CALT-TH 2023-052

MIT-CTP/5659

Beyond the Holographic Entropy Cone via Cycle Flows

Temple He1, Sergio Hernández-Cuenca2, Cynthia Keeler3

1Walter Burke Institute for Theoretical Physics, California Institute of
Technology,
Pasadena, CA 91125 USA
2Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139 USA

3Department of Physics, Arizona State University,
Tempe, AZ 85281 USA

Motivated by bit threads, we introduce a new prescription for computing
entropy vectors outside the holographic entropy cone. By utilizing cycle flows
on directed graphs, we show that the maximum cycle flow associated to any
subset of vertices, which corresponds to a subsystem, manifestly obeys
purification symmetry. Furthermore, we prove that the maximum cycle flow obeys
both subadditivity and strong subadditivity, thereby establishing it as a
viable candidate for the entropy associated to the subsystem. Finally, we
demonstrate how our model generalizes the entropy vectors obtainable via
conventional flows in undirected graphs and hypergraphs.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and Conventions
    1. 2.1 Quantum Information Theory
    2. 2.2 Graph Theory
  3. 3 Cycle Flow Entropy Cone
    1. 3.1 Entropy as Cycle Flows
    2. 3.2 Subadditivity
    3. 3.3 Strong Subadditivity
  4. 4 Discussion
  5. A Remarks on the Dual Program
    1. A.1 The Linear Program and its Dual
    2. A.2 Vertex Cuts
    3. A.3 Alternative Proofs of SA

##  1 Introduction

The analysis of the holographic entropy cone (HEC) was initiated in [1], and
its success has relied crucially on the fact that the entanglement entropy of
the boundary conformal field theory (CFT) is captured geometrically by Ryu-
Takayanagi (RT) surfaces in the bulk for static geometries [2].111The
covariant generalization of RT surfaces is given in [3]. In particular, given
any boundary region $A$ of the CFT, the RT surface associated to $A$ is the
surface homologous to $A$ in the bulk anti-de Sitter (AdS) spacetime with the
minimum area. The entanglement entropy between $A$ and its complement on the
boundary is then proportional to the area of the RT surface. This can of
course be easily generalized to computing the entanglement entropy between
multiple disjoint subregions on the boundary. We will henceforth always
implicitly assume that the subregions are disjoint.

A crucial insight made in [1] was that the geometrization of entanglement
entropy for CFTs implies that the structure of entanglement entropy for the
various subregions can be described in terms of a discrete graph. More
precisely, suppose we are interested in the entanglement entropy of the
boundary subregions $A_{1},A_{2},\ldots A_{{\sf{N}}}$, and denote
$A_{{\sf{N}}+1}=(A_{1}\cup A_{2}\cup\dots A_{\sf{N}})^{c}$ as the purifier,
where the superscript $c$ denotes the complement. Then there exists a discrete
undirected graph with a subset of its vertices labeled as
$A_{1},A_{2},\ldots,A_{{\sf{N}}+1}$, which we call boundary vertices, such
that the min cut of any subset of the boundary vertices gives the entanglement
entropy associated to the union of corresponding boundary subregions. We will
call all other vertices internal vertices.

The recasting of holographic entropies into min cuts of discrete graphs was
enormously fruitful, as it allowed for the utilization of various graph
theoretic tools [4, 5, 6, 7, 8, 9, 10]. For instance, the max flow-min cut
theorem was used to give a “bit thread” interpretation of holographic
entropies [11, 12, 13, 14, 15, 16]. Under this perspective, rather than
thinking of entanglement entropy as given by a minimum cut in the graph, it is
given by a maximal flow between a subset of boundary vertices and its
complement. In the original boundary CFT, we can interpret the entanglement
entropy of a subregion as the maximum flux of a vector field that flows
through the AdS bulk geometry out of the subregion.

Despite the various successes of studying entropies of holographic states,
there are many quantum states, such as the 4-party GHZ state, that are not
holographic. It is then a natural question if there is some generalization of
undirected graphs that can perhaps capture a wider class of states. The most
obvious choice may be to allow the graph to have directed edges. However, if
the entropy associated to a subset of boundary vertices is given by the max
flow, then there is no guarantee that it is equal to the max flow out of the
complement of the subset of boundary vertices, thereby violating purification
symmetry of entanglement entropy. Nevertheless, rather than introducing
directed edges, there have been various such generalizations in the
literature, including both hypergraphs [17] and links [18].222In particular,
stabilizer states provide a well-studied class of states which are not all
holographic. The papers cited above use the stabilizer states as an example.
For more on the entropies available to stabilizer states see [17, 19, 20, 21,
22, 23]; for similar explorations on Dicke states, see [24]. Such
generalizations automatically respect purification symmetry as there is no
directionality in their structures, while also allowing for min cut structures
that are disallowed in the original undirected graphs, and there has been
fruitful progress in using them to explore entropy vectors of non-holographic
states [19, 20, 25, 26].

In this paper, we present an alternative way of moving beyond entropies of
holographic states by using instead another notion of flows. Simply stated,
our protocol to compute the entropy ${\texttt{S}}({\underline{I}})$ from a
graph is the following.

  * •

Begin with a balanced directed network.333For us, we choose to call a network
a graph with edge weights. This is defined precisely later.

  * •

Identify the set of boundary vertices $\partial V$ which represent boundary
regions.

  * •

Partition the boundary vertices $\partial V$ into the set ${\underline{I}}$
and its complement, ${\underline{I}}^{c}\coloneqq\partial
V\setminus{\underline{I}}$.

  * •

Consider closed paths (cycles) in the graph which pass through at least one
vertex in ${\underline{I}}$ and at least one vertex in ${\underline{I}}^{c}$.

  * •

The entropy ${\texttt{S}}({\underline{I}})$ is given by the magnitude of the
maximum simultaneous cycle flow along all such paths.

Note that this protocol uses cycle flows, which are flows with neither sources
nor sinks, on balanced directed networks, which are graphs with directed edges
obeying the condition where at every vertex the total weight of ingoing edges
equals that of outgoing edges. Additionally, the specific flow pattern is
important; we need to know which cycles have flow turned on, not just which
edges. As we shall see, this generalization to directed graphs under a cycle
flow protocol allows access to a broader set of entropy vectors than the
undirected or hyperedge graph protocols. Additionally, the cycle flow protocol
retains some of the topological features of the link picture.

Importantly, this cycle flow protocol respects purification symmetry, since
the vertex set ${\underline{I}}$ and its complement ${\underline{I}}^{c}$ play
the same role. As we show below, entropies calculated from the cycle flow
protocol always obey both subadditivity (SA) and strong subadditivity (SSA),
which are necessary conditions for identifying the flow as entanglement
entropy. Furthermore, we will also show that in general, the flow will not
satisfy monogamy of mutual information (MMI), which is a holographic entropy
inequality [27], thereby implying entropies represented by such cycle flows
represent non-holographic states as well.

Our paper is structured as follows. In Section 2, we give a brief review of
relevant background information from both quantum information theory and graph
theory. We will also use the opportunity to establish the various conventions
used throughout. In Section 3.1, we will formally introduce our prescription
for identifying the maximum cycle flow with the entropy. We then prove in
Section 3.2 and 3.3 that our prescription satisfies both SA and SSA, giving
evidence that such maximum cycle flows can indeed be thought of as von Neumann
entropies. Finally, we conclude with a discussion and open questions in
Section 4. In Appendix A, we remark on the dual program and how to prove SA
using it.

##  2 Preliminaries and Conventions

In this section, we review relevant background information from both graph
theory and quantum information theory. The quantum information theory portion
is reviewed in Section 2.1, and the graph theory portion is reviewed in
Section 2.2.

###  2.1 Quantum Information Theory

The systematic study of the entropy cone associated to a general quantum
system is organized in terms of the number of parties.444For the holographic
entropy cone, each party corresponds to a spatial boundary subregion of the
CFT, and all the subregions are disjoint. If there are ${\sf{N}}$ parties
$A_{1},A_{2},\ldots,A_{\sf{N}}$, we can specify the entanglement structure of
the system by a vector in $2^{{\sf{N}}}-1$ dimensions, with each component
being ${\texttt{S}}(I)$, the entropy of the subsystem $\bigcup_{i\in I}A_{i}$,
for every nonempty subset
$I\subseteq[{\sf{N}}]\coloneqq\\{1,2,\ldots,{\sf{N}}\\}$. For small number of
parties, we can also label the parties $A,B,\ldots$. The vector space in which
this entropy vector lives is called the entropy space. For example, for
${\sf{N}}=3$ parties, with the parties labeled as $A,B$, and $C$, the entropy
vector lives in the $7$-dimensional entropy space and is given by

$\displaystyle\vec{\texttt{S}}=\big{(}{\texttt{S}}(A),\;{\texttt{S}}(B),\;{\texttt{S}}(C),\;{\texttt{S}}(AB),\;{\texttt{S}}(AC),\;{\texttt{S}}(BC),\;{\texttt{S}}(ABC)\big{)}.$
(2.1)

In general, we will label the components of the entropy vector in
lexicographical order. There is the additional purifier party
$A_{{\sf{N}}+1}\coloneqq(A_{1}\cup A_{2}\cup\cdots\cup A_{\sf{N}})^{c}$ (or
$O$ if we are labeling the parties using the alphabet), but by purification
symmetry we have

$\displaystyle{\texttt{S}}({\underline{I}})={\texttt{S}}({\underline{I}}^{c}),$
(2.2)

for any ${\underline{I}}\subseteq[{\sf{N}}+1]$ and
${\underline{I}}^{c}\coloneqq[{\sf{N}}+1]\setminus{\underline{I}}$ is the
complement. We will always use $I,J,\ldots$ to denote subsets of $[{\sf{N}}]$,
and ${\underline{I}},{\underline{J}},\ldots$ to denote subsets of
$[{\sf{N}}+1]$, which includes the purifier. Thus, the entropy of any subset
of parties involving the purifier $A_{{\sf{N}}+1}$ can be rewritten as the
entropy of its complement, which does not involve $A_{{\sf{N}}+1}$ and is
hence a component of the entropy vector as defined above.

Not every point in entropy space is realizable by quantum states. For
instance, the positivity of entropy implies that for all
${\underline{I}}\subseteq[{\sf{N}}+1]$, we have

$\displaystyle{\texttt{S}}({\underline{I}})\geq 0.$ (2.3)

This means that entropy vectors corresponding to some quantum state must live
in the region of entropy space where all its components are positive.
Furthermore, we also know that entropies obey certain linear inequalities,
which for $I,J,K\subseteq[{\sf{N}}]$ are given by

Subadditivity (SA):
$\displaystyle{\texttt{S}}(I)+{\texttt{S}}(J)\geq{\texttt{S}}(IJ)$ (2.4)
Araki-Lieb (AL):
$\displaystyle{\texttt{S}}(I)+{\texttt{S}}(IJ)\geq{\texttt{S}}(J)$ (2.5)
Strong subadditivity (SSA):
$\displaystyle{\texttt{S}}(IJ)+{\texttt{S}}(JK)\geq{\texttt{S}}(J)+{\texttt{S}}(IJK)$
(2.6) Weak monotonicity (WM):
$\displaystyle{\texttt{S}}(IJ)+{\texttt{S}}(JK)\geq{\texttt{S}}(I)+{\texttt{S}}(K),$
(2.7)

where $IJ\coloneqq I\cup J$ is shorthand for the union of the two subset of
regions, and $IJK$ is similarly defined for three regions. In the above
inequalities, we did not utilize the purifier entropy
${\texttt{S}}({\sf{N}}+1)$. It is an easy exercise to see that AL and WM are
related to SA and SSA by purification symmetry (2.2), respectively. Therefore,
we can consolidate the four types of inequalities above into the following two
types:

Subadditivity (SA):
$\displaystyle{\texttt{S}}({\underline{I}})+{\texttt{S}}({\underline{J}})\geq{\texttt{S}}({\underline{I}}{\underline{J}})$
(2.8) Strong subadditivity (SSA):
$\displaystyle{\texttt{S}}({\underline{I}}{\underline{J}})+{\texttt{S}}({\underline{J}}{\underline{K}})\geq{\texttt{S}}({\underline{J}})+{\texttt{S}}({\underline{I}}{\underline{J}}{\underline{K}}).$
(2.9)

Henceforth, SA and SSA will refer to the inequalities (2.8) and (2.9), which
can involve the purifier entropy ${\texttt{S}}({\sf{N}}+1)$.

For any choice of ${\underline{I}},{\underline{J}},{\underline{K}}$, the
inequalities (2.8) and (2.9) each define a half-space in entropy space. The
intersection of all such half-spaces then forms a polyhedral cone in which
entropy vectors corresponding to realizable quantum states live. We will refer
to this cone as the SSA entropy cone. The quantum entropy cone (QEC) is
defined to be the topological closure of the set of all the entropy vectors
realizable by quantum states, and was shown in [28] to be a convex cone. Thus,
because all such entropy vectors must obey SA and SSA, the QEC lives inside
the SSA entropy cone. It is known that for ${\sf{N}}=3$ the QEC actually
coincides with the SSA entropy cone. However, for ${\sf{N}}\geq 4$, besides
the existence of infinitely many constrained inequalities (inequalities that
only apply on subspaces of codimension higher than zero) [29, 30], very little
is known about the QEC.555In particular, it remains an open question whether
the QEC is even polyhedral for higher parties.

Holographic states are a subset of quantum states. Therefore, the holographic
entropy cone (HEC), which is a convex, polyhedral cone consisting of the set
of all entropy vectors realizable by holographic states [1, 6], lies within
the QEC. Indeed, we know that holographic states satisfy additional
inequalities that are not necessarily satisfied by all quantum states. These
are known as holographic entropy inequalities (HEIs), and one such inequality
is the monogamy of mutual information (MMI) [27, 13], which is given by

$\displaystyle\begin{split}&\text{Monogamy of mutual information (MMI):}\\\
&\qquad\qquad{\texttt{S}}({\underline{I}}{\underline{J}})+{\texttt{S}}({\underline{I}}{\underline{K}})+{\texttt{S}}({\underline{J}}{\underline{K}})\geq{\texttt{S}}({\underline{I}})+{\texttt{S}}({\underline{J}})+{\texttt{S}}({\underline{K}})+{\texttt{S}}({\underline{I}}{\underline{J}}{\underline{K}}).\end{split}$
(2.10)

Notice that MMI and SA together imply SSA, since adding (2.8) to (2.10) yields
(a relabeling of) (2.9). MMI is the only HEI for ${\sf{N}}<5$, but additional
HEIs are known for ${\sf{N}}\geq 5$ [1, 31, 32, 10, 33].

It is known that a model for describing entropy vectors in the HEC uses
undirected graphs [1]. More specifically, given any entropy vector in HEC
involving ${\sf{N}}$ parties (excluding the purifier), there exists an
undirected graph ${\mathcal{G}}=(V,E)$ such that a subset of vertices
$\partial V\coloneqq\\{1,2,\ldots,{\sf{N}}+1\\}\subseteq V$, which we call
boundary vertices, correspond to the ${\sf{N}}$ parties plus the purifier, and
the min cut of any subset of boundary vertices equals the entropy associated
to the corresponding subset of parties. Entropy as defined by such graph
models satisfy the HEIs by construction. Because our goal is to explore beyond
the HEC (but still remain within the QEC), we would like to determine an
alternative model to describe a class of entropy vectors that satisfy SA (2.8)
and SSA (2.9) without having to satisfy HEIs like MMI (2.10). We will
introduce such a model in Section 3, but before continuing, let us first
review some preliminaries from graph theory in the next subsection.

###  2.2 Graph Theory

We begin by introducing notation and establishing conventions for relevant
graph theory concepts. Consider first a discrete directed graph
${\mathcal{G}}\coloneqq(V,E)$, where $V$ is the collection of vertices and $E$
is the collection of directed edges.666Note that undirected graphs are a
special case of directed graphs where between every two vertices
$v_{1},v_{2}\in V$ there exists a directed edge from $v_{1}$ to $v_{2}$ and
another directed edge from $v_{2}$ to $v_{1}$. Thus, it suffices for us to
study directed graphs only. Associated to every directed edge $e\in E$ is a
capacity function $c:\,E\to\mathbb{R}_{\geq 0}$, which maps every edge to a
non-negative real number known as the edge’s capacity or weight. We can then
define a network ${\mathcal{N}}\coloneqq(V,E,c)$ to be a graph $(V,E)$ with a
capacity function $c$.

For every directed edge $e$, it is useful to define $s(e)\in V$ to be the
source vertex from which the edge originates from, and $t(e)\in V$ to be the
target vertex towards which the edge is directed. Using the capacity function,
we will now introduce the following notion of cycle flows, which will be
useful for our purposes.

######  Definition 1 (Simple cycle flow).

Given a network ${\mathcal{N}}=(V,E,c)$, we define a edge cycle to be an
ordered sequence of distinct edges
$E_{\mathcal{C}}\coloneqq\\{e_{1},e_{2},\ldots,e_{n}\\}$ such that
$t(e_{i})=s(e_{i+1})$ for $i=1,\ldots,n$ with $e_{n+1}\coloneqq e_{1}$. Notice
that $E_{\mathcal{C}}$ is defined only up to cyclic permutations. A simple
cycle flow is then an ordered pair
${\mathcal{C}}\coloneqq(E_{\mathcal{C}};{\lambda})$, where $E_{\mathcal{C}}$
is any edge cycle and ${\lambda}\in\mathbb{R}_{\geq 0}$ is a non-negative
number, satisfying the property

$\displaystyle c(e)\geq{\lambda}\quad\text{for every $e\in E_{\mathcal{C}}$.}$
(2.11)

We refer to ${\lambda}$ as the magnitude, or flux, of the simple cycle flow
${\mathcal{C}}$.

Unlike typical flows in network theory involving sources and sinks, which have
arisen in quantum information literature as bit threads [11], simple cycle
flows have not been connected to any notion of entropy. However, we will
introduce a proposal in the next section that relates cycle flows to
entanglement entropy beyond holographic states. It would be interesting to see
if there are any relations between the cycle flows we are considering and the
more well-understood bit threads (see Footnote 13).777For an exploration of
the more conventional flows involving sources and sinks, and how they pertain
to the study of the holographic entropy cone in the context of bit threads, we
refer the reader to [11] and in particular Section 6 of [13].

In order to connect cycle flows to entropy, we will need to define a notion of
magnitude for such flows. Typically in network theory, the magnitude of a flow
is measured by the flux from vertices that are sources to vertices that are
sinks. However, cycle flows have zero flux, as there are no vertices that are
sources or sinks. Hence, the flux of a cycle flow into any vertex equals that
out of the vertex. Nevertheless, given the set of boundary vertices $\partial
V=[{\sf{N}}+1]\subseteq V$, we can introduce an alternative notion of the
magnitude of a flow through some subset of boundary vertices. As was
introduced in Section 2.1, subsets of boundary vertices will be denoted using
${\underline{I}},{\underline{J}}$, and ${\underline{K}}$.

######  Definition 2.

Given a network ${\mathcal{N}}=(V,E,c)$, let $\partial V=[{\sf{N}}+1]\subseteq
V$ be a subset of vertices called boundary vertices. For any simple cycle flow
${\mathcal{C}}=(E_{\mathcal{C}};\lambda)$ on ${\mathcal{N}}$, define

$\displaystyle V({\mathcal{C}})\coloneqq\bigg{(}\bigcup_{e\in
E_{\mathcal{C}}}t(e)\bigg{)}\cap\partial V$ (2.12)

to be the boundary vertices part of the cycle flow ${\mathcal{C}}$. Letting
${\underline{I}},{\underline{J}}\subseteq\partial V$ denote any two disjoint
subsets of boundary vertices, we define the function

$\displaystyle
f_{\mathcal{C}}({\underline{I}},{\underline{J}})=\begin{cases}\lambda&V({\mathcal{C}})\cap{\underline{I}}\neq\emptyset\quad\text{and}\quad
V({\mathcal{C}})\cap{\underline{J}}\neq\emptyset\\\
0&\text{otherwise.}\end{cases}$ (2.13)

Notice that $f_{\mathcal{C}}$ is symmetric with respect to its two arguments,
and we call $f_{\mathcal{C}}({\underline{I}},{\underline{J}})$ the flux
between ${\underline{I}}$ and ${\underline{J}}$ due to simple cycle
${\mathcal{C}}$. Thus, the flux between ${\underline{I}}$ and
${\underline{J}}$ is nonzero only if ${\mathcal{C}}$ involves at least one
vertex in ${\underline{I}}$ and one in ${\underline{J}}$.

Utilizing simple cycles, we can now define the analog of multi-flows for cycle
flows, which we call multi-cycle flows.

######  Definition 3 (Multi-cycle flow).

Given a network ${\mathcal{N}}=(V,E,c)$, a multi-cycle flow is a set of simple
cycle flows
${\mathfrak{C}}\coloneqq\\{{\mathcal{C}}_{i}=(E_{i};{\lambda}_{i})\\}_{i=1,\ldots,m}$
obeying the following condition. For every $e\in E$, defining $u_{i}(e)=1$ if
$e\in E_{i}$ and 0 otherwise, we have

$\displaystyle\sum_{i=1}^{m}u_{i}(e){\lambda}_{i}\leq c(e).$ (2.14)

Furthermore, the set of boundary vertices which are part of the multi-cycle
flow is given by

$\displaystyle
V({\mathfrak{C}})\coloneqq\bigcup_{{\mathcal{C}}\in{\mathfrak{C}}}V({\mathcal{C}}).$
(2.15)

Lastly, letting ${\underline{I}},{\underline{J}}\subseteq\partial V$ denote
any two disjoint subsets of boundary vertices, the flux between
${\underline{I}}$ and ${\underline{J}}$ due to ${\mathfrak{C}}$ is defined to
be

$\displaystyle
f_{\mathfrak{C}}({\underline{I}},{\underline{J}})=\sum_{i=1}^{m}f_{{\mathcal{C}}_{i}}({\underline{I}},{\underline{J}}),$
(2.16)

where $f_{{\mathcal{C}}_{i}}({\underline{I}},{\underline{J}})$ is the flux
between ${\underline{I}}$ and ${\underline{J}}$ due to simple cycle flow
${\mathcal{C}}_{i}$, as defined in (2.13).

Given any network ${\mathcal{N}}$ and any subset of boundary vertices
${\underline{I}}\subseteq\partial V$, it is clear that the maximum flux
between ${\underline{I}}$ and its complement
${\underline{I}}^{c}\coloneqq\partial V\setminus{\underline{I}}$ is bounded
above by the sum of capacities of edges emanating from ${\underline{I}}$, that
is,

$\displaystyle\max_{{\mathfrak{C}}}f_{\mathfrak{C}}({\underline{I}},{\underline{I}}^{c})\leq\sum_{e\in
E:\,s(e)\in{\underline{I}}}c(e).$ (2.17)

In particular, this means the maximum flux between ${\underline{I}}$ and
${\underline{I}}^{c}$ is well-defined, and we denote it as

$\displaystyle{\texttt{S}}({\underline{I}})\coloneqq\max_{{\mathfrak{C}}}f_{\mathfrak{C}}({\underline{I}},{\underline{I}}^{c}).$
(2.18)

In the following section, we will identify the maximum flux of a multi-cycle
flow through some boundary vertices ${\underline{I}}$ of a _balanced_ network
(defined in 7) as the entropy, hence the suggestive notation of denoting it as
${\texttt{S}}({\underline{I}})$. Furthermore, for convenience, we will
henceforth denote

$\displaystyle f_{\mathcal{C}}({\underline{I}})\coloneqq
f_{\mathcal{C}}({\underline{I}},{\underline{I}}^{c}),\qquad
f_{\mathfrak{C}}({\underline{I}})\coloneqq
f_{\mathfrak{C}}({\underline{I}},{\underline{I}}^{c}).$ (2.19)

Let us illustrate the notion of flux and multi-cycle flows with a simple
example. In the figure below, we draw two possible multi-cycle flows, where
${\mathfrak{C}}_{1}$ is on the left figure and ${\mathfrak{C}}_{2}$ is on the
right figure. The multi-cycle flows ${\mathfrak{C}}_{1}$ and
${\mathfrak{C}}_{2}$ each consists of two simple cycles colored by red and
blue. We can easily compute

$\displaystyle\begin{split}f_{{\mathfrak{C}}_{1}}(A)=2,\qquad
f_{{\mathfrak{C}}_{1}}(B)=2,\qquad f_{{\mathfrak{C}}_{1}}(AB)=0,\\\
f_{{\mathfrak{C}}_{2}}(A)=2,\qquad f_{{\mathfrak{C}}_{2}}(B)=2,\qquad
f_{{\mathfrak{C}}_{2}}(AB)=1.\end{split}$ (2.20)

In particular, notice that the flux out of $AB$ in ${\mathfrak{C}}_{1}$
differs from that in ${\mathfrak{C}}_{2}$.

$O$$B$$A$$O$$B$$A$ Figure 1: All the edges implicitly are weight 1. We
illustrate two multi-cycle flows on the graph. In both cases, the multi-cycle
flow consists of two simple cycles, which are colored red and blue in the
figure. Notice that the two multi-cycle flows give rise to different fluxes
through the subsets of vertices.

For future convenience, it is actually useful to define a notion of a
“minimal” simple cycle that does not use unnecessary edges. To make this
definition precise, we begin by defining the natural notion of a subcycle of a
simple cycle flow.

######  Definition 4 (Subcycle flow).

Given a simple cycle flow ${\mathcal{C}}=(E_{\mathcal{C}};\lambda)$, where
$E_{{\mathcal{C}}}=\\{e_{1},\ldots,e_{n}\\}$, a subcycle flow is any simple
cycle flow ${\mathcal{C}}^{\prime}=(E_{\mathcal{C}}^{\prime};\lambda)$, with
$E_{{\mathcal{C}}^{\prime}}$ being a subsequence of $E_{\mathcal{C}}$.

From 2, it is clear that the flux between two sets of boundary vertices
${\underline{I}}$ and ${\underline{J}}$ for a simple cycle flow
${\mathcal{C}}$ will be unchanged if ${\mathcal{C}}$ is replaced by any
subcycle flow ${\mathcal{C}}^{\prime}$ so long as the latter also traverses
vertices in both ${\underline{I}}$ and ${\underline{J}}$. For this reason, it
makes sense to define a notion of minimality of simple cycle flows relative to
vertex subsets as follows.

######  Definition 5 (Minimal simple cycle flow).

A simple cycle flow ${\mathcal{C}}$ is minimal with respect to a collection of
boundary vertex subsets
$\\{{\underline{I}}_{1},{\underline{I}}_{2},\ldots,{\underline{I}}_{m}\\}$ if,
for any subcycle flow ${\mathcal{C}}^{\prime}$ with
$E_{{\mathcal{C}}^{\prime}}\subset E_{{\mathcal{C}}}$ , the flux
$f_{{\mathcal{C}}^{\prime}}({\underline{I}}_{i},{\underline{I}}_{j})<f_{{\mathcal{C}}}({\underline{I}}_{i},{\underline{I}}_{j})$
for some $i,j\in[m]$.

$B$$A_{1}$$A_{2}$$C$4123$B$$A_{1}$$A_{2}$$C$326145 Figure 2: In the figure, we
have a partition of the four vertices of the graph into $\\{A_{1},A_{2}\\}$,
$\\{B\\}$, and $\\{C\\}$. All the edges have capacity 1, and the circled
numbers indicate the ordering of the edge cycle in the cycle flow. On the
left, we have a minimal cycle flow that is colored in red. On the right we
have a simple cycle flow that is not a minimal cycle flow, since we can remove
the subcycle involving edges labeled by 2 and 3 without changing the flux
between any pair of vertex subsets $\\{A_{1},A_{2}\\}$, $\\{B\\}$, and
$\\{C\\}$.

We illustrate the difference between a simple cycle flow that is a minimal
cycle flow and one that is not in Figure 2. Effectively, minimal cycles allow
us to capture the flux between subsets of boundary vertices without “wasting”
any flows on edges that do not contribute to the flux. Similarly, we can
readily generalize the notion of “minimality” to multi-cycle flows.

######  Definition 6 (Minimal multi-cycle flow).

Given a partition of the boundary vertices $\partial
V=\\{{\underline{I}}_{1},\ldots,{\underline{I}}_{n}\\}$, a multi-cycle flow
${\mathfrak{C}}$ is minimal with respect to
$\\{{\underline{I}}_{1},\ldots,{\underline{I}}_{n}\\}$ if all the simple cycle
flows comprising ${\mathfrak{C}}$ are minimal with respect to
$\\{{\underline{I}}_{1},\ldots,{\underline{I}}_{n}\\}$.

We conclude by proving a result regarding minimal cycles that will later be
useful. Henceforth, we will only focus on a class of networks known as
balanced networks, which are defined as follows.

######  Definition 7.

A network ${\mathcal{N}}=(V,E,c)$ is balanced if at every vertex $v\in V$, the
total capacity of ingoing edges equals that of outgoing edges. In other words,

$\displaystyle\sum_{e\in E:\,s(e)=v}c(e)=\sum_{e\in
E:\,t(e)=v}c(e)\quad\text{for all $v\in V$.}$ (2.21)

Recall now the notion of a residual network given a network
${\mathcal{N}}=(V,E,c)$ and a multi-cycle flow
${\mathfrak{C}}=\\{(E_{i};{\lambda}_{i})\\}_{i=1,\ldots,m}$. Utilizing the
definition of $u_{i}$ given in 3, the residual graph is given by

$\displaystyle{\text{Res}}({\mathcal{N}},{\mathfrak{C}})\coloneqq\bigg{(}V,E,c-\sum_{i=1}^{m}u_{i}\lambda_{i}\bigg{)}.$
(2.22)

Notice that because of (2.14), the capacity function of the residual network
is always non-negative. If the capacity of an edge in
${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$ is 0, then we simply remove the
edge. Intuitively, the residual network is the remaining capacity of the edges
of ${\mathcal{N}}$ given there is a multi-cycle flow ${\mathfrak{C}}$ already
flowing on ${\mathcal{N}}$, and is useful for many reasons, including the
following proposition.

######  Proposition 1.

Given any (not necessarily balanced) network ${\mathcal{N}}=(V,E,c)$ and a
multi-cycle flow ${\mathfrak{C}}$, if ${\mathcal{C}}$ is a simple cycle flow
in ${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$, then
${\mathfrak{C}}\cup{\mathcal{C}}$ is a multi-cycle flow in the original
network ${\mathcal{N}}$.

###### Proof.

Denoting
${\mathfrak{C}}=\\{{\mathcal{C}}_{i}=(E_{i};\lambda_{i})\\}_{i=1,\ldots,m}$,
the capacity of every edge of the residual network
${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$ is given by

$\displaystyle c(e)-\sum_{i=1}^{m}u_{i}(e)\lambda_{i},$ (2.23)

where $u_{i}(e)=1$ if $e\in E_{i}$ and 0 otherwise, as was introduced in 3.
Since ${\mathcal{C}}=(E_{\mathcal{C}};\lambda)$ is a cycle flow in
${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$, this means by 1

$\displaystyle c(e)-\sum_{i=1}^{m}u_{i}(e)\lambda_{i}\geq\lambda\quad\text{for
every $e\in E_{\mathcal{C}}$}.$ (2.24)

It is trivial then to see that

$\displaystyle c(e)\geq u(e)\lambda+\sum_{i=1}^{m}u_{i}(e)\lambda_{i},$ (2.25)

where $u(e)=1$ if $e\in E_{\mathcal{C}}$ and 0 otherwise. This proves that
${\mathfrak{C}}\cup{\mathcal{C}}$ is a cycle flow in the original network
${\mathcal{N}}$. ∎

Furthermore, for the special case where the network ${\mathcal{N}}$ is
balanced, we have the following nice property.

######  Proposition 2.

Given a balanced network ${\mathcal{N}}=(V,E,c)$ and a multi-cycle flow
${\mathfrak{C}}$, the residual network
${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$ is a balanced network.
Furthermore, there exists a multi-cycle flow ${\mathfrak{C}}$ such that the
residual network has $c(e)=0$ for every $e\in E$.

###### Proof.

Because the multi-cycle flow ${\mathfrak{C}}$ is balanced, this means for
every $v\in V$, the total capacity reduction of the edges going into $v$
equals the total capacity reduction of the edges coming out of $v$, which
proves the claim ${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$ is balanced.

To prove the second part of the claim, consider some multi-cycle flow
${\mathfrak{C}}^{\prime}$ in ${\mathcal{N}}$. It is obvious that the capacity
of every edge in
${\mathcal{N}}_{1}\coloneqq{\text{Res}}({\mathcal{N}},{\mathfrak{C}}^{\prime})$
is less than or equal to that in ${\mathcal{N}}$, with the capacity of at
least one edge in ${\mathcal{N}}_{1}$ being strictly less than that in
${\mathcal{N}}$. If $c(e)>0$ for some $e\in E$ in ${\mathcal{N}}_{1}$, we can
construct a simple cycle ${\mathcal{C}}_{1}$ containing $e$ by first
constructing an edge cycle $E_{{\mathcal{C}}_{1}}$ explicitly starting from
$e$; this is always possible because ${\mathcal{N}}_{1}$ is balanced. The
magnitude of ${\mathcal{C}}_{1}$ is then given to be $\text{min}_{e\in
E_{{\mathcal{C}}_{1}}}c(e)$. Notice this means the residual graph
${\mathcal{N}}_{2}\coloneqq{\text{Res}}({\mathcal{N}}_{1},{\mathcal{C}}_{1})$
has at least one fewer edge than ${\mathcal{N}}_{1}$. We can then repeat this
process with ${\mathcal{N}}_{1}$ and ${\mathcal{N}}$ and ${\mathcal{N}}_{2}$
in place of ${\mathcal{N}}_{1}$ and obtain a simple cycle flow
${\mathcal{C}}_{2}$. Repeating this until the resulting residual graph has no
edges, i.e. $c(e)=0$ for all edges, it then follows
${\mathfrak{C}}\coloneqq{\mathfrak{C}}^{\prime}\cup{\mathcal{C}}_{1}\cup\cdots\cup{\mathcal{C}}_{n}$
is a multi-cycle flow where the residual graph
${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$ has $c(e)=0$ for every edge. ∎

We next prove that cycle flows on a balanced network have a crucial property
known as nesting.888This claim is analogous to the result from [11], where
they prove the existence of nested flows for conventional flows involving
sources and sinks. It would be interesting to further explore connections
between cycle flows and conventional flows, especially in the context of
multi-commodity flows. As we will see in the next section, this property will
be repeatedly used to prove that (2.18) obeys both SA and SSA, thereby
motivating us to identify it with the entropy.

######  Theorem 1.

Given a balanced network ${\mathcal{N}}=(V,E,c)$ with boundary vertices
$\partial V\subseteq V$, and given two disjoint subsets of boundary vertices
${\underline{I}},{\underline{J}}$ where
${\underline{I}}\cap{\underline{J}}=\emptyset$, let
${\underline{K}}\coloneqq\partial V\setminus{\underline{I}}{\underline{J}}$
denote the rest of the boundary vertices. Then there exists a minimal multi-
cycle flow ${\mathfrak{C}}({\underline{I}},{\underline{J}})$, with respect to
the boundary vertex partition $\partial
V=\\{{\underline{I}},{\underline{J}},{\underline{K}}\\}$ whose magnitude
through ${\underline{I}}$ and ${\underline{I}}{\underline{J}}$ is
simultaneously maximized.

###### Proof.

The proof proceeds by identifying sets of minimal cycles which can be
simultaneously combined to construct a multi-cycle flow with the desired
properties. We start from a multi-cycle flow
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}{\underline{K}}}$ that
maximizes the flux through ${\underline{I}}$ conditioned on only using minimal
cycle flows that each include at least one boundary vertex in all
${\underline{I}},{\underline{J}}$, and ${\underline{K}}$. Note that this means
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}{\underline{K}}}$ will already
automatically include all contributions to the flux through
${\underline{I}}{\underline{J}}$ which involve at least one vertex in
${\underline{I}}$. By construction, the residual network
${\mathcal{N}}_{1}\coloneqq{\text{Res}}({\mathcal{N}},{\mathfrak{C}}_{{\underline{I}}{\underline{J}}{\underline{K}}})$
no longer contains any minimal cycles involving vertices in all
${\underline{I}},{\underline{J}}$, and ${\underline{K}}$.

Next, on ${\mathcal{N}}_{1}$, let
${\mathfrak{C}}_{{\underline{I}}{\underline{K}}}$ be the multi-cycle flow that
maximizes the flux through ${\underline{I}}$ and consists only of minimal
cycle flows that each include at least one boundary vertex in both
${\underline{I}}$ and ${\underline{K}}$. By construction, this means that the
residual network
${\mathcal{N}}_{2}\coloneqq{\text{Res}}({\mathcal{N}}_{1},{\mathfrak{C}}_{{\underline{I}}{\underline{K}}})$
does not have any minimal cycles simultaneously connecting vertices in both
${\underline{I}}$ and ${\underline{K}}$. In particular, this means any vertex
in ${\underline{J}}$ that is part of a minimal cycle with a vertex in
${\underline{K}}$ cannot be connected to any vertices in ${\underline{I}}$,
and similarly any vertex in ${\underline{J}}$ that is part of a minimal cycle
involving vertices in ${\underline{I}}$ cannot be connected to any vertex in
${\underline{K}}$. It follows that we can simultaneously construct multi-cycle
flows ${\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$ and
${\mathfrak{C}}_{{\underline{J}}{\underline{K}}}$ on ${\mathcal{N}}_{2}$ such
that ${\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$ only involve minimal
cycles that maximize the flux between ${\underline{I}}$ and ${\underline{J}}$,
and ${\mathfrak{C}}_{{\underline{J}}{\underline{K}}}$ only involve minimal
cycle flows that maximize the flux between ${\underline{J}}$ and
${\underline{K}}$.

Because the networks ${\mathcal{N}}$, ${\mathcal{N}}_{1}$, and
${\mathcal{N}}_{2}$ are all balanced, there are no longer any directed edges
left in
${\mathcal{N}}_{3}\coloneqq{\text{Res}}({\mathcal{N}}_{2},{\mathfrak{C}}_{IJ}\cup{\mathfrak{C}}_{JK})$
connecting ${\underline{I}}$, ${\underline{J}}$, or ${\underline{K}}$. The
remaining minimal cycles in ${\mathcal{N}}_{3}$ do not contribute to any
fluxes through ${\underline{I}}$, ${\underline{J}}$, or ${\underline{K}}$.
Hence, it is clear that
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}{\underline{K}}}\cup{\mathfrak{C}}_{{\underline{I}}{\underline{K}}}\cup{\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$
maximizes the flux through ${\underline{I}}$, and
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}{\underline{K}}}\cup{\mathfrak{C}}_{{\underline{I}}{\underline{K}}}\cup{\mathfrak{C}}_{{\underline{J}}{\underline{K}}}$
maximizes the flux through ${\underline{I}}{\underline{J}}$. By our argument
in the previous paragraph, since we can simultaneously have
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$ and
${\mathfrak{C}}_{{\underline{J}}{\underline{K}}}$, this means the multi-cycle
flow
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}{\underline{K}}}\cup{\mathfrak{C}}_{{\underline{I}}{\underline{K}}}\cup{\mathfrak{C}}_{{\underline{I}}{\underline{J}}}\cup{\mathfrak{C}}_{{\underline{J}}{\underline{K}}}$
simultaneously maximizes the flux through both ${\underline{I}}$ and
${\underline{I}}{\underline{J}}$. ∎

Before continuing, we remark that it was crucial in the proof above that we
are constructing the multi-cycle flows using _minimal cycle flows_ and not
just simple cycle flows. To see why, consider the two simple cycle flows in
Figure 2 with ${\underline{I}}=\\{A_{1},A_{2}\\}$, ${\underline{J}}=\\{B\\}$,
and ${\underline{K}}=\\{C\\}$. In both cases, the red simple cycle flow
maximizes the flux through ${\underline{I}}$ while utilizing at least one
vertex in ${\underline{I}},{\underline{J}}$, and ${\underline{K}}$. However,
the right figure, which is not a _minimal cycle flow_ , prevents the addition
of the minimal cycle involving just the vertices $A_{2}$ and $B$ when
constructing ${\mathfrak{C}}_{IJ}$. Therefore, the inclusion of the non-
minimal simple cycle depicted in the right network in the construction given
in the proof of Theorem 1 will not lead to a multi-cycle flow that maximizes
the flux through ${\underline{I}}=\\{A_{1},A_{2}\\}$.

We conclude this section by noting that the above theorem crucially relies on
the fact the network has the balanced property. For instance, consider the
unbalanced network below, where every edge has capacity 1. It is easy to see
that the maximal flux through $ABC$ is 1, and the maximal flux through $C$ is
2. However, there is only one cycle flow that maximizes the flux through
$ABC$, which is shown in red in the figure below, and it does not maximize the
flux through $C$. Hence, nesting fails for this unbalanced network.

$O$$C$$B$$A$ Figure 3: An unbalanced network, where all the edges have
capacity 1, that does not satisfy the nesting property. We have
${\texttt{S}}(ABC)=1$ and ${\texttt{S}}(C)=2$. The red highlighted edges
denote the only cycle flow that maximizes the flux through $ABC$, and it does
not simultaneously maximize the flux through $C$.

##  3 Cycle Flow Entropy Cone

In this section, we give a prescription for using the fluxes of the cycle
flows of networks to represent entropies. This is introduced in Section 3.1.
We will then prove that the entropies defined via this prescription satisfy SA
in Section 3.2 and SSA in Section 3.3 by using various properties of minimal
cycle flows.

###  3.1 Entropy as Cycle Flows

Given a balanced network ${\mathcal{N}}=(V,E,c)$, we designate a subset of its
vertices $\partial V=[{\sf{N}}+1]\subseteq V$, which we call boundary
vertices, such that they each correspond to the party $A_{i}$ for
$i\in[{\sf{N}}+1]$ in our system, with $A_{{\sf{N}}+1}$ being the purifier. We
will then identify the entropy of the parties
$\bigcup_{i\in{\underline{I}}}A_{i}$ to be the maximum magnitude of a multi-
cycle flow through ${\underline{I}}\subseteq\partial V$, as defined in (2.18).
In other words, recalling (2.18), we define

$\displaystyle\text{Entropy of
$\bigg{(}\bigcup_{i\in{\underline{I}}}A_{i}$}\bigg{)}:={\texttt{S}}({\underline{I}})=\max_{{\mathfrak{C}}}f_{{\mathfrak{C}}}({\underline{I}},{\underline{I}}^{c}).$
(3.1)

Before we check various properties of this definition, it is useful to give
some examples of how to associate an entropy vector to a balanced network.
First, we claim that the prescription above describes the entropy vectors of
all holographic states. To see why, notice that any undirected graph
${\mathcal{G}}=(V,E)$ with weighted edges can be represented as a balanced
network such that the maximum flow out of any subset of vertices remain
unchanged. For every edge $e\in E$ with capacity $c(e)$ connecting vertices
$v_{1}$ and $v_{2}$, we can replace it with two directed edges such that one
directed edge goes from $v_{1}$ to $v_{2}$ with capacity $c(e)$ and the other
goes from $v_{2}$ to $v_{1}$ with the same capacity $c(e)$ (see Figure 4).
This gives rise to a balanced network ${\mathcal{N}}$, and the flow out of any
vertex stays the same as that of the original graph ${\mathcal{G}}$.

$c(e)$$c(e)$$c(e)$ Figure 4: The left graph represents the entropy of a Bell
pair in the original graph model. We can equally represent the entropy vector
using cycle flows via the network on the right.

We now prove that the cycle flow model we are proposing in (3.1) is sufficient
to describe all original holographic entropy vectors describable by graph
models.

######  Proposition 3.

All holographic entropy vectors can be represented using the max cycle flux of
balanced networks given in (3.1).

###### Proof.

Given a holographic entropy vector, it can be represented by some undirected
graph ${\mathcal{G}}=(V,E)$, with vertices $V$ and weighted edges $E$ [1]. In
particular, there exists a subset of vertices $\partial
V=[{\sf{N}}+1]\subseteq V$, which we call boundary vertices, and these
vertices represent the ${\sf{N}}$ parties and the purifier. The entropy
associated to any subset ${\underline{I}}\subseteq\partial V$ is then maximum
flow out of ${\underline{I}}$ to its complement ${\underline{I}}^{c}$, and as
we mentioned above such a flow still exists in the directed network
constructed from ${\mathcal{G}}$ by replacing each undirected edge with a pair
of oppositely directed edges shown in Figure 4. We can easily augment the flow
into a cycle flow by simply using the edges going in the opposite direction to
go from ${\underline{I}}^{c}$ back to the original vertices ${\underline{I}}$.
The cycle flux in this case just equals the flux out of ${\underline{I}}$, so
the max flux out of ${\underline{I}}$ in ${\mathcal{G}}$ equals the max cycle
flux out of ${\underline{I}}$ in the corresponding network, completing the
proof. ∎

However, our goal is to move beyond holographic entropy vectors, and it is
instructive to see an example. Consider the balanced network in Figure 5,
where the vertices are labeled by $A,B,C$, and purifier $O$, and all edges
have capacity $1$.

$O$$C$$B$$A$$e_{1}$$e_{2}$$e_{3}$$e_{4}$ Figure 5: A balanced network, where
every edge implicitly has capacity 1, representing a 4-party GHZ state.

For this network, every vertex is a boundary vertex. Furthermore, given any
subset of boundary vertices, the multi-cycle flow with maximum magnitude
through any subset of boundary vertices is the minimal cycle flow
${\mathcal{C}}=(\\{e_{1},e_{2},e_{3},e_{4}\\};1)$, which is the clockwise flow
with magnitude $1$. By our definition, the magnitude of this cycle flow
through all subsets of boundary vertices is

$\displaystyle{\texttt{S}}(A)={\texttt{S}}(B)={\texttt{S}}(C)={\texttt{S}}(AB)={\texttt{S}}(AC)={\texttt{S}}(BC)={\texttt{S}}(ABC)=1.$
(3.2)

This set of entropies precisely matches those of the 4-party GHZ state. The
GHZ state is not holographic, since it clearly violates MMI (2.10). Thus, we
see that by identifying maximum flux associated to cycle flows as entropy, we
are able to obtain entropy vectors of non-holographic states.999It was proven
in Theorem 5 of [13] that for directed inner-superbalanced networks (which
includes balanced networks), if we identify the max flow out of a subset of
boundary vertices ${\underline{I}}$ as the entropy $S_{\underline{I}}$, then
MMI is obeyed. However, since we identify entropy $S_{\underline{I}}$ not as
the max flow out of ${\underline{I}}$, but as the max flux of a multi-cycle
flow between ${\underline{I}}$ and ${\underline{I}}^{c}$, the results of [13]
do not apply here. For instance, the standard max flow out of vertices
$\\{A,C\\}$ in Figure 5 is $2$, whereas the max flux of a cycle flow through
$\\{A,C\\}$ is $1$.

$A_{1}$$A_{2}$$A_{n}$$A_{3}$$A_{n-1}$ Figure 6: An $n$-party GHZ state, which
is also representable as a hypergraph.

From the above example, we can easily construct a balanced network
corresponding to a $n$-party GHZ state. Labeling the regions
$A_{1},\ldots,A_{n}$, we depict this state in Figure 6. As in the 4-party GHZ
network, every vertex is a boundary vertex, and the multi-cycle flow
maximizing the magnitude through any subset of boundary vertices is again the
clockwise cycle flow with magnitude $1$. Thus we find

$\displaystyle{\texttt{S}}({\underline{I}})=1\quad\text{for any
${\underline{I}}\subset[n]$}.$ (3.3)

Notice that this also captures the notion of a general hypergraph edge
explored in [17]. For instance, we can represent hypergraph 7 given in Figure
4 of [17] as a cycle flow in Figure 7 below. We simply replace each 4-vertex
hyperedge with the 4-vertex cyclic network shown in Figure 5 and every
undirected edge with the two directed edges shown in Figure 4. We leave it as
an exercise for the reader to check that given our prescription (3.1), we
indeed reproduce the entropy vector of the hypergraph. Furthermore, we prove
below that in general, entropy vectors representable by hypergraphs can be
captured by cycle flows.

$A$$B$$C$$D$$O$ Figure 7: This balanced network reproduces the entropy vector
represented by hypergraph 7 in Figure 4 of [17]. All edges have capacity $1$.

######  Proposition 4.

All entropy vectors representable by hypergraphs can be represented using the
prescription given in (3.1) on balanced networks.

###### Proof.

Given a hypergraph model, consider replacing every undirected edge by two
oppositely directed edges shown in Figure 4, and every hyperedge involving $n$
vertices by a $n$-cyclic network like the one shown in Figure 6 with the same
capacity for each edge.101010The choice of orientation for these directed edge
cycles that replace hyperedges will not matter. Neither does the order of the
boundary vertices around each cycle. For any subset of vertices
${\underline{I}}$, its entropy in the hypergraph model is given by the min cut
of the hypergraph. We now construct a multi-cycle flow ${\mathfrak{C}}$ as
follows. For every edge in the min cut, we will include in ${\mathfrak{C}}$
the minimal cycle that traverses the corresponding oppositely directed edges
to a vertex in ${\underline{I}}^{c}$ with magnitude equal to the capacity of
the min cut edge. If the min cut involves a hyperedge connecting $n$ vertices,
then we include in ${\mathfrak{C}}$ the cycle flow through the corresponding
$n$-cyclic network with magnitude saturating the capacity of the edges. The
cycle flow we constructed has flux exactly equal to the min cut magnitude, and
so the flux of the max cycle flow through ${\underline{I}}$ must be at least
equal to the min cut magnitude of ${\underline{I}}$. However, the min cut
separates the boundary vertices ${\underline{I}}$ from ${\underline{I}}^{c}$,
which implies by our construction that the vertices in ${\underline{I}}$ are
disconnected from those in ${\underline{I}}^{c}$ in the residual graph
${\text{Res}}({\mathcal{N}},{\mathfrak{C}})$. As there exists no more edges
between ${\underline{I}}$ and ${\underline{I}}^{c}$ in the residual graph, we
conclude ${\mathfrak{C}}$ must also be a max cycle flow, proving the flux of
the max cycle flow through ${\underline{I}}$ corresponds to the min cut of
${\underline{I}}$ in the hypergraph. ∎

As a more involved example, consider the balanced network given in Figure 8.
This network is of particular interest as the entropic pattern it realizes
cannot be captured by hypergraphs. In other words, it demonstrates that the
proposed cycle flow prescription generalizes beyond the hypergraph models of
[17]. The entropy vector that this network yields corresponds precisely to
that of the stabilizer “graph state 15” of [20], which proved that the
entanglement structure of such a stabilizer state is unattainable by
hypergraphs. It was later shown in [18] that the entropy vector of graph state
15 can be obtained by providing a min cut model on topological links while
still realizing all hypergraph entropies, albeit obscuring any potentially
dual flow perspective. The topological character of the models of [18], and
their realization of graph state $15$, served as motivation for our
construction of the state using directed networks in Figure 8, as well for our
cycle flow prescription for computing entropy in general. Although proving an
equivalence between our prescription and that of [18] at the level of entropy
cones is beyond the scope of this paper, the fact that we are able to
reproduce the subsystem entropies of graph state $15$ does demonstrate that
cycle flows share key topological features with link models.

$A$$B$$D$$E$$C$$O$$2$$2$$2$ Figure 8: This balanced network reproduces the
entropy vector represented by “graph state 15” in Figure 1 of [20]. Its
associated entropy vector is $(11111;2222212212;2222222222;22221;1)$, where we
have ordered the components in lexicographic order.

Having illustrated how to compute the maximum flux out of a subset of vertices
given a cycle flow, we now turn to the properties of such max cycle flows. It
is easy to see the prescription (3.1) obeys purification symmetry.

######  Proposition 5.

Given the identification (3.1) between entanglement entropy for a subset of
vertices and the maximum flux out of the vertices in a cycle flow,
purification symmetry is satisfied.

###### Proof.

Using (3.1), we have

$\displaystyle{\texttt{S}}({\underline{I}})=\max_{{\mathfrak{C}}}f_{{\mathfrak{C}}}({\underline{I}},{\underline{I}}^{c})=\max_{{\mathfrak{C}}}f_{{\mathfrak{C}}}({\underline{I}}^{c},{\underline{I}})={\texttt{S}}({\underline{I}}^{c}),$
(3.4)

where we simply used the fact the maximization of flux between
${\underline{I}}$ and ${\underline{I}}^{c}$ is identical to that between
${\underline{I}}^{c}$ and ${\underline{I}}$. ∎

In the next two subsections, we will show that the prescription (3.1) also
satisfies SA and SSA. Although the proofs are more involved, the spirit of the
proofs is almost identical to those using bit threads found in [11].

###  3.2 Subadditivity

We will now prove that our prescription for entropy (3.1) satisfies
subadditivity (SA). Our approach follows closely that used in [11].

######  Theorem 2.

The entropy defined via cycle flows given in (3.1) satisfies subadditivity.

###### Proof.

Consider a graph with boundary vertices $\partial
V={\underline{I}}\cup{\underline{J}}\cup{\underline{K}}$, where
${\underline{I}},{\underline{J}}$, and ${\underline{K}}$ are mutually
disjoint. We will prove

$\displaystyle{\texttt{I}}({\underline{I}}:{\underline{J}})={\texttt{S}}({\underline{I}})+{\texttt{S}}({\underline{J}})-{\texttt{S}}({\underline{I}}{\underline{J}})\geq
0.$ (3.5)

To this end, we begin by computing the conditional entropy

$\displaystyle{\texttt{H}}({\underline{I}}|{\underline{J}})\equiv{\texttt{S}}({\underline{I}}{\underline{J}})-{\texttt{S}}({\underline{J}}).$
(3.6)

To compute ${\texttt{S}}({\underline{I}}{\underline{J}})$, let
${\mathfrak{C}}({\underline{J}},{\underline{I}})$ be a minimal multi-cycle
flow that maximizes the magnitude through both ${\underline{J}}$ and
${\underline{I}}{\underline{J}}$ simultaneously; such a multi-cycle flow
exists by Theorem 1. It then follows

$\displaystyle\begin{split}{\texttt{S}}({\underline{I}}{\underline{J}})&=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}\neq\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}}{\underline{J}},{\underline{K}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}}{\underline{J}},{\underline{K}})\\\
&=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{K}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}}).\end{split}$
(3.7)

To obtain the second equality, we first noted that the first sum can be
rewritten as follows:

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}\neq\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}}{\underline{J}},{\underline{K}})=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}\neq\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}},{\underline{K}})=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{K}}),\end{split}$
(3.8)

where the first equality follows from the fact the minimal simple cycles we
are summing over all involve boundary vertices ${\underline{J}}$ already, and
the second equality follows from the fact additional minimal simple cycles
included in the last sum do not involve ${\underline{J}}$, and hence do not
contribute to the sum. As for the second sum in (3.7), we used the fact it
obeys the equality

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}}{\underline{J}},{\underline{K}})=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}}),\end{split}$
(3.9)

since the minimal simple cycles we are summing over do not involve boundary
vertices ${\underline{J}}$.

Likewise, we have

$\displaystyle{\texttt{S}}({\underline{J}})=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}})}f_{\mathcal{C}}({\underline{J}},{\underline{I}}{\underline{K}})=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{{\mathcal{C}}}({\underline{J}},{\underline{I}}{\underline{K}}),$
(3.10)

where the second equality simply follows from the fact the minimal multi-flow
${\mathfrak{C}}({\underline{J}},{\underline{I}})$ also maximizes the flux
between ${\underline{J}}$ and
${\underline{J}}^{c}={\underline{I}}{\underline{K}}$. Putting everything
together, we obtain

$\displaystyle\begin{split}{\texttt{H}}({\underline{I}}|{\underline{J}})&=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{K}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}})-\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{I}}{\underline{K}})\\\
&=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{K}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}})-\left[\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}},{\underline{I}}{\underline{K}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{K}}\neq\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}},{\underline{I}}{\underline{K}})\right]\\\
&=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{K}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}})-\left[\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}},{\underline{I}})+\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{J}},{\underline{K}})\right]\\\
&=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}},{\underline{I}}),\end{split}$
(3.11)

where the second equality follows from decomposing last sum in the first line
into two disjoint sums, and the third equality follows from relations
analogous to (3.8) and (3.9). Using the definition of mutual information, we
then have

$\displaystyle\begin{split}{\texttt{I}}({\underline{I}}:{\underline{J}})&={\texttt{S}}({\underline{I}})-{\texttt{H}}({\underline{I}}|{\underline{J}})\\\
&=\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{I}},{\underline{J}})}f_{{\mathcal{C}}}({\underline{I}},{\underline{J}}{\underline{K}})-\left[\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}},{\underline{I}})\right]\\\
&\geq\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{I}},{\underline{J}})}f_{{\mathcal{C}}}({\underline{I}},{\underline{J}}{\underline{K}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{K}})\\\
&\geq\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{I}},{\underline{J}})}f_{{\mathcal{C}}}({\underline{I}},{\underline{J}}{\underline{K}})-\sum_{{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}},{\underline{I}})}f_{\mathcal{C}}({\underline{I}},{\underline{J}}{\underline{K}})\\\
&\geq 0,\end{split}$ (3.12)

where the third line follows from the fact
$f_{\mathcal{C}}({\underline{J}},{\underline{I}})\geq 0$ for all minimal
simple cycles; the fourth line follows from the fact
$f_{\mathcal{C}}({\underline{I}},{\underline{K}})\leq
f_{\mathcal{C}}({\underline{I}},{\underline{J}}{\underline{K}})$ for all
minimal simple cycles, and also we are no longer restricting the sum with the
condition $V({\mathcal{C}})\cap{\underline{J}}=\emptyset$; and the last
inequality uses the fact the first sum in the penultimate line is the maximal
flux between ${\underline{I}}$ and ${\underline{I}}^{c}$ for any multi-cycle
flow, whereas the second sum is the flux between ${\underline{I}}$ and
${\underline{I}}^{c}$ for the particular multi-cycle flow
${\mathfrak{C}}({\underline{J}},{\underline{I}})$. This argument completes the
proof of subadditivity. ∎

###  3.3 Strong Subadditivity

We now turn to prove strong subadditivity (SSA).

######  Theorem 3.

The entropy defined via cycle flows given in (3.1) satisfies strong
subadditivity.

###### Proof.

Consider a graph with boundary vertices $\partial
V={\underline{I}}\cup{\underline{J}}\cup{\underline{K}}\cup{\underline{L}}$,
where ${\underline{I}},{\underline{J}},{\underline{K}}$, and ${\underline{L}}$
are mutually disjoint. We will prove

$\displaystyle{\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})={\texttt{S}}({\underline{I}}{\underline{K}})+{\texttt{S}}({\underline{J}}{\underline{K}})-{\texttt{S}}({\underline{K}})-{\texttt{S}}({\underline{I}}{\underline{J}}{\underline{K}})\geq
0,$ (3.13)

where ${\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})$ is the
conditional mutual information. First, we observe that we can write
${\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})$ as

$\displaystyle{\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})={\texttt{H}}({\underline{I}}|{\underline{K}})-{\texttt{H}}({\underline{I}}|{\underline{J}}{\underline{K}}).$
(3.14)

Using (3.11), we have

$\displaystyle\begin{split}{\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})&=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{J}}{\underline{L}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{K}},{\underline{I}})\\\
&\qquad-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}}{\underline{K}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{L}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{J}}{\underline{K}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}}{\underline{K}},{\underline{I}})\\\
&=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{J}}{\underline{L}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{K}},{\underline{I}})\\\
&\qquad-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{L}})+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}}{\underline{K}},{\underline{I}}),\end{split}$
(3.15)

where the second equality follows from us choosing a particular minimal multi-
cycle flow that maximizes the flux through
${\underline{K}},{\underline{I}}{\underline{K}}$ simultaneously in the first
two sums (namely one that also maximizes the flux through
${\underline{I}}{\underline{J}}{\underline{K}}$), and similarly choosing
another particular minimal multi-cycle flow that maximizes the flow through
${\underline{J}}{\underline{K}}$ and
${\underline{I}}{\underline{J}}{\underline{K}}$ in the last two sums (namely
one that also maximizes the flux through ${\underline{K}}$). Now, we observe
that the first sum satisfies

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{J}}{\underline{L}})&\geq\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{J}}{\underline{L}})=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{L}}),\end{split}$
(3.16)

where the inequality follows from the fact we are further restricting the sum
over minimal simple cycles, and the equality follows from the same reasoning
as (3.9). Likewise, the fourth sum in (3.15) satisfies

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}}{\underline{K}},{\underline{I}})\geq\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{J}}{\underline{K}},{\underline{I}})=\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{K}},{\underline{I}}).\end{split}$
(3.17)

Substituting (3.16) and (3.17) into (3.15), we obtain the inequality

$\displaystyle\begin{split}{\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})&\geq\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{L}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{K}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{I}},{\underline{L}})\\\
&\qquad+\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{J}},{\underline{I}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{K}},{\underline{I}})-\sum_{\begin{subarray}{c}{\mathcal{C}}\in{\mathfrak{C}}({\underline{K}},{\underline{I}},{\underline{J}})\\\
V({\mathcal{C}})\cap{\underline{J}}{\underline{L}}=\emptyset\end{subarray}}f_{\mathcal{C}}({\underline{K}},{\underline{I}}).\end{split}$
(3.18)

Now, the first sum is the maximum flux of multi-cycle flows that are only
between ${\underline{I}}$ and ${\underline{L}}$ while the second sum is the
minimum such flux, both subject to the constraint we are maximizing the flux
through ${\underline{K}}$ and ${\underline{I}}{\underline{J}}{\underline{K}}$.
The reason is because minimal simple cycle flows between ${\underline{I}}$ and
${\underline{L}}$ will contribute to the maximization of the flux through
${\underline{I}}{\underline{K}}$ and
${\underline{I}}{\underline{J}}{\underline{K}}$ in the first sum, whereas they
only contribute to the maximization of the flux through
${\underline{I}}{\underline{J}}{\underline{K}}$ in the second sum. As the
maximum cannot be less than the minimum, the first two sums combine to be non-
negative. Likewise, the third sum is the maximum flux of multi-cycle flows
that are only between ${\underline{I}}$ and ${\underline{K}}$ while the second
sum is the minimum of such flux, both subject to the constraint we are
maximizing the flux through ${\underline{K}}$ and
${\underline{I}}{\underline{J}}{\underline{K}}$. Similar to before, the reason
is because minimal simple cycle flows between ${\underline{I}}$ and
${\underline{K}}$ contribute to the maximization of the flux through
${\underline{J}}{\underline{K}}$ and
${\underline{I}}{\underline{J}}{\underline{K}}$ in the third sum, whereas they
only contribute to the maximization through
${\underline{I}}{\underline{J}}{\underline{K}}$ in the last sum. Thus the last
two sums also combine to be non-negative. Thus,

$\displaystyle{\texttt{I}}({\underline{I}}:{\underline{J}}|{\underline{K}})\geq
0,$ (3.19)

and so SSA is satisfied. ∎

##  4 Discussion

In this paper, we introduced a potential way of extending the map between
undirected graph models and holographic entropy vectors to more general
quantum entropy vectors. In the well-established holographic case, the
undirected graph model has a clear geometric interpretation on a fixed time
slice of the corresponding bulk AdS geometry [1]. The boundary vertices
correspond to boundary subregions of the CFT, the internal vertices correspond
to bulk subregions carved out by the RT surfaces, and each edge corresponds to
the portion of the RT surface dividing the two bulk subregions, with the
capacity of the edge given by the area of that portion. It is clear then that
the entanglement entropy associated some subset of boundary regions is given
by the min cut associated to the corresponding boundary vertices in the graph.
Utilizing the max flow min cut theorem, [11] demonstrated how the entanglement
entropy can equivalently be recast as max flow out of the corresponding
boundary vertices.

In the generalization that we proposed in this paper, we do not have such a
clean geometric picture. Rather than generalizing the notion of graphs to
undirected hypergraphs and links, as was previously explored in [17, 18], we
show that it suffices to consider max flows on directed graphs. However,
rather than considering traditional flows with sources and sinks, we need to
consider cycle flows, as this was crucial to ensure that purification symmetry
is manifest. Utilizing cycle flows, we are able to show that our prescription
for computing entanglement entropy satisfies both SA and SSA. Furthermore, we
straightforwardly showed that both the conventional graph model representing
holographic entropy states and the hypergraph model are special cases of our
cycle flow model. Importantly, the entropy vector associated to the stabilizer
state that could not be reproduced by the hypergraph model has a very simple
realization in our cycle flow model (see Figure 8).

We would still like to obtain some form of geometric interpretation for the
cycle flow model. The usual intuition that undirected edges capture geometric
connectivity seems subtle to extend to directed edges. However, the work of
[34] did successfully provide a geometric picture for the GHZ-like
entanglement that the hypergraphs of [17] capture, even though hyperedges are
at first glance not geometric in the sense standard edges are.111111Indeed,
what geometrizes hyperedges are patched Euclidean geometries which meet at a
space of higher codimension. As such, these are non-locally Euclidean spaces
(thus not manifolds). We thank Tom Hartman for pointing out this geometric
interpretation of GHZ-like entropies. Perhaps one particularly tantalizing
clue is that the entropies of GHZ states involving $n$ parties are represented
simply by a directed loop involving $n$ vertices in our cycle flow model.

Furthermore, in the conventional “bit thread” model involving max flows, one
only cared about the thread configuration making up the final multicommodity
flow, and not the individual flows. However, as is evidenced by Figure 1, for
us the simple cycles making up a multi-cycle flow are actually important, and
two multi-cycle flows with exactly the same flow along each edge actually
correspond to different entropy vectors depending on their respective simple
cycles composing them. Additionally, these individual cycles are directed. For
an undirected graph, we can transform it into a directed graph by taking every
undirected edge and replacing it with two oppositely directed edges, as is
illustrated in Figure 4, and the cycle flow is obtained from the max flow
traversing some set of edges and coming back along the corresponding
oppositely directed edges. However, for general balanced graphs, the path we
take “going there” can be different from the one we take “coming back.” It
would be both interesting and worthwhile to understand the physical
significance of these minimal cycle flows.

Lastly, it is important to point out that we have only proved necessary, but
not sufficient, conditions for the cycle flow model to describe quantum
entropy vectors. It is possible that there exist entropy vectors that obey all
instances of SA and SSA, and yet are not realizable (or approximated
arbitrarily well) by any quantum state. In other words, if we define an
entropy cone given by the entropy vectors realizable by cycle flows, it is not
clear whether or not this cone coincides with the quantum entropy cone,
contains the quantum entropy cone, or neither. We leave the complete
exploration of the interplay between the entropy vectors realizable using our
cycle flow model and the quantum entropy cone to future work.

## Acknowledgements

We would like to thank Nina Anikeeva, Newton Cheng, Tom Hartman, Matthew
Headrick, Oliver Kosut, Shinsei Ryu, Vincent Su, Brian Swingle, and Tadashi
Takayanagi for useful conversations. We would especially like to thank Ning
Bao, Sebastian Fischetti, and William Munizzi, who were involved in earlier
iterations of this work. T.H. is supported by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2817,
the U.S. Department of Energy grant DE-SC0011632, and the Walter Burke
Institute for Theoretical Physics. S.H-C. is supported by the U.S. Department
of Energy Award DE-SC0021886. C.K. is supported by the U.S. Department of
Energy under grant number DE-SC0019470 and by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2818.

##  Appendix A Remarks on the Dual Program

The max cycle flow prescription we proposed in (3.1) for computing
entanglement entropy is a linear program. Consequently, there exists a dual
program. As we shall see, the dual program cannot be simply described via a
min edge cut prescription; nevertheless, we will find it provides a
complementary approach to the max flow prescription. In Section A.1, we will
set up the linear program and its dual. Next, in Section A.2, we explain the
analogous notion of a vertex cut. Lastly, we demonstrate in Section A.3 how to
use the dual program to provide alternative proofs of SA. We assume reasonable
familiarity with convex duality, and refer the reader to [35] for an excellent
exposition on this subject.

###  A.1 The Linear Program and its Dual

To construct a linear program, we need to specify a set of variables, a set of
linear constraints, and the extremization goal. In this subsection, we will
assume our goal is to compute ${\texttt{S}}({\underline{I}})$, but the
generalization to multipartite regions should be clear.

We begin by constructing the primal program. Consider a balanced directed
network ${\mathcal{N}}=(V,E,c)$ with boundary vertices $\partial V\subseteq V$
partitioned as $\partial V=\\{{\underline{I}},{\underline{I}}^{c}\\}$.
Consider all possible edge cycles $E_{i}$ in ${\mathcal{N}}$,121212We remind
the reader an edge cycle $E_{i}$ is an ordered sequence of distinct edges
$\\{e_{1},e_{2},\ldots,e_{n}\\}$ such that $t(e_{i})=s(e_{i+1})$ for
$i=1,\ldots,n$ with $e_{n+1}\coloneqq e_{1}$ (see 1). for
$i=1,\ldots,M_{\underline{I}}$, such that each $E_{i}$ contains both a vertex
in ${\underline{I}}$ and a vertex in ${\underline{I}}^{c}=\partial
V\setminus{\underline{I}}$. In particular, we are _not_ restricting our list
of edge cycles to those corresponding to _minimal_ simple cycles. Our primal
program for the flow is then given by the following:

  * •

Our variables are $\lambda_{i}$ for $i=1,\ldots,M_{\underline{I}}$, which are
the fluxes associated to each edge cycle $E_{i}$.

  * •

Our primal objective is to maximize

$\sum_{i=1}^{M_{\underline{I}}}\lambda_{i}.$ (A.1)

  * •

The linear constraints in the program are as follows. First, we have

$\displaystyle\lambda_{i}\geq 0\qquad\text{for all
$i=1,\ldots,M_{\underline{I}}$,}$ (A.2)

since the flux through any edge cycle is non-negative. Furthermore, for every
edge $e\in{\mathcal{N}}$, there is the capacity constraint (2.14), which
states

$\sum_{i=1}^{M_{\underline{I}}}u_{i}(e)\lambda_{i}\leq c(e)\qquad\text{for all
$e\in E$,}$ (A.3)

where $u_{i}(e)=1$ if $e$ is an edge contained in the edge cycle $i$ and $0$
otherwise, and $c(e)$ is the capacity of the edge.

Notice that although this program appears very similar to the program of
finding the max flow of a discrete directed graph, there is a crucial
difference. In the usual max flow program, the variables are the fluxes
through each edge. Accordingly, given a multi-flow consisting of flows
${\mathcal{F}}_{1},{\mathcal{F}}_{2},\ldots$, it does not matter which
${\mathcal{F}}_{i}$ is contributing to the flux. On the other hand, for us,
knowing the flux through each edge is not sufficient, as is evidenced in
Figure 1, where both multi-cycle flows has exactly the same flux through the
edges. Thus, we must keep track of not just the fluxes through each edge, but
rather the fluxes associated to the _particular cycles_ themselves. As we
shall see below, it is precisely this complication that renders the dual
program more nontrivial than the usual min cut program.131313Nevertheless, we
note the rewriting of our primal program in terms of all possible cycles on
the network ${\mathcal{N}}$ is similar to the rewriting of the bit threads
picture in terms of all possible paths from one region of the boundary back to
another, as per Section 5.1.1 of [16], although our cycles are directed and
discrete. We leave an exploration of possible connections for future work.

We now want to construct the corresponding dual program, which amounts to
switching the variables and the constraints. In our case, the number of edge
cycles $M_{\underline{I}}$ in the network ${\mathcal{N}}$ corresponds to the
number of variables in the primal program, and thus to the number of
constraints in the dual. Similarly, if we treat the constraints (A.2) as
implicit constraints, the number of explicit constraints (A.3) is the number
of edges $e$ in the network ${\mathcal{N}}$, which corresponds to the number
of variables in the dual program. We will denote these dual variables $d(e)$,
and the dual program is given as follows:

  * •

The dual variables, one per edge, are $d(e)$.

  * •

The dual objective is to _minimize_

$\sum_{e\in E}c(e)d(e).$ (A.4)

  * •

As for the case of the primal linear constraints, there is the dual non-
negativity constraint

$\displaystyle d(e)\geq 0\qquad\text{for all $e\in E$.}$ (A.5)

Furthermore, the linear constraints in the dual program are

$\sum_{e\in E}u_{i}(e)d(e)\geq 1\quad\text{for
$i=1,\ldots,M_{\underline{I}}$.}$ (A.6)

Thus, there are $M_{\underline{I}}$ constraints, as expected.

We finish by observing that the dual program is precisely equivalent to the
primal program by strong duality, which follows from Slater’s condition. To
check that Slater’s condition is satisfied, we simply need to show the
existence of primal variables such that the constraints (A.2) and (A.3) are
strictly satisfied. A simple choice is to set $\lambda_{i}={\varepsilon}$ for
a sufficiently small ${\varepsilon}>0$ for all $i=1,\ldots,M_{\underline{I}}$.

###  A.2 Vertex Cuts

As we remarked earlier, the fact our primal variables are the fluxes on cycle
flows, and not edges, makes it difficult to interpret the dual variables
$d(e)$ as edge cuts in the usual “min cut” perspective.141414However, as is
the case with standard source-sink directed flow min cuts, we suspect there
exists a feasible minimization of (A.4) where $d(e)$ is either 0 or 1.
Intuitively, if we view every simple cycle as a rubber band, whose magnitude
corresponds to the “thickness” of the rubber band, setting $d(e)=1$
corresponds to cutting enough rubber bands at edge $e$ so that the thickness
of the cut rubber bands equals the capacity of the edge. In this sense, the
dual variables $d(e)$ are “cycle cuts.” Instead, we will introduce the notion
of a vertex cut, which we define as follows. For any network
${\mathcal{N}}=(E,V,c)$, a vertex cut ${\mathcal{V}}$ is a subset of (not
necessarily boundary) vertices, with its magnitude given by

$\displaystyle|{\mathcal{V}}|\coloneqq\max_{{\mathfrak{C}}}f_{{\mathfrak{C}}}\big{(}{\mathcal{V}},V\setminus{\mathcal{V}}\big{)}.$
(A.7)

Note that as defined, we have $|{\mathcal{V}}|=|V\setminus{\mathcal{V}}|$.
Now, given some subset of boundary vertices ${\underline{I}}\in\partial V$, we
will consider the set of all vertex cuts that separate ${\underline{I}}$ from
the rest of the boundary vertices ${\underline{I}}^{c}=\partial
V\setminus{\underline{I}}$. In other words, a vertex cut associated to
${\underline{I}}$ can contain any number of internal vertices in addition to
${\underline{I}}$, as long as no further boundary vertices are included. We
will then show that ${\texttt{S}}({\underline{I}})$ is given by the minimum
magnitude of all such cuts.

For every multi-cycle flow ${\mathfrak{C}}({\underline{I}})$ maximizing the
flux through boundary vertices ${\underline{I}}$, there exists a vertex cut
${\mathcal{V}}_{{\underline{I}}}$ whose magnitude is precisely the max flow
${\texttt{S}}({\underline{I}})$, which we shall construct as follows. From
${\mathfrak{C}}({\underline{I}})$, we can construct a reduced multi-cycle flow
${\mathfrak{C}}^{r}({\underline{I}})$ by removing any simple cycle flow
${\mathcal{C}}\in{\mathfrak{C}}({\underline{I}})$ that does not contain a
vertex in both ${\underline{I}}$ and ${\underline{I}}^{c}$. Notice that
${\mathfrak{C}}^{r}({\underline{I}})$ still maximizes the flow out of the
${\underline{I}}$ vertices; it just does not contain any extraneous simple
cycle flows. Using ${\mathfrak{C}}^{r}({\underline{I}})$, we can construct the
residual graph
${\text{Res}}({\mathcal{N}},{\mathfrak{C}}^{r}({\underline{I}}))$, as defined
in (2.22).

In the residual graph
${\text{Res}}({\mathcal{N}},{\mathfrak{C}}^{r}({\underline{I}}))$, the
vertices in ${\underline{I}}$ are disconnected from those in
${\underline{I}}^{c}$. Otherwise, either the network ${\mathcal{N}}$ is
unbalanced, or the flux through the multi-cycle flow
${\mathfrak{C}}^{r}({\underline{I}})$ could be increased, which is not
possible as it is a max cycle flow through ${\underline{I}}$. Consequently, we
associate to ${\mathfrak{C}}({\underline{I}})$ the vertex cut
${\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})}$ consisting of
${\underline{I}}$ and any other vertices (both internal and boundary) still
connected to a vertex in ${\underline{I}}$ in the residual graph
${\text{Res}}({\mathcal{N}},{\mathfrak{C}}^{r}({\underline{I}}))$. As for the
magnitude $|{\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})}|$ of the cut, note
that by (A.7), it cannot be simply the sum of the capacities of all edges
leaving the cut, as the standard “min cut” perspective would
suggest.151515Consider the 4-party GHZ state in Figure 5. If we choose our
vertex cut to consist of vertices $\\{A,C\\}$, then by (A.7)
$|{\mathcal{V}}(\\{A,C\\})|=1$. However, if we just summed the capacity of the
edges which leave the vertex cut, the flow would instead be $2$. Instead, by
(A.7), we see that

$|{\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})}|=\max_{{\mathfrak{C}}}f_{\mathfrak{C}}\big{(}{\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})},V\setminus{\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})}\big{)}.$
(A.8)

Notice that this value coincides with the magnitude of the max flow
$f_{{\mathfrak{C}}({\underline{I}})}({\underline{I}},{\underline{I}}^{c})$,
since it was built from the residual network
${\text{Res}}({\mathcal{N}},{\mathfrak{C}}^{r}({\underline{I}}))$. Indeed, if
$|{\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})}|$ were larger than
$f_{{\mathfrak{C}}({\underline{I}})}({\underline{I}},{\underline{I}}^{c})$,
then the magnitude of the original multi-cycle flow
${\mathfrak{C}}({\underline{I}})$ could have been increased, which is
impossible since it is a max flow arrangement. By strong duality, the
magnitude of the max flow, which as we just argued is
$|{\mathcal{V}}_{{\mathfrak{C}}({\underline{I}})}|$, is precisely the minimum
value for (A.4).

###  A.3 Alternative Proofs of SA

In this subsection, we will give two related ways to prove SA. First, we will
prove it using the above described vertex cuts. Then, we will show how we can
also prove SA by directly using the dual variables $d(e)$. For simplicity, we
will use ${\mathcal{V}}({\underline{I}})$ throughout to refer to the vertex
cut constructed using the max flow ${\mathfrak{C}}({\underline{I}})$ described
above, and keep the dependence on the multi-cycle flow
${\mathfrak{C}}({\underline{I}})$ implicit. Furthermore, in this subsection
only, we will adopt the notation ${\underline{I}}^{c}\coloneqq
V\setminus{\underline{I}}$, so that the superscript $c$ indicates the
complement with respect to _all_ the vertices, and not just the boundary
vertices.

Let ${\underline{I}},{\underline{J}}$ be two disjoint subsets of the boundary
vertices, and construct the corresponding vertex cuts
${\mathcal{V}}({\underline{I}})$ and ${\mathcal{V}}({\underline{J}})$ using
the max flow procedure. As we argued in the previous subsection, by strong
duality the magnitudes of the vertex cuts are equal to
${\texttt{S}}({\underline{I}})$ and ${\texttt{S}}({\underline{J}})$,
respectively, so that

$\displaystyle{\texttt{S}}({\underline{I}})+{\texttt{S}}({\underline{J}})=|{\mathcal{V}}({\underline{I}})|+|{\mathcal{V}}({\underline{J}})|.$
(A.9)

Now, ${\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}})$ is
obviously a vertex cut for ${\underline{I}}{\underline{J}}$. Furthermore,
every simple cycle between
${\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}})$ and
$({\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}}))^{c}$ is
either a simple cycle between ${\mathcal{V}}({\underline{I}})$ and
${\mathcal{V}}({\underline{I}})^{c}$ or one between
${\mathcal{V}}({\underline{J}})$ and ${\mathcal{V}}({\underline{J}})^{c}$.
Therefore, given a max multi-cycle flow
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$ maximizing the flux through
${\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}})$, we can
rewrite this as

$\displaystyle{\mathfrak{C}}_{{\underline{I}}{\underline{J}}}=\big{\\{}{\mathfrak{C}}_{\underline{I}},{\mathfrak{C}}_{\underline{J}}\big{\\}},$
(A.10)

where ${\mathfrak{C}}_{\underline{I}}$ is the set of simple cycles in
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$ that involves
${\mathcal{V}}({\underline{I}})$ and ${\mathcal{V}}({\underline{I}})^{c}$, and
${\mathfrak{C}}_{\underline{J}}$ are the remaining simple cycles in
${\mathfrak{C}}_{{\underline{I}}{\underline{J}}}$ that necessarily involves
${\mathcal{V}}({\underline{J}})$ and ${\mathcal{V}}({\underline{J}})^{c}$. It
follows

$\displaystyle\begin{split}|{\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}})|&=f_{{\mathfrak{C}}_{\underline{I}}}\big{(}{\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}}),({\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}}))^{c}\big{)}+f_{{\mathfrak{C}}_{\underline{J}}}\big{(}{\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}}),({\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}}))^{c}\big{)}\\\
&=f_{{\mathfrak{C}}_{\underline{I}}}\big{(}{\mathcal{V}}({\underline{I}}),{\mathcal{V}}({\underline{I}})^{c}\big{)}+f_{{\mathfrak{C}}_{\underline{J}}}\big{(}{\mathcal{V}}({\underline{J}}),{\mathcal{V}}({\underline{J}})^{c}\big{)}\\\
&\leq|{\mathcal{V}}({\underline{I}})|+|{\mathcal{V}}({\underline{J}})|,\end{split}$
(A.11)

where the second equality followed from the way we grouped the simple cycles
in (A.10), and the last inequality follows from the fact the magnitude
$|{\mathcal{V}}({\underline{I}})|$ is the max flow out of
${\mathcal{V}}({\underline{I}})$, and likewise for
$|{\mathcal{V}}({\underline{J}})|$. Further using the fact that the vertex cut
${\mathcal{V}}({\underline{I}}{\underline{J}})$ is the one that has the
minimum magnitude among all possible vertex cuts of
${\underline{I}}{\underline{J}}$, implying that
$|{\mathcal{V}}({\underline{I}}{\underline{J}})|\leq|{\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}})|$,
we thus have

$\displaystyle
S({\underline{I}})+S({\underline{J}})=|{\mathcal{V}}_{\underline{I}}|+|{\mathcal{V}}_{\underline{J}}|\geq|{\mathcal{V}}({\underline{I}})\cup{\mathcal{V}}({\underline{J}})|\geq|{\mathcal{V}}({\underline{I}}{\underline{J}})|={\texttt{S}}({\underline{I}}{\underline{J}}),$
(A.12)

thereby giving our first alternative proof of SA.

Similarly, we can also prove SA directly using the dual variables $d(e)$. Let
$d_{I}(e)$ and $d_{J}(e)$ be the dual variables chosen to minimize the dual
objectives (A.4) corresponding to ${\underline{I}}$ and ${\underline{J}}$
while satisfying the constraints (A.6), so that

$\displaystyle\begin{split}\sum_{e\in E}u_{i}^{I}(e)d_{I}(e)&\geq
1\quad\text{for $i=1,\ldots,M_{\underline{I}}$}\\\ \sum_{e\in
E}u_{i}^{J}(e)d_{J}(e)&\geq 1\quad\text{for
$i=1,\ldots,M_{\underline{J}}$},\end{split}$ (A.13)

where $u_{i}^{I}(e)=1$ if $e$ is contained in the $i$th cycle between
${\underline{I}}$ and ${\underline{I}}^{c}$ and 0 otherwise, and likewise for
$u_{i}^{J}$. It is then obvious that by the non-negativity of the dual
variables $d(e)$ that

$\displaystyle\begin{split}\sum_{e\in E}u_{i}^{I}(e)(d_{I}(e)+d_{J}(e))\geq
1\quad\text{for $i=1,\ldots,M_{\underline{I}}$}\\\ \sum_{e\in
E}u_{i}^{J}(e)(d_{I}(e)+d_{J}(e))\geq 1\quad\text{for
$i=1,\ldots,M_{\underline{J}}$}.\end{split}$ (A.14)

Since every simple cycle between ${\underline{I}}{\underline{J}}$ and
$({\underline{I}}{\underline{J}})^{c}$ must be a simple cycle between
${\underline{I}}$ and ${\underline{I}}^{c}$ or one between ${\underline{J}}$
and ${\underline{J}}^{c}$, this means $d_{I}(e)+d_{J}(e)$ is a dual variable
associated to the simple cycle flows between ${\underline{I}}{\underline{J}}$
and $({\underline{I}}{\underline{J}})^{c}$. Thus, by the dual program, if we
denote $d_{IJ}(e)$ to be the dual variable that minimizes (A.4) given the
corresponding constraint (A.6) involving cycle flows between
${\underline{I}}{\underline{J}}$ and
$V\setminus({\underline{I}}{\underline{J}})$, we get

$\displaystyle\begin{split}{\texttt{S}}({\underline{I}})+{\texttt{S}}({\underline{J}})=\sum_{e\in
E}c(e)(d_{I}(e)+d_{J}(e))\geq\sum_{e\in
E}c(e)d_{IJ}(e)={\texttt{S}}({\underline{I}}{\underline{J}}),\end{split}$
(A.15)

thereby proving SA in yet another way.

Of course, we are also interested to prove SSA from the dual perspective. At
the moment, it is not clear which of the two perspectives, or perhaps a
combination of the two, is the best way to prove SSA. We leave such
explorations to future work.

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