sagecode

title:  Persistent hypergraph homology and its applications

keywords: Topological data analysis, persistent homology, hypergraphs, face-
to-face contacts, population behavior.

authors:

  * •

Yaru Gao, School of Mathematical Sciences, Dalian University of Technology,
Dalian, Liaoning, P.R.China, 116024

<EMAIL_ADDRESS>
  * •

Yan Xu, School of Mathematical Sciences, Dalian University of Technology,
Dalian, Liaoning, P.R.China, 116024

<EMAIL_ADDRESS>
  * •

Fengchun Lei, School of Mathematical Sciences, Dalian University of
Technology, Dalian, Liaoning, P.R.China, 116024

<EMAIL_ADDRESS>
corresponding author: Yaru Gao

Abstract:  Persistent homology theory is a relatively new but powerful method
in data analysis. Using simplicial complexes, classical persistent homology is
able to reveal high dimensional geometric structures of datasets, and
represent them as persistent barcodes. However, many datasets contain complex
systems of multi-way interactions, making these datasets more naturally and
faithfully modeled by hypergraphs. In this article, we investigate the
persistent hypergraph model, an important generalization of the classical
persistent homology on simplicial complexes. We introduce a new homology,
$\hat{H}$, on hypergraphs and an efficient algorithm to compute both
persistent barcodes and $\hat{H}$ barcodes. As example, our theory is
demonstrated by analyzing face-to-face interactions of different populations.
The datasets that we select consist of baboons in primate center, people from
rural Malawi, scientific conference, workplace and high school.

##  1\. Introduction

Topological data analysis (TDA) is a rapidly developing component in modern
data science. Its strength has been demonstrated in a variety of area such as
image recognition1, feature identification2, protein folding prediction3,
medicine design4-7 and much more. Persistent homology, a new branch of
algebraic topology, has been proposed to bridge traditional topology and
geometry. The essential idea is to introduce a filtration process and measure
homology groups by their “lifespans” during the process. Different from
traditional topological models, the “lifespan” measurement provides a family
of geometric characterizations of the topological invariants. Recently, there
has been a burgeoning movement in extending persistent homology theory in
various directions to suit broader applications. Notably, Carlsson develops
multi-dimensional persistent homology theory8, Meehan connects quiver
representation with persistent homology9, the theory also gets combined with
sheaf theory10, etc11-15.

This article investigates an important generalization called the hypergraph
model, and apply it in analysis of contact patterns. The persistent hypergraph
theory is originally introduced by Wu and others, together with theoretical
results including its stability16-20. In this paper, we introduce a new
homology, $\hat{H}$, as a complement to the embedded homology. The $\hat{H}$
homology characterizes those higher dimensional simplex whose boundary does
not exist. We also provide an efficient algorithm that computes persistent
embedded homology and $\hat{H}$ homology simultaneously for hypergraph model.

Measuring and analyzing interactions between individuals provide key
information on social contact behaviors in communities. Recently developed
technology of wearable sensors makes the collection of reliable contact data a
relatively easy task21-26. Previously, contact data is mostly analyzed using
graphic models where each vertex represents an individual and each edge
represents an interaction. The graphic model efficiently demonstrates 0 and 1
dimensional features, in the topological sense, such as connectivity. However,
it does not contain any information from higher dimensions. As a
generalization, the hypergraph model consists of connectivity information of
all dimensions, that is all group meetings of more than 2 individuals are
recorded. Moreover, our algorithm efficiently computes the topological
features, namely persistent $H^{\inf}$ and $\hat{H}$ homology, from the
persistent hypergraph model. Consequently, we obtain new features of contact
patterns that are inaccessible and largely neglected in the past.

##  2\. Background: Classical persistent homology

###  2.1. From point clouds to complexes.

The most obvious way to convert a point cloud $X$ in a metric space into a
topological object is to use the point cloud as the vertices of a
combinatorial graph whose edges are determined by proximity (vertices within
some specified distance $r$). Such a graph, while capturing connectivity data,
ignores a wealth of higher-order features beyond clustering. These features
can be accurately discerned by thinking of the graph as a scaffold for a
higher-dimensional object. Specifically, one completes the graph to a
simplicial complex — a space built from simple pieces (simplices) identified
combinatorially along faces. The choice of how to fill in the higher-
dimensional simplices of the proximity graph allows for different global
representations. The most natural method is to generate Rips complexes,
defined as follows.

######  Definition 1.

A point configuration is a finite subset of $\mathbb{R}^{n}$. Let $X$ be a
point configuration, and let $r>0$, the Rips complex, $R(X,r)$ is the
simplicial complex whose $k$-simplices are
$\\{\\{x_{0},x_{1},\dots,x_{k}\\}:x_{i}\in X,d(x_{i},x_{j})<r\\}$, where $d$
is the Euclidean metric on $\mathbb{R}^{n}$.

To analyze simplicial complexes, algebraic topology offers a mature set of
tools for counting and collating holes and other topological features in
spaces and maps between them. In the context of high-dimensional data,
algebraic topology works like a telescope, revealing objects and features not
visible to the naked eye. In what follows, we concentrate on homology for its
balance between ease of computation and topological resolution27.

Let $C_{*}(X,r)$ be the chain complex associated to $R(X,r)$ i.e. we have

$C_{0}\xleftarrow{\partial_{1}}C_{1}\xleftarrow{\partial_{2}}C_{2}\xleftarrow{\partial_{3}}\cdots$

where $C_{k}$ is the free abelian group generated by $k$-simplices of $R(X,r)$
and $\partial_{k}$ is the $k$-th boundary map. We denote the cycles by
$Z_{k}(X,r)=\ker(\partial_{k})$, the boundaries by
$B_{k}(X,r)=\text{img}(\partial_{k+1})$, the homology by
$H_{k}(X,r)=Z_{k}(X,r)/B_{k}(X,r)$ and the number of generators of homology
group $H_{k}$ is the Betti number $\beta_{k}$.

###  2.2. Persistence

Despite being both computable and insightful, the homology of a complex
associated to a point cloud at a particular $r$ is insufficient: it is a
mistake to ask which value of $r$ is optimal. One requires a means of
declaring which holes are essential and which can be safely ignored.

Persistence, as introduced by Edelsbrunner, Letscher, and Zomorodian29 and
refined by Carlsson and Zomorodian30, is a rigorous response to this problem.

######  Definition 2.

Let $0<r_{1}<r_{2}<\cdots$, the persistent complex $\mathscr{C}(X)$ is the set
of chain complexes $C_{*}(X,r_{1}),C_{*}(X,r_{2}),\dots$ together with chain
maps $f^{i}:C_{*}(X,r_{i})\to C_{*}(X,r_{i+1})$ where $f^{i}$ consists of the
natural embeddings
$f^{i}_{k}:C_{k}(X,r_{i})\xhookrightarrow{}C_{k}(X,r_{i+1})$.

To reveal which features persist, we examine the induced inclusion maps
$f_{*}^{i}:H_{*}(X,r_{i})\to H_{*}(X,r_{i+1})$.

######  Definition 3.

Let $0<r_{1}<r_{2}<\cdots$ and $i<j$, the $(i,j)$ persistent homology of
$\mathscr{C}(X)$, denoted $H^{i,j}_{*}(X)$ is defined to be the image of the
induced homomorphism $f_{*}^{i,j}:H_{*}(X,r_{i})\to H_{*}(X,r_{j})$. In the
other words, $H_{k}^{i,j}(X)=Z_{k}(X,r_{i})/(Z_{k}(X,r_{i})\cap
B_{k}(X,r_{j}))$. The number of generators of homology group $H_{k}^{i,j}$ is
called the persistent Betti number $\beta_{k}^{i,j}$.

We can visualize the persistent homology by barcodes. For example, we have 21
points in $\mathbb{R}^{2}$. Figure 1 represents persistent homology of the
Rips complexes based on these points. The first row represents the 21 points
with different radii. In the persistent barcodes representation, the
horizontal axis corresponds to radius and the bars represent unordered
generators of homology.

Figure 1. VR complexes and persistent barcodes

##  3\. Persistent hypergraphs homology

In this section, we establish the mathematical background for persistent
theory of hypergraphs. More details about topology can be found in classical
textbooks27,28. The base ring $R$ could be any field. For simplicity, we will
assume $R$ is of characteristic 2 in the computations.

###  3.1. Hypergraphs

######  Definition 4.

A hypergraph $\mathcal{H}$ is a pair $(V,E)$ where $V=\\{1,2,\dots,|V|\\}$ is
a finite set and $E\subseteq 2^{V}$. Elements in $E$ are called hyperedges.
Let $\mathcal{H}_{n}=R$-span$\\{e\in E:|e|=n+1\\}$ and let $C$ be the
simplicial complex $(V,2^{V})$, $C_{n}$ be the collection of all $n$-dim
simplices. we view $\mathcal{H}_{n}$ as a subspace of $C_{n}$.

Throughout this subsection, let $\partial_{n}:C_{n}\to C_{n-1}$ be the
boundary map of $C_{*}$. Consider the following complexes based on the
hypergraph $\mathcal{H}$.

######  Definition 5.

  1. (1)

The infimum chain complex of $\mathcal{H}$, denoted by
$\mathcal{H}_{*}^{\inf}$, is defined as
$\mathcal{H}_{n}^{\inf}=\partial_{n}^{-1}(\mathcal{H}_{n-1})\cap\mathcal{H}_{n}$.

  2. (2)

The supremum chain complex of $\mathcal{H}$, denoted by
$\mathcal{H}_{*}^{\sup}$, is defined as
$\mathcal{H}_{n}^{\sup}=\partial_{n+1}(\mathcal{H}_{n+1})+\mathcal{H}_{n}$.

  3. (3)

The associated simplicial complex
$\Delta\mathcal{H}=(V,\Delta_{\mathcal{H}}(E))$ where
$\Delta_{\mathcal{H}}(E)=\\{\sigma\in 2^{V}:\sigma\subseteq e\text{ for some
}e\in E\\}$.

  4. (4)

The lower-associated simplicial complex
$\delta\mathcal{H}=(V,\delta_{\mathcal{H}}(E))$ where
$\delta_{\mathcal{H}}(E)=\\{\sigma\in E:e\in E\text{ for all
}e\subseteq\sigma\\}$.

Let $\Delta\mathcal{H}_{*}$ and $\delta\mathcal{H}_{*}$ denote the chain
complexes associated to $\Delta\mathcal{H}$ and $\delta\mathcal{H}$
respectively. The boundary maps for $\mathcal{H}^{\inf}_{*}$,
$\mathcal{H}^{\sup}_{*}$, $\Delta\mathcal{H}_{*}$ and $\delta\mathcal{H}_{*}$
are the restricted maps obtained from $\partial_{n}$. In particular, we denote
$\partial_{n}|_{\mathcal{H}_{n}^{\inf}}$ and
$\partial_{n}|_{\mathcal{H}_{n}^{\sup}}$ by $\partial_{n}^{\inf}$ and
$\partial_{n}^{\sup}$ respectively. The $n$-th homology of
$\mathcal{H}^{\inf}_{*}$, $\mathcal{H}^{\sup}_{*}$, $\Delta\mathcal{H}_{*}$
and $\delta\mathcal{H}_{*}$ are denoted by $H_{n}^{\inf}$, $H_{n}^{\sup}$,
$\Delta H_{n}$ and $\delta H_{n}$ respectively.

Wu and others prove the following insightful result.

######  Theorem 6 (Wu and others17).

For a hypergraph $\mathcal{H}$, one has
$\partial_{n}(\mathcal{H}_{n}^{\inf})\subseteq\mathcal{H}_{n-1}^{\inf}$ and
$\partial_{n}(\mathcal{H}_{n}^{\sup})\subseteq\mathcal{H}_{n-1}^{\sup}$.
Hence, $\mathcal{H}_{*}^{\inf}$ and $\mathcal{H}_{*}^{\sup}$ are indeed chain
complexes. Moreover, $H_{n}^{\inf}\cong H_{n}^{\sup}$.

From their result, $H_{n}^{\inf}$ is called the embedded homology of
$\mathcal{H}$.

It is not hard to see that
$\ker(\partial_{n}^{\inf})=\ker(\partial_{n}|_{\mathcal{H}_{n}})$ and
$\mathrm{img}(\partial_{n}^{\sup})=\partial_{n}(\mathcal{H}_{n})$. To better
understand the embedded homology, we study $\ker(\partial_{n}^{\sup})$,
$\mathrm{img}(\partial_{n}^{\inf})$ and their quotient. This quotient space,
denoted by
$\hat{H}_{n}=\ker(\partial_{n}^{\sup})/\mathrm{img}(\partial_{n+1}^{\inf})$ is
called the $\hat{H}$ homology of $\mathcal{H}$.

Consider the following commutative diagram

$0$$\mathcal{H}^{\sup}_{0}$$\mathcal{H}^{\sup}_{1}$$\mathcal{H}^{\sup}_{2}$$\cdots$$0$$\mathcal{H}^{\inf}_{0}$$\mathcal{H}^{\inf}_{1}$$\mathcal{H}^{\inf}_{2}$$\cdots$$\partial_{1}^{\sup}$$\partial_{2}^{\sup}$$\partial_{3}^{\sup}$$\partial_{1}^{\inf}$$\partial_{2}^{\inf}$$\partial_{3}^{\inf}$$\iota_{0}$$\iota_{1}$$\iota_{2}$

where $\iota_{n}:\mathcal{H}_{n}^{\inf}\to\mathcal{H}_{n}^{\sup}$ is the
natural embedding.

For each $n$, since
$\partial_{n}^{\sup}\circ(\iota_{n}\circ\partial_{n+1}^{\inf})=\iota_{n-1}\circ\partial_{n}^{\inf}\circ\partial_{n+1}^{\inf}=0$,
we have the following chain complex

$\mathcal{H}^{n}_{*}:\cdots\xrightarrow{\partial_{n+3}^{\inf}}\mathcal{H}_{n+2}^{\inf}\xrightarrow{\partial_{n+2}^{\inf}}\mathcal{H}^{\inf}_{n+1}\xrightarrow{\iota_{n}\circ\partial_{n+1}^{\inf}}\mathcal{H}^{\sup}_{n}\xrightarrow{\partial_{n}^{\sup}}\mathcal{H}^{\sup}_{n-1}\xrightarrow{\partial_{n-1}^{\sup}}\cdots.$

Under this setting, the $\hat{H}_{n}$ homology is just the $n$-th homology of
$\mathcal{H}^{n}_{*}$. In general, $H_{n}^{\inf}$, $\hat{H}_{n}$, $\Delta
H_{n}$ and $\delta H_{n}$ could be mutually distinct.

###  3.2. Morphisms and filtrations

######  Definition 7.

A hypergraph morphism $\phi:(V_{1},E_{1})\to(V_{2},E_{2})$ is a map
$\phi:V_{1}\to V_{2}$ such that for any hyperedge $e\in E_{1}$, its image
$\\{\phi(v):v\in e\\}$ is a hyperedge in $E_{2}$. We say $\phi$ is injective
if the map on vertices $\phi:V_{1}\to V_{2}$ is injective. In particular,
$|e|=|\phi(e)|$ for all hyperedges $e\in E_{1}$. We say $\phi$ is an embedding
if $\phi$ is injective and $\phi(e)=\phi(e^{\prime})$ implies $e=e^{\prime}$
for all hyperedges $e,e^{\prime}\in E_{1}$.

An injective hypergraph morphism $\phi:\mathcal{H}\to\mathcal{K}$ between
hypergraphs $\mathcal{H}$ and $\mathcal{K}$ induces morphisms on chain
complexes

  * •

$\phi_{*}^{\inf}:\mathcal{H}_{*}^{\inf}\to\mathcal{K}_{*}^{\inf}$,

  * •

$\phi_{*}^{\sup}:\mathcal{H}_{*}^{\sup}\to\mathcal{K}_{*}^{\sup}$,

  * •

$\Delta\phi_{*}:\Delta\mathcal{H}_{*}\to\Delta\mathcal{K}_{*}$,

  * •

$\delta\phi_{*}:\delta\mathcal{H}_{*}\to\delta\mathcal{K}_{*}$.

Consequently, $\phi$ induces morphisms of the homology spaces $\Delta H_{*}$,
$\delta H_{*}$, $H_{*}^{\inf}$ and $\hat{H}_{*}$.

######  Definition 8.

Fix a finite set $V=\\{1,2,\dots,|V|\\}$. A hypergraph filtration is a map
$f:2^{V}\to\mathbb{N}\cup\\{\infty\\}$ such that $f(\emptyset)=0$. A
hypergraph filtration is a simplicial complex filtration if $f(e)\leq
f(e^{\prime})$ whenever $e\subseteq e^{\prime}$.

A filtration induces a family of hypergraphs
$\\{\mathcal{H}^{t}=(V,E^{t}):t\in\mathbb{N}\\}$ with $E^{t}=\\{e\in
2^{V}:f(e)\leq t\\}$. For $t<r$, one has the natural embedding of hypergraphs
$\phi^{t,r}:\mathcal{H}^{t}\to\mathcal{H}^{r}$ that further induces morphisms
on chain complexes. The $t,r$-persistent homology is defined as

$\overline{H}_{n}^{t,r}=\ker\left(\partial_{n}|_{\overline{\mathcal{H}}_{n}^{t}}\right)/\left(\ker\left(\partial_{n}|_{\overline{\mathcal{H}}_{n}^{t}}\cap\partial_{n+1}\left(\overline{\mathcal{H}}_{n+1}^{r}\right)\right)\right)$

for $\overline{\mathcal{H}}=\Delta\mathcal{H},\delta\mathcal{H}$ or
$\mathcal{H}^{\inf}$. In the other words, let
$\left(\overline{\phi}^{t,r}\right)_{n}^{*}:\overline{H}_{n}^{t}\to\overline{H}_{n}^{r}$
be the induced morphism on homology $\overline{H}=\Delta H,\delta H,H^{\inf}$
or $\hat{H}$, then
$\overline{H}_{n}^{t,r}=\mathrm{img}\left(\left(\overline{\phi}^{t,r}\right)_{n}^{*}\right)$.

When $f$ is a simplicial complex filtration, $f$ induces the classical
persistent homology $H_{*}^{t,r}$ of simplicial complex as introduced by
Zomorodian and Carlsson30.

###  3.3. Persistence module

A persistence module is a family of $R$-modules $\\{M_{i}:i\in\mathbb{N}\\}$
with $R$-linear maps $\phi_{i}:M_{i}\to M_{i+1}$. In this paper, we only
consider the case where $R$ is a field of characteristic 2 and $M_{i}$ are
generated by finitely many hyperedges. In this setting, $(M_{i},\phi_{i})$
forms a persistence module of finite type and has an elegant decomposition.

Each persistence module $(M_{i},\phi_{i})$ can be identified as a graded
$R[t]$-module $\displaystyle\bigoplus_{i\geq 0}M_{i}$ where the action of $t$
is given by

$t\cdot(m_{0},m_{1},m_{2},m_{3},\dots)=(0,\phi_{0}(m_{0}),\phi_{1}(m_{1}),\phi_{2}(m_{2}),\dots).$

Since $R[t]$ is a principal ideal domain, we have the following decomposition,
followed from the classical structural theorem in commutative algebra.

######  Theorem 9.

$\bigoplus_{i\geq
0}M_{i}=\left(\bigoplus_{i=1}^{n}\Sigma^{\alpha_{i}}R[t]\right)\oplus\left(\bigoplus_{j=1}^{m}\Sigma^{\beta_{j}}R[t]/(t^{\gamma_{j}})\right)$

where $\Sigma^{a}$ means shifting the grading up by $a$.

In the decomposition, each summand can be interpreted by a pair
$(i,j)\in\mathbb{N}\times\mathbb{N}\cup\\{\infty\\}$ as
$\Sigma^{\alpha_{i}}R[t]\mapsto(\alpha_{i},\infty)$ and
$\Sigma^{\beta_{j}}R[t]/(t^{\gamma_{j}})\mapsto(\beta_{j},\beta_{j}+\gamma_{j})$.
The multiset $\\{(\alpha_{i},\infty),(\beta_{j},\beta_{j}+\gamma_{j}):1\leq
i\leq n,1\leq j\leq m\\}$ is called the barcodes representation of the
persistent module $(M_{i},\phi_{i})$.

##  4\. Computational experiments and data analysis

###  4.1. Dataset

The data we use in this paper is provided by SocioPatterns30 sensing program.
As is shown in Table 1, the data consists of face-to-face interactions of the
following social groups

  1. (1)

13 Guinea baboons living at CNRS primate center25;

  2. (2)

86 individuals from a village in rural Malawi26;

  3. (3)

403 participants of a scientific conference24;

  4. (4)

217 people at a workplace24;

  5. (5)

327 students from a high school in Marseilles22.

data sets | number of individuals | total number of contacts | duration of data collection
---|---|---|---
Baboon | 13 | 63095 | 4 weeks
Malawi | 86 | 102293 | 2 weeks
Conference | 403 | 70261 | 2 days
Workplace | 217 | 78249 | 2 weeks
High school | 327 | 188508 | 1 week
Table 1. Characteristics of the data sets.

All of these face-to-face contact data sets are collected and recorded in
unsupervised fashion by using wearable sensors called Radio-Frequency
Identification Devices (RFID). The sensors send signal once every 20 seconds,
and they can receive signals from other devices within the radius of 1.5
meter. The sensors are weared on the chest of each individuals so that their
devices will receive and record signals when they are facing each other. If a
pair of devices both receive signal from each other within a period of 20
seconds, we say the two individuals are in face-to-face contact in that
period. It is shown that wearable sensors detect face-to-face interactions
with more than 99% accuracy.

###  4.2. Persistent hypergraph model

For each set of data, we construct a persistent hypergraph. For each hyperedge
$e$ consisting of $k$ individuals, we record $T_{e}$, the total number of
periods during which every pair among them are in face-to-face contact. In the
other words, $T_{e}$ is the total length of time that these $k$ people are in
a group meeting. It should be noted that if there is a group meeting, only the
hyperedge consisting of the whole group counts. We ignore all the subgroups to
avoid repeated counting.

The filtration for a hyperedge $e$ is given by $M-\log(T_{e})$ where
$M=\log(\max\\{T_{f}:f\text{ is hyperedge}\\})$ is the fixed constant ensuring
the non-negativity of the filtration. By convention, the filtration of $e$ is
set to infinity if $T_{e}=0$.

To better explain our model, we give an easy but complete example. Suppose our
data consists of six individuals. The interactions are given in Figure 2 in
terms of graphs i.e. in $t_{k}$, an edge $(i,j)$ means there is an interaction
between $i$ and $j$ at time $k$.

ABCDEF$t_{1}$

ABCDEF$t_{2}$

ABCDEF$t_{3}$

ABCDEF$t_{4}$

ABCDEF$t_{5}$

ABCDEF$t_{6}$

ABCDEF$t_{7}$

ABCDEF$t_{8}$

ABCDEF$t_{9}$

Figure 2. Interactions between six individuals, an edge $(i,j)$ in $t_{k}$
means an interaction takes place between $i$ and $j$ at time $k$.

Then, the filtration function and persistent hypergraphs are given in Figure
3.

$e$ | AB | DF | CD | CF | AC | BC | ABC | DEF
---|---|---|---|---|---|---|---|---
$T_{e}$ | 4 | 3 | 2 | 2 | 1 | 1 | 3 | 2
$f(e)$ | 0 | $\log 4/3$ | $\log 2$ | $\log 2$ | $\log 4$ | $\log 4$ | $\log 4/3$ | $\log 2$

$t=0$$t=\log 4/3$$t=\log 2$$t=\log 4$Barcodes for $H_{0}^{\inf}$ in dark red,
and $H_{1}^{\inf}$ in dark blueAdditional barcodes for $\hat{H}_{1}$ in dark
blue

Figure 3. Hyperedges $e$, duration of interactions $T_{e}$, filtrations
$f(e)$, persistent hypergraphs and persistent barcodes

After these preparations, persistent embedded homology and $\hat{H}$ homology
of dimension 0 and 1 are computed using our algorithm.

###  4.3. Algorithm

In this section, we provide an algorithm of computing barcodes for hypergraphs
from a given filtration. This algorithm is inspired by the one for simplicial
complex introduced by Zomorodian and Carlsson30. When the filtration satisfies
the condition of a persistent simplicial complex, our algorithm yields the
classical barcodes for simplicial complex.

Suppose we fix a filtration $f:2^{V}\to\mathbb{N}\cup\\{\infty\\}$, we compute
the dimension $k$ persistent $H^{\inf}$ and $\hat{H}$ barcodes simultaneously
as follows:

  * •

Step 1. Let $E_{k}=\\{e\in 2^{V}:|e|=k+1\\}$ be the set of hyperedges of
dimension $k$. Sort $E_{k}$ decreasingly according to the filtration. Let
$E_{k+1}=\\{e\in 2^{V}:|e|=k+2,f(e)<\infty\\}$ be the set of hyperedges of
dimension $k+1$ whose filtration is less than infinity. Sort $E_{k+1}$
increasingly according to the filtration.

  * •

Step 2. Construct a boundary matrix $M=(M_{ij})$ whose rows correspond to
$E_{k}$ and columns correspond to $E_{k+1}$. The entry $M_{ij}=1$ if the
$i$-th hyperedge in $E_{k}$ is contained in the $j$-th hyperedge in $E_{k+1}$,
and $M_{ij}=0$ otherwise.

  * •

Step 3. Compute pivots from boundary matrix $M$ using the pseudo-codes

endendendchange column $k$ to column $k$ $-$ column $j$;if $k$ is not a pivot
column and $M_{i,k}=1$for $k=j+1$ to number of columnsadd $(i,j)$ to
Pivot;find the smallest $j$ such that $j$ is not a pivot column and
$M_{i,j}=1$;for $i=1$ to number of rowsinitialize
Pivot$=\emptyset$;ComputePivot($M$)10987654321

  * •

Step 4. For each row, record an ordered pair (birth, death) where birth is the
filtration of the hyperedge corresponding to the row. If there is a pivot at
column $j$, then death is the filtration of the hyperedge corresponding to
column $j$. If the row has no pivot, then death is set to be infinity.

  * •

Step 5. For each ordered pair (birth, death), if birth $<$ death, then the
pair corresponds to a barcode in persistent embedded homology $H_{n}^{\inf}$.
If birth $>$ death, then the pair (death, birth) corresponds to a barcode in
persistent homology $\hat{H}$.

The complexity of our algorithm is at the level of $n^{k+2}$ where $n$ is the
number of individuals and $k$ is the dimension. This is sufficiently efficient
for our purpose. For the network of a conference in which there are 403
individuals and 70261 face-to-face contacts, the computation for dimension 0
and 1 barcodes is done within a few seconds on a regular personal computer.The
software used to compute the results in this section is available at
https://github.com/Gao-Yaru/Interaction-Persistent-Hypergraph.

There are certainly ways to improve this algorithm. For example, when
constructing the boundary matrix, instead of taking all hyperedges for rows,
we only need to take those that are subsets of existing column hyperedges. But
we will not discuss it further in this paper.

###  4.4. Results: 0-dimensional persistent hypergraph homology

We begin by analyzing results in dimension 0. In principle, dimension 0
barcodes always begin at 0. Those bars ending at infinity correspond to
connected components in the connectivity graph, that is, groups of individuals
such that any pair of individuals from different groups never conduct face-to-
face interaction. The persistent hypergraph model provides an efficient
computation on the number of connected components. Among data sets we use,
Malawi has two connected components, meaning that the local residents form two
groups, and there is no contact between the two groups during the two weeks of
data collection. Our computation also show that all of the other data sets are
connected. Figure 4 provides a comparison between barcodes.

BaboonMalawiConferenceWorkplaceHigh school Figure 4. Dimension 0 barcodes of
the datasets.

While we can read the number of mutually unrelated groups from persistent
barcodes, they do not contain the information of the exact members of the
groups. Finding the members requires additional efforts.

###  4.5. Results: 1-dimensional persistent hypergraph homology

Much more interesting results appear in dimension 1. In this case, there are
two types of barcodes: the persistent $H^{\inf}$ and $\hat{H}$ barcodes.
Theoretically, persistent $H^{\inf}$ barcodes correspond to holes in the
connectivity graphs, that is a loop whose interior is not filled. In
particular, a bar ending at infinity means there exists a group of
individuals, say 3 individuals for example, that experience pair-wise face-to-
face interaction, but they never meet together simultaneously. On the other
hand, an $\hat{H}$ bar corresponds to an anti-hole i.e. a filled interior with
some missing edges on its boundary. In particular, an $\hat{H}$ bar ending at
infinity means there is a group meeting of 3 individuals among which some
pairs never interact privately. As examples, we show these barcodes calculated
from the datasets in Figure 5.

BaboonMalawiConferenceWorkplaceHigh school Figure 5. Dimension 1 persistent
$H^{\inf}$ (green) and $\hat{H}$ (blue) barcodes of the datasets.

Table 2 provides computation results of the number of persistent $H^{\inf}$
and $\hat{H}$ barcodes and those ending at infinity. We emphasize that the
number of barcodes ending at infinity is a new and useful statistic quantity
in analysis of population behaviors. More precisely, we focus on the
proportions $n/(N+\hat{N})$ and $\hat{n}/(N+\hat{N})$ of the bars ending at
infinity among all bars. These results are visualized in Figure 6.

data sets | $N$ | $n$ | $\hat{N}$ | $\hat{n}$
---|---|---|---|---
Baboon | 78 | 14 | 0 | 0
Malawi | 332 | 270 | 10 | 4
Conference | 7733 | 7290 | 1363 | 1072
Workplace | 4048 | 3537 | 142 | 112
High school | 5096 | 3797 | 473 | 281
Table 2. Statistics on barcodes of dimension 1: $N$ is the number of
persistent $H^{\inf}$ barcodes, $n$ is the number of persistent $H^{\inf}$
barcodes ending at infinity, $\hat{N}$ is the number of persistent $\hat{H}$
barcodes, $\hat{n}$ is the number of persistent $\hat{H}$ barcodes ending at
infinity.

Roughly speaking, a high proportion of persistent barcodes ending at infinity
reflects that a high proportion of the population tends to contact privately
rather than communicate in groups. In contrast, higher proportion of $\hat{H}$
barcodes ending at infinity represents that population are more open to
strangers in the sense that individuals are more active in participating in
group meetings with those they have not contacted.

$0.897$

$3.947$

$4.007$

$4.221$

$3.409$

$0$

$0.146$

$0.589$

$0.134$

$0.252$

BaboonMalawiConferenceWorkplaceHigh schoolproportion of barcodes for
$H^{\inf}_{1}$ ending at infinityproportion of additional barcodes for
$\widehat{H}_{1}$ ending at infinity

Figure 6. Proportion of dimension 1 barcodes ending at infinity.

For the data set of baboons, most persistent barcodes end before infinity.
This result describes that baboons like to meet in groups with other
individuals that they usually contact with. On the other hand, there is no
$\hat{H}$ barcodes, showing that baboons never attend any group meetings if
there is someone they do not and will not contact inside the group. All other
data sets consist of human beings, and they all have relatively high
proportion of persistent barcodes ending at infinity, meaning most pair-wise
interactions do not end up with group discussion. The data set of conference
holds the highest proportion of $\hat{H}$ barcodes ending at infinity,
followed by the data set of high school. This phenomenon can be explained by
that collaborations of more than 2 scientists or high school students are more
common than those in rural Malawi and workplace. Moreover, such collaborations
often do not require that individuals are familiar with each other in the
sense of private connections.

In the end of this section, we want to point out that there are many other
potential statistics one can derive from persistent hypergraph model. For
example, the lengths of dimension 1 barcodes reflect the differences in length
between group meetings and pair-wise private meetings. The persistent betti
numbers count the number of holes or anti-holes with each particular period
length. Furthermore, higher dimensional barcodes reveal the contact
information of group of more than 3 individuals. These statistic figures could
potentially help understanding more deeply the behavior of face-to-face
interactions under different environments.

##  5\. Conclusion

The main contribution of this article is the introduction of persistent
hypergraph model in data analysis. We introduce a new homology $\hat{H}$, and
we extend classical algorithm of persistent homology to hypergraphs that
computes both persistent $H^{\inf}$ and $\hat{H}$ barcodes simultaneously and
efficiently. The persistent hypergraph model fits particularly well in the
data sets of interaction networks. Both $H^{\inf}$ and $\hat{H}$ homology have
real-world meanings, they represent connected components in the connection
graph, cycles of pair-wise interactions and group meetings of at least 3
individuals.

The commonly used statistics features on face-to-face interaction analysis
include connection graphs, mean values, standard deviations, distributions and
their modifications. These features are essentially 0 or 1 dimensional in the
topological viewpoint. Topological data analysis provides a very efficient way
to reveal features in not only 0 and 1, but also higher dimensions. As
demonstrated in the experiment, persistent hypergraph model reveals connected
components, cycles, group meetings and other new and potentially useful
information for understanding contact patterns.

Furthermore, our algorithm for persistent hypergraph model gives both global
features and local details, including information of the exact locations and
the sizes of the cycles. Therefore, topological data analysis can be used as
an important complementary to classical data analysis methods. Ideally, global
and local results can be further combined with such as principal component
analysis and machine learning to obtain a more complete and precise picture of
the data sets.

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††thanks<EMAIL_ADDRESS>
# Proposal for optomagnonic teleportation and entanglement swapping

Zhi-Yuan Fan  Interdisciplinary Center of Quantum Information, State Key
Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key
Laboratory of Quantum Technology and Device, School of Physics, Zhejiang
University, Hangzhou 310027, China  Xuan Zuo  Interdisciplinary Center of
Quantum Information, State Key Laboratory of Modern Optical Instrumentation,
and Zhejiang Province Key Laboratory of Quantum Technology and Device, School
of Physics, Zhejiang University, Hangzhou 310027, China  Hang Qian
Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics, Zhejiang University,
Hangzhou 310027, China  Jie Li  Interdisciplinary Center of Quantum
Information, State Key Laboratory of Modern Optical Instrumentation, and
Zhejiang Province Key Laboratory of Quantum Technology and Device, School of
Physics, Zhejiang University, Hangzhou 310027, China

###### Abstract

A protocol for realizing discrete-variable quantum teleportation in an
optomagnonic system is provided. Using optical pulses, an arbitrary photonic
qubit state encoded in orthogonal polarizations is transferred onto the joint
state of a pair of magnonic oscillators in two macroscopic yttrium-iron-garnet
(YIG) spheres that are placed in an optical interferometer. We further show
that optomagnonic entanglement swapping can be realized in an extended dual-
interferometer configuration with a joint Bell-state detection. Consequently,
magnon Bell states are prepared. We analyze the effect of the residual thermal
occupation of the magnon modes on the fidelity in both the teleportation and
entanglement swapping protocols.

##  I Introduction

As an indispensable building block of quantum information science, quantum
teleportation refers to the process of transferring an unknown quantum state
at one location onto another quantum system some distance away. Ever since it
was first proposed by Bennett et al. Bennett (1993), quantum teleportation has
been successfully realized in various physical systems over the past few
decades. These include photons Bouwmeester (1997); Pan (1998); Lombardi
(2002); Gisin (2003), nuclear spins Nielsen (1998), trapped ions Barrett
(2004); Riebe (2004), atomic ensembles CYA (2008); BaoXH (2012), solid-state
systems Steffen (2013), high-frequency vibration phonons DuanLM (2016), and
optomechanical systems Simon (2021), among many others. These successful
demonstrations lay the foundation for realizing many other quantum protocols,
such as quantum repeaters Gisin (2011), fault-tolerant quantum computation
Gottesman (1999), etc.

Here we provide an optomagnonic quantum teleportation protocol which can
transfer an arbitrary photonic qubit state to a dual-rail encoding magnonic
system of two yttrium-iron-garnet (YIG) spheres. We adopt an optical
interferometer configuration, of which each arm contains a YIG sphere
supporting an optomagnonic system. We use the Stokes type scattering of the
magnon-induced Brillouin light scattering (BLS) to create an optomagnonic EPR
state. A subsequent Bell-state measurement onto the input photonic qubit and
the output Stokes photon from the interferometer enables such a photon-to-
magnon quantum state transfer. The magnon state can be read out by activating
the anti-Stokes scattering realizing the optomagnonic state-swap operation,
from which the magnonic qubit state is retrieved onto the anti-Stokes photon.
We further propose an optomagnonic entanglement swapping protocol based on an
extended dual-interferometer configuration, which realizes the transfer of the
optomagnonic entanglement to a dual-rail encoding two-qubit magnonic system
involving four YIG spheres and thus prepares a magnonic Bell state.

In what follows, we first introduce the basic optomagnonic interactions in
Section II, and describe our two protocols respectively in Section III and
Section IV. We then analyze the effect of the magnon thermal excitations on
the fidelity of the two protocols in Section V and finally conclude in Section
VI.

##  II Optomagnonic interaction

In the past decade, hybrid magnonic systems based on collective spin
excitations in ferrimagnetic materials, such as YIG, have gained significant
attention due to their excellent ability to coherently couple with a variety
of physical systems, including microwave photons Huebl (2013); Tabuchi (2014);
Zhang (2014), optical photons Naka (2016); HX (2016); Haigh (2016); Usami
(2018); ZhuNa (2020), vibration phonons Zhang (2016); JL (2018); Yu (2020);
Potts (2021); Shen (2022) and superconducting qubits Tabuchi (2015); Naka
(2017, 2020); Wolski (2020); Xu (2023). The emerging field of hybrid quantum
magnonics provides a platform not only for studying strong interactions
between light and matter, but also for developing novel quantum technologies
to be applied in quantum information processing, quantum sensing and quantum
networks Nake (2019); Yuan (2022); LiYi (2020); JL (2021). In particular, the
coupling between magnons and optical photons, namely the optomagnonic
interaction Naka (2016); HX (2016); Haigh (2016); Usami (2018); ZhuNa (2020),
is indispensable for building a magnonic quantum network JL (2021), where the
transmission of the information between remote quantum nodes is realized by
optical photons. Such an optomagnonic interaction has been exploited in many
proposals to cool magnons Bauer (2018), prepare magnon Fock SVK (2018), cat
QYH (2021), path-entangled WWJ (2021) states and magnon-photon entangled
states Xie (2022); JL (2021), and realize magnon laser Xiong (2020, 2022),
frequency combs Liu (2022), photon blockade xie (2022), polarization-state
engineering Wu (2023), etc.

Figure 1: (a) A typical optomagnonic system: a YIG sphere supports a magnon
mode and two optical WGMs. (b) Mode frequencies corresponding to the
optomagnonic Stokes scattering, which leads to the PDC interaction. (c) Mode
frequencies associated with the optomagnonic anti-Stokes scattering, which
yields the state-swap (BS) interaction.

The typical optomagnomic system, as depicted in Figure 1(a), consists of a YIG
sphere, which supports both a magnetostatic mode (i.e., the magnon mode) and
optical whispering gallery modes (WGMs). The WGM resonator is near the surface
of the YIG sphere and the input optical field is evanescently coupled to the
WGM, e.g., via a tapered fiber Naka (2016); HX (2016) or prism Haigh (2016);
Usami (2018). The optomagnonic interaction in such a system is embodied by the
magnon-induced BLS, where the photons of a WGM are scattered by lower-
frequency magnons, typically in GHz Naka (2016); HX (2016); Haigh (2016);
Usami (2018), giving rise to sideband photons with their frequency shifted by
the magnon frequency and their polarization changed. When the frequency of the
scattered photons matches another WGM, namely the triple resonance condition,
the optomagnonic scattering probability is maximized. This can be easily
achieved by tuning the magnon frequency via changing the strength of the bias
magnetic field ($B_{0}$). Typically, there are two types of magnon-induced
BLSs, i.e., the Stokes and anti-Stokes scatterings, corresponding to
optomagnonic parametric down conversion (PDC) and state-swap (beam-splitter,
BS) interactions, respectively JL (2021); WWJ (2021). In the optomagnonic
scattering, the angular momenta of the WGM photons and magnons are conserved,
which leads to a selection rule and prominent asymmetry in the Stokes and
anti-Stokes scattering strengths Sharma (2017); Pantazopoulos (2017); Osada
(2018); Haigh (2018). This allows us to select a particular type of the
optomagnonic interactions (either PDC or BS) on demand.

The Hamiltonian accounting for the optomagnonic interaction in such a three-
mode system reads

$\begin{split}H/\hbar=&\
\omega_{v}a^{\dagger}a+\omega_{h}b^{\dagger}b+\omega_{m}m^{\dagger}m\\\
&+g_{0}(a^{\dagger}bm+ab^{\dagger}m^{\dagger})+H_{\rm{dri}}/\hbar,\end{split}$
(1)

where $j\,\,{=}\,\,a,b$ ($j^{\dagger}$) and $m$ ($m^{\dagger}$) are the
annihilation (creation) operators of the two WGMs and the magnon mode,
respectively, and $\omega_{k}$ ($k\,\,{=}\,\,v,h,m$) correspond to their
resonance frequencies, satisfying the relations
$|\omega_{v}-\omega_{h}|=\omega_{m}$ and $\omega_{m}\ll\omega_{v,h}$. Here the
subscripts $v$ and $h$ represent two orthogonal polarizations of the two WGMs,
i.e., the transverse-electric (TE) mode and the transverse-magnetic (TM) mode.
The optomagnonic interaction is a three-wave process and $g_{0}$ is the
corresponding single-photon coupling strength. The last term denotes the
driving Hamiltonian $H_{\rm{dri}}=i\hbar
E_{j}(j^{\dagger}e^{-i\omega_{0}t}-je^{i\omega_{0}t})$, where
$E_{j}=\sqrt{\kappa_{j}P_{j}/(\hbar\omega_{0})}$ is the coupling strength
between the WGM $j$ (with decay rate $\kappa_{j}$) and the optical drive field
with frequency $\omega_{0}$ and power $P_{j}$. To enhance the optomagnonic
coupling strength, a strong drive field is used to resonantly pump one of the
WGMs, i.e., $\omega_{0}=\omega_{v}$ or $\omega_{h}$, depending on which type
of the optomagnonic interactions (either PDC or BS) is desired.

For the case where the TE WGM ($a$) is pumped, i.e., $\omega_{0}=\omega_{v}$
and $\Delta_{b}\equiv\omega_{h}-\omega_{0}=-\omega_{m}$, cf. Figure 1(b), the
strongly driven WGM $a$ can be treated classically as a number
$\alpha\equiv\langle a\rangle$ JL (2021), and the effective optomagnonic
Hamiltonian then becomes the PDC form, $H_{\rm{int}}^{S}=\hbar
g(bm+b^{\dagger}m^{\dagger})$, with $g=g_{0}\alpha$ being the effective
optomagnonic coupling strength. This interaction corresponds to the Stokes
scattering, where the TE-WGM photons convert into lower-frequency sideband
photons (resonant with the TM WGM $b$) by creating magnon excitations. Such a
PDC interaction can be used to generate optomagnonic entangled states Xie
(2022); JL (2021). Specifically, the WGM $b$ and magnon mode $m$ are prepared
in a two-mode squeezed state (unnormalized)

$|\psi\rangle_{b,m}=|00\rangle_{b,m}+\\!\\!\sqrt{P}|11\rangle_{b,m}+{\cal
O}(P)\,\,,$ (2)

where $P$ is the probability for a single Stokes scattering event to occur and
${\cal O}(P)$ denotes the higher-excitation terms, of which the probabilities
are equal to or smaller than $P^{2}$. The scattering probability increases
with the power of the drive field, and when the power is sufficiently weak,
the scattering probability $P\ll 1$. In this weak-coupling limit, the
probability of generating two-magnon (photon) state $|2\rangle_{m(b)}$ and
higher-excitation states $|n\rangle_{m(b)}$ ($n\,\,{>}\,\,2$) is negligibly
small. Such a low scattering probability of creating an entangled pair of
single excitations was adopted in Ref. WWJ (2021), which suggests an
optomagnonic variant of the Duan-Lukin-Cirac-Zoller protocol Duan (2001). It
was also used in cavity optomechanical experiments for creating entangled
states of single photons and phonons Simon (2016, 2018).

Similarly, when the TM WGM ($b$) is resonantly pumped, i.e.,
$\omega_{0}=\omega_{h}$ and
$\Delta_{a}\equiv\omega_{v}-\omega_{0}=\omega_{m}$, cf. Figure 1(c), the anti-
Stokes scattering is activated and the effective optomagnonic Hamiltonian
becomes the BS type, $H_{\rm{int}}^{AS}=\hbar g(a^{\dagger}m+am^{\dagger})$,
with the effective coupling $g=g_{0}\beta$ and $\beta\equiv\langle b\rangle$.
This interaction realizes the state-swap operation between the magnon mode and
the WGM $a$, accompanied with the process that TM-WGM photons convert into
higher-frequency anti-Stokes photons (resonant with the TE WGM $a$) by
eliminating magnon excitations. As will be shown later, this interaction is
used to read out the magnon state.

Figure 2: Sketch of the optomagnonic quantum teleportation protocol. It
consists of three steps: preparation of the optomagnonic EPR state, Bell-state
measurement and readout of the magnonic state. In the EPR-state preparation
setup, an optical interferometer configuration is adopted and its each arm
contains a YIG sphere which supports an optomagnonic system of a magnon mode
and two optical WGMs. See text for detailed descriptions.

##  III Optomagnonic quantum teleportation

We now proceed to describe our optomagnonic teleportation protocol, which is
able to transfer a photonic qubit state (in polarization encoding) onto a
magnonic system consisting of two optomagnonic devices placed in two arms of
an optical interferometer, cf. Figure 2. The two magnon oscillators are
subject to a simultaneous excitation using a weak pulse that drives the TE WGM
to activate the optomagnonic Stokes scattering. After a Bell-state measurement
of the input photon with the Stokes photon from the optomagnonic devices, the
input photonic qubit state is then transferred onto the dual-rail encoding
magnonic system.

For simplicity, we assume at most single excitations in the optomagnonic
devices. This is the case of using a weak pulse, where the probability of
creating higher-excitation states $|n\rangle_{m(b)}$ ($n\,\,{\geq}\,\,2$) in
the Stokes scattering is negligible. Since the vacuum component of the state
(2) will not trigger any coincidences in the Bell-state detection, leading to
unsuccessful trials for the teleportation, the protocol can be described using
a simplified model Li (2020), where a TE-polarized single-photon pulse is sent
onto a 50/50 BS to drive the optomagnonic devices in the interferometer. After
the BS, an optical path-entangled state,
$\frac{1}{\sqrt{2}}(\ket{01}_{AB}+\ket{10}_{AB})$, is generated in the two
outputs (i.e., the upper path A and lower path B). In each path, the pulse
resonantly drives the TE WGM of the YIG sphere to activate the Stokes
scattering. By selecting trials with successful scattering events, the PDC
interaction prepares an optomagnonic Bell state in the form of

$\ket{\phi^{+}}_{bm}=\frac{1}{\sqrt{2}}\left(\ket{H}_{b}\ket{L}_{m}+\ket{V}_{b}\ket{U}_{m}\right),$
(3)

where $\ket{L}_{m}$ ($\ket{U}_{m}$) denotes the generated single magnon is at
the lower path B (the upper path A), and $\ket{H}_{b}$ ( $\ket{V}_{b}$)
represents the accompanied Stokes photon is in the horizontal (vertical)
polarization. The Stokes photon, with equal probability in path A or B, then
couples to the nanofiber or the prism coupler and enters the first polarizing
beam splitter (PBS1) before going to the next stage of the Bell-state
detection. Note that the polarization of the Stokes photon in the upper path
is changed from $H$ to $V$ after passing through a half-wave plate (HWP). We
remark that the unsuccessful scattering events leave the single photons
remaining in the vertical polarization. These photons eventually enter the
other output of the PBS1 and thus have no impact on the subsequent Bell-state
measurement. We also assume that the magnon modes are initially in their
quantum ground state, which is the case for GHz magnons at a low temperature,
e.g., of 10 mK. Nevertheless, in Section V we shall discuss the effect of the
residual thermal occupation on the fidelity of the teleportation.

The input photonic qubit state to be teleported is an arbitrary superposition
of two polarization modes, i.e.,
$\ket{\chi}_{c}=\alpha\ket{H}_{c}+\beta\ket{V}_{c}$, with the complex
coefficients $\alpha$ and $\beta$ satisfying $|\alpha|^{2}+|\beta|^{2}=1$.
Such an arbitrary qubit state can be constructed on the surface of the
polarization Poincare sphere by using a HWP and a quarter-wave plate (QWP).
This input photon, being resonant with the TM WGM, goes into one input of the
PBS2 and meanwhile the output of the PBS1 enters the other input port.
Thereby, the joint state before the Bell-state measurement is as follows:

$\begin{split}\ket{\phi^{+}}_{bm}\otimes\ket{\chi}_{c}\\!&=\\!\frac{1}{\sqrt{2}}\big{(}\alpha\ket{H}_{b}\ket{H}_{c}\ket{L}_{m}+\beta\ket{H}_{b}\ket{V}_{c}\ket{L}_{m}\\\
&\quad\
+\alpha\ket{V}_{b}\ket{H}_{c}\ket{U}_{m}+\beta\ket{V}_{b}\ket{V}_{c}\ket{U}_{m}\big{)}.\end{split}$
(4)

The Bell-state measurement is performed onto the input photonic qubit ($c$)
and the output Stokes photon ($b$) from the interferometer, and the detection
setup consists of a HWP, a PBS and two single-photon detectors in each output
of the PBS2, cf. Figure 2. A coincidence measurement projects the optical
modes onto the polarized Bell states
$\ket{\phi^{\pm}}_{bc}=\frac{1}{\sqrt{2}}(\ket{H}_{b}\ket{H}_{c}\pm\ket{V}_{b}\ket{V}_{c})$.
The fast axis of the HWP is set at $22.5^{\circ}$, which acts as a Hadamard
operation on the polarization of the photons passing through PBS2. The Bell
states $\ket{\phi^{\pm}}_{bc}$ correspond to different coincidence
measurements, as can be seen from the following

$\begin{split}\ket{\phi^{+}}_{bc}&\xrightarrow{\mathrm{PBS2\,\&\,HWP}}\frac{1}{\sqrt{2}}\left(a_{3,h}^{\dagger}a_{4,h}^{\dagger}+a_{3,v}^{\dagger}a_{4,v}^{\dagger}\right)\ket{\rm{vac}},\\\
\ket{\phi^{-}}_{bc}&\xrightarrow{\mathrm{PBS2\,\&\,HWP}}\frac{1}{\sqrt{2}}\left(a_{3,h}^{\dagger}a_{4,v}^{\dagger}+a_{3,v}^{\dagger}a_{4,h}^{\dagger}\right)\ket{\rm{vac}},\end{split}$
(5)

where the operator $a_{i,h}^{\dagger}$ ($a_{i,v}^{\dagger}$) denotes the
detection of a single horizontally (vertically)-polarized photon at one of the
two outputs of the PBS$i$ ($i=3,4$), and $\ket{\rm{vac}}$ means the vacuum
state. Note that, in addition to the Bell states $\ket{\phi^{\pm}}_{bc}$, the
other two types of Bell states
$\ket{\psi^{\pm}}_{bc}=\frac{1}{\sqrt{2}}(\ket{H}_{b}\ket{V}_{c}\pm\ket{V}_{b}\ket{H}_{c})$
can be realized by using photon-number-resolving detectors, and we disregard
these cases for simplicity.

The measurement that projects the optical modes $b$ and $c$ onto the Bell
state $\ket{\phi^{+}}_{bc}$ projects the magnonic system onto the state

$\ket{\chi^{\prime}}_{m}=\alpha\ket{L}_{m}+\beta\ket{U}_{m},$ (6)

which indicates the successful teleportation of the input photonic qubit state
$\ket{\chi}_{c}$ to a dual-rail encoding magnonic system, and corresponds to
the ideal quantum teleportation without requiring additional correction
operations in the readout step. On the other hand, the measurement associated
with the Bell state $\ket{\phi^{-}}_{bc}$ projects the magnonic system onto
the state

$\ket{\chi^{\prime\prime}}_{m}=\alpha\ket{L}_{m}-\beta\ket{U}_{m}.$ (7)

It has a $\pi$-phase difference with respect to the input optical state
$\ket{\chi}_{c}$, which can be corrected in the readout step using a feed-
forward operation by applying a phase shift in the optical interferometer Li
(2020); Simon (2021).

To verify the successful teleportation, we need to retrieve the teleported
magnonic qubit state. To achieve this, we exploit the optomagnonic state-swap
(BS) interaction, as introduced in Section II, where a TM-polarized weak pulse
is sent to the interferometer to activate the anti-Stokes scattering and the
magnonic state is then transferred to the anti-Stokes photon. Due to the
orthogonal polarization with respect to the Stokes photon (produced in the
Stokes scattering), the anti-Stokes photon leaves from the other output of the
PBS1, and enters the state-readout setup (cf. Figure 2) Simon (2021), from
which the magnonic qubit state is retrieved.

Figure 3: Sketch of the optomagnonic entanglement swapping protocol. It is
based on a dual-interferometer configuration, combined with a joint Bell-state
detection and magnonic state readout devices.

##  IV Optomagnonic entanglement swapping

Quantum entanglement plays an essential role in all the quantum-teleportation
related protocols. Naturally, entangled states can be obtained in directly
coupled systems, such as the EPR state produced in the optomagnonic Stokes
scattering. For two systems that have no direct interaction, quantum mechanics
also manifests its unique capabilities to establish quantum entanglement
between them, one of which is referred to the entanglement swapping Bennett
(1993); Pan (1998); Zukowski (1993). In this section, we show that the
entanglement swapping protocol allows us to prepare the magnon modes in space-
separated YIG spheres into an entangled Bell state.

The detailed entanglement swapping protocol is shown in Figure 3, which
consists of two optical interferometer setups used in Section III, a joint
Bell-state detection and the associated state-readout devices. Similarly to
the teleportation protocol, in each interferometer setup containing two YIG
spheres, a TE-polarized single-photon pulse is sent to activate the Stokes
scattering, which prepares an optomagnonic Bell entangled state
$\ket{\phi^{+}}_{bm}$, as in Equation (3). Therefore, the overall state of the
two interferometer setups before performing a joint Bell-state measurement on
the scattered Stokes photons reads

$\begin{split}\ket{\Psi}_{\rm{total}}=&\
\ket{\phi^{+}}_{b_{1}m_{1}}\otimes\ket{\phi^{+}}_{b_{2}m_{2}}\\\ =&\
\frac{1}{2}\big{(}\ket{H}_{b_{1}}\ket{L}_{m_{1}}+\ket{V}_{b_{1}}\ket{U}_{m_{1}}\big{)}\\\
&\times\big{(}\ket{H}_{b_{2}}\ket{L}_{m_{2}}+\ket{V}_{b_{2}}\ket{U}_{m_{2}}\big{)},\end{split}$
(8)

where the subscripts $1$ and $2$ are used to distinguish the two
interferometers, and the notation is the same as in Equation (3). The above
state can be rewritten in the basis of Bell states as

$\begin{split}\ket{\Psi}_{\rm{total}}=&\
\frac{1}{2}\Big{(}\ket{\psi^{+}}_{b_{1}b_{2}}\ket{\psi^{+}}_{m_{1}m_{2}}+\ket{\psi^{-}}_{b_{1}b_{2}}\ket{\psi^{-}}_{m_{1}m_{2}}\\\
&+\ket{\phi^{+}}_{b_{1}b_{2}}\ket{\phi^{+}}_{m_{1}m_{2}}+\ket{\phi^{-}}_{b_{1}b_{2}}\ket{\phi^{-}}_{m_{1}m_{2}}\Big{)},\end{split}$
(9)

where $\ket{\phi^{\pm}}_{b_{1}b_{2}}$ and $\ket{\psi^{\pm}}_{b_{1}b_{2}}$ are
the four Bell states of the Stokes photons in the two interferometers, which
take the same form as $\ket{\phi^{\pm}}_{bc}$ and $\ket{\psi^{\pm}}_{bc}$ in
Section III. $\ket{\phi^{\pm}}_{m_{1}m_{2}}$ and
$\ket{\psi^{\pm}}_{m_{1}m_{2}}$ are the four Bell states of a pair of dual-
rail encoding magnonic systems, defined as
$\ket{\phi^{\pm}}_{m_{1}m_{2}}=\frac{1}{\sqrt{2}}(\ket{L}_{m_{1}}\ket{L}_{m_{2}}\pm\ket{U}_{m_{1}}\ket{U}_{m_{2}})$
and
$\ket{\psi^{\pm}}_{m_{1}m_{2}}=\frac{1}{\sqrt{2}}(\ket{L}_{m_{1}}\ket{U}_{m_{2}}\pm\ket{U}_{m_{1}}\ket{L}_{m_{2}})$.

From Equation (9), it is clear to see that a Bell-state measurement on the
Stokes photons from the outputs of the two interferometers projects the
magnonic systems onto a corresponding Bell state. Specifically, a coincidence
measurement corresponding to the optical Bell state
$\ket{\phi^{+}}_{b_{1}b_{2}}$ ($\ket{\phi^{-}}_{b_{1}b_{2}}$) projects the
magnonic systems onto the Bell state $\ket{\phi^{+}}_{m_{1}m_{2}}$
($\ket{\phi^{-}}_{m_{1}m_{2}}$). This implies that the magnonic systems
establish the same form of entanglement as the corresponding optical Bell
state. Similarly as in the teleportation protocol, we disregard the other two
types of the Bell-state measurements associated with
$\ket{\psi^{\pm}}_{b_{1}b_{2}}$.

The entanglement swapping can also be understood in the framework of quantum
teleportation in the sense that it transfers an optomagnonic quantum
correlation (instead of a photonic qubit state $\ket{\chi}_{c}$) to the
magnonic systems. This can be seen by comparing Eqs. (4) and (8). This further
reflects the versatility of the quantum teleportation protocol, which can
transfer not only a qubit state but also multi-qubit states, e.g., quantum
correlations. Replacing the optomagnonic system with an optomechanical system
Simon (2018) in one of the interferometers also allows us to prepare a hybrid
magnon-phonon Bell state, which may find potential applications in hybrid
quantum networks JL (2021).

##  V Effect of magnonic thermal excitations

In the teleportation and entanglement swapping protocols, we neglect the
dissipation of the magnon modes. This is the case of using fast optical pulses
Simon (2016) such that the magnon dissipation can be assumed to be negligible
during an experimental run. We also assume that the magnon modes are initially
prepared in their quantum ground state. This is a good approximation for the
magnon modes at $\sim$ GHz frequency Naka (2016); HX (2016); Haigh (2016);
Usami (2018); ZhuNa (2020) and at low temperature of, e.g., 10 mK. However,
the optical pulses may heat the magnon modes due to the optical absorption of
the YIG, causing the magnon modes to be at a thermal state. To include this
heating effect in practical situations, we assume that the magnon modes are
initially prepared in a thermal state
$\rho_{th}=(1-s)\sum_{n=0}^{\infty}s^{n}\ket{n}\bra{n}$, with
$s=\bar{n}_{0}/(\bar{n}_{0}+1)$ and a thermal occupation $\bar{n}_{0}\ll 1$.
Note that the frequencies of the magnon modes are assumed to be (nearly)
identical and thus they have equal thermal occupation, i.e., they are in the
same thermal state. For a small $\bar{n}_{0}<0.2$, $s<0.17$, $s^{2}<0.03$ and
$s^{3}<0.005$, and the total probability of higher-excitation terms $\ket{n}$
($n>2$) is less than $0.5\%$. Thus we can safely approximate
$\rho_{th}\simeq(1-s)\left(\ket{0}\bra{0}+s\ket{1}\bra{1}+s^{2}\ket{2}\bra{2}\right)$.
The density matrix of the two magnon modes (in path A and B) in the
teleportation scheme is then

$\rho_{m}=\rho_{A}\otimes\rho_{B}\simeq(1-s)^{2}\sum_{n_{A},n_{B}=0}^{2}s^{n_{A}+n_{B}}\ket{n_{A}n_{B}}\bra{n_{A}n_{B}},$
(10)

which is a probabilistic mixture of nine pure states $\ket{n_{A}n_{B}}$
($n_{A},n_{B}=0,1,2$). This mixed initial state eventually leads to the
following teleported magnonic state after the Bell-state measurement
associated with $\ket{\phi^{+}}_{bc}$:

$\rho^{\prime}_{m}=(1-s)^{2}\sum_{n_{A},n_{B}=0}^{2}s^{n_{A}+n_{B}}\ket{\varphi_{n_{A}n_{B}}}\bra{\varphi_{n_{A}n_{B}}},$
(11)

where
$\ket{\varphi_{n_{A}n_{B}}}=\alpha\ket{n_{A},n_{B}+1}_{AB}+\beta\ket{n_{A}+1,n_{B}}_{AB}$
is the teleported magnonic state corresponding to the pure inital state
$\ket{n_{A}n_{B}}$ in Equation (10). Clearly,
$\ket{\varphi_{00}}=\alpha\ket{01}_{AB}+\beta\ket{10}_{AB}\equiv\alpha\ket{L}_{m}+\beta\ket{U}_{m}$
corresponds to the ideal initial state $\ket{00}_{AB}$ considered in Section
III. For the Bell-state detection related to $\ket{\phi^{-}}_{bc}$, we obtain
the same $\rho^{\prime}_{m}$ as in Equation (11) but with $\beta$ replaced by
$-\beta$ in $\ket{\varphi_{n_{A}n_{B}}}$. All other states in
$\rho^{\prime}_{m}$ are orthogonal to the desired state $\ket{\varphi_{00}}$,
and result in a reduction of the teleportation fidelity, which is

$\mathcal{F}_{1}=\bra{\varphi_{00}}\rho^{\prime}_{m}\ket{\varphi_{00}}=1/\big{(}1+s+s^{2}\big{)}^{2}.$
(12)

The solid line in Figure 4 clearly shows a declining fidelity versus the
thermal occupation $\bar{n}_{0}$. For a genuine quantum teleportation with the
fidelity $\mathcal{F}_{1}>2/3$ Pope , a small $\bar{n}_{0}\leq\,\sim\,$0.2 is
required. This is similar to the finding in the optomechanical teleportation
Li (2020).

Figure 4: Fidelity in the teleportation (solid) and entanglement swapping
(dashed) protocol versus the thermal occupation $\bar{n}_{0}$ of the magnon
modes.

By contrast, in the entanglement swapping protocol, four magnon modes (in path
A, B, C and D, respectively) are involved, cf. Figure 3. Following the same
approach, we obtain the final joint state of the magnonic systems, after the
Bell-state measurement associated with $\ket{\phi^{\pm}}_{b_{1}b_{2}}$, given
by

$\rho^{\prime}_{m_{1}m_{2}}=(1-s)^{4}\\!\\!\sum_{n_{A(B)}=0}^{2}\sum_{n_{C(D)}=0}^{2}s^{n_{A}+n_{B}}s^{n_{C}+n_{D}}\ket{\varphi_{m_{1}m_{2}}}\bra{\varphi_{m_{1}m_{2}}},$
(13)

where $A$, $B$, $C$ and $D$ are used to distinguish the magnon modes via their
path information, and
$\ket{\varphi_{m_{1}m_{2}}}=\frac{1}{\sqrt{2}}\left(\ket{n_{A},n_{B}+1,n_{C},n_{D}+1}_{m_{1}m_{2}}\pm\ket{n_{A}+1,n_{B},n_{C}+1,n_{D}}_{m_{1}m_{2}}\right)$
is the teleported magnonic state corresponding to the pure state
$\ket{n_{A}\,n_{B}\,n_{C}\,n_{D}}$ in the mixed initial state
$\rho_{m_{1}m_{2}}=\rho_{m_{1}}\otimes\rho_{m_{2}}$, cf. Equation (10). For
the ideal case of the initial ground state considered in Section IV, we obtain
the desired states
$\ket{\varphi_{m_{1}m_{2}}}=\frac{1}{\sqrt{2}}\left(\ket{0101}_{m_{1}m_{2}}\pm\ket{1010}_{m_{1}m_{2}}\right)\equiv\ket{\phi^{\pm}}_{m_{1}m_{2}}$
after the entanglement swapping. The other additional terms in
$\rho^{\prime}_{m_{1}m_{2}}$ are related to the residual thermal excitations
in the magnonic initial state, which are unwanted and reduce the fidelity in
the entanglement swapping protocol. The corresponding fidelity is given by

$\mathcal{F}_{2}\,{=}\,_{m_{2}m_{1}}\\!\\!\bra{\phi^{\pm}}\rho^{\prime}_{m_{1}m_{2}}\ket{\phi^{\pm}}_{m_{1}m_{2}}=1/\big{(}1+s+s^{2}\big{)}^{4}.$
(14)

The dashed line in Figure 4 shows a decreasing fidelity $\mathcal{F}_{2}$ as
the thermal occupation $\bar{n}_{0}$ increases. It reduces more rapidly with
respect to the fidelity $\mathcal{F}_{1}$ in the teleportation protocol
because of the relation $\mathcal{F}_{2}=\mathcal{F}_{1}^{2}$, as seen from
Equations (12) and (14).

##  VI Conclusion

We present two protocols for realizing optomagnonic quantum teleportation and
entanglement swapping, respectively, adopting YIG spheres and an optical
interferometer configuration. The optomagnonic Stokes and anti-Stokes
scatterings are the essential elements for preparing optomagnonic EPR states
and optically reading out the magnonic states. A Bell-state detection enables
the transfer of an arbitrary photonic qubit state to a dual-rail encoding
magnonic system in the former protocol, and the transfer of the optomagnonic
entanglement to the magnon modes in a dual-interferometer configuration which
are prepared in a Bell state in the latter protocol. We further discuss the
effect of the residual thermal excitations on the fidelity in both the
protocols. Our work suggests that optomagnonic systems could become a new
platform for realizing quantum teleportation and entanglement swapping where
quantum superposition and entangled states of macroscopic objects (e.g., YIG
spheres with diameter of hundreds of microns Naka (2016); HX (2016); Haigh
(2016); Usami (2018)) could be generated. The work may also find applications
in quantum information processing and hybrid quantum networks based on
magnonics.

## Acknowledgments

This research was funded by National Key Research and Development Program of
China (Grant no. 2022YFA1405200) and National Natural Science Foundation of
China (Grant no. 92265202).

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