not introduce a so called
integrality gap: the two program will have the same optimal configuration.
This is the content of the Hoffman-Kruskal Theorem. A totally unimodular
matrix is one which every square submatrix has determinant $-1,0,1$.
Interestingly, our constraints though infinite in number have this property.
Sadly, the absence of an integrality gap can not be guaranteed in the case of
a convex objective. resulting in a further relaxed program which is convex:

$\min_{\\{\psi_{i}\\}}\sum_{i}\int_{M}\sqrt{g}|\nabla_{\mu}\psi_{i}|\quad\text{s.t.}\quad\left.\psi_{i}\right|_{A_{j}}=\delta_{ij},\quad\left.\sum_{i}\psi_{i}\right|_{\mathcal{O}}=1\,$
(39)

The consequence of this relaxation is an increase in the space of surfaces
which are considered when minimizing. Namely, the relaxation now allows
surface where regions of $\mathcal{O}$ are shared in a constrained way between
the different regions $\mathcal{A}$. Generically, the two programs (38) and
(39) are not the same and may have different minima. In fact we can find
explicit examples where the two programs disagree. We do not know a complete
set of sufficient and necessary conditions to guarantee agreement, but it is
true that in many cases of interest we find that the original surface proposal
and our relaxed program give the same result. This is essentially a
consequence of convexity of the relevant surfaces which results from the
negative curvature of the manifold. As we continue we will assume we are
working in such cases when the two programs (38) and (39) are in fact
equivalent and relegate further details and discussion of the integer
relaxation to B.

We wish to comment however that the relaxed program with its increased
feasible space of surfaces represents an interesting class of geometric
surfaces in the bulk. It is entirely possible that this generalization
captures multipartite EOP as a separate proposal (which agrees with the
current one in certain cases) or could represent another class of surfaces
with a potential dual information theoretic interpretation.

####  3.1.2 Dualization

We proceed with the dualization of (39) by imposing each constraint with a
Lagrange multiplier and find the Lagrangian:

$\begin{split}L(\\{\psi\\},\\{\phi\\},\\{v^{\mu}\\},\\{\gamma\\},\alpha)&=\sum_{i}\int_{M}\sqrt{g}\left[|\phi_{\mu
i}|+v^{\mu}_{i}(\phi_{\mu i}-\nabla_{\mu}\psi_{i})\right]\\\
&+\int_{A_{i}}\sqrt{h}\left[\gamma_{ii}(\psi_{i}-1)+\sum_{i\neq
j}\gamma_{ij}\psi_{j}\right]+\int_{\mathcal{O}}\sqrt{h}(-\alpha)(\psi_{i}-1).\end{split}$
(40)

For convenience we define the covectors $\phi_{\mu i}=\nabla_{\mu}\psi_{i}$
the equivalence of which is imposed by the Lagrange multipliers $v^{\mu}_{i}$.
These vector fields will be the dual variables which are our flows. We first
dualize with respect to $\\{\phi\\}$ and get the standard bulk flow norm bound
constraints:

$\phi_{\mu i}v^{\mu}_{i}+|\phi_{\mu i}|=0,\quad|v^{\mu}_{i}|\leq 1$ (41)

Utilizing these and performing integration by parts the Lagrangian becomes

$\begin{split}\sum_{i}\int_{M}\sqrt{g}\psi_{i}\nabla_{\mu}v^{\mu}_{i}+&\int_{A_{i}}\sqrt{h}\left[\gamma_{ii}(\psi_{i}-1)+\sum_{i\neq
j}\gamma_{ij}\psi_{j}+\sum_{j}n_{\mu}v^{\mu}_{j}\psi_{j}\right]\\\
+&\int_{\mathcal{O}}\sqrt{h}\left[-\alpha(\psi_{i}-1)+\sum_{j}n_{\mu}v^{\mu}_{j}\psi_{j}\right].\end{split}$
(42)

Finally we dualize with respect to the remaining primal variables $\\{\psi\\}$

$\begin{split}&\text{On $M$: }\nabla_{\mu}v^{\mu}_{i}=0\\\ &\text{On $A_{i}$:
}\gamma_{ii}+n_{\mu}v^{\mu}_{i}=0,\quad\gamma_{ij}+n_{\mu}v^{\mu}_{j}=0\\\
&\text{On $\mathcal{O}$: }-\alpha+n_{\mu}v^{\mu}_{i}=0.\\\ \end{split}$ (43)

This gives the dual program which requires us to maximize over a set of flows
$\mathcal{V}$ and the function $\alpha$ on $\mathcal{O}$:

$\begin{split}E_{w}(\mathcal{A})=\max_{\mathcal{V},\alpha}\left(\sum_{i}\int_{A_{i}}\sqrt{h}n_{\mu}v^{\mu}_{i}+\int_{\mathcal{O}}\sqrt{h}\alpha\right)\quad\text{s.t.}\\\
\nabla_{\mu}v_{i}^{\mu}=0,\;|v_{i}|\leq
1,\;\left.n_{\mu}v^{\mu}_{i}\right|_{\mathcal{O}}=\alpha\,.\end{split}$ (44)

This flow program for the holographic multipartite EOP proposal is our main
result.

|  |
---|---|---
Figure 14: An example of a maximal flow configuration $\mathcal{V}$ to (44)
for three regions. Note the local flux on $\mathcal{O}$ which we call $\alpha$
is the same for all three of the flows. In the example each surface
$b_{p}(A_{i})$ can support three threads while each surface $\Sigma(A_{i})$
can support four.

The program (44) involves several flows, one for each boundary region $A_{i}$.
These flows interact with each other only through the constraint
$n_{\mu}v_{i}^{\mu}|_{\mathcal{O}}=\alpha$. The flow lines can be thought of
as different-colored threads, or as living on different copies of the manifold
that are glued together along $\mathcal{O}$.777This perspective has
tantalizing visual overlap and possible connections to notions of wormhole
geometries Dutta:aa ; Bao:2018ac ; Bao:2018ab . As an example of an optimal
configuration to provide intuition see fig. 14.

An important feature of (44) is that at no point were we required to define a
partition of $\mathcal{O}.$888Since $\mathcal{O}=m(\mathcal{A})$ is the
minimal surface homologous to $\mathcal{A}^{c}$ any threads on $\mathcal{O}$
can always nonuniquely be extended back to $\mathcal{A}^{c}$. In this sense
one could view the flows as elements of the full manifold $M$. Moving forward
we will not adopt this viewpoint. That is the threads find the correct
location of $\Sigma$ (where they saturate) naturally as a result of the
maximization. This is in contrast to the cut side where we were required to
explicitly minimize over both the area of surfaces and choice of partition. In
a sense the lack of knowledge of the optimal partition is what makes this
problem hard. Given the optimal partition $E_{w}(\mathcal{A})$ can be
calculated trivially using standard flows which calculate entanglement
entropies. This however hides the rich structure and multipartite nature which
we will demonstrate.

For the reader familiar with the notion of a _multiflow_ Cui:2018aa , we
emphasize that these flows do _not_ define a multiflow since they do not
necessarily obey a _joint_ norm bound. (See appendix A for a review and
discussion of multiflows.)

###  3.2 Flow decomposition

In order to interpret the maximal flows of our program (44), we find it
convenient to make some restrictions on the space of solutions. To do so we
place additional constraints on the flow program which act to steer us towards
a particular class of solutions. These solutions have the property of
isolating the bipartite contribution from an additional term which is truly
multipartite. This separation leads to a drastic simplification.

Let $\mathcal{S}\subset\mathcal{F}$ be the space of maximal flow
configurations, which is a subset of all of the feasible points of the convex
program. We now define the subspace $\mathcal{S}^{\prime}\subset\mathcal{S}$
with the following constraints: The flux on the boundary $\mathcal{O}$ is
everywhere positive: $\alpha=n_{\mu}v^{\mu}_{i}|_{\mathcal{O}}\geq 0$. The
total flux on the boundary $\mathcal{O}$ is as small as possible. The flow
configuration contains no threads which contribute zero to the objective. This
includes loops and threads which connect regions where the flux is not being
counted.

The last of these constraints is simply implemented for convenience and can
always be done. It allows us to discuss “simple” solutions which only contain
threads which count towards the objective. Given a maximal thread
configuration which contains such spurious threads we define a new solution by
deleting them.

The result of this appropriate gauging of the solution is that the holographic
multipartite EOP proposal has a natural decomposition into the number of
threads connecting the regions $A_{i}$ to each other, which equals the total
bipartite homology region cross section
$\sum\operatorname{area}(b_{p}(A_{i}))$, and an additional term counting the
number of threads on $\mathcal{O}$, which is fundamentally multipartite. This
leads naturally to the conjecture that the actual multipartite EOP may exhibit
a similar decomposition:

$E_{p}(\mathcal{A})=\sum_{i}E_{p}(A_{i}:\cup_{i\neq j}A_{j})+M(\mathcal{A})$
(45)

where $M(\mathcal{A})$ is the multipartite contribution.

Our task is to implement each of the above constraints in the convex
maximization program and understand how they respond to dualization. This will
allow us to define a modified dual which implements the restriction to
$\mathcal{S}^{\prime}$. We will argue that each of these constraints does not
affect the optimal value of our program.

####  3.2.1 Forcing $\alpha\geq 0$

The first restriction we would like to add to the flow program (44) is
$\alpha\geq 0$:

$\begin{split}\tilde{E}_{w}(\mathcal{A})=\max_{\mathcal{V},\alpha}\left(\sum_{i}\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}+\int_{\mathcal{O}}\sqrt{h}\,\alpha\right)\quad\text{s.t.}\\\
\nabla_{\mu}v_{i}^{\mu}=0\,,\;|v_{i}|\leq
1\,,\;\left.n_{\mu}v^{\mu}_{i}\right|_{\mathcal{O}}=\alpha\,,\;\alpha\geq
0\,.\end{split}$ (46)

In the dualization presented in subsection 3.1.2 above, $\alpha$ arose as the
Lagrange multiplier for the equality constraint
$\sum\psi_{i}|_{\mathcal{O}}=1$. According to the rules of convex dualization,
an _inequality_ constraint is enforced by a _non-negative_ Lagrange
multiplier, so changing this to an inequality constraint
$\sum\psi_{i}|_{\mathcal{O}}\geq 1$ simply leads to the additional inequality
constraint $\alpha\geq 0$ on the dual, which is just what we want. Thus the
dual of (46) is as follows:

$\tilde{E}_{w}(\mathcal{A})=\min_{\\{\psi_{i}\\}}\sum_{i}\int_{M}\sqrt{g}|\nabla_{\mu}\psi_{i}|\quad\text{s.t.}\quad\left.\psi_{i}\right|_{A_{j}}=\delta_{ij},\quad\left.\sum_{i}\psi_{i}\right|_{\mathcal{O}}\geq
1\,.$ (47)

We will now argue that these two programs are equivalent to (44) and (39),
i.e. that the new constraint $\alpha\geq 0$ and the new relaxation
$\sum\psi_{i}|_{\mathcal{O}}\geq 1$ have no affect on the optimal value of
these programs, and thus $\tilde{E}_{w}(\mathcal{A})=E_{w}(\mathcal{A})$.

We have two arguments for this equivalence. First, as argued in appendix B, we
don’t expect that in the solution to (39) the regions where the different
$\psi_{i}$s are non-zero will overlap (at least in geometries relevant for
AdS/CFT dualities), and the same argument applies to (47). In this case, we
can show that the solution must have $\psi_{i}\leq 1$ everywhere: if
$\psi_{i}>1$ somewhere, setting it to 1 reduces the value of the objective
while maintaining the boundary conditions. Therefore $\sum\psi_{i}\leq 1$
everywhere, and in particular on $\mathcal{O}$ the sum equals 1.

The second argument applies complementary slackness to the new inequality
constraint. This says that, if at a given point $x\in\mathcal{O}$, there
exists an optimal flow configuration with $\alpha(x)>0$, then for _all_
optimal configurations of (47), $\sum\psi_{i}(x)=1$. Given the well-known non-
uniqueness of max flows, it would be surprising if there was no way to have a
non-zero flux at some given point on $\mathcal{O}$.

####  3.2.2 Minimizing flux on $\mathcal{O}$

In the program (46), the flow $v_{i}$ has non-negative flux on $\mathcal{O}$.
Therefore, in any feasible configuration, its flux on $A_{i}$ is bounded above
by the bipartite homology-region cross section:

$\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}\leq\operatorname{area}(b_{p}(A_{i}))\,.$
(48)

We will now show that the bounds (48) are collectively tight: there exist
optimal configurations in which they are simultaneously saturated for all $i$.

In the space $\mathcal{S}$ of solutions of (46), the different terms in the
objective do not necessarily take fixed values: a given term can take
different values on different solutions, although of course the total value of
the objective is the same in all the solutions. We are interested in solutions
in which the first term is as large as possible and the second term (the flux
on $\mathcal{O}$) as small as possible. To steer the solution in that
direction, we add to the objective a term $\epsilon\sum\int_{A_{i}}v_{i}$,
where $\epsilon$ is small and positive, yielding the following modified
program:

$\begin{split}\tilde{E}^{\epsilon}_{w}(\mathcal{A})=\max_{\mathcal{V},\alpha}\left((1+\epsilon)\sum_{i}\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}+\int_{\mathcal{O}}\sqrt{h}\,\alpha\right)\quad\text{s.t.}\\\
\nabla_{\mu}v_{i}^{\mu}=0\,,\;|v_{i}|\leq
1\,,\;\left.n_{\mu}v^{\mu}_{i}\right|_{\mathcal{O}}=\alpha\,,\;\alpha\geq
0\,.\end{split}$ (49)

In general, for concave functions $f$ and $g$,

$\max(f+\epsilon g)=\max f+\epsilon\max_{\mathcal{S}}g$ (50)

(to first order in $\epsilon$), where $\mathcal{S}$ is the set of maximizers
of $f$. In this case, we have

$\tilde{E}^{\epsilon}_{w}(\mathcal{A})=\tilde{E}_{w}(\mathcal{A})+\epsilon\max_{\mathcal{S}}\sum_{i}\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}\,.$
(51)

We will show that the optimal value is bounded below as follows:

$\tilde{E}_{w}^{\epsilon}(\mathcal{A})\geq\tilde{E}_{w}(\mathcal{A})+\epsilon\sum_{i}\operatorname{area}(b_{p}(A_{i}))\,,$
(52)

so

$\max_{\mathcal{S}}\sum_{i}\int_{A_{i}}\sqrt{h}\,n_{\mu}v^{\mu}_{i}\geq\sum_{i}\operatorname{area}(b_{p}(A_{i}))\,.$
(53)

Given the inequalities (48), this can only hold if there exist solutions to
(46) in which all of those inequalities are saturated (and thus (53) is also
saturated).

To show (52), we first note that, by a straightforward dualization, (49) is
equivalent to a version of (47) with a modified boundary condition:

$\tilde{E}^{\epsilon}_{w}(\mathcal{A})=\min_{\\{\psi_{i}\\}}\sum_{i}\int_{M}\sqrt{g}\,|\nabla_{\mu}\psi_{i}|\quad\text{s.t.}\quad\left.\psi_{i}\right|_{A_{j}}=(1+\epsilon)\delta_{ij}\,,\quad\left.\sum_{i}\psi_{i}\right|_{\mathcal{O}}\geq
1\,.$ (54)

Consider now a feasible configuration of (54). We can turn it into a feasible
configuration for (47) as follows: Wherever $\psi_{i}<0$, set it to 0, and
wherever $\psi_{i}>1$, set it to 1. Since it is a feasible configuration for
(47), the objective has value no less than $\tilde{E}_{w}(\mathcal{A})$. This
operation reduces the objective because it removes level sets. (Recall that,
by the co-area formula, the term $\int|\nabla\psi_{i}|$ in the objective
equals the area of the level set $\\{x:\psi_{i}(x)=p\\}$, integrated over
$p$.) In particular, the level sets for $1<\psi_{i}<1+\epsilon$ are homologous
to $A_{i}$ relative to $\mathcal{O}$, and therefore have area at least
$b_{p}(A_{i})$. So the reduction in the objective is at least

$\epsilon\sum_{i}\operatorname{area}(b_{p}(A_{i}))\,,$ (55)

which establishes the inequality (52).

We will denote the set of solutions to (49) that saturate (48)
$\mathcal{S}^{\prime}$. These decompose the holographic multipartite EOP into
bipartite and “truly multipartite” contributions.999Because of the non-
uniqueness of thread configurations there is a freedom to convert some of the
bipartite threads into threads on $\mathcal{O}$ which we would here refer to
as multipartite contributions. Such thread configurations are not in
$\mathcal{S^{\prime}}$. Taking this idea seriously there is some freedom due
to space in the geometry to convert between bipartite and multipartite
contributions. This freedom could have an interesting information theoretic
interpretation in the boundary CFT, but we leave such exploration to future
work. Importantly, by restricting to $\mathcal{S}^{\prime}$ we are maximizing
the number of bipartite threads. It is in this sense that the remainder of the
flux, which must be on $\mathcal{O}$, _must_ be “truly multipartite”. From now
on we will assume that all of the solutions of (44) we consider are in
$\mathcal{S}^{\prime}$ unless otherwise stated.

###  3.3 $\mathcal{V}$ flows

Figure 15: A portion of the manifold from $A_{1}$. In the example
$b_{p}(A_{1})$ can support 3 threads. Left: The flow $v_{A_{1}}$. Right: The
incoming flow from another region is limited by the required flux on
$\mathcal{O}$.

To understand the behavior of $\mathcal{V}$ consider one of the flows
$v_{A_{i}}$. It is always possible to send enough flow from $A_{i}$ through
$b_{p}(A_{i})$ such that the flow saturates on $b_{p}(A_{i})$. This is clear
from the multiflow construction of $b_{p}(\mathcal{A})$. For the flow to also
saturate on $\Sigma(A_{i})$ there must be additional flux from the portion of
$\mathcal{O}$ between $b_{p}(A_{i})$ and $\Sigma(A_{i})$ of the amount
$\operatorname{area}(\Sigma(A_{i}))-\operatorname{area}(b_{p}(A_{i}))$.
However, because of the condition
$\alpha=n_{\mu}v^{\mu}_{i}|_{\mathcal{O}}=n_{\mu}v^{\mu}_{j}|_{\mathcal{O}}$
this means the same amount of flux must be present for every other flow. These
threads must go back to one of the boundary regions, for concreteness say
$A_{i}$, and as a result cross $b_{p}(A_{i})$. Note that doing so limits the
amount of flux that the other flows can send back to $A_{i}$ (see fig 15). It
is through this mechanism of balancing the amount of flux that leaves
$\mathcal{O}$ conditioned on maximizing the flux each region can send that the
flows find the correct location of $\Sigma$ and value of $E_{w}(\mathcal{A})$.
For illustration, a numerical example is presented in fig. 16.

|
---|---
$v_{A}$ | $v_{B}$
$v_{C}$
Figure 16: An example of maximal flows $\mathcal{V}$ for a choice of boundary
regions $A,B,C$ on a flat square. The region $A$ is in red, $B$ in blue, and
$C$ in green. The minimal surface $\Sigma$ is given by the orange triangle.
The flows was calculated by latticizing the manifold and using _Mathematica_
’s built in FindMaximum function.

###  3.4 $\mathcal{U}$ and $w$ vector fields

Consider a change of variables by defining new vector fields

$w^{\mu}=\frac{1}{n}\sum_{i}v_{i}^{\mu},\quad
u^{\mu}_{i}=v^{\mu}_{i}-w^{\mu}.$ (56)

The program (44) becomes101010The condition $\sum_{i}u^{\mu}_{i}=0$ is not
necessary. Consider a set of maximal vector fields such that
$\sum_{i}u^{\mu}_{i}\neq 0$. Letting $\frac{1}{n}\sum_{i}u_{i}=x$ we can
create a new set of fields $u^{\prime}_{i}=u_{i}-x$ and $w^{\prime}=w+x$ which
are maximal, but satisfy $\sum_{i}u^{\prime\mu}_{i}=0$.

$\max_{\mathcal{U},w}\;\sum_{i}\int_{A_{i}}\sqrt{h}n_{\mu}u^{\mu}_{i}\;\text{s.t.
}\nabla_{\mu}u^{\mu}_{i}=0,\;\nabla_{\mu}w^{\mu}=0,\;|u_{i}+w|\leq
1,\;n_{\mu}u^{\mu}_{i}|_{\mathcal{O}}=0,\;\sum_{i}u^{\mu}_{i}=0.$ (57)

This redefinition comes with a trade off: at the cost of introducing a new
vector field $w$ which does not contribute to the objective and modifying the
norm bounds we have the benefit that $u_{i}$ cannot end on $\mathcal{O}$. It
is important to note it is not possible to generically set $w=0$. As we will
see this is a distinguishing feature from the bipartite case. As a result we
do not have the constraint $|u_{i}|\leq 1$, but rather $|u_{i}+w|\leq 1$, that
is the constitutes of $\mathcal{U}$ are _not_ flows. For this reason we
explicitly and pedantically refer to $(\mathcal{U},w)$ as vector fields.

In order for $E_{w}(\mathcal{A})$ to have a larger maximum than
$b_{p}(\mathcal{A})$ it must be possible to transport additional flux through
the manifold: this is exactly the purpose of $w$. Because of the norm bound
$|u_{i}+w|\leq 1$, $w$ has the property of modifying the flux that $u_{i}$ can
transport. Whenever $u_{i}$ and $w$ are anti-aligned it is possible to send
twice as much flux than would normally be possible, but if they are aligned
$u_{i}$ must locally be zero . As such the behavior of $w$ is to originate
from $\mathcal{O}$ and travel out to $\mathcal{A}$. This way each $u_{i}$ does
not encounter the bottle neck at $b_{p}(A_{i})$. In doing so, however every
other vector field $u_{j_{\neq}i}$ can send less flux through to $A_{i}$. This
naturally limits the maximum flux of $w$ out to $\mathcal{A}$.

One way to interpret this is to imagine we have chosen a configuration for
$w$. Then each $u_{i}$ will experience a different effective geometry
$g_{\mu\nu}^{i}$. When $u_{i}$ goes against $w$ there is twice as much space,
but when $u_{i}$ tries to flow with $w$ there is none. In the effective
geometry each $u_{i}$ can be renormalized to be a flow, but it is impossible
to do this for the vector fields as a whole.

An interesting feature of this framework is that all of the multipartite
features have been isolated to $w$, that is one vector field gives all the
information needed to understand the deviation from the area of the bipartite
surfaces. This can be seen in the numerical example fig. 17 which is fig. 16
shown in terms of the $u$ and $w$ vector fields.

|
---|---
$u_{A}$ | $u_{B}$
|
$u_{C}$ | $w$
Figure 17: The same configuration shown in fig. 16, in terms of the vector
fields $\mathcal{U}$ and $w$

###  3.5 The bipartite case

Having used our intuition and general properties of MFMC to define the flow
formulation for $E_{w}(A:B)$ we will now provide a direct proof using convex
duality which follows as a special case our multipartite derivation. Consider
two regions $A$ and $B$ with RT surface $\mathcal{O}=m(AB)$ such that
$\partial r(AB)=A\cup B\cup\mathcal{O}$. The minimal homology region cross
section is given by

$\min_{b_{p}(A:B)\sim A\,\text{rel}\,m(AB)}\operatorname{area}(b_{p}(A:B)).$
(58)

Figure 18: The bipartite homology region cross section.

Since the surface $b_{p}(A:B)$ naturally splits $\partial r(AB)$ into two
regions $\tilde{A}$ and $\tilde{B}$ we can equivalently calculate the minimal
surface among all partitions of $\mathcal{O}$

$\min_{\tilde{A}\tilde{B}}\operatorname{area}(b^{\min}_{AB})=\operatorname{area}(b_{p}(A:B))$
(59)

where $b^{\min}_{AB}$ is the minimal surface for a particular (not necessarily
minimal) partition (see fig. 18).

Setting $n=2$ in (44) we can write the flow program as

$\begin{split}\max_{v^{\mu}_{A}\;v^{\mu}_{B}}\int_{A}\sqrt{h}n_{\mu}v^{\mu}_{A}+\int_{B}\sqrt{h}n_{\mu}v^{\mu}_{B}+\frac{1}{2}\int_{\mathcal{O}}\sqrt{h}n_{\mu}\left(v_{A}^{\mu}+v_{B}^{\mu}\right)\;\text{s.t.
}\\\ \nabla_{\mu}v_{A}^{\mu}=0,\;\nabla_{\mu}v_{B}^{\mu}=0,\;|v_{A}|\leq
1,\;|v_{B}|\leq
1,\;n_{\mu}v^{\mu}_{A}|_{\mathcal{O}}=n_{\mu}v^{\mu}_{B}|_{\mathcal{O}}.\end{split}$
(60)

Figure 19: An example of a general flows $v_{A}$ and $v_{B}$. Such flows are
not in $\mathcal{S}^{\prime}$.

The minimal homology region cross section is calculated by maximizing the flux
of two flows $v^{\mu}_{A}$ and $v^{\mu}_{B}$. These flows have the property
that they are allowed to have flux on $\mathcal{O}$, but the local flux on
$\mathcal{O}$ of the two flows is constrained to be the same. Such boundary
threads are only counted with half the contribution of threads between $A$ and
$B$ (see fig. 19). Note the regions of $A$ and $B$ are large enough to source
enough flow to saturate the minimal surface. As such restricting to
$\mathcal{S^{\prime}}$ amounts to considering those flows for which the flux
on $\mathcal{O}$ is zero.

In order to make this freedom explicit we can use Stoke’s theorem and the
divergencelessness of the flows to find

$\frac{1}{2}\int_{\mathcal{O}}\sqrt{h}n_{\mu}\left(v_{A}^{\mu}+v_{B}^{\mu}\right)=-\frac{1}{2}\left(\int_{A}\sqrt{h}n_{\mu}\left(v_{A}^{\mu}+v_{B}^{\mu}\right)+\int_{B}\sqrt{h}n_{\mu}\left(v_{A}^{\mu}+v_{B}^{\mu}\right)\right).$
(61)

Substituting this in the objective becomes

$=\frac{1}{2}\left(\int_{A}\sqrt{h}n_{\mu}\left(v_{A}^{\mu}-v_{B}^{\mu}\right)+\int_{B}\sqrt{h}n_{\mu}\left(v_{B}^{\mu}-v_{A}^{\mu}\right)\right).$
(62)

To continue we make a change of variables by defining new vector fields

$w^{\mu}=\frac{1}{2}\left(v_{A}^{\mu}+v_{B}^{\mu}\right),\quad
u_{A}^{\mu}=v_{A}^{\mu}-w^{\mu},\quad u_{B}^{\mu}=v_{B}^{\mu}-w^{\mu}$ (63)

where by definition

$u^{\mu}_{A}+u^{\mu}_{B}=0.$ (64)

Figure 20: An example of vector fields $u$ and $w$.

The program becomes (see fig. 20):

$\begin{split}\max_{u_{A},u_{B},w}\;&\int_{A}\sqrt{h}n_{\mu}u_{A}^{\mu}+\int_{B}\sqrt{h}n_{\mu}u_{B}^{\mu}\;\text{s.t.
}\\\
&\nabla_{\mu}u_{A}^{\mu}=0,\;\nabla_{\mu}u_{B}^{\mu}=0,\;\nabla_{\mu}w^{\mu}=0,\;|u_{A}+w|\leq
1,\;|u_{B}+w|\leq 1,\;n_{\mu}u^{\mu}|_{\mathcal{O}}=0.\end{split}$ (65)

Because $u^{\mu}_{A}=-u^{\mu}_{B}$ a result of (64) the symmetry of the
problem dictates $w$ can not pass through the minimum surface and therefore is
unable to help increase the maximum. Thus, we can freely set $w^{\mu}=0$ when
we maximize (doing so restricts us to $\mathcal{S}^{\prime}$). Accounting for
normalization we have the final program

$\max_{u_{A}}\;\int_{A}\sqrt{h}n_{\mu}u_{A}^{\mu}\;\text{s.t.
}\nabla_{\mu}u_{A}^{\mu}=0,\;|u_{A}|\leq
1,\;n_{\mu}u_{A}^{\mu}|_{\mathcal{O}}=0$ (66)

which matches (19).

###  3.6 Weak duality

To understand the role of the surface $\Sigma(\mathcal{A})$ consider an
arbitrary set of vector fields $(\mathcal{U},w)$ which are not necessarily
maximal. By relative homology we know flux on the boundary for each $u_{i}$
must be the same as that of $\Sigma(A_{i})$. Furthermore, because
$\Sigma(\mathcal{A})$ is a closed surface we must have
$\int_{\Sigma(\mathcal{A})}w=0$. Thus

$\sum_{i}\int_{A_{i}}u_{i}=\sum_{i}\int_{\Sigma(A_{i})}u_{i}=\sum_{i}\int_{\Sigma(A_{i})}u_{i}+\int_{\Sigma(\mathcal{A})}w=\sum_{i}\int_{\Sigma(A_{i})}u_{i}+w\,.$
(67)

Because of the norm bound $|u_{i}+w|\leq 1$ we have
$\int_{\Sigma(A_{i})}u_{i}+w\leq\operatorname{area}(\Sigma(A_{i}))$. So we
have established weak duality:

$\sum_{i}\int_{A_{i}}u_{i}\leq\operatorname{area}(\Sigma(\mathcal{A}))\quad\forall\;(\mathcal{U},w)\,.$
(68)

###  3.7 The flow from $\mathcal{O}$ is truly multipartite

We would now like to interpret the quantity

$\int_{\mathcal{O}}\alpha=\int_{\mathcal{O}}w=\int_{\mathcal{O}}v_{i}$ (69)

for maximal flow configurations of (44). By considering only solutions in
$\mathcal{S}^{\prime}$ we guarantee every thread of $w$ is purposefully
utilized by $\mathcal{U}$ to achieve the maximum. That is there are no
extraneous threads of $w$. This is obvious as a reduction in
$\int_{\mathcal{O}}w$ further by deleting integral curves of $w$ would
necessarily mean such a collection of vector fields was not maximal. If $w$ is
zero we know $\mathcal{U}$ are flows and $b_{p}(\mathcal{A})$ is the
bottleneck. In this case $u_{i}$ saturates on $b_{p}(A_{i})$. When $w$ is
nonzero, because the configuration $(\mathcal{U},w)\in\mathcal{S}^{\prime}$,
for each thread of $w$ which crosses $b_{p}(A_{i})$, an additional thread of
$u_{i}$ can be transported across. This means the maximum flux across
$b_{p}(\mathcal{A})$ is given by

$\operatorname{area}(b_{p}(\mathcal{A}))+\int_{\mathcal{O}}w=\sum_{i}\int_{b_{p}(A_{i})}u_{i}.$
(70)

For maximal vector field configurations, $|u_{i}+w|$ saturates on
$\Sigma(A_{i})$, so that

$\operatorname{area}(\Sigma(\mathcal{A}))=\int_{\Sigma(\mathcal{A})}w+\sum_{i}\int_{\Sigma(A_{i})}u_{i}=\sum_{i}\int_{\Sigma(A_{i})}u_{i}$
(71)

where in the last step we have used $\int_{\Sigma(\mathcal{A})}w=0$. Now since
$b_{p}(\mathcal{A})$ is homologous to $\Sigma(\mathcal{A})$ (relative to
$\mathcal{O}$), we have shown

$\int_{\mathcal{O}}w=\operatorname{area}(\Sigma(\mathcal{A}))-\operatorname{area}(b_{p}(\mathcal{A})).$
(72)

This means the flux on $\mathcal{O}$ through the manifold for a given geometry
is a fixed quantity for all maximal thread configurations limited to the
solution space $\mathcal{S}^{\prime}$. The magnitude of the flux on
$\mathcal{O}$ measures precisely the difference between the multipartite and
bipartite holographic EOP proposals. In other words this flux represents how
much of the holographic multipartite EOP is truly multipartite.

###  3.8 Thread orientation

So far we have described bit threads as oriented integral curves of a vector
field (usually a flow). We would now like to show that it is possible to
instead maximize over all valid thread configurations without reference to
orientation. This can be done essentially because it is always possible to
impose a standard orientation which will maximize a given thread
configuration. Consider a maximal set of flows
$\mathcal{V}\in\mathcal{S^{\prime}}$ which calculates $E_{w}(\mathcal{A})$.
Now focusing on a single flow $v_{i}$ forget about the orientation. There are
two classes of threads to consider: those between $A_{i}$ and another interval
$A_{j}$ and those between $\mathcal{A}$ and $\mathcal{O}$.

For threads between $A_{i}$ and $A_{j}$ since the flux in (44) is being
counted on $A_{i}$ the orientation should always be imposed so that the thread
exits $A_{i}$ and ends on $A_{j}$. Each such thread contributes $1$ to the
objective.

Since the flux on $\mathcal{O}$ must be the same for all of $\mathcal{V}$ any
thread on $\mathcal{O}$ must have the same orientation for each of the flows.
Suppose the threads on $\mathcal{O}$ are oriented outward and travel to
$\mathcal{A}$. For $v_{i}$ the thread will avoid going back to $A_{i}$ as this
would contribute $\frac{1}{n}-1$ to the objective so instead the thread will
go out to end on some $A_{j}$ this way each thread contributes $\frac{1}{n}$.
The other possibility is that the threads connecting to $\mathcal{O}$ are
oriented to end there. This allows a thread of $v_{i}$ to begin on $A_{i}$ and
end on $\mathcal{O}$ and contribute $1-\frac{1}{n}$. The issue is that such a
thread must necessarily pass through $b_{p}(A_{i})$ meaning one less thread
can go between $A_{i}$ and $A_{j}$ which can never be maximal.

What we have shown is that for any maximal set of flows $\mathcal{V}$ the
correct orientation is to have all threads flow out $A_{i}$ and $\mathcal{O}$
for each $v_{i}$. Therefore we can simply count the number of threads leaving
these regions. Letting $N_{R}$ be the number of threads on region $R$ then the
holographic multipartite EOP proposal can be written as

$E_{w}(\mathcal{A})=\max\sum_{A_{i}}N_{A_{i}}|_{v^{\mu}_{i}}+N_{\mathcal{O}}$
(73)

where the maximization is over valid flow configurations. Note that even
though the boundary threads “appear” for each flow they are only counted once.

###  3.9 Bounds

We now prove several bounds for the holographic multipartite EOP proposal
using bit threads. For convenience and to give readers a feel for each
construction we will freely switch between the $\mathcal{V}$ flows and the
$\mathcal{U}$ and $w$ vector fields.

##### Properties

Let $\mathcal{A}$ be a pure state that is $\cup_{i}A_{i}=\partial M$. In this
case $\mathcal{O}=\emptyset$. Thus when maximizing (44) each flow $v_{i}$ is
geometrically limited by $m(A_{i})$ so that the flux calculates $S(A_{i})$.
Thus:

$\text{For pure states: }E_{w}(\mathcal{A})=\sum_{i}S(A_{i}).$ (74)

We will now show that $E_{w}(\mathcal{A})$ can only be smaller after
discarding part of one of the subregions. This can be thought of intuituvely
as reducing the amount of space the threads can occupy. Let
$\mathcal{A^{\prime}}=\\{X\cup A_{1},A_{2},...\\}$ we then have the two
programs

$\begin{split}E_{w}(\mathcal{A^{\prime}})=&\max_{\mathcal{V},\alpha}\sum_{i}\int_{A_{i}}\sqrt{h}n_{\mu}v^{\mu}_{i}+\int_{\mathcal{O^{\prime}}}\sqrt{h}\alpha\;s.t.\;n_{\mu}v^{\mu}_{i}|_{\mathcal{O^{\prime}}}=\alpha\;\text{on
$r(\mathcal{A^{\prime}})$}\\\
E_{w}(\mathcal{A})=&\max_{\mathcal{V},\alpha}\sum_{i}\int_{A_{i}}\sqrt{h}n_{\mu}v^{\mu}_{i}+\int_{\mathcal{O}}\sqrt{h}\alpha\;s.t.\;n_{\mu}v^{\mu}_{i}|_{\mathcal{O}}=\alpha\;\text{on
$r(\mathcal{A})$}.\end{split}$ (75)

Our goal will be to write the second of these as a program on $r(A^{\prime})$
with additional constraints. This can be done by viewing the program as being
on $r(\mathcal{A^{\prime}})$ and imposing the additional constraint that any
threads must remain in or end on the boundary of $r(\mathcal{A})$. This is
always possible since $r(\mathcal{A})\subset r(\mathcal{A}^{\prime})$.
Therefore using Theorem 1

$E_{w}(\mathcal{A})\leq E_{w}(\mathcal{A^{\prime}}).$ (76)

##### Upper bound

Consider the maximal flow $v_{A_{i}}$ which saturates on $\Sigma(A_{i})$ with
flux $\operatorname{area}(\Sigma(A_{i}))$. In order for this flux to count
maximally to the objective, the flow will want to end on
$\mathcal{A}\backslash A_{i}$ which is limited by the geometric obstruction
$\sum_{j\neq i}b_{p}(A_{j})$. The flux is further limited by the boundary
threads from the other flows in the amount $\sum_{j\neq
i}\operatorname{area}(\Sigma(A_{j}))-\sum_{j\neq
i}\operatorname{area}(b_{p}(A_{j}))$. From these considerations we have the
bound

$E_{w}(\mathcal{A})\leq 2\min_{i}\sum_{j\neq
i}\operatorname{area}(b_{p}(A_{j})).$ (77)

Figure 21: In this case
$\operatorname{area}(\Sigma(A_{1}))=\operatorname{area}(\Sigma(A_{2}))+\operatorname{area}(\Sigma(A_{3}))$
and the flux on $\mathcal{O}$ will be zero.

Note such a surface is in the same relative homology class as the surfaces we
have been considering and is thus an allowed candidate when minimizing (see
fig. 21). This inequality saturates when one boundary region has enough
capacity to send flow and saturate the bipartite surfaces of the other regions
by itself. In such cases there is no flux on $\mathcal{O}$ (put another way
$w=0$).

For a pure state this bound reduces to

$E_{w}(\mathcal{A})\leq\min_{i}(S(A_{i})+\ldots+S(A_{1}\ldots
A_{i-1},A_{i+1}\ldots A_{n})+\ldots+S(A_{n}))$ (78)

which matches the bound given by Umemoto2018 . Note that part of
$S(A_{1}\ldots A_{i-1},A_{i+1}\ldots A_{n})$ is part of $\mathcal{O}$ in the
non pure case. In our construction this surface can never be counted
indicating this bound is only saturated in the case of purity.

##### Lower bounds

Figure 22:  Minimal surfaces are shown as dashed lines and flows as solid
lines. The maximum multiflow the manifold can support is represented here, but
not all flow lines are shown as each dashed line is saturated. To create a
valid flow configuration in the feasible set of (44) we delete each thread
shown in red.

As in the bipartite case, we will derive a lower bound on the holographic
multipartite EOP proposal via an explicit construction. As previously
described the maximum total flux a multiflow can support is the sum of the
entropies of the regions whose union is $\partial M$. To get a valid flow
configuration of $E_{w}(\mathcal{A})$ we can choose to look at multiflows
which remain in $r(\mathcal{A})$. The simplest way (though not optimal) to
create such a flow is to delete each thread which passes through
$m(\mathcal{A})$. The number of times such threads leave $r(\mathcal{A})$ is
exactly $S(A_{1}\ldots A_{n})$ which on the boundary are each counted twice
(see fig. 22). That is, we have shown the holographic multipartite EOP
proposal is bounded below by the multipartite information 111111The explicit
construction of this flow provides evidence that an information theoretic
quantity the squashed entanglement will always be saturated holographically.
See D for more details.

$E_{w}(\mathcal{A})\geq I(\mathcal{A})\coloneqq\sum_{i}S(A_{i})-S(A_{1}\ldots
A_{n}).$ (79)

To get another lower bound consider the set of flow configurations of (57). If
we restrict ourselves to only flow configurations with $w=0$ then the
geometric barrier $b_{p}(\mathcal{A})$ is unavoidable. Thus, if one explicitly
imposes the condition $w=0$ to (57) the resulting program will calculate
$b_{p}(\mathcal{A})$. So by Theorem 1 we have the bound

$E_{w}(\mathcal{A})\geq\operatorname{area}(b_{p}(\mathcal{A}))$ (80)

which was first shown in Umemoto2018 .

##  4 Future directions

##### Flux from boundary and bulk intersections

In this article we have utilized convex duality to derive flow descriptions of
the holographic bipartite and multipartite EOP proposals. In the latter case
we find the addition of a coupled boundary flow which after restricting to
$\mathcal{S}^{\prime}$ is truly multipartite. Upon examination of the
dualization process this is not a unique feature, but rather should be present
in the flow dual to any surface with nontrivial boundary intersection.
Similarly, as can be seen in C, bulk intersection points will generically lead
to a bulk flux as indicated by the vector fields having a coupled non-zero
divergence. It would be interesting to explore the generality of these
features in more detail.

##### Dynamics

The maximin construction of Wall_2014  allows for covariant HRT surfaces
Hubeny_2007  to be calculated by minimizing a surface on a fixed time slice
and then maximizing over all time slices. Since our construction of
holographic EOP proposal holds for a fixed time slice it follows one could
define analogously the covariant EOP by calculating a maximal flow and then
maximizing over all possible time slices. This seems to indicate that many of
the features of the holographic EOP proposal should easily generalize to the
covariant case. It would be worthwhile to work out these features in detail
and describe them in the context of the full covariant construction of bit
threads cbt .

##### Bit threads and entanglement distillation

For the standard entanglement entropy a bit thread is normally described as
representing a distilled EPR pair between the boundary regions it connects.
This holds too for the multiflow where the collection of flows can be
simultaneously placed on the manifold. The question then arises how one should
interpret the flows of (44) for the holographic multipartite EOP proposal
which should presumably be related to a distillation of the multipartite
entanglement. While the bipartite contributions may retain their
interpretation as EPR pairs it is less clear how to represent the threads of
$\mathcal{V}$ which connect to $\mathcal{O}$ (or equivalently the ancillary
vector field $w$). This is because even though each $v_{i}$ will have the same
number of such threads the region on which they end can and must change
amongst the various elements of $\mathcal{V}$. The need for multiple threads
to describe a multipartite state is not dissimilar to the original description
of the GHZ state given in Freedman2017 . Possibly one could interpret each
$v_{i}$ as a distinct purification such that a collection of purifications is
needed to describe $E_{p}(\mathcal{A})$. It would be very interesting to see
if direct CFT calculations of $E_{p}(\mathcal{A})$ would give rise to similar
features.

##### Holographically multipartite entanglement is inefficient

We have established the bounds

$\operatorname{area}(b_{p}(\mathcal{A}))\leq E_{w}(\mathcal{A})\leq
2\min_{i}\sum_{j\neq i}\operatorname{area}(b_{p}(A_{j}))$ (81)

for the holographic multipartite EOP. Note that the flow configurations in
both extremes have $\alpha=0$. That is, in the language of our framework,
these configurations have no multipartite entanglement. This is not so
surprising as the threads on $\mathcal{O}$ are rather inefficient. This is
because one thread of each “color” is required to make a “unit” contribution.
This is in contrast to the bipartite threads connecting $A_{i}$ and $A_{j}$
which only require one. The multipartite threads simply take up more space in
the geometry. It would be interesting to understand why it is “hard” to get
multipartite entanglement. Potentially an understanding of the relation
between the choice of boundary configurations and the maximum multipartite
contribution could shed light on this feature.

##### Towards proving the conjecture

It has been firmly established that the homology region cross section and its
multipartite generalization are interesting bulk surfaces with possible
information theoretic content. What remains to be done is to firmly establish
the role these surfaces play in our understanding of AdS/CFT. Throughout this
paper we have worked under the assumption
$E_{p}(\mathcal{A})=E_{w}(\mathcal{A})$ following evidence such as the
original proposals Umemoto_2018 ; Nguyen2018 ; Umemoto2018 ; Bao:2018aa  and
work towards a proof Bao:2018ac ; Guo:2019aa ; bao:aa . Still there is reason
to proceed with some healthy skepticism. Alternate proposals exist on both
sides: Dutta:aa  calculates the bipartite homology region cross section using
a canonical purification whose geometry is that of a wormhole. In this
formalism explicit CFT calculations were able to be done using the replica
trick. Other duals such as the logarithmic negativity Kudler-Flam:2018aa  and
the odd entanglement entropy Tamaoka_2019  have appeared as well. Similarly
the “bipartite dominance” conjecture of Cui:2018aa  predicts that
holographically $E_{p}(A:B)=\frac{1}{2}I(A:B)$ always.

Almost all of the work done up to this point has focused on the bipartite case
where things are much simpler. However from the bit thread perspective it is
only when one is working the multipartite case that interesting features
appear specifically the coupling of flows due to the boundary intersection of
the minimal surface. It is our hope that this work and the characterization
$\Sigma$ in terms of bit threads may be elucidating to others and provide
inertia towards a complete holographic characterization that answers these
many questions.

###### Acknowledgements.

The work of J.H. is supported in part by the National Science Foundation under
the IGERT: Geometry and Dynamics Award No. 1068620 and in part by the Simons
Foundation through _It from Qubit: Simons Collaboration on Quantum Fields,
Gravity, and Information_. The work of M.H. is supported in part by the Simons
Foundation through _It from Qubit: Simons Collaboration on Quantum Fields,
Gravity, and Information_ and in part by the Department of Energy Office of
High-Energy Physics through Award DE-SC0009987. We would like to thank Cesar
Agón, Ning Bao, Bartlomiej Czech, Jesse Held, Veronika Hubeny, Charles Stine,
Koji Umemoto, and Mark Van Raamsdonk for useful discussion. We would also like
to thank Harsha Hampapura and Jesse Held for useful comments on an earlier
draft. J.H. would like to thank the Instituto Balseiro, the International
Centre for Theoretical Sciences (ICTS), the Institute for Advanced Study
(IAS), the Israel Institute for Advanced Study (IIAS), and the Yukawa
Institute for Theoretical Physics (YITP) for hospitality during various stages
of this work.

##  Appendix A Multiflows

In this appendix we define the multiflow and then for convenience state its
most important properties including a flow proof of monogamy of mutual
information. The entirety of this construction is due to the authors of
Cui:2018aa  where additional details can be found.

######  Definition 1 (Multiflow).

On a Riemannian manifold $M$ with boundary $\partial M$, split $\partial M$
into n non overlapping regions $\mathcal{A}=\\{A_{i}\\}$ . A multiflow is a
collection of vector fields $\mathcal{V}_{m}=\\{v_{ij}^{\mu}\\}$ which satisfy
the following properties: $n_{\mu}v^{\mu}_{ij}=0\;\text{on }A_{k},\;k\neq i,j$
$v_{ij}=-v_{ij}$ $\nabla_{\mu}v_{ij}^{\mu}=0$
$\sum\limits_{i<j}\left|v_{ij}\right|\leq 1.$

We will use $\mathcal{V}$ to indicate a set of flows, but we will reserve the
notation $\mathcal{V}_{m}$ to indicate a multiflow which satisfies all the
conditions of 1.

######  Definition 2 (Subflow).

Given a multiflow $\mathcal{V}_{m}$ consider the union of any number of
boundary regions in $\mathcal{A}$ call it $B$. The subflow of $B$ is the sum
of all flows leaving $B$

$v_{B}=\sum\limits_{i\text{ s.t }A_{i}\in B,\;j}v_{ij}$ (82)

######  Theorem 2 (Existence of a maximal multiflow).

There exists a multiflow $\mathcal{V}_{m}$ such that all single interval
subflows and any one additional subflow, call the associated boundary region
$B$, are maximal. That is

$S(A_{i})=\int_{A_{i}}v_{A_{i}}\quad S(B)=\int_{B}v_{B}$ (83)

for some $\mathcal{V}_{m}$. In general all other subflows will not be maximal.

######  Theorem 3 (Monogamy of mutual information (MMI)).

Consider the case of $4$ boundary regions $A,B,C,D$

$-I_{3}(A:B:C)=S(AB)+S(AC)+S(BC)-S(A)-S(B)-S(C)-S(ABC)\geq 0$ (84)

###### Proof.

The proof follows almost immediately by writing out the subflows and using
existence of a maximal multiflow. Consider the two interval subflows
$v_{AB},v_{AC},v_{BC}$. In general these can not all be made maximal thus

$S(AB)+S(BC)+S(AC)\geq\int_{AB}v_{AB}+\int_{AC}v_{AC}+\int_{BC}v_{BC}.$ (85)

Expanding out the subflows and using maximality of the single interval
subflows one has

$=\int_{A}v_{A}+\int_{B}v_{B}+\int_{C}v_{C}-\int_{ABC}v_{D}=S(A)+S(B)+S(C)+S(ABC)$
(86)

Subtracting the two sides gives MMI. ∎

##  Appendix B The integer relaxation

Figure 23: An allowed surface configuration after the integer relaxation. |  |
---|---|---
Figure 24: The corresponding configuration of $\psi_{i}$’s.

In our derivation of the the flow program for the holographic multipartite EOP
we were required to make the the following relaxation of the primal program

$\text{On $\mathcal{O}$:
}\psi_{i}\in\\{0,1\\}\Rightarrow\psi_{i}\in\mathbb{R}.$ (87)

Notably this allows solutions where each $\psi_{i}$ changes in more than one
location and certain portions of $\mathcal{O}$ are shared by multiple
$\psi_{i}$’s (for example consider the configuration of surfaces and the
corresponding $\psi_{i}$’s given by figs. 23 and 24).

Figure 25: An integer surface configuration.

It is possible to construct a geometry (though granted a highly contrived one)
which “forces” the lengths of these surfaces to be small making it the correct
minimum to the program (39) which would be different from the optimal
configuration of (35) (which is shown in fig. 25) . This demonstrates that our
relaxed program is no longer generically the same as the original proposal.

We would now like to prove that for a $2+1d$ bulk with everywhere negative
curvature and three boundary regions the relaxed program (39) will always be
the same as the original proposal (35). That is we can not do any better by
allowing the $\psi_{i}$’s to share $\mathcal{O}$ and the optimal solution will
in fact be integer.

For this setup the surface $\mathcal{O}$ are geodesics of the manifold.
Because the space is negatively curved we are guaranteed that the distance
between points on different geodesics is a convex function of the position on
each geodesic, parametrized by distance along the geodesic. We consider the
family of all possible solutions with two points on $\mathcal{O}$ where the
points are labeled by their location on (or affine distance along)
$\mathcal{O}$. The objective is the weighted area of the six surfaces
connecting these points

$\operatorname{area}(\Sigma_{2})=\frac{1}{2}(d_{16}+d_{63}+d_{32}+d_{25}+d_{54}+d_{41})$
(88)

which is a convex function in all six variables of the locations on
$\mathcal{O}$. We then consider the analogous problem with only one point on
each part of $\mathcal{O}$ and furthermore pick the points
$1^{\prime},2^{\prime},3^{\prime}$ such that we are at a the global minimum

$\operatorname{area}(\Sigma)=d_{1^{\prime}3^{\prime}}+d_{3^{\prime}5^{\prime}}+d_{5^{\prime}1^{\prime}}.$
(89)

Thus taking $1=2=1^{\prime},\;3=4=3^{\prime},\;5=6=5^{\prime}$ we find this
configuration is also a minimum of $\operatorname{area}(\Sigma_{2})$. Since we
are at a minimum the gradient is zero thus deforming the surfaces by moving
any one of the end points can only increase the value of the objective.
Convexity demands that this is the global minimum of
$\operatorname{area}(\Sigma_{2})$.

This argument has obvious generalizations to more complex configurations of
$\psi_{i}$’s and larger number of boundary regions and shows for these cases
(39) is exactly dual to (35). It would be interesting to try to generalize
this reasoning to higher dimensions, but we leave this to future work.121212It
may possible to prove a form of such a result by utilizing the quotient
geometry induced by the relative homology.

##  Appendix C Multipartite surfaces with bulk intersection

Figure 26: An example of a class of surfaces in the bulk which allow for bulk
intersection points.

Another surface similar to the one which gives the holographic multipartite
EOP proposal has appeared in the literature Bao:2018aa

$E_{m}(\mathcal{A})=\min\operatorname{area}(\text{Surfaces which split
$r(\mathcal{A})$ into $n$ regions homologous to each $A_{i}$}).$ (90)

The key distinction being the allowance of a bulk intersection point (see fig.
26).

Following convex dualization we find some interesting parallels in the flow
descriptions between $E_{w}(\mathcal{A)}$ and $E_{m}(\mathcal{A})$. For
simplicity we will dualizing assuming the boundary is pure and then generalize
to the case of nonpurity. We dualize by first performing a convex and integer
relaxation. We will ignore analogous complications due to the possible
inequivalence of the programs due to the integer relaxation. Note that because
the minimal surface meets in the bulk we get a constraint on the scalar fields
which also must be imposed in the bulk.

$\min\sum_{i}\int_{M}\sqrt{g}|\nabla_{\mu}\psi_{i}|\quad\text{s.t.}\quad\psi_{i}|_{A_{j}}=\delta_{ij},\quad\sum_{i}\psi_{i}|_{M}=1$
(91)

We get the Lagrangian

$\begin{split}L(\\{\psi\\},\\{\phi\\},\\{v^{\mu}\\},\\{\gamma\\},\\{\beta\\})&=\int_{M}\sqrt{g}\left[\sum_{i}\left(|\phi_{i}|+v^{\mu
i}(\phi_{\mu i}-\nabla_{\mu}\psi_{i}))+\beta\psi_{i}\right)-\beta\right]\\\
&+\sum_{i}\int_{A_{i}}\sqrt{h}\left[\gamma_{ii}(\psi_{i}-1)+\sum_{i\neq
j}\gamma_{ij}\psi_{j}\right].\end{split}$ (92)

Dualizing with respect to $\\{\phi\\}$ we get norm constraints:

$\phi_{\mu i}v^{\mu i}+|\phi_{\mu i}|=0,\quad|v^{\mu i}|\leq 1$ (93)

which can be substituted in. After performing integration by parts we have

$\int_{M}\sqrt{g}\left[\sum_{i}\left(\psi_{i}\nabla_{\mu}v^{\mu
i}+\beta\right)-\beta\right]+\sum_{i}\int_{A_{i}}\sqrt{h}\left[\gamma_{ii}(\psi_{i}-1)+\sum_{i\neq
j}\gamma_{ij}\psi_{j}+\sum_{j}n_{\mu}v^{\mu j}\psi_{j}\right].$ (94)

Next we dualize with respect to $\\{\psi\\}$.

$\begin{split}&\text{On $M$: }\nabla_{\mu}v^{\mu}_{i}=-\beta\\\ &\text{On
$A_{i}$:
}\gamma_{ii}+n_{\mu}v^{\mu}_{i}=0,\quad\gamma_{ij}+n_{\mu}v^{\mu}_{j}=0\end{split}$
(95)

Note that we do not get the normal divergencelessness constraints, flux will
be created and destroyed in the bulk. The resulting dual is

$\max_{\mathcal{V}}\sum_{i}\left(\int_{A_{i}}\sqrt{h}n_{\mu}v^{\mu}_{i}+\frac{1}{n}\int_{M}\sqrt{h}\nabla_{\mu}v^{\mu}_{i}\right).$
(96)

We can once again redefine our vector fields

$w^{\mu}=\frac{1}{n}\sum_{i}v_{i}^{\mu}\quad u^{\mu}_{i}=v^{\mu}_{i}-w^{\mu}$
(97)

with the program becoming

$\max_{\mathcal{U},w}\;\sum_{i}\int_{A_{i}}\sqrt{h}n_{\mu}u^{\mu}_{i}\;\text{s.t.
}\nabla_{\mu}u^{\mu}_{i}=0,\;\nabla_{\mu}w^{\mu}=-\beta,\;|u_{i}+w|\leq
1,\;\sum_{i}u^{\mu}_{i}=0.$ (98)

Comparing to the holographic multipartite EOP proposal essentially what has
happened is the entire bulk $M$ has become an unlimited source of $w$ by
virtue of the fact that $w$ is not divergenceless and its divergence is an
unconstrained function in the bulk when maximizing. It is as if the entire
bulk is geometrically connected to the boundary and acts as the purifier. This
gives the flows a tremendous amount of freedom when attempting to maximize
pushing the minimal surface deep into the bulk. The dualization argument
presented can be generalized by also considering a non-pure boundary with
region $\mathcal{O}$ as in the case of the multipartite EOP. In doing so all
we must change is $w$ in addition to having flux in the bulk can begin and end
on $\mathcal{O}$. From this we see starting with this program we can impose
$\nabla_{\mu}w^{\mu}=0$ as an explicit constraint and make use of Theorem 1 to
find

$E_{m}(\mathcal{A})\geq E_{w}(\mathcal{A}).$ (99)

The mechanism for the flows associated with these two surface is almost
identical the only difference being whether $w$ is sourced on the boundary,
the bulk or both. In general nonzero divergence in the bulk has been
associated with quantum corrections to entanglement entropy. While it is not
know what information theoretic quantity, if any, this surface is dual to in
the boundary CFT, it seems suggestive that quantum corrections may play a
role. At the very least these surfaces are intimately connected by the common
mechanism for how the corresponding flows are generated. As previously
mentioned we expect this to be a general feature of any flow program dual to a
bulk surface with intersections.

##  Appendix D Squashed entanglement

Another notable entanglement measure is the squashed entanglement
Christandl:aa ; Tucci:aa  and its multipartite generalization:

$E_{sq}(\mathcal{A})=\min I(\mathcal{A}|E).$ (100)

Here the minimization is taken over all extensions
$\rho_{\mathcal{A}}=\operatorname{Tr}[\rho_{\mathcal{A}E}]$. Most notably the
multipartite squashed entanglement has the property

$E_{sq}(\mathcal{A})\leq I(\mathcal{A})\leq E_{p}(\mathcal{A}).$ (101)

There are two important points to consider: One, calculating a conditioned
entropy measure is implemented using bit threads as a change in the region
that a maximal thread configuration can occupy. This can be thought of as
imposing additional boundary constraints on the convex program which force the
threads to remain in the region. Two, using the surface/state correspondence
Miyaji:2015aa  we can guarantee any choice of extension can be holographically
represented as a surface outside the homology region of $\mathcal{A}$. Thus,
the most restrictive space we could utilize is $r(\mathcal{A})$ itself. This
can also be motivated using MMI which implies utilizing an extension can never
be advantageous Umemoto2018 .

With these considerations in mind we can make the following observation:
during our discussion of the multipartite EOP we were able to explicitly
construct a multiflow which calculated $I(\mathcal{A})$ and was contained
entirely in the homology region $r(\mathcal{A})$ (see fig. 22). This shows
that holographically the squashed entanglement will be saturated which we view
as additional evidence for the conjecture of Umemoto2018

$E_{sq}(\mathcal{A})=I(\mathcal{A}).$ (102)

## References

  * (1) T. Takayanagi and K. Umemoto, _Entanglement of purification through holographic duality_ , _Nature Phys._ 14 (2018) 573–577, [1708.09393].
  * (2) P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel and B. Swingle, _Entanglement of purification: from spin chains to holography_ , _JHEP_ 01 (2018) 098, [1709.07424].
  * (3) S. Dutta and T. Faulkner, _A canonical purification for the entanglement wedge cross-section_ , 1905.00577.
  * (4) J. Kudler-Flam and S. Ryu, _Entanglement negativity and minimal entanglement wedge cross sections in holographic theories_ , 1808.00446.
  * (5) K. Tamaoka, _Entanglement Wedge Cross Section from the Dual Density Matrix_ , _Phys. Rev. Lett._ 122 (2019) 141601, [1809.09109].
  * (6) M. Miyaji and T. Takayanagi, _Surface/State Correspondence as a Generalized Holography_ , _PTEP_ 2015 (2015) 073B03, [1503.03542].
  * (7) N. Bao, _Minimal Purifications, Wormhole Geometries, and the Complexity=Action Proposal_ , 1811.03113.
  * (8) W.-Z. Guo, _Entanglement of Purification and Projective Measurement in CFT_ , 1901.00330.
  * (9) N. Bao, A. Chatwin-Davies, J. Pollack and G. N. Remmen, _Towards a Bit Threads Derivation of Holographic Entanglement of Purification_ , 1905.04317.
  * (10) B. M. Terhal, M. Horodecki, D. W. Leung and D. P. DiVincenzo, _The entanglement of purification_ , _J. Math. Phys._ 43 (2002) 4286–4298, [quant-ph/0202044].
  * (11) A. Bhattacharyya, T. Takayanagi and K. Umemoto, _Entanglement of Purification in Free Scalar Field Theories_ , _JHEP_ 04 (2018) 132, [1802.09545].
  * (12) P. Caputa, M. Miyaji, T. Takayanagi and K. Umemoto, _Holographic Entanglement of Purification from Conformal Field Theories_ , _Phys. Rev. Lett._ 122 (2019) 111601, [1812.05268].
  * (13) A. Bhattacharyya, A. Jahn, T. Takayanagi and K. Umemoto, _Entanglement of Purification in Many Body Systems and Symmetry Breaking_ , _Phys. Rev. Lett._ 122 (2019) 201601, [1902.02369].
  * (14) S. Bagchi, _Monogamy, polygamy, and other properties of entanglement of purification_ , _Physical Review A_ 91 (2015) .
  * (15) P. Hayden, M. Headrick and A. Maloney, _Holographic Mutual Information is Monogamous_ , _Phys. Rev._ D87 (2013) 046003, [1107.2940].
  * (16) M. Headrick, _General properties of holographic entanglement entropy_ , _JHEP_ 03 (2014) 085, [1312.6717].
  * (17) C. A. Agon, J. De Boer and J. F. Pedraza, _Geometric Aspects of Holographic Bit Threads_ , _JHEP_ 05 (2019) 075, [1811.08879].
  * (18) M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang and Y.-T. Zhou, _The connection between holographic entanglement and complexity of purification_ , 1902.02475.
  * (19) J. Kudler-Flam, I. MacCormack and S. Ryu, _Holographic entanglement contour, bit threads, and the entanglement tsunami_ , 1902.04654.
  * (20) K. Umemoto and Y. Zhou, _Entanglement of Purification for Multipartite States and its Holographic Dual_ , _JHEP_ 10 (2018) 152, [1805.02625].
  * (21) N. Bao and I. F. Halpern, _Conditional and Multipartite Entanglements of Purification and Holography_ , _Phys. Rev._ D99 (2019) 046010, [1805.00476].
  * (22) S. Boyd and L. Vandenberghe, _Convex Optimization_.  Cambridge University Press, 2004, 10.1017/cbo9780511804441.
  * (23) M. Freedman and M. Headrick, _Bit threads and holographic entanglement_ , _Commun. Math. Phys._ 352 (2017) 407–438, [1604.00354].
  * (24) M. Headrick and V. E. Hubeny, _Riemannian and Lorentzian flow-cut theorems_ , _Class. Quant. Grav._ 35 (2018) 10, [1710.09516].
  * (25) D.-H. Du, C.-B. Chen and F.-W. Shu, _Bit threads and holographic entanglement of purification_ , 1904.06871.
  * (26) S. Ryu and T. Takayanagi, _Aspects of Holographic Entanglement Entropy_ , _JHEP_ 08 (2006) 045, [hep-th/0605073].
  * (27) N. Bao and I. F. Halpern, _Holographic Inequalities and Entanglement of Purification_ , _JHEP_ 03 (2018) 006, [1710.07643].
  * (28) S. X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, _Bit Threads and Holographic Monogamy_ , 1808.05234.
  * (29) N. Bao, A. Chatwin-Davies and G. N. Remmen, _Entanglement of Purification and Multiboundary Wormhole Geometries_ , _JHEP_ 02 (2019) 110, [1811.01983].
  * (30) A. C. Wall, _Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy_ , _Class. Quant. Grav._ 31 (2014) 225007, [1211.3494].
  * (31) V. E. Hubeny, M. Rangamani and T. Takayanagi, _A Covariant holographic entanglement entropy proposal_ , _JHEP_ 07 (2007) 062, [0705.0016].
  * (32) M. Headrick and V. Hubeny, “Covariant bit threads.” To appear.
  * (33) M. Christandl and A. Winter, _”squashed entanglement” - an additive entanglement measure_ , _J. Math. Phys. Vol 45, No 3, pp. 829-840 (2004)_ (2003) , [quant-ph/0308088].
  * (34) R. R. Tucci, _Entanglement of distillation and conditional mutual information_ , quant-ph/0202144.