a]Computational Science Initiative, Brookhaven National Lab, Upton, NY, 11973,
USA b]Center for Theoretical Physics, Department of Physics, University of
California, Berkeley, CA 94720, USA c]Department of Physics, University of
California, Santa Barbara, CA 93106, USA

# Topological link models of multipartite entanglement

Ning Bao<EMAIL_ADDRESS>Newton Cheng<EMAIL_ADDRESS>Sergio
Hernández-Cuenca<EMAIL_ADDRESS>Vincent Paul Su<EMAIL_ADDRESS>[  [
[

###### Abstract

We introduce a novel model of multipartite entanglement based on topological
links, generalizing the graph/hypergraph entropy cone program. We demonstrate
that there exist link representations of entropy vectors which provably cannot
be represented by graphs or hypergraphs. Furthermore, we show that the
contraction map proof method generalizes to the topological setting, though
now requiring oracular solutions to well-known but difficult problems in knot
theory.

##  1 Introduction

Originally motivated by relations obeyed by the entropy function in holography
[1], [2, 3, 4] studied a graph model of entanglement, wherein the von Neumann
entropies of quantum subsystems are calculated by minimum cuts separating
those subsystems from their complements in a corresponding weighted graph. The
combinatorial properties of minimum cuts on graphs allowed for the development
of a method to prove entropy inequalities obeyed by holographic states. While
powerful, this technology was limited to graph models intended to describe the
subset of holographic states, and therefore did not capture the entanglement
structure of more general quantum states. In follow-up work, the graph model
was generalized to a hypergraph model [5], with the aim of reproducing the
entropies of a wider class of states while retaining the ability to prove
entropy inequalities via a generalization of the proof technology. For four
parties, this was indeed the case: the hypergraph cone coincided with that of
stabilizer states, which is larger than the holographic one of graphs, but
still not large enough to capture general quantum states.111The full four-
party quantum entropy cone is unknown [6], though it is known to properly
contain the stabilizer entropy cone as defined in [7].

The hypergraph model also provided interesting new results intimately related
to the properties of ranks of linear subspaces and the entropies of stabilizer
states in quantum mechanics [5]. Further explorations of their connection to
the latter revealed that the hypergraph model was able to encode the entropies
of a subset of stabilizer states [8, 9], and that this containment was proper
[10]. The latter result relied on the discovery of a particular stabilizer
state whose entropy vector, referred to as ray $15$ in [10], violates an
inequality proven to be valid for all hypergraphs in the hypergraph model.

By contrast, another avenue of study has been the study of topology as an
efficient encoding of certain patterns of entanglement. Some work in this
direction include [11, 12, 13, 14, 15, 16]. Because all graph and hypergraph
models of entanglement can be captured by topological links (as will be
discussed in detail later in this work), using topological links to generalize
encodings of entanglement patterns in a systematic way is an intriguing avenue
of exploration to describe the entanglement of more general quantum states.

In this paper, we introduce such a generalization of the graph and hypergraph
models of quantum entanglement based on topological links. Similar to the
(hyper)graph case, entropies will be given by the weights of minimal cuts on
links, and used to define the entropy cone of link models as the set of all
entropy vectors realizable by such models. In Section 2, we review the graph
and hypergraph models and introduce the contraction map method that was
applied to prove entropy inequalities on such models. In Section 3, we then
formally introduce link models as a generalization of hypergraph models. We
demonstrate that such models strictly contain the hypergraph entropy cone by
exhibiting a link whose entropies capture those of ray 15. We then prove a
generalized contraction map condition on such models that, subject to access
to an oracle whose abilities we carefully define, provides a sufficient
condition for the validity of a given entropy inequality on a link model. We
also motivate that link models may usefully capture the entropies of quantum
states by showing that the entropy cones of link models and quantum states are
identical for $n\leq 3$ parties. A more detailed discussion of how link models
can be applied to further characterize entanglement structures of quantum
states is provided in Section 4.

##  2 Preliminaries

###  2.1 Graph models

We first review the definition of the graph model and the intuition behind it,
as originally introduced and studied in [2]. For any positive integer $k$, let
$[k]=\\{1,\dots,k\\}$. A graph model consists of an undirected graph $G=(V,E)$
with nonnegative edge weights specified by a map $w:E\to\mathbb{R}_{\geq 0}$.
A subset of vertices $\partial V\subseteq V$ are called external vertices and
the rest are referred to as internal vertices. For $\partial V$ of cardinality
$n+1$, a bijective coloring $\partial V\to[n+1]$ is used to identify internal
vertices with partitions of a quantum state: $G$ represents the entropies of a
mixed $n$-party state on the vertices $[n]$, with vertex $n+1$ representing
the purification of the state, and all internal vertices encoding the
entanglement structure. In this setting, there are $2^{n}-1$ subsystems of the
quantum state one can consider, corresponding to every non-empty subset
$I\subseteq[n]$. The entropy of any one of these is defined as the weight of
the minimum cut or min-cut separating vertices $I$ from $\partial
V\smallsetminus I$.

Given an arbitrary graph model, one may intuitively expect that the min-cut
prescription should place restrictions on the entropies of states that can be
faithfully represented by such models. These restrictions turn out to take the
form of linear entropy inequalities, which can be canonically written as

$\sum_{l=1}^{L}\alpha_{l}S(I_{l})\geq\sum_{r=1}^{R}\beta_{r}S(J_{r}),$ (2.1)

where $\alpha_{l},\beta_{r}>0$, and $I_{l},J_{r}$ are subsets of external
vertices. In other words, we arrange terms into a left-hand side (LHS) and a
right-hand side (RHS) such that they all appear with positive coefficients.
The entropies of any state which can be represented by a graph model will obey
inequalities of the general form of (2.1).

These inequalities on graph models can be proven via a technique known as
proof-by-contraction. The method functions by essentially formalizing the
intuition that the weights of any clever construction of cuts corresponding to
a given choice of external vertices are always lower bounded by the weights of
the corresponding min-cut.222See [17] for a description of the contraction-map
proof method in the geometric setting that originally inspired it, based on
the holographic Ryu-Takayanagi formula [1]. To give the precise statement of
the proof-by-contraction method, let us first introduce some basic notation.
For each $i\in[n+1]$, let $x^{(i)}$ and $y^{(i)}$ respectively be bitstrings
of lengths $L$ and $R$, with entries given by $x^{(i)}_{l}=\delta(i\in I_{l})$
and $y^{(i)}_{r}=\delta(i\in J_{r})$, where $\delta$ is a Boolean indicator
function yielding 1 or 0 depending on whether its argument is true or false,
respectively. These are often referred to as _occurrence bitstrings_ , as they
simply encode the occurrence of the $i^{\text{th}}$ party in $I_{l}$ or
$J_{r}$. Given a vector $\gamma\in\mathbb{R}^{k}$, we also introduce a
weighted Hamming norm on $\mathbb{R}^{k}$ as
$\norm{x}_{\gamma}=\sum_{j=1}^{k}\gamma_{j}|x_{j}|$. We can now state the
proof-by-contraction method, a sufficient condition for the validity of a
given entropy inequality [2]:

######  Theorem 1.

Let $f:\\{0,1\\}^{L}\to\\{0,1\\}^{R}$ be a
$\norm{\cdot}_{\alpha}$-$\norm{\cdot}_{\beta}$ contraction, i.e.

$\norm{x-x^{\prime}}_{\alpha}\geq\norm{f(x)-f(x^{\prime})}_{\beta}\qquad\forall
x,x^{\prime}\in\\{0,1\\}^{L}.$ (2.2)

If $f\left(x^{(i)}\right)=y^{(i)}$ for all $i=1,\ldots,n+1$, then (2.1) is a
valid entropy inequality on graphs.

The final requirement on occurrence bitstrings is a simple consistency
condition that only the appropriate external vertices from LHS min-cuts are
included by $f$ in forming cuts for each RHS set $J_{r}$. If $f$ is a function
satisfying the contraction condition in Theorem 1, we call $f$ a contraction
map. For the detailed proof, we refer the interested reader to [2, 4], but we
briefly review the elements here. At the core of the proof is constructing a
partitioning of the underlying graph into $2^{L}$ subsets of vertices as
determined by inclusion/exclusion with respect to each of the $L$ min-cuts of
LHS terms in (2.1). Each of these $2^{L}$ subsets is conveniently labeled by a
bitstring $x\in\\{0,1\\}^{L}$. Given a bitstring $x$, one can reconstruct the
associated vertex subset via $W(x)=\bigcap_{l=1}^{L}W_{l}^{x_{l}}$, where
$W_{l}^{1}=W_{l}$ is the min-cut for the $l^{\text{th}}$ region, and
$W_{l}^{0}={W_{l}}^{\complement}$ is its complement. This may also be inverted
to recover the cuts themselves via $W_{l}=\bigcup_{x_{l}=1}W(x)$, where the
union runs over all $x\in\\{0,1\\}^{L}$ subject to $x_{l}=1$. While the
entropy will depend on the actual weights of the edges in a graph, the
construction of these subsets $W(x)$ allows one to keep track of how many
times we count edge weights coming out of that subset when computing
entropies.

The contraction map $f$ takes bitstrings of length $L$ to bitstrings of length
$R$, and in this way utilizes the subsets $W(x)$ to build (not necessarily
minimal) cuts $U_{r}$ for each of the RHS subsystems. More explicitly, these
cuts are specified in terms of $f$ by $U_{r}=\bigcup_{f(x)_{r}=1}W(x)$ (cf.
the reconstruction of the LHS cuts $W_{l}$ above). By minimality of cuts, the
sum of edge weights involved in every $U_{r}$ will upper bound the true
entropy of terms on the RHS. Thus, $f$ can be thought of as giving a sort of
naive overestimate of the entropies of terms on the RHS, as long as the cuts
$U_{r}$ are appropriately faithful (e.g., vertices in a cut for $I$ are not
all excluded from vertices in a cut for $I\cup J$).

The contraction condition on $f$, while looking cryptic and abstract since it
is defined on bitstrings, enforces that edges coming out of the $W(x)$ are
counted more times on the LHS of the inequality than on the RHS. Because it
holds for every pair of bitstrings, which correspond to different sets $W(x)$,
it says that edges between different $W(x)$ were counted no more times on the
RHS than on the LHS, hence proving the inequality (2.1) valid regardless of
the edge weights in the underlying graph. Determining the existence of a
contraction map for a given entropy inequality can be accomplished via greedy
algorithms which are guaranteed to find one if it exists [2]. Unfortunately
though, this rapidly becomes computationally intractable, since one is
effectively searching in a space of all possible $2^{R\times(2^{L}-(n+1))}$
maps $f$ compatible with the last line of Theorem 1 as specified by their
image bitstrings.333See [4] for a more refined study of the space of possible
contraction maps.

###  2.2 Generalizing to hypergraphs

In [5], this was generalized to hypergraph models, where, again, entropies
were declared to be calculated by weights of min-cuts. The sets $W(x)$ were
again useful, but because hyperedges can connect more than two vertices at a
time, one had to consider not just pairwise interactions between the $W(x)$,
but also general $k$-edges, i.e., hyperedges connecting $k$ vertices.
Demanding that $k$-edges on the LHS were counted no fewer times than on the
RHS, essentially imposed additional criteria on the contraction map $f$, which
neatly nested into a sum over the rank $k$ of the involved hyperedges. This is
captured in a generalization of the contraction condition:

######  Theorem 2.

Let $f:\\{0,1\\}^{L}\to\\{0,1\\}^{R}$ be an $i^{k}_{\alpha}$-$i^{k}_{\beta}$
contraction:

$i^{k}_{\alpha}(x^{1},\ldots,x^{k})\geq
i^{k}_{\beta}(f(x^{1}),\ldots,f(x^{k}))\qquad\forall
x^{1},\ldots,x^{k}\in\\{0,1\\}^{L}.$ (2.3)

If $f\left(x^{(i)}\right)=y^{(i)}$ for all $i\in[n+1]$, then (2.1) is a valid
inequality on rank-$k$ hypergraphs.

Here,
$i^{k}_{\alpha}(x^{1},\ldots,x^{k})=\sum_{l}\alpha_{l}i^{k}(x^{1}_{l},\ldots,x^{k}_{l})$
and $i^{k}:\\{0,1\\}^{k}\to\\{0,1\\}$ is an indicator function that returns 0
if the input bitstring is all 0s or all 1s, and 1 otherwise. Because a valid
contraction map $f$ for hypergraphs must obey additional constraints, an
inequality that is true for graphs may no longer hold for hypergraphs. Thus,
the hypergraph entropy cone has fewer constraints and is larger. That this
containment is strict was shown in [5].

##  3 Link models

###  3.1 Introduction

Inspired by the previous models, we now introduce the link model. In this
paper, a link consists of standard $1$-dimensional knots in $3$-dimensional
Euclidean space. For our purposes, knots will often be trivial (the unknot)
and just referred to as loops. Letting $\mathcal{L}$ denote the finite set of
loops, we distinguish a subset $\partial\mathcal{L}\subseteq\mathcal{L}$ as
external and call the others internal. All loops are assigned some nonnegative
weight. Notice that links no longer carry distinct vertex/edge-like components
– loops will implement both functionalities now. In analogy with previous
discussions, a bijection $\partial\mathcal{L}\to[n+1]$ once again identifies
the external loops as parties of a quantum state.

To define the notion of loop cuts on a link model in analogy with edge cuts on
graph models, we will need the following basic idea:

######  Definition 1.

An $n$-loop link $L$ embedded in $\mathbb{S}^{3}$ is said to be _topologically
linked_ if for any choice of embedded disks
$\\{D_{i}\subset\mathbb{S}^{3}\\}_{i=1}^{n}$ such that
$L=\bigcup_{i=1}^{n}\partial D_{i}$, the set $\;\bigcup_{i=1}^{n}D_{i}$ is
topologically connected. Otherwise $L$ is said to be _topologically unlinked_.

Given a link with loop set $\mathcal{L}$, any subset of loops
$\tilde{\mathcal{L}}\subseteq\mathcal{L}$ defines a _sublink_. Similar to the
notion of connected components in a topological space, one easily sees that a
general link $L$ may always be decomposed into _connected sublinks_ , namely,
maximal sublinks which are topologically linked. Hence we can equivalently
understand a link $L$ as being topologically unlinked if and only if it
consists of more than one connected sublink. Clearly, topological linking is a
property which can be lost when some subset of loops is removed from a link,
in the sense that the remaining loops can break into multiple connected
sublinks. Following this intuition, we will say that two loops of a link $L$
are topologically unlinked in $L$ if they belong to distinct connected
sublinks, and similarly that two subsets of loops are topologically unlinked
in $L$ if every pair of respective loops belong to distinct connected
sublinks.444The converses are clearly as follows: two loops of a link $L$ are
topologically linked in $L$ if they belong to the same connected sublink, and
similarly two subsets of loops are topologically linked in $L$ if all their
loops belong to the same connected sublink. However, let us emphasize an
important subtlety: two loops of a link $L$ which are topologically linked in
$L$ need not be so if taken to define a link on their own. For instance, if
$L$ is a Brunnian link, any two loops in it are topologically linked in $L$,
but if taken to define a link on their on (cf. deleting the other loop), they
would be topologically unlinked (indeed, they would define a $2$-component
unlink). We will come back to this in Section 3.3.

We use the above to define a _loop cut_ $C$ for a subsystem $I\subseteq[n]$ as
any set of internal loops
$C\subseteq\mathcal{L}\smallsetminus\partial\mathcal{L}$ which, if removed,
give a link where the loops in the subsystem $I$ are topologically unlinked
from all other external loops in $\partial\mathcal{L}$. The weight of $C$ is
given by the sum of its loop weights, and denoted by $\absolutevalue{C}$. The
entropy of $I$ is defined as the minimum weight of all possible loop cuts $C$
for $I$. Any such _loop min-cut_ for $I$ will be denoted by $C_{I}$.555When
there exist multiple min-cuts with the same cut weight, one simply makes a
choice and sticks to it. In the spirit of previous work, we posit that the
entropy of a subsystem $I$ is given by the weight of a loop min-cut $C_{I}$.
Note that both the graph and hypergraph models are easily reproducible in the
link model by converting vertices to infinite-weight loops and $k$-edges to
finite weight loops, all Hopf-linked to their vertex loops, as illustrated in
Figure 1.

Figure 1: Link construction from graph and hypergraph models. One can
reproduce all min-cut weights appropriately by making the following
substitutions. Vertices do not contribute to the cut weight, so they should be
replaced with loops of infinite weight. Loops of finite weight can be used to
replace both edges and hyperedges where the vertices are Hopf-linked to the
edge loop. Note that link models inherently no longer distinguish between
vertices and edges.

###  3.2 Capturing the entropies of ray 15

To motivate the construction of the link model, we share an explicit quantum
state whose subsystem entropies can be represented by the link model yet
provably fail to be realizable using hypergraph models [5, 10]. The pertinent
entropy vector, henceforth referred to as ray 15 following [10], comes from a
particular 5-party mixed graph state,666Note that the graph state described
here is in the context of [18], and not a state whose entropy vector can be
realized by graph representation. where the vertices denote spin systems and
edges represent Ising-type interactions [18]. Explicitly, ray 15 reads

$\vec{S}_{15}=(1\,1\,1\,1\,1\,;1\,2\,2\,2\,2\,2\,2\,2\,2\,1\,;2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,;2\,2\,1\,2\,2\,;1),$
(3.1)

where entropies are listed in lexicographical order, with subsystems of
increasing number of parties separated by a semicolon. Notably, all graph
states are stabilizer states, and hence, ray 15 must lie within the 5-party
stabilizer entropy cone.

Figure 2: The topological link whose entropies reproduce ray 15 from [10].
$A$, $B$, $C$, $D$, and $E$ are the parties in question, with $F$ taken to be
the purifier. The min-cut of any single external loop has weight $1$, as do
the pairs $AB$, $ED$, and $CF$. All other pairs of external loops have min-cut
weight $2$, and so do all triples. The min-cut weight of any other subsystem
can be obtained from that of its complement with respect to all external
loops.

In attempting to determine whether the hypergraph and stabilizer entropy cones
coincide, the authors found that the answer is negative [10], indicating the
strict containment of the hypergraph entropy cone within the stabilizer
entropy cone [8]. Indeed, ray 15 violates a linear entropy inequality that all
hypergraph models were shown to obey in [10]:

$S_{AB}+S_{DE}+S_{ACD}+2S_{ACE}+S_{BCD}+S_{ABDE}\geq
S_{AC}+S_{AE}+S_{BD}+2S_{ABCD}+S_{ACDE}.$ (3.2)

In Figure 2, we give an explicit link model that reproduces ray 15, thereby
demonstrating that there exist link models which reproduce entropy vectors
inaccessible by hypergraphs alone. Moreover, because all hypergraph models can
be reproduced in the link model, the entropy cone of links properly contains
the hypergraph entropy cone.

This construction motivates us to investigate the degree to which link models
can represent multipartite entanglement of generic quantum states. We will
consider whether the contraction-map technology can be generalized to link
models, allowing us to extend the entropy cone of hypergraphs to that of link
models. This will pave the way for exploring whether the link model entropy
cone can possibly contain the stabilizer entropy cone of [7] or even the full
quantum entropy cone [6]. One can consider a program of using contraction maps
to directly prove candidate entropy inequalities on link models, and exploring
such inequalities as candidate entropy inequalities for the currently unknown
higher-party stabilizer and quantum entropy cones (e.g. by testing whether
they are violated by the entropies of known stabilizer or quantum states). In
this way, the contraction map methods and associated graph/link models serve
as an immediate application and useful set of tools in the study of quantum
state entropies and their constraints. We further discuss the application of
link models to classifying the structure of entanglement of quantum states in
Section 4.

###  3.3 The link model toolkit

Studying general link models will require the introduction of various
definitions and constructs, to which we now turn. Given some link, we say that
a sublink $\tilde{\mathcal{L}}$ of it is an _irreducible sublink_ if it is
intrinsically topologically linked, i.e. if it is topologically linked once
all loops in $\mathcal{L}\smallsetminus\tilde{\mathcal{L}}$ are
deleted.777Note that a sublink of some link $L$ may not be irreducible even if
its constituent loops happen to be topologically linked in $L$. – see footnote
4 for an example. A simple class of irreducible sublinks are the connected
sublinks into which any link may be decomposed.

Any loop cut $C$ for $I$ naturally defines two additional subsets of loops.
Let $W_{C}\subseteq\mathcal{L}\smallsetminus C$ be the minimal-cardinality
subset of loops with $W_{C}\cap\partial\mathcal{L}=I$ and such that, once the
loops in $C$ are removed, form a collection of connected sublinks. In
practice, $W_{C}$ can be found by collecting all loops in the link defined by
$\mathcal{L}\smallsetminus C$ that are topologically linked to the pertinent
external loops $I$. The resulting set $W_{C}$ is clearly unique and referred
to as the cut interior of $C$. Similarly,
$\overline{W}_{C}=\mathcal{L}\smallsetminus(W_{C}\cup C)$ is also unique and
referred to as the cut exterior of $C$. The cut exterior contains all external
loops $\partial\mathcal{L}\smallsetminus I$ which are absent from $W_{C}$, and
consists of connected sublinks as well. When talking about a loop min-cut
$C_{I}$, we will denote its associated cut interior as $W_{I}$ for simplicity.
As an example, consider looking at subsystem $I=AC$ in Figure 2: a valid loop
min-cut $C_{AC}$ for it is the weight-$2$ loop which, when removed, defines
the cut interior $W_{AC}=\\{A,C\\}$ alone – the corresponding cut exterior
$\overline{W}_{AC}$ consists of the three weight-$1$ loops and the
complementary external loops $B$, $D$, $E$ and $F$.

To refine our analysis of loop cuts, we introduce a notion of prunability. For
a given subsystem $I$, the loop min-cut $C_{I}$ by definition contains loops
which topologically unlink $W_{I}$ from $\overline{W}_{I}$. Consider a sublink
$B_{I}\subseteq\mathcal{L}$ containing $k$ loops that is irreducible and has
nontrivial intersection with all of $W_{I},\overline{W}_{I},$ and $C_{I}$. Let
us refer to such a sublink as a bridge across the loop min-cut $C_{I}$. By
definition, bridges clearly have cardinality $k\geq 3$. We call a bridge
prunable if it remains a bridge upon removal of any nontrivial subset of its
loops. We say a bridge is minimal if it cannot be pruned. A minimal bridge of
loop cardinality $k$ is called a $k$-bridge and denoted by $B^{k}_{I}$. These
turn out to be crucial objects for link models of entanglement to not
trivialize down to hypergraphs. Within this framework, hypergraphs consists
only of Hopf-linked $3$-bridges. One can explicitly construct a link model
from any hypergraph model by replacing vertices with infinite weight loops and
(hyper)edges with loops of the corresponding weight – see Figure 1 for an
illustration. The richer entanglement structure that link models capture is
due to $3$-bridges with different topologies as well as the existence of
$k$-bridges with $k\geq 4$. As an example of a topologically interesting
$3$-bridge, consider the link model in Figure 2. If we take $I=BCD$, the loop
min-cut consists of the weight-$2$ loop. An associated $3$-bridge consists of
$B$ in the cut interior, the weight-$1$ loop in the cut exterior, and the
weight-$2$ loop in the loop min-cut. This 3-bridge does not involve Hopf-
linked loops – instead, as a sublink, it is a 3-component Brunnian link.

Loops in $C_{I}$ can be classified by the minimal cardinality of the
$k$-bridges for $I$ that they contribute to. In particular, we say that
$\ell\in C_{I}$ is a $k$-loop for $I$ if, among all minimal bridges $B\ni\ell$
for $I$, the smallest bridge cardinality is $k$. For instance, one can easily
verify that the weight-$2$ loop in Figure 2 is a 3-loop for $I=BCD$. We can
stratify the loops $\ell\in C_{I}$ into subsets $C^{(k)}_{I}\subseteq C_{I}$
such that $\ell$ is a $k$-loop for $I$ if and only if $\ell\in C^{(k)}_{I}$.
Importantly, the subsets $C^{(k)}_{I}$ are disjoint for different $k$ and
define a finite spanning partition of $C_{I}$ for any link with finite
$\absolutevalue{\mathcal{L}}$, i.e.,
$\bigcup_{k=3}^{\absolutevalue{\mathcal{L}}}C^{(k)}_{I}=C_{I}$. When computing
the total weight of a min-cut $C_{I}$, we thus have

$\absolutevalue{C_{I}}=\sum_{k=3}^{\absolutevalue{\mathcal{L}}}\absolutevalue{C^{(k)}_{I}}=\sum_{k=3}^{\absolutevalue{\mathcal{L}}}\sum_{\ell\in
C^{(k)}_{I}}w(\ell),$ (3.3)

where $w(\ell)$ is the weight of $\ell$.

###  3.4 A topological min-cut bottleneck

In previous work, the simple connectivity structure of graphs and hypergraphs
implied that a cut, given by a subset of vertices, automatically specified the
(hyper)edges whose weights contributed to the cut weight. When it comes to
link models, however, loops subsume both the role of edges and vertices. While
powerful for capturing a wider class of entropy vectors, this generality comes
with a price. The analogous cut interior $W_{C}$ now no longer specifies
uniquely the loop cut $C$ that gave rise to it and whose weight would
determine the cut weight.

In order to address this new issue, one must consider the topological link
classification problem [19]; that is, given a topological link, determining
whether a sublink is topologically linked to the remainder of the link, or
whether they are, in fact, unlinked. This problem is an outstanding problem in
topology which we do not propose a solution to. Instead, we will build our
formalism on top of a posited solution. We parameterize our ignorance in the
form of an oracle that essentially does the difficult work of obtaining the
min-cut structure of a general link.

Although such an oracle already performs a large task, we attempt to pare its
power to the minimal necessary for our purposes. Specifically, consider an
oracle called Link-Min-Cut that takes as input a link $\mathcal{L}$ and a set
of external loops $I\subseteq\partial\mathcal{L}$, and outputs its
corresponding loop min-cut $C_{I}$. One can then construct the cut interior
$W_{I}$ as described above by taking the set of external loops $I$ and finding
the smallest number of connected sublinks that contain them after $C_{I}$ is
removed.

Building on the Link-Min-Cut routine, we employ the following additional
subroutine:

$\texttt{Bridge-Oracle}(\mathcal{L},I,B)=\begin{cases}1&\quad\text{if $B$ is a
$\absolutevalue{B}$-bridge across $C_{I}$},\\\
0&\quad\text{otherwise.}\end{cases}$ (3.4)

where $B$ is a set of loops, a candidate minimal bridge for $I$, and the
required knowledge of $C_{I}$ is obtained from the output of Link-Min-Cut for
the same $\mathcal{L}$ and $I$ arguments. The ultimate purpose of Bridge-
Oracle is to solve a decision problem, namely, whether $B$ serves as a
$|B|$-bridge for the set of external loops in $I$. In the generalization of
the contraction-map proof method, this oracle will be used for determining
which loops contribute to the entropy of the link model.

###  3.5 Generalization of contraction maps

The contraction-map method to prove entropy inequalities involves a clever
partitioning of the vertex set in (hyper)graph models. For a given inequality
as in (2.1) with indices $\\{I_{l}\subseteq[n]\\}_{l\in[L]}$ on its LHS, every
vertex could be labeled by a bitstring of length $L$ where each bit encoded
whether it was contained in the min-cut for each $I_{l}$. For the link model,
we need to generalize this addressing to loops in terms of loop min-cuts,
which behave differently. Our goal in this subsection will be to prove that
such a generalization exists, given access to an appropriate oracle for
bridges.

We first introduce some notation and useful constructions. Pick a loop min-cut
$C_{l}$ for each index $I_{l}$ on the LHS of the entropy inequality, and let
$W_{l}$ and $\overline{W}_{l}$ be respectively its cut interior and exterior.
To each loop $\ell\in\mathcal{L}$, we assign an upgraded bitstring
$x\in\\{-1,0,1\\}^{L}$ via

$x_{l}=\delta(\ell\in W_{l})-\delta(\ell\in\overline{W}_{l}).$ (3.5)

In other words, $x_{l}$ is set to $-1$, $0$ or $1$ if $\ell$ is in
$\overline{W}_{l}$, $C_{l}$, or $W_{l}$, respectively. We can now partition
the loop set $\mathcal{L}$ into disjoint subsets of loops via

$W(x)=\bigcap_{l\in[L]}W_{l}^{x_{l}},$ (3.6)

where we are declaring $W_{l}^{1}\equiv W_{l}$, $W_{l}^{0}\equiv C_{l}$ and
$W_{l}^{-1}\equiv\overline{W}_{l}$. There will be $3^{L}$ such sets of loops
(i.e., as many as bitstrings), with some possibly empty. Note that we can
reconstruct the cut interior $W_{l}$ of any $I_{l}$ in terms of the $W(x)$
subsets via

$W_{l}=\bigcup_{x:x_{l}=1}W(x),$ (3.7)

and similarly for the loop min-cuts using

$C_{l}=\bigcup_{x:x_{l}=0}W(x).$ (3.8)

In (hyper)graph models, one was easily able to determine the edges
contributing to a min-cut (cf. $C_{l}$) by just knowing the vertex min-cut
(cf. $W_{l}$). In contrast, in the link model the cut interior $W_{l}$ does
not immediately determine what the loop min-cut $C_{l}$ is in any obvious way
for a general link.888Recall that it was $C_{l}$ which defined $W_{l}$, and
not the converse. This explains why we needed an upgraded bitstring addressing
encoding these two objects separately. In fact, we expect an explicit
specification of such a map $W_{l}\mapsto C_{l}$ to be a highly nontrivial
problem in knot theory which we will not try to solve completely. Instead, we
will try to break down this map into pieces suitable for a proof by
contraction. However, with these objects defined, we can now give the formal
statement of the generalized contraction map:

######  Proposition 1.

An entropy inequality in the form of (2.1) is valid on link models if:

  1. 1.

There exists a map $f:\\{-1,0,1\\}^{L}\to\\{-1,0,1\\}^{R}$ from min-cuts for
regions on the LHS of (2.1) to valid cuts for regions on the RHS of (2.1);
that is, the regions defined by

$U_{r}=\bigcup_{x:f(x)_{r}=1}W(x)$ (3.9)

are (not necessarily minimal) cuts for $J_{r}$.

  2. 2.

For every $3\leq n\leq k\leq|\mathcal{L}|$, $f$ obeys the cut-dependent
contraction property

$\sum_{l=1}^{L}\alpha_{l}F^{(n,k)}_{l}(x^{1},\ldots,x^{n})\geq\sum_{r=1}^{R}\beta_{r}\delta(f(x^{1})_{r}=0)\sum_{l=1}^{L}F^{(n,k)}_{l}(x^{1},\ldots,x^{n}),$
(3.10)

where $F_{l}^{(n,k)}:\\{-1,0,1\\}^{L\times n}\to\\{0,1\\}$ is an oracular
indicator function that returns 1 if the input $n$-tuple of bitstrings
corresponds to a $k$-bridge.999Double-counting of loop contributions is
avoided by constructing $F_{l}^{(n,k)}$ as in Algorithm 1.

We will need a few intermediate ingredients to prove this result, as we
describe next. To begin, we will think of the loop cut $C_{l}$ as a collection
of loops independently satisfying some property with respect to $W_{l}$.
Paralleling (3.3), we begin by decomposing $C_{l}$ into sets of $k$-loops
through

$C_{l}=\bigcup_{k=3}^{\absolutevalue{\mathcal{L}}}C_{l}^{(k)}.$ (3.11)

By definition, a loop $\ell\in C_{l}$ is a $k$-loop $\ell\in C_{l}^{(k)}$ for
the smallest $k$ such that it belongs to a $k$-bridge $B_{l}^{k}$, where
$B_{l}^{k}$ by construction intersects all of $W_{l}$, $\overline{W}_{l}$, and
$C_{l}$ nontrivially. In particular, notice that if $\ell\in C_{l}$, then it
must be that $\ell\in W(x)$ for some bitstring with $x_{l}=0$. However, this
decomposition is not fine-grained enough for our purposes: for a contraction
condition to hold, we need to be able to sum over bitstrings, and the
decomposition above does not contain enough information. Recall (3.6), which
defines $W(x)$ for a bitstring $x$ in terms of an intersections of vertex
subsets $W_{l}$ or their complements. However, a given $k$-bridge will
generally intersect multiple $W(x)$, and the decomposition into $k$-loops does
not let us determine which bridge $B_{l}^{k}$ or set $W(x)$ it belongs to.
That is, we are unable to pass directly from $k$-loops to a condition on
bitstrings, because we cannot identify bitstrings with specific $k$-loops that
contribute to a cut.

To handle this, we will introduce a further decomposition of a $k$-loop.
Firstly, assuming access to an appropriate oracle for validating the existence
of a bridge, we show that it is possible to decompose a cut $C_{l}$ such that
each loop in $C_{l}$ belongs to a distinct bridge:

######  Lemma 1.

Consider a loop min-cut $C_{l}$ and $k$-bridge $B_{l}^{k}$ for the boundary
region $I_{l}$. Then $|B_{l}^{k}\cap C_{l}|=1$.

###### Proof.

We prove this by contradiction. Suppose $|B_{l}^{k}\cap C_{l}|>1$, and let
$\ell_{i}\in B_{l}^{k}\cap C_{l}$. By definition of bridge minimality,
$B_{l}^{k}$ is not prunable, so removing $\ell_{i}$ will cause $B_{l}^{k}$ to
no longer be a bridge. This means that $B_{l}^{k}\setminus\\{\ell_{i}\\}$ is
either no longer irreducible, does not have nontrivial intersection with
$W_{l}$, $\overline{W}_{l}$, and $C_{l}$, or both. However, because
$\ell_{i}\in C_{l}$ and there exists at least one $\ell_{j}\in C_{l}$ with
$j\neq i$, the last two options are not possible. Hence, it must be the case
that $B_{l}^{k}\setminus\\{\ell_{i}\\}$ is now reducible, meaning that the
sublink defined by the remaining loops contains more than one connected
sublink.

If one of the resulting connected sublinks still has nontrivial intersection
with $W_{l}$, $\overline{W}_{l}$, and $C_{l}$, then the resulting loop set is
a $k^{\prime}$ bridge with $k^{\prime}<k$, meaning $B_{l}^{k}$ was prunable
and not minimal, yielding a contradiction. If, instead, none of the resulting
connected sublinks has non-trivial intersection with $W_{l}$,
$\overline{W}_{l}$, and $C_{l}$, then at least one of
$(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap W_{l}$,
$(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap\overline{W}_{l}$, or
$(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap C_{l}$ is topologically unlinked from
the other two loop sets. If $(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap C_{l}$
becomes topologically unlinked from the other two loop sets, then it must be
the case that, in fact, all three loop sets are topologically unlinked from
each other, for otherwise $C_{l}$ would have not defined a cut for the
pertinent subsystem in the first place. Hence, it must be the case that
$(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap W_{l}$ and
$(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap\overline{W}_{l}$ are topologically
unlinked, regardless of what happens to
$(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap C_{l}$. However, this implies $C_{l}$
was not minimal in weight, as we could instead define an alternative cut
$C_{l}^{\prime}=C_{l}\setminus\\{\ell_{j}\in C_{l}:j\neq i\\}$ for the same
subsystem, with $(B_{l}^{k}\setminus\\{\ell_{i}\\})\cap C_{l}$ now being part
of either the new interior or exterior. As noted above, removing
$C_{l}^{\prime}$ topologically unlinks the new interior and exterior, so it is
a valid cut. We also see that removing $C^{\prime}_{l}$ makes $B_{l}^{k}$
topologically unlinked, and the weight of $C^{\prime}_{l}$ is strictly smaller
than that of $C_{l}$, because $C^{\prime}_{l}\subset C_{l}$. We conclude that
we must have $|B_{l}^{k}\cap C_{l}|=1$. ∎

Lemma 1 will allow us to reduce our analysis of a cut $C_{l}$ to an analysis
of the distinct bridges to make up each of its loops. Applying the lemma, we
can separate the loops within a $k$-bridge $B_{l}^{k}$ into the loop in
$C_{l}$, and the remaining $k-1$ loops in $B_{l}^{k}$ that belong to either
$W_{l}$ or $\overline{W}_{l}$. Those loops in $B_{l}^{k}\cap W_{l}$
($B_{l}^{k}\cap\overline{W}_{l}$) will necessarily belong to some $W(x)$ sets
with $x_{l}=1$ ($x_{l}=-1$). However, notice that these additional $k-1$ loops
need not all belong to distinct $W(x)$ sets – instead, in general there will
be between $2$ and $k-1$ such sets containing them, as mentioned previously.
We now introduce the further decomposition of loop-cuts
$C_{l}^{(n,k)}\subseteq C_{l}^{(k)}$ consisting of those $k$-loops in $C_{l}$
for which the smallest number of $W(x)$ sets needed to cover any one of its
associated $k$-bridges is $n$. Enumerating bitstrings by a superscript as
$x^{i}$, this means that every $\ell\in C_{l}^{(n,k)}$ is a $k$-loop with
$\bigcup_{i=1}^{n}W(x^{i})$ containing $B_{k}$ minimally in the sense that $n$
is the smallest across all associated $k$-bridges. Hence:

######  Corollary 1.

Every $C_{l}^{(k)}$ can be further decomposed as

$C_{l}^{(k)}=\bigcup_{n=3}^{k}C_{l}^{(n,k)}$ (3.12)

where

$C_{l}^{(n,k)}=\left\\{\ell\in
W(x^{1})\,:\,x^{1}_{l}=0,\;\bigcup_{i=1}^{n}W(x^{i})\supseteq
B_{k}\ni\ell\text{ minimally}\right\\}_{(x^{1},\ldots,x^{n})},$ (3.13)

where the subscript $(x^{1},\ldots,x^{n})$ implies iteration over all
$n$-tuples of bitstrings in $\\{-1,0,1\\}^{L}$, and the choice $x^{1}_{l}=0$
is without loss of generality.

Notice that $C_{l}^{(n,k)}$ is nontrivial only if $n,k\geq 3$, for otherwise
the conditions that define it are impossible to satisfy. At the level of
bitstrings, we can see that the condition in (3.13) that
$\bigcup_{i=1}^{n}W(x^{i})$ contain a $k$-bridge minimally necessarily means
that $\sum_{i=1}^{n}|x^{i}_{l}|=n-1$ (i.e., only $x^{1}_{l}=0$), and that
$\absolutevalue{\sum_{i=1}^{n}x^{i}_{l}}\leq n-3$ (i.e., there are both $+1$
and $-1$ bits). While necessary, these bitstrings conditions are not
sufficient to guarantee those in (3.13) – this issue will reappear when
formulating the contraction map. In the contraction-map method, we want to
turn a sum over the $k$-tuples of bitstrings satisfying the conditions in
(3.13) into a sum over all $k$-tuples of bitstrings, with some indicator
function specifying which tuples do participate in each $C_{l}^{(n,k)}$;
however, we cannot formulate the necessary and sufficient conditions for a
$k$-bridge purely in terms of elementary properties of a bitstring. Hence, we
will need access to an oracle:

######  Definition 2 (Oracular Indicator).

Consider a candidate entropy inequality in the canonical form of (2.1) on an
$n$-party link model. The _oracular indicator_ is a Boolean map on bitstrings,
$F^{(n,k)}_{l}:\\{-1,0,1\\}^{nL}\to\\{0,1\\}$, which returns 1 when all of the
necessary and sufficient conditions for a $k$-bridge to be minimally contained
in $\bigcup_{i=1}^{n}W(x^{i})$ are satisfied, and 0 otherwise.

Note that, following the convention of (3.13), we may without loss of
generality choose to have $F_{l}^{(n,k)}(x^{1},\ldots,x^{n})=0$ when
$x^{1}_{l}\neq 0$. The function also has one additional feature beyond
checking the $k$-bridge conditions: it keeps track of what loops have already
contributed to the cut weight and may be part of multiple $k$-bridges.101010In
the context of the link given in Figure 2, notice that the weight-2 loop
contributes to forming 3- but also 4-bridges, but because of the coloring
strategy described below it is only counted once as a 3-loop. This can be
implemented by associating each loop in the cut set with the $k$-bridge of
smallest $k$ in which it participates, with an additional ordering for
$k$-bridges of the same valence. An example for how such a function could be
implemented, assuming access to Link-Min-Cut and Bridge-Oracle is:

Calculate min-cut $C_{l}$ for subsystem $I_{l}$ via Link-Min-Cut;

Color all loops in $C_{l}$ gray and set all values of
$F^{(n,k)}_{l}(x^{1},\ldots,x^{n})$ to zero;

$k\coloneqq 3$;

while  _$C_{l}$ contains gray loops_ do

$n\coloneqq 3$;

while  _$n\leq k$_ do

for  _each $n$-tuple of bitstrings $(x^{1},\ldots,x^{n})$ with $x_{l}^{1}=0$_
do

$\mathbf{W}\coloneqq\bigcup_{i=1}^{n}W(x^{i})$;

for  _each subset of $k$ loops $B\subseteq\mathbf{W}$_ do

if  _Bridge-Oracle $(\mathcal{L},I_{l},B)$_ then

$\ell\coloneqq\text{loop in }B\bigcap W(x^{1})$;

if  _$\ell$ exists and is gray _ then

Color $\ell$ green;

Set $F^{(n,k)}_{l}(x^{1},\ldots,x^{n})=1$;

end if

end if

end for

end for

$n\coloneqq n+1$;

end while

$k\coloneqq k+1$;

end while

Algorithm 1 Implementation of $F^{(n,k)}_{l}$

To be explicit, the oracular nature of $F_{l}^{(n,k)}$ manifests purely in
determining the sufficient conditions for $B$ to be a $k$-bridge – the other
features are for presentation/convenience, and can be implemented as separate
functions on bitstrings.

Finally, we comment on the first condition appearing in the statement of
Proposition 1, which is an extra requirement on any possible contraction map
$f$, in addition to the contraction property. Recall that for a cut $C_{l}$
for a region $I_{l}$ appearing on the LHS of a candidate entropy inequality
(2.1), we were able to decompose the cut as in (3.8). In order to relate the
weights of these cuts to those on the RHS, we need to perform a similar
decomposition of the RHS cuts. However, the caveat is that the decomposition
must still be in terms of the $W(x)$ (or else the comparison would be too
difficult), and the $W(x)$ are defined only in terms of the regions appearing
on the LHS of (3.7). That is, given a candidate contraction map
$f:\\{-1,0,1\\}^{L}\to\\{-1,0,1\\}^{R}$, when we construct the corresponding
vertex subsets for the RHS as in (3.9), each such $U_{r}$ must be a valid cut
for the $r^{\text{th}}$ region appearing on the RHS. This means
$U_{r}\cap\partial\mathcal{L}=J_{r}$ and $U_{r}$ is a cut interior for some
(not necessarily minimal) loop cut $C_{r}$ for $J_{r}$. If such an initial
condition cannot be satisfied, then the contraction map method has simply
failed. That such a function exists is immediate in the hypergraph case, due
to the bijective relation between cut vertices and edges. However, it is not
clear that the same is true here. We interpret this fact as simply a
manifestation of another stricter condition that is required for the
contraction-map method to succeed.

With these tools in hand, we now provide a proof of Proposition 1:

###### Proof of Proposition 1.

Consider a candidate $n$-party entropy inequality in the canonical form of
(2.1). Let $C_{l}$ be the minimal loop cut for the $l^{\text{th}}$ region
appearing on the LHS. Then using the decomposition of $C_{l}$ into
$C_{l}^{(n,k)}$ from Corollary 1 and the oracular indicator $F_{l}^{(n,k)}$,
we can write the LHS as a sum over all bitstrings:

$\sum_{l=1}^{L}\alpha_{l}|C_{l}|=\sum_{l=1}^{L}\alpha_{l}\sum_{k=3}^{|\mathcal{L}|}\sum_{n=3}^{k}|C_{l}^{(n,k)}|=\sum_{l=1}^{L}\alpha_{l}\sum_{k=3}^{|\mathcal{L}|}\sum_{(x^{1},\ldots,x^{n})}^{3\leq
n\leq k}F_{l}^{(n,k)}(x^{1},\ldots,x^{n})|W(x^{1})|.$ (3.14)

We continue by considering the RHS of (2.1). By the first assumption in the
statement of the proposition, there exists a function
$f:\\{-1,0,1\\}^{L}\to\\{-1,0,1\\}^{R}$ such that

$U_{r}=\bigcup_{x:f(x)_{r}=1}W(x),$ (3.15)

is a valid cut for the $r^{\text{th}}$ region appearing on the RHS. Notice
that generally there are many implicit conditions $f$ will have to satisfy. In
particular, the zeros of $f$ already specify RHS loop cuts, which themselves
uniquely specify all RHS cut interiors and exteriors. This implies that all
the $\pm 1$ entries of $f$ cannot be chosen freely, but are fixed by the
zeroes of $f$. The latter also determine the weights of the RHS cuts $U_{r}$
given by (3.9), which can be expressed as follows:

$|U_{r}|=\sum_{k=3}^{|\mathcal{L}|}\sum_{(x^{1},\ldots,x^{n})}^{3\leq n\leq
k}\sum_{l=1}^{L}\delta(f(x^{1})_{r}=0)F^{(n,k)}_{l}(x^{1},\ldots,x^{n})|W(x^{1})|.$
(3.16)

This expression simply counts when loops from the LHS cuts end up contributing
to the RHS. Compared to (3.14), which counts loop weights in the minimum cut
for each of the LHS subsystems $I_{l}$, this expression counts the weight of
those loops that contribute to a cut for the RHS subsystem $J_{r}$ via the
assignment specified by $f$. Recall that the $r^{\text{th}}$ bit of $f(x^{1})$
equaling $0$ means that $W(x^{1})$ is being assigned to be part of the loop
cut of $J_{r}$. Altogether, for the RHS we can write

$\sum_{r=1}^{R}\beta_{r}|U_{r}|=\sum_{r=1}^{R}\beta_{r}\sum_{k=3}^{|\mathcal{L}|}\sum_{(x^{1},\ldots,x^{n})}^{3\leq
n\leq
k}\sum_{l=1}^{L}\delta(f(x^{1})_{r}=0)F^{(n,k)}_{l}(x^{1},\ldots,x^{n})|W(x^{1})|.$
(3.17)

Assuming the cut specified by $f$ as in (3.9) is indeed valid, then this will,
by definition, upper bound the true value of the min-cut associated with the
RHS. Hence, (2.1) would then be satisfied if

$\displaystyle\sum_{k=3}^{|\mathcal{L}|}\sum_{(x^{1},\ldots,x^{n})}^{3\leq
n\leq k}$
$\displaystyle\left[\sum_{l=1}^{L}\alpha_{l}F^{(n,k)}_{l}(x^{1},\ldots,x^{n})\right]|W(x^{1})|$
(3.18)
$\displaystyle\geq\sum_{k=3}^{|\mathcal{L}|}\sum_{(x^{1},\ldots,x^{n})}^{3\leq
n\leq
k}\left[\sum_{r=1}^{R}\beta_{r}\delta(f(x^{1})_{r}=0)\sum_{l=1}^{L}F^{(n,k)}_{l}(x^{1},\ldots,x^{n})\right]|W(x^{1})|.$

A simple sufficient condition for this to hold is to demand that the
inequality be satisfied for every $n$-tuple of bitstrings with $3\leq n\leq k$
for each value of $k$. Were this to be the case, then all possible equations
of the following form would hold:

$\sum_{l=1}^{L}\alpha_{l}F^{(n,k)}_{l}(x^{1},\ldots,x^{n})\geq\sum_{r=1}^{R}\beta_{r}\delta(f(x^{1})_{r}=0)\sum_{l=1}^{L}F^{(n,k)}_{l}(x^{1},\ldots,x^{n}).$
(3.19)

Note that this is the second condition given in the statement of Proposition
1. If the above is true, then the given entropy inequality is also true, by
the minimality property of min-cuts.111111Namely, since by construction for
the LHS we have min-cuts with $S(I_{l})=|C_{l}|$, whereas for the RHS we have
simply constructed cuts which are not necessarily minimal and thus obey
$S(J_{r})\leq|C_{r}|$. ∎

We remark that we obtain a contraction condition which is cut-dependent, in
the sense that it depends on the oracular indicator $F_{l}^{(n,k)}$. This cut-
dependence of the contraction map is a new feature of link models that was not
present in (hyper)graphs, arising from the new topological character of min-
cuts.

###  3.6 Subadditivity and Strong Subadditivity

Although we have a generalized contraction condition in hand, an important
consistency check that link models are relevant for quantum mechanics is
whether the two universal quantum inequalities of subadditivity and strong
subadditivity hold for topological links models. Indeed, it is known that
these two inequalities completely determine the entropy cone of all quantum
states for $n\leq 3$ parties.

It turns out that these properties do indeed easily follow by the definition
of min-cuts on links, without requiring the contraction-map technology. In the
case of subadditivity, $S(A)+S(B)\geq S(AB)$, the cuts that separate $A$ from
its complement and the cuts that separate $B$ from its complement will by
definition separate $AB$ from its complement, and not necessarily optimally.
This proves subadditivity. For strong subadditivity, $S(AB)+S(BC)\geq
S(B)+S(ABC)$, the argument is much the same: the loop cuts that separate $AB$
from its complement and $BC$ from its complement can also be used to separate
$B$ from its complement and $ABC$ from its complement. In addition, the
subsets of loops of the $AB$ and $BC$ cuts that get recycled to build loop
cuts for each $B$ and $ABC$ can be chosen to be disjoint, thus preventing
double-counting on the RHS without accompanying double-counting on the LHS.
The converse inequality is clearly false: loops in cuts for $B$ and $ABC$ do
not always allow one to build a loop cut for e.g. $AB$, as in particular $A$
can still be linked with $C$. This type of reasoning is very much the same one
involved in the proof by contraction.

The above arguments show that the entropy cone of link models is contained
within the $n$-party quantum cone for $n\leq 3$. We could show containment in
the other direction by exhibiting links that realize the extreme rays of the
corresponding cones. This is immediate, since any hypergraph entropies can be
realized by links as explained in Figure 1, and it is known that hypergraphs
realize all such extreme rays [5]. We hence find that entropy cone of link
models is equivalent to the quantum entropy cone for $n\leq 3$ parties.

###  3.7 Potential Alternative Formulation: Super-Loops

Here, we briefly discuss a more natural generalization of the contraction-map
method to loops, in the sense that it better captures how (hyper)graphs work.
We are unable to prove the statement, so we leave it as a conjecture.

An intuitive way to understand the $W(x)$ sets in the graph case is as “super-
vertices” – because one is only interested in the edges that enter and leave
$W(x)$, the connections within a given $W(x)$ are unimportant, and one can
think of each $W(x)$ as simply being a large vertex connected to other super-
vertices by appropriate edges. Indeed, it is precisely this intuition that is
used in the proofs of polyhedrality of the graph and hypergraph entropy cones
[2, 5].

One would like to generalize this to the loop case, letting $W(x)$ denote a
“super-loop”, and redefining the notion of a $k$-bridge to correspond to a set
of $k$-many $W(x)$ sets. This would immediately simplify the proof and
notation of the contraction condition above: we would no longer need to
separately sum over $n$-many $W(x)$ sets to find all possible $k$-bridges.
Instead, we would simply have to sum over all $k$-many subsets of the $W(x)$
sets directly.

Unfortunately, the barrier to doing so lies in, once again, the fact that
there is no longer a unique way to connect loops. The key in applying the
super-vertex method is that cut entropies on the new graph involving super-
vertices are identical to those on the old graph. This is immediate on
hypergraphs due to the simplicity of hyperedges. However, it is not clear
whether the same is true for loop models. The freedom and variety of ways to
connect loop sets together means that the preservation of cut entropies in the
new super-loop model is not obviously guaranteed. We leave it as an open
question whether such a generalization is possible.

##  4 Discussion

In this work, we extend the program of building models of multipartite
entanglement based on min-cuts initiated in [2] for graphs and in [10, 5, 8]
for hypergraphs to topological links. In particular, we motivate a specific
topological generalization of the min-cut to a model of entanglement involving
links where the entanglement entropy of subsystems is given by weights of
linked loops. We have demonstrated the usefulness of this definition by
showing that link models allow for the construction of entropy vectors which
violate known inequalities for hypergraph states– see, for example, the link
model in Figure 2. Additionally, we generalized the contraction-map method for
proving inequalities, although this required the inclusion of certain oracular
functions. The inherent difficulty of working with the topological properties
of these link models of entanglement leaves many open directions for future
work.

Firstly, a more direct version of the contraction-map method with less
reliance on oracles would be desirable in order to turn the proof method
proposed here into a combinatorial one that can be implemented by computer.
Likely this would involve formulating the necessary conditions in a cut-
independent way. Alternatively, were this not to be possible, it would be
interesting to prove that oracles are necessary and thus that this method just
cannot be easily turned into a combinatorial one.

Relatedly, while we are able to show that link models reach beyond the
hypergraph cone, we are unable to obtain insight into the geometric structure
of the link cone, e.g. we do not know if the entropy cone of link models is
polyhedral. It is possible that insights into the full structure of the cone
may enable a more efficient or direct proof approach. We remark that lack of
polyhedrality is not necessarily an issue from a conceptual standpoint: it is
unknown if the quantum entropy cone is polyhedral for $n>3$ parties, and it
may well be that it is not.

Next, a heuristic strategy of constructing more “interesting” topological link
models that are provably not realizable by hypergraphs would be desirable.
This would require a more complete understanding of how to produce general
topological links whose entropies, like those of the link in Figure 2, go
beyond the limits of the hypergraph paradigm. Ultimately, one would hope to
find a method for, given an entropy vector, determining whether there exists a
link model realizing it and, if so, which form it takes. This would precisely
correspond to a generalization of the powerful methods developed in [4] for
graphs (also used in [5, 10] for hypergraphs) via integer linear programs. For
link models, it is possible that nascent machine learning methods in
topological link/knot classification à la those of [20] will be useful in
heuristically generating these nontrivial link structures.

Moreover, even if there does exist a link model realizing an entropy vector,
it may not be the only one – there may be multiple inequivalent link
representations of a given entropy vector. This is similar to (hyper)graph
models, where it is also possible to find arbitrarily many inequivalent graph
representations of a given entropy vector. Hence, we do not believe that this
non-uniqueness impacts the comparison of the link model entropy cone with the
stabilizer and quantum entropy cones. It would, of course, be interesting to
find “minimal” links for each entropy vector, analogous to the notion of
minimality for graph representatives of entropy vectors defined by [4].
Rigorously defining a sharp notion of minimality on links would certainly be
useful for distilling which properties of links characterize entanglement
structures unreachable by hypergraphs. Along similar lines, it would be
interesting to determine what, if any, implications the extra information
contained in inequivalent representations of the same entropy vector has for
physical quantum states.

Finally, and perhaps most importantly, we would like to further strengthen the
connections between topological link models and physical quantum state
entropies. As argued above, topological link models obey the only two
inequalities known to hold on arbitrary quantum states: subadditivity and
strong subadditivity (and symmetry permutations thereof). Though necessary as
conditions for topological link models to capture precisely the entropies of
quantum states, these are not sufficient. The following logically independent
questions remain:

  * •

_Do topological link models obey all other quantum inequalities?_ If the
answer to this were to be affirmative, its proof would be of paramount
importance, and correspondingly a very ambitious challenge. The reason for
this is that other quantum inequalities are strongly believed to exists, but
none are actually known! Obtaining an inequality valid on link models through
the machinery presented in this work would shed very exciting light on this.
If instead the answer were to be negative, the finding of a counterexample
would be extremely interesting: it would provide a set of link entropies
consistent with subadditivity and strong subadditivity, yet inconsistent with
quantum mechanics, thereby pointing at the existence of novel quantum entropy
inequalities that rule out such entropies.

  * •

_Do topological link models reproduce all quantum entropies?_ If true, a proof
of this in the affirmative would require either an explicit construction of
link models for given quantum entropy vectors, or a proof of nonexistence of
valid link inequalities stronger than quantum ones, both of which seem beyond
the reach of current tools. If false, however, it would only take the proof of
some inequality on links that does not hold quantum mechanically, which may be
more achievable using the tools in this paper.

As can be seen, the landscape of possibilities is vast and worth exploring. A
YY/YN/NY/NN answer to the questions above would imply that the link entropy
cone is equal to/a proper subset of/a proper superset of/incomparable to the
quantum entropy cone. While the YY option would be particularly tantalizing,
we believe that any of them would be of interest in advancing our tools for
classifying multipartite quantum entanglement. If YN, link models would
continue the program started in the context of holography in the obvious way:
the most basic patterns of multipartite entanglement would be those realizable
by graphs, then those realizable by hypergraphs but not graphs, then those
realizable by topological links but not (hyper)graphs, etc. In fact, if NN,
the same classification would still apply, since as explained around Figure 1
all hypergraph entropies are realizable by links. Of course, in this latter NN
case there would however be “unphysical” link models realizing entropies which
are not quantum mechanical. One may nonetheless expect that some structural
constraint on link models could be implemented to avoid this, potentially
bringing one back to the interesting YN case. In the NY case, an analogous
constraint would also be very interesting, as it could return one to the ideal
YY case.

Let us emphasize why such a classification of multipartite entanglement into
graphs, hypergraphs, links, etc. is particularly exciting. Graph models of
entanglement originated naturally in the context of holography [2], where the
entanglement of boundary quantum states is geometrized in the bulk in a rather
explicit way. This geometrization is possible precisely because of the local
character of quantum correlations in holographic states. We may thus
characterize quantum states having entropies realizable by graphs as
possessing only _locally_ entangled subsystems. Hypergraph models break this
notion of locality by allowing for $k$-local correlations, implemented in the
form of $k$-edges [5]. As a result, hypergraphs are further capturing quantum
states which may also involve subsystems which have _non-locally_ entanglement
in a multi-local sense. The additional extension to links brings us quite
naturally to another layer of complexity in the structural form of
entanglement where multi-locality is superseded by a more global notion of
_topological_ entanglement. We thus see link models as an appealing extension
of the classification of multipartite entanglement initiated in the context of
holography in terms of graphs and followed by hypergraphs.

Given the promise demonstrated by the topological link model in going beyond
the entropies describable by hypergraphs, and also given the large range of
open questions for future exploration in this direction, we anticipate that
this will be an open area of exploration in the area of multipartite
entanglement for years to come.

## Acknowledgements

We would like thank Aaron Elersich, Greg Kuperberg, Jonathan Harper, Nils
Baas, and Matthew Headrick for useful discussions. For parts of the completion
of this work, N.B. was supported by the Department of Energy under grant
number DE-SC0019380, and is supported by the Computational Science Initiative
at Brookhaven National Laboratory, and by the U.S. Department of Energy
QuantISED Quantum Telescope award. N.C. and V.P.S. were supported in part by
the National Science Foundation (award number PHY-1521446), and by the U.S.
Department of Energy under contract DE-AC02-05CH11231 and award DE-SC0019380.
V.P.S. gratefully acknowledges support by the NSF Graduate Research Fellowship
Program under Grant No. DGE 1752814. During the completion of this work,
S.H.C. was supported by NSF grant PHY-1801805 and by funds from the University
of California Santa Barbara, and is supported by NSF grant PHY-2107939 and by
a Len DeBenedictis Graduate Fellowship.

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