# Practically Enhanced Hyperentanglement Concentration for Polarization-
spatial Hyperentangled Bell States with Linear Optics and Common Single-photon
Detectors

Gui-Long Jiang,1 Wen-Qiang Liu,2 and Hai-Rui Wei1<EMAIL_ADDRESS>1 School
of Mathematics and Physics, University of Science and Technology Beijing,
Beijing 100083, China
2 Center for Quantum Technology Research and Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurements (MOE), School of Physics,
Beijing Institute of Technology, Beijing 100081, China

(Received 22 September 2022; revised 10 December 2022; accepted 13 February
2023; published 14 March 2023)

###### Abstract

Hyperentanglement, defined as the simultaneous entanglement in several
independent degrees of freedom (DOFs) of a quantum system, is a fascinating
resource in quantum information processing with its outstanding merits. Here
we propose heralded hyperentanglement concentration protocols (hyper-ECPs) to
concentrate an unknown partially less polarization-spatial hyperentangled Bell
state with available linear optics and common single-photon detectors. By
introducing time-delay DOFs, the schemes are highly efficient in that the
success of the scheme can be accurately heralded by the detection signatures,
and postselection techniques or photon-number-resolving detectors, necessary
for previous experiments, are not required. Additionally, our linear optical
architectures allow certain states, where concentration fails, to be
recyclable, and a trick makes the success probabilities of our schemes higher
than those of previous linear optical hyper-ECPs.

###### pacs:

03.67.Hk, 03.65.Ud, 03.67.Mn, 03.67.Pp

##  I Introduction

Entanglement, as an extremely characteristic quantum mechanical phenomenon,
plays a significant part in quantum information processing and has been widely
applied to quantum communication and quantum computation 1 ; 2 . The entangled
photons, which can carry quantum information through various qubitlike degrees
of freedom (DOFs), have been recognized as the optimum candidates in long-
distance quantum communication tasks, such as quantum key distribution 3 ; 4 ;
5 , quantum teleportation 6 ; 7 , quantum dense coding 8 ; 9 , quantum secret
sharing 10 ; 11 ; 12 , and quantum secure direct communication 13 ; 14 ; 15 .
Encoding qubits in the polarization DOF of photons has been particularly
appealing, for their arbitrary single-qubit manipulation can be easily
achieved with two quarter-wave plates and one half-wave plate in an extremely
fast and accurate manner single-qubit . Recently, hyperentanglement, defined
as the simultaneous entanglement in multiple independent DOFs of a quantum
system, has captured much attention as it can considerably increase the
capacity of quantum communication and speed up quantum computation 17 ; 18 ;
momentum1 ; momentum2 ; OAM1 ; OAM2 ; OAM3 ; HPQC1 ; HPQC2 ; HPQC3 ; HBSA1 .
Diverse kinds of hyperentanglement have been experimentally demonstrated in
detail, e.g., polarization-spatial 17 , polarization-spatial-time 18 ,
polarization-momentum momentum1 ; momentum2 , and polarization-orbital-
angular-momentum OAM1 ; OAM2  DOFs.

The hyperentangled states can break the channel capacity limit of superdense
coding OAM3  and provide substantial applications in hyperparallel optical
quantum computing HPQC1 ; HPQC2 ; HPQC3  and hyperparallel communication, such
as hyperentanglement swapping HBSA1 ; HBSA2 , hyperentangled Bell states
analysis 17 ; HBSA1 ; HBSA2 ; HBSA3 ; HBSA4 ; HBSA-wei , and hyperentanglement
purification HEPP1 ; HEPP2 ; HEPP3 . Moreover, hyperentangled states can be
used to accomplish some specific tasks that are impossible to achieve in
single-DOF systems, such as complete and deterministic Bell states analysis
with linear optics momentum2 ; BSA1 ; BSA2 ; BSA3 . However, the entangled
photons will inevitably be affected by environment noise during transmission
and storage processes in long-distance quantum communication, resulting in a
great probability of degradation of the fidelity and security of protocols.
One effective method to prevent the degradation of entanglement in photon
systems is entanglement concentration protocol (ECP), which can distill
maximally entangled states from less-entangled pure states. Another approach
is entanglement purification protocol (EPP), which is applied to extract
maximally entangled states with high fidelity from a number of mixed entangled
states EPP1 ; EPP2 ; EPP3 ; EPP4 ; EPP5 . In 1996, Bennett _et al._ ECP1
proposed an ECP by using the Schmidt projection method, and various
interesting ECPs and improvements were later proposed and experimentally
demonstrated ECP2 ; ECP2-experiment1 ; ECP3 ; ECP4 ; ECP5 ; ECP6 ; ECP7 .

The current ECPs are mostly focused on single-DOF systems. In 2013, Ren _et
al._ hyper-ECP1  proposed an interesting polarization-spatial hyper-ECP for
known hyperentangled Bell states with linear optics by using the parameter-
splitting structure. In 2015, Li and Ghose hyper-ECP2  presented two hyper-
ECPs for hyperentangled states in the polarization and time-bin DOFs with
known and unknown parameters by using linear elements with similar methods.
Later, Ren and Wang _et al._ hyper-ECP3  designed polarization-spatial-time-
bin hyper-ECPs for hyperentangled Bell states and hyperentangled Greenberger-
Horne-Zeilinger (GHZ) states hyper-ECP4  with linear optical elements. In
2019, Li and Shen hyper-ECP02  reported asymmetrical hyper-ECPs for unknown
and known photon systems entangled in polarization and orbital angular
momentum DOFs. For less-entangled states with known parameters, parameter
splitting is the current optimal method for entanglement concentration with
linear optics hyper-ECP1 . While for less-entangled states with unknown
parameters, polarization beam splitters (PBSs) are usually employed to
complete parity-check measurement on the polarization photon pair hyper-ECP2 ;
hyper-ECP3 ; hyper-ECP4 ; hyper-ECP02 , and postselections or sophisticated
photon-number-resolving detectors are necessary to accomplish the scheme
exactly with linear optics. In addition, some deterministic hyper-ECPs were
proposed assisted by nonlinear platforms, such as cross-Kerr mediates hyper-
ECP5 , quantum dots hyper-ECP6 , or diamond nitrogen-vacancy centers hyper-
ECP7 ; hyper-ECP8 , and these methods are challenged by impracticality in
experiments with the current technique.

In this paper, we present two practical hyper-ECPs for partially
hyperentangled states with unknown parameters. A maximally polarization-
spatial-based hyperentangled Bell (GHZ) state is exactly extracted from a
partially less hyperentangled Bell (GHZ) state by using linear optics and
common single-photon detectors. Using the method of introducing temporal DOF
in Ref. BSA3 , we orchestrate unbalanced interferences that allow
postselection principles or sophisticated photon-number-resolving detectors to
be avoided and the success of the schemes to be completely heralded by the
detection signatures. As the heralded concentration equipment is composed only
of linear optical elements and has less apparatus than the previous schemes,
our schemes are quite applicable and practical with current technology.
Besides, the success probability of our hyper-ECPs can be higher than that of
the existing hyper-ECPs with linear optics.

##  II Hyper-ECP for photon systems in unknown hyperentangled pure states

In this section, we construct heralded hyper-ECPs for partially less
hyperentangled Bell states and GHZ states in both polarization and spatial
DOFs with unknown parameters by using linear optics. The basic principles of
our hyper-ECPs for the hyperentangled Bell and GHZ states are shown in Fig. 1
and Fig. 3, respectively.

###  II.1 Hyper-ECP for partially hyperentangled Bell states with unknown
parameters

Assume that two initial two-photon states $|\phi\rangle_{AB}$ and
$|\phi\rangle_{A^{\prime}B^{\prime}}$ partially entangled in polarization and
spatial DOFs simultaneously are generated from $S_{1}$ and $S_{2}$,
respectively. Here $|\phi\rangle_{AB}$ and
$|\phi\rangle_{A^{\prime}B^{\prime}}$ are given by the following forms:

$\displaystyle\begin{split}&|\phi\rangle_{AB}=\frac{1}{2}(|HH\rangle+|VV\rangle)\otimes(|a_{1}b_{1}\rangle+|a_{2}b_{2}\rangle),\\\
&|\phi\rangle_{A^{\prime}B^{\prime}}=\frac{1}{2}(|HH\rangle+|VV\rangle)\otimes(|a_{1^{\prime}}b_{1^{\prime}}\rangle+|a_{2^{\prime}}b_{2^{\prime}}\rangle).\end{split}$
(1)

where $H$ ($V$) denotes the horizontal (vertical) polarized photon. The
$|a_{1}\rangle$ and $|a_{2}\rangle$ ($|a_{1^{\prime}}\rangle$ and
$|a_{2^{\prime}}\rangle$) are two spatial modes of photon $A$ ($A^{\prime}$)
and the $|b_{1}\rangle$ and $|b_{2}\rangle$ ($|b_{1^{\prime}}\rangle$ and
$|b_{2^{\prime}}\rangle$) are those of photon $B$ ($B^{\prime}$). Then, an
initial four-photon state of the four-photon system consisting of photons $A$,
$B$, $A^{\prime}$, and $B^{\prime}$ is given by

$\displaystyle\begin{split}|\Phi_{0}\rangle=&|\phi\rangle_{AB}\otimes|\phi\rangle_{A^{\prime}B^{\prime}}\\\
=&\frac{1}{4}(|HH\rangle+|VV\rangle)\otimes(|HH\rangle+|VV\rangle)\\\
&\otimes(|a_{1}b_{1}\rangle+|a_{2}b_{2}\rangle)\otimes(|a_{1^{\prime}}b_{1^{\prime}}\rangle+|a_{2^{\prime}}b_{2^{\prime}}\rangle).\end{split}$
(2)

As shown in Fig. 1, photon $B$ in the spatial mode $|b_{1}\rangle$
($|b_{2}\rangle$) and photon $A^{\prime}$ in the spatial mode
$|a_{1^{\prime}}\rangle$ ($|a_{2^{\prime}}\rangle$) first pass through a 50:50
beam splitter BS1 (BS2), where the balanced BS1 or BS2 matrix is given by

$\displaystyle\text{BS}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\\\ \end{array}\right)$ (5)

in the spatial mode $\\{|b_{i}\rangle,|a_{i^{\prime}}\rangle\\}$ ($i=1,2$)
basis. After passing through BS1 and BS2, $|\Phi_{0}\rangle$ is transformed
into

$\displaystyle\begin{split}|\Phi_{1}\rangle=&\frac{1}{8}(|HH\rangle+|VV\rangle)\otimes(|HH\rangle+|VV\rangle)\\\
&\otimes[|a_{1}\rangle(|b_{1}\rangle+|a_{1^{\prime}}\rangle)+|a_{2}\rangle(|b_{2}\rangle+|a_{2^{\prime}}\rangle)]\\\
&\otimes[(|b_{1}\rangle-|a_{1^{\prime}}\rangle)|b_{1^{\prime}}\rangle+(|b_{2}\rangle-|a_{2^{\prime}}\rangle)|b_{2^{\prime}}\rangle].\end{split}$
(6)

Then, photons $A$ and $A^{\prime}$ ($B$ and $B^{\prime}$) are sent to Alice
(Bob), and owing to the influence of noise, $|\Phi_{1}\rangle$ may decay to a
partially less entangled state

$\displaystyle\begin{split}|\Phi_{2}\rangle=&\frac{1}{2}(\alpha|HH\rangle+\beta|VV\rangle)\otimes(\alpha|HH\rangle+\beta|VV\rangle)\\\
&\otimes[\gamma|a_{1}\rangle(|b_{1}\rangle+|a_{1^{\prime}}\rangle)+\delta|a_{2}\rangle(|b_{2}\rangle+|a_{2^{\prime}}\rangle)]\\\
&\otimes[\gamma(|b_{1}\rangle-|a_{1^{\prime}}\rangle)|b_{1^{\prime}}\rangle+\delta(|b_{2}\rangle-|a_{2^{\prime}}\rangle)|b_{2^{\prime}}\rangle].\end{split}$
(7)

Here coefficients satisfy the normalization conditions
$|\alpha|^{2}+|\beta|^{2}=1$ and $|\gamma|^{2}+|\delta|^{2}=1$, and they are
unknown to the parties of communication.

The purpose of the two distant parties is to distill the maximally
hyperentangled Bell state

$\displaystyle\begin{split}|\phi^{++}\rangle_{AB}=\frac{1}{2}(|HH\rangle+|VV\rangle)\otimes(|a_{1}b_{1}\rangle+|a_{2}b_{2}\rangle)\end{split}$
(8)

from $|\Phi_{2}\rangle$. For clarity of exposition of the working principles
of our ECP, base on the Hong-Ou-Mandel effect, we rewrite $|\Phi_{2}\rangle$
in the following normalized form:

$\displaystyle\begin{split}|\Phi_{2}\rangle=&\frac{1}{\sqrt{2}}(\alpha^{2}|HHHH\rangle+\beta^{2}|VVVV\rangle)\\\
&\otimes[\gamma^{2}|a_{1}b_{1^{\prime}}\rangle(|b_{1}b_{1}\rangle-|a_{1^{\prime}}a_{1^{\prime}}\rangle)\\\
&+\delta^{2}|a_{2}b_{2^{\prime}}\rangle(|b_{2}b_{2}\rangle-|a_{2^{\prime}}a_{2^{\prime}}\rangle)]\\\
&+\frac{\alpha\beta}{2}(|HVHV\rangle+|VHVH\rangle)\\\
&\otimes[\gamma^{2}|a_{1}b_{1^{\prime}}\rangle(|b_{1}b_{1}\rangle-|a_{1^{\prime}}a_{1^{\prime}}\rangle-|b_{1}a_{1^{\prime}}\rangle\\\
&+|a_{1^{\prime}}b_{1}\rangle)+\delta^{2}|a_{2}b_{2^{\prime}}\rangle(|b_{2}b_{2}\rangle-|a_{2^{\prime}}a_{2^{\prime}}\rangle\\\
&-|b_{2}a_{2^{\prime}}\rangle+|a_{2^{\prime}}b_{2}\rangle)]\\\
&+\frac{\gamma\delta}{2}(\alpha^{2}|HHHH\rangle+\beta^{2}|VVVV\rangle\\\
&+\alpha\beta|HVHV\rangle+\beta\alpha|VHVH\rangle)\\\
&\otimes[|a_{1}b_{2^{\prime}}\rangle(|b_{1}b_{2}\rangle-|a_{1^{\prime}}a_{2^{\prime}}\rangle-|b_{1}a_{2^{\prime}}\rangle+|a_{1^{\prime}}b_{2}\rangle)\\\
&+|a_{2}b_{1^{\prime}}\rangle(|b_{2}b_{1}\rangle-|a_{2^{\prime}}a_{1^{\prime}}\rangle-|b_{2}a_{1^{\prime}}\rangle+|a_{2^{\prime}}b_{1}\rangle)].\end{split}$
(9)

Figure 1: (Color online) Schematic diagram of the hyper-ECP for a
polarization-spatial-based hyperentangled Bell state with unknown parameters.
$S_{1}$ and $S_{2}$ are partial hyperentanglement sources for
$|\phi\rangle_{AB}$ and $|\phi\rangle_{A^{\prime}B^{\prime}}$, respectively.
$\textrm{PBS}_{i,i^{\prime}}(i=1,2,\cdots,6)$ represents a polarization beam
splitter, which transmits the horizontal polarization state $|H\rangle$ and
reflects the vertical polarization state $|V\rangle$.
$\textrm{HWP}^{45^{\circ}}$ represents a half-wave plate oriented at
$45^{\circ}$, which is used to exchange the vertical and horizontal
polarization, i.e., $|H\rangle\rightarrow|V\rangle$,
$|V\rangle\rightarrow|H\rangle$. $\textrm{HWP}^{22.5^{\circ}}_{i}(i=1,2)$ is
used to perform a Hadamard operation on polarization, i.e.,
$|H\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$,
$|V\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle)$.
$\textrm{BS}_{i,i^{\prime}}(i=1,2,\cdots,5)$ denotes the 50:50 beam splitter.
$D_{i}$ $(i=1,2,\cdots,8)$ represents the common single-photon detector.
$Z^{S}$ and $Z^{P}$ are the classical feed-forward operations, which can be
achieved by setting a PS and a $\textrm{HWP}^{0^{\circ}}$, respectively.

As shown in Fig. 1, Bob flips firstly the polarization and spatial states of
photon $B^{\prime}$, which causes $|\Phi_{2}\rangle$ to change into

$\displaystyle\begin{split}|\Phi_{3}\rangle=&\frac{1}{\sqrt{2}}(\alpha^{2}|HVHH\rangle+\beta^{2}|VHVV\rangle)\\\
&\otimes[\gamma^{2}|a_{1}b_{2^{\prime}}\rangle(|b_{1}b_{1}\rangle-|a_{1^{\prime}}a_{1^{\prime}}\rangle)\\\
&+\delta^{2}|a_{2}b_{1^{\prime}}\rangle(|b_{2}b_{2}\rangle-|a_{2^{\prime}}a_{2^{\prime}}\rangle)]\\\
&+\frac{\alpha\beta}{2}(|HHHV\rangle+|VVVH\rangle)\\\
&\otimes[\gamma^{2}|a_{1}b_{2^{\prime}}\rangle(|b_{1}b_{1}\rangle-|a_{1^{\prime}}a_{1^{\prime}}\rangle-|b_{1}a_{1^{\prime}}\rangle\\\
&+|a_{1^{\prime}}b_{1}\rangle)+\delta^{2}|a_{2}b_{1^{\prime}}\rangle(|b_{2}b_{2}\rangle-|a_{2^{\prime}}a_{2^{\prime}}\rangle\\\
&-|b_{2}a_{2^{\prime}}\rangle+|a_{2^{\prime}}b_{2}\rangle)]\\\
&+\frac{\gamma\delta}{2}(\alpha^{2}|HVHH\rangle+\beta^{2}|VHVV\rangle\\\
&+\alpha\beta|HHHV\rangle+\beta\alpha|VVVH\rangle)\\\
&\otimes[|a_{1}b_{1^{\prime}}\rangle(|b_{1}b_{2}\rangle-|a_{1^{\prime}}a_{2^{\prime}}\rangle-|b_{1}a_{2^{\prime}}\rangle+|a_{1^{\prime}}b_{2}\rangle)\\\
&+|a_{2}b_{2^{\prime}}\rangle(|b_{2}b_{1}\rangle-|a_{2^{\prime}}a_{1^{\prime}}\rangle-|b_{2}a_{1^{\prime}}\rangle+|a_{2^{\prime}}b_{1}\rangle)],\end{split}$
(10)

where the polarization-based bit-flip operation is realized by two half-wave
plates oriented at $45^{\circ}$ (HWP${}^{45^{\circ}}$s), and the spatial-based
bit-flip operation is realized by the block composed of BS3, BS4, and one
phase shifter (PS). The PS and the HWP${}^{45^{\circ}}$ matrices are given by

$\displaystyle\text{PS}=\left(\begin{array}[]{cc}-1&0\\\ 0&-1\\\
\end{array}\right),\;\;\text{HWP}^{45^{\circ}}=\left(\begin{array}[]{cc}0&1\\\
1&0\\\ \end{array}\right),$ (15)

in the $\\{|H\rangle,|V\rangle\\}$ basis, respectively.

Secondly, as shown in Fig. 1, by using four unbalanced interferometers, each
consisting of two PBSs, Alice introduces time delays $t$ to
$|H\rangle|a_{1^{\prime}}\rangle$, $2t$ to $|V\rangle|a_{1^{\prime}}\rangle$,
$3t$ to $|V\rangle|a_{2^{\prime}}\rangle$, and $4t$ to
$|H\rangle|a_{2^{\prime}}\rangle$, respectively. While Bob introduces time-
delays $t$ to $|V\rangle|b_{2}\rangle$, $2t$ to $|H\rangle|b_{2}\rangle$, $3t$
to $|H\rangle|b_{1}\rangle$, and $4t$ to $|V\rangle|b_{1}\rangle$. The time
delays can be achieved by lengthening optical circuits, and we assume that
phase $\omega t=2n\pi$, where $n$ is the nonzero integer and the time for
photons to pass through the entire device is much less than the two-photon
coherence time. We can verify that after photons $A^{\prime}$ and $B$ are
emitted from the four interferometers, $|\Phi_{3}\rangle$ becomes

$\displaystyle|\Phi_{4}\rangle=|\Omega_{0}\rangle+|\Omega_{1}\rangle+|\Omega_{2}\rangle+|\Omega_{3}\rangle,$
(16)

where

$\displaystyle\begin{split}|\Omega_{0}\rangle=&\frac{\alpha\beta}{2}[\gamma^{2}(|HH\rangle|a_{1}b_{2^{\prime}}\rangle+|VV\rangle|a_{1}b_{2^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|b_{1}b_{1}\rangle|3t,4t\rangle-|HV\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle|t,2t\rangle)\\\
&+\delta^{2}(|HH\rangle|a_{2}b_{1^{\prime}}\rangle+|VV\rangle|a_{2}b_{1^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|b_{2}b_{2}\rangle|2t,t\rangle-|HV\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle|4t,3t\rangle)\\\
&+\gamma^{2}(|HH\rangle|a_{1}b_{2^{\prime}}\rangle-|VV\rangle|a_{1}b_{2^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|a_{1^{\prime}}b_{1}\rangle|t,4t\rangle-|HV\rangle|b_{1}a_{1^{\prime}}\rangle|3t,2t\rangle)\\\
&+\delta^{2}(|HH\rangle|a_{2}b_{1^{\prime}}\rangle-|VV\rangle|a_{2}b_{1^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|a_{2^{\prime}}b_{2}\rangle|4t,t\rangle-|HV\rangle|b_{2}a_{2^{\prime}}\rangle|2t,3t\rangle)],\end{split}$
(17)
$\displaystyle\begin{split}|\Omega_{1}\rangle=&\frac{\gamma\delta}{2}[\alpha^{2}(|HV\rangle|a_{1}b_{1^{\prime}}\rangle+|HV\rangle|a_{2}b_{2^{\prime}}\rangle)\\\
&\otimes(|HH\rangle|b_{1}b_{2}\rangle|3t,2t\rangle-|HH\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle|t,4t\rangle)\\\
&+\alpha^{2}(|HV\rangle|a_{1}b_{1^{\prime}}\rangle-|HV\rangle|a_{2}b_{2^{\prime}}\rangle)\\\
&\otimes(|HH\rangle|a_{1^{\prime}}b_{2}\rangle|t,2t\rangle-|HH\rangle|b_{1}a_{2^{\prime}}\rangle|3t,4t\rangle)\\\
&+\beta^{2}(|VH\rangle|a_{1}b_{1^{\prime}}\rangle+|VH\rangle|a_{2}b_{2^{\prime}}\rangle)\\\
&\otimes(|VV\rangle|b_{1}b_{2}\rangle|4t,t\rangle-|VV\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle|2t,3t\rangle)\\\
&+\beta^{2}(|VH\rangle|a_{1}b_{1^{\prime}}\rangle-|VH\rangle|a_{2}b_{2^{\prime}}\rangle)\\\
&\otimes(|VV\rangle|a_{1^{\prime}}b_{2}\rangle|2t,t\rangle-|VV\rangle|b_{1}a_{2^{\prime}}\rangle|4t,3t\rangle)],\end{split}$
(18)
$\displaystyle\begin{split}|\Omega_{2}\rangle=&\frac{\alpha\beta\gamma\delta}{2}[(|HH\rangle|a_{1}b_{1^{\prime}}\rangle+|VV\rangle|a_{2}b_{2^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|b_{1}b_{2}\rangle|3t,t\rangle-|HV\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle|t,3t\rangle)\\\
&+(|HH\rangle|a_{2}b_{2^{\prime}}\rangle+|VV\rangle|a_{1}b_{1^{\prime}}\rangle)\\\
&\otimes(|VH\rangle|b_{1}b_{2}\rangle|4t,2t\rangle-|VH\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle|2t,4t\rangle)\\\
&+(|HH\rangle|a_{1}b_{1^{\prime}}\rangle-|VV\rangle|a_{2}b_{2^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|a_{1^{\prime}}b_{2}\rangle|t,t\rangle-|HV\rangle|b_{1}a_{2^{\prime}}\rangle|3t,3t\rangle)\\\
&+(|HH\rangle|a_{2}b_{2^{\prime}}\rangle-|VV\rangle|a_{1}b_{1^{\prime}}\rangle)\\\
&\otimes(|HV\rangle|a_{2^{\prime}}b_{1}\rangle|4t,4t\rangle-|HV\rangle|b_{2}a_{1^{\prime}}\rangle|2t,2t\rangle)].\end{split}$
(19)
$\displaystyle\begin{split}|\Omega_{3}\rangle=&\frac{1}{\sqrt{2}}[\alpha^{2}\gamma^{2}|HV\rangle|a_{1}b_{2^{\prime}}\rangle\\\
&\otimes(|HH\rangle|b_{1}b_{1}\rangle|3t,3t\rangle-|HH\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle|t,t\rangle)\\\
&+\beta^{2}\gamma^{2}|VH\rangle|a_{1}b_{2^{\prime}}\rangle\\\
&\otimes(|VV\rangle|b_{1}b_{1}\rangle|4t,4t\rangle-|VV\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle|2t,2t\rangle)\\\
&+\alpha^{2}\delta^{2}|HV\rangle|a_{2}b_{1^{\prime}}\rangle\\\
&\otimes(|HH\rangle|b_{2}b_{2}\rangle|2t,2t\rangle-|HH\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle|4t,4t\rangle)\\\
&+\beta^{2}\delta^{2}|VH\rangle|a_{2}b_{1^{\prime}}\rangle\\\
&\otimes(|VV\rangle|b_{2}b_{2}\rangle|t,t\rangle-|VV\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle|3t,3t\rangle)],\end{split}$
(20)

Based on Eqs. (17-20), we can see that $|\Omega_{0}\rangle$ and
$|\Omega_{1}\rangle$ indicate the failure of the protocol, that is, the scheme
is terminated; $|\Omega_{2}\rangle$ has the potential to yield the maximal
hyperentangled Bell state since it has the same coefficient for each item,
whereas $|\Omega_{3}\rangle$ is promising for recycling to improve the success
probability as it can yield a partially hyperentangled Bell state hyper-ECP5 ;
hyper-ECP6 ; hyper-ECP7 . The rigorous argument is provided below.

Thirdly, Alice (Bob) performs spatial-based and polarization-based Hadamard
operations on photon $A^{\prime}$ ($B$) by $\text{BS}_{5^{\prime}}$ and
$\text{HWP}_{1^{\prime},2^{\prime}}^{22.5^{\circ}}$ ($\text{BS}_{5}$ and
$\text{HWP}_{1,2}^{22.5^{\circ}}$), respectively. Here the
$\text{HWP}^{22.5^{\circ}}$ matrix is given by

$\displaystyle\text{HWP}^{22.5^{\circ}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\\\ \end{array}\right),$ (23)

in the $\\{|H\rangle,|V\rangle\\}$ basis. Here, $\text{BS}_{5^{\prime}}$ and
$\text{BS}_{5}$ complete the transformations

$\displaystyle\begin{split}&|a_{1^{\prime}}\rangle\rightarrow\frac{1}{\sqrt{2}}(|a_{1^{\prime}}\rangle+|a_{2^{\prime}}\rangle),\;\;|b_{1}\rangle\rightarrow\frac{1}{\sqrt{2}}(|b_{1}\rangle+|b_{2}\rangle),\\\
&|a_{2^{\prime}}\rangle\rightarrow\frac{1}{\sqrt{2}}(|a_{1^{\prime}}\rangle-|a_{2^{\prime}}\rangle),\;\;|b_{2}\rangle\rightarrow\frac{1}{\sqrt{2}}(|b_{1}\rangle-|b_{2}\rangle).\end{split}$
(24)

For the desired state $|\Omega_{2}\rangle$ shown in Eq. (19), after photon
$A^{\prime}$ ($B$) passes through $\text{BS}_{5^{\prime}}$ ($\text{BS}_{5}$)
and $\text{HWP}_{1^{\prime},2^{\prime}}^{22.5^{\circ}}$
($\text{HWP}_{1,2}^{22.5^{\circ}}$) in succession, if both photons have the
same polarization, spatial mode, and time interval, they will interfere
constructively or destructively with each other, e.g.,
$|HH\rangle|a_{1^{\prime}}b_{1}\rangle|t,t\rangle-|HH\rangle|a_{1^{\prime}}b_{1}\rangle|3t,3t\rangle\rightarrow
0$,
$|HV\rangle|a_{1^{\prime}}b_{1}\rangle|t,t\rangle+|VH\rangle|b_{1}a_{1^{\prime}}\rangle|3t,3t\rangle\rightarrow
2|HV\rangle|a_{1^{\prime}}b_{1}\rangle D(0,0)$, and
$|HV\rangle|b_{1}b_{2}\rangle|3t,t\rangle+|HV\rangle|b_{1}b_{2}\rangle|4t,2t\rangle\rightarrow
2|HV\rangle|b_{1}b_{2}\rangle D(2t,0)$ HBSA-wei ; BSA3 . Here $D(2t,0)$
represents that the first photon has a relative time delay of $2t$ compared
with the second photon, while $D(0,0)$ means that the two photons have no
relative time delay. Therefore, Eq. (20) is converted into

$\displaystyle\begin{split}|\Omega_{2}^{\prime}\rangle=&\frac{\alpha\beta\gamma\delta}{4}[|\phi_{0}^{++}\rangle_{AB^{\prime}}\otimes(|HH\rangle|b_{1}b_{1}\rangle-|HH\rangle|b_{2}b_{2}\rangle\\\
&-|HH\rangle|b_{1}b_{2}\rangle+|HH\rangle|b_{2}b_{1}\rangle-|VV\rangle|b_{1}b_{1}\rangle\\\
&+|VV\rangle|b_{2}b_{2}\rangle+|VV\rangle|b_{1}b_{2}\rangle-|VV\rangle|b_{2}b_{1}\rangle\\\
&-|HH\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle+|HH\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle-|HH\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle\\\
&+|HH\rangle|a_{2^{\prime}}a_{1^{\prime}}\rangle+|VV\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle-|VV\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle\\\
&+|VV\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle-|VV\rangle|a_{2^{\prime}}a_{1^{\prime}}\rangle)D(2t,0)\\\
&+|\phi_{0}^{--}\rangle_{AB^{\prime}}\otimes(|VH\rangle|b_{1}b_{1}\rangle-|VH\rangle|b_{2}b_{2}\rangle\\\
&-|VH\rangle|b_{1}b_{2}\rangle+|VH\rangle|b_{2}b_{1}\rangle+|HV\rangle|b_{2}b_{2}\rangle\\\
&-|HV\rangle|b_{1}b_{1}\rangle+|HV\rangle|b_{1}b_{2}\rangle-|HV\rangle|b_{2}b_{1}\rangle\\\
&-|HV\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle+|HV\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle-|HV\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle\\\
&+|HV\rangle|a_{2^{\prime}}a_{1^{\prime}}\rangle+|VH\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle-|VH\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle\\\
&+|VH\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle-|VH\rangle|a_{2^{\prime}}a_{1^{\prime}}\rangle)D(2t,0)\\\
&+2|\phi_{0}^{+-}\rangle_{AB^{\prime}}\otimes(|HH\rangle|a_{2^{\prime}}b_{1}\rangle-|HH\rangle|a_{1^{\prime}}b_{2}\rangle\\\
&-|VV\rangle|a_{2^{\prime}}b_{1}\rangle+|VV\rangle|a_{1^{\prime}}b_{2}\rangle)D(0,0)\\\
&+2|\phi_{0}^{-+}\rangle_{AB^{\prime}}\otimes(|VH\rangle|a_{1^{\prime}}b_{1}\rangle-|HV\rangle|a_{1^{\prime}}b_{1}\rangle\\\
&-|VH\rangle|a_{2^{\prime}}b_{2}\rangle+|HV\rangle|a_{2^{\prime}}b_{2}\rangle)D(0,0)],\end{split}$
(25)

where

$\displaystyle\begin{split}&|\phi_{0}^{+\pm}\rangle_{AB^{\prime}}=\frac{1}{2}(|HH\rangle+|VV\rangle)\otimes(|a_{1}b_{1^{\prime}}\rangle\pm|a_{2}b_{2^{\prime}}\rangle),\\\
&|\phi_{0}^{\pm+}\rangle_{AB^{\prime}}=\frac{1}{2}(|HH\rangle\pm|VV\rangle)\otimes(|a_{1}b_{1^{\prime}}\rangle+|a_{2}b_{2^{\prime}}\rangle).\end{split}$
(26)

For the recycling state $|\Omega_{3}\rangle$ shown in Eq. (20), we can infer
that after the spatial-based and polarization-based Hadamard operations are
applied on photons $A^{\prime}$ and $B$, it is transformed into

$\displaystyle\begin{split}|\Omega_{3}^{\prime}\rangle=&\frac{\sqrt{(|\alpha|^{4}+|\beta|^{4})(|\gamma|^{4}+|\delta|^{4})}}{2\sqrt{14}}\\\
&\times[|\phi_{1}^{++}\rangle_{AB^{\prime}}\otimes(|HH\rangle|b_{1}b_{1}\rangle+|HH\rangle|b_{2}b_{2}\rangle\\\
&+|VV\rangle|b_{1}b_{1}\rangle+|VV\rangle|b_{2}b_{2}\rangle\\\
&-|HH\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle-|HH\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle\\\
&-|VV\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle-|VV\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle)D(0,0)\\\
&+2|\phi_{1}^{-+}\rangle_{AB^{\prime}}\otimes(|HV\rangle|b_{1}b_{1}\rangle+|HV\rangle|b_{2}b_{2}\rangle\\\
&-|HV\rangle|a_{1^{\prime}}a_{1^{\prime}}\rangle-|HV\rangle|a_{2^{\prime}}a_{2^{\prime}}\rangle)D(0,0)\\\
&+2|\phi_{1}^{+-}\rangle_{AB^{\prime}}\otimes(|HH\rangle|b_{1}b_{2}\rangle+|VV\rangle|b_{1}b_{2}\rangle\\\
&-|HH\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle-|VV\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle)D(0,0)\\\
&+2|\phi_{1}^{--}\rangle_{AB^{\prime}}\otimes(|HV\rangle|b_{1}b_{2}\rangle+|HV\rangle|b_{2}b_{1}\rangle\\\
&-|HV\rangle|a_{1^{\prime}}a_{2^{\prime}}\rangle-|HV\rangle|a_{2^{\prime}}a_{1^{\prime}}\rangle)D(0,0)],\end{split}$
(27)

where

$\displaystyle\begin{split}|\phi_{1}^{+\pm}\rangle_{AB^{\prime}}=&(\alpha^{\prime}|HV\rangle+\beta^{\prime}|VH\rangle)\otimes(\gamma^{\prime}|a_{1}b_{2^{\prime}}\rangle\\\
&\pm\delta^{\prime}|a_{2}b_{1^{\prime}}\rangle),\\\
|\phi_{1}^{\pm+}\rangle_{AB^{\prime}}=&(\alpha^{\prime}|HV\rangle\pm\beta^{\prime}|VH\rangle)\otimes(\gamma^{\prime}|a_{1}b_{2^{\prime}}\rangle\\\
&+\delta^{\prime}|a_{2}b_{1^{\prime}}\rangle),\end{split}$ (28)

with $\alpha^{\prime}=\frac{\alpha^{2}}{\sqrt{|\alpha|^{4}+|\beta|^{4}}}$,
$\beta^{\prime}=\frac{\beta^{2}}{\sqrt{|\alpha|^{4}+|\beta|^{4}}}$,
$\gamma^{\prime}=\frac{\gamma^{2}}{\sqrt{|\gamma|^{4}+|\delta|^{4}}}$, and
$\delta^{\prime}=\frac{\delta^{2}}{\sqrt{|\gamma|^{4}+|\delta|^{4}}}$.

Finally, the polarization and spatial information of photons $A^{\prime}$ and
$B$ are measured by the PBSs and the common single-photon detector $D_{i}$
$(i=1,2,\cdots,8)$.

For $|\Omega_{0}\rangle$ or $|\Omega_{1}\rangle$, after passing through
$\text{BS}_{5}$, $\text{BS}_{5^{\prime}}$, $\text{HWP}^{22.5^{\circ}}_{1,2}$,
and $\text{HWP}^{22.5^{\circ}}_{1^{\prime},2^{\prime}}$, this results in the
arbitrary single-photon pair $(D_{i},D_{j})$ $(i,j=1,2,\cdots,8)$ triggered
with the time interval of $t$ or $3t$, which means the scheme is failed.

For $|\Omega_{2}^{\prime}\rangle$, as shown by Tab. 1, this results in the
arbitrary single-photon pair $(D_{i},D_{j})$ fired with the time interval of
$2t$, or Alice and Bob both having one detector triggered without time
interval, and $|\Omega_{2}^{\prime}\rangle$ collapses to
$|\phi_{0}^{\pm\pm}\rangle_{AB^{\prime}}$. After some classical feed-forward
operations (see Tab. 1) are performed on photon $B^{\prime}$, Alice and Bob
obtain the desired maximal hyperentangled Bell state
$|\phi_{0}^{++}\rangle_{AB^{\prime}}$ with a success probability of
$P_{1}=4|\alpha\beta\gamma\delta|^{2}$.

As for $|\Omega_{3}^{\prime}\rangle$, this results in only one common single-
photon detector $D_{i}$ fired or two single-photon detectors from one side,
Alice or Bob, fired simultaneously. Meanwhile, $|\Omega_{2}^{\prime}\rangle$
collapses to one of the states given in Eq. (28). After implementing the
corresponding feed-forward operations on photon $B^{\prime}$, Alice and Bob
obtain the state $|\phi_{1}^{++}\rangle_{AB^{\prime}}$ with a probability of
$(|\alpha|^{4}+|\beta|^{4})(|\gamma|^{4}+|\delta|^{4})$.

Table 1: The relations between the detection signatures and the classical feed-forward operations to complete the hyper-ECP for hyperentangled Bell state. The spatial-based feed-forward operation $Z^{S}=|b_{1^{\prime}}\rangle\langle b_{1^{\prime}}|-|b_{2^{\prime}}\rangle\langle b_{2^{\prime}}|$ can be achieved by using a PS, and the polarization-based $Z^{P}=|H\rangle\langle H|-|V\rangle\langle V|$ can be accomplished by employing $\text{HWP}^{0^{\circ}}$. The superscript $2t$ on the detector $D_{i}$ indicates a relative time delay of 2t. $|\phi_{0}^{\pm\pm}\rangle_{AB^{\prime}}$ and $|\phi_{1}^{\pm\pm}\rangle_{AB^{\prime}}$ are the desired and recyclable outcomes, respectively. |  Single- |  Outcomes |  Feed- |  Success
---|---|---|---|---
Time |  photon |  of |  forward |  proba-
interval |  detectors |  $A$ and $B^{\prime}$ |  on $B^{\prime}$ |  bility
0 |  $D_{1},D_{2},D_{3},D_{4}$ |  $|\phi_{1}^{++}\rangle_{AB^{\prime}}$ |  none |
|  $D_{5},D_{6},D_{7},D_{8}$ |  |  |
|  $(D_{1},D_{2}),(D_{3},D_{4})$ |  $|\phi_{1}^{-+}\rangle_{AB^{\prime}}$ |  $Z^{P}$ |  $(|\alpha|^{4}$
|  $(D_{5},D_{6}),(D_{7},D_{8})$ |  |  |  $+|\beta|^{4})$
|  $(D_{1},D_{4}),(D_{2},D_{3})$ |  $|\phi_{1}^{+-}\rangle_{AB^{\prime}}$ |  $Z^{S}$ |  $\times(|\gamma|^{4}$
|  $(D_{5},D_{8}),(D_{6},D_{7})$ |  |  |  $+|\delta|^{4})$
|  $(D_{1},D_{3}),(D_{2},D_{4})$ |  $|\phi_{1}^{--}\rangle_{AB^{\prime}}$ |  $Z^{S},Z^{P}$ |
|  $(D_{5},D_{7}),(D_{6},D_{8})$ |  |  |
0 |  $(D_{1},D_{8}),(D_{2},D_{7})$ |  $|\phi_{0}^{+-}\rangle_{AB^{\prime}}$ |  $Z^{S}$ |  $4|\alpha\beta\gamma\delta|^{2}$
|  $(D_{3},D_{6}),(D_{4},D_{5})$ |  |  |
|  $(D_{1},D_{6}),(D_{2},D_{5})$ |  $|\phi_{0}^{-+}\rangle_{AB^{\prime}}$ |  $Z^{P}$ |
|  $(D_{3},D_{8}),(D_{4},D_{7})$ |  |  |
$2t$ |  $(D_{1}^{2t},D_{2}),(D_{1},D_{2}^{2t})$ |  $|\phi_{0}^{--}\rangle_{AB^{\prime}}$ |  $Z^{S},Z^{P}$ |
|  $(D_{3}^{2t},D_{4}),(D_{3},D_{4}^{2t})$ |  |  |
|  $(D_{1}^{2t},D_{3}),(D_{1},D_{3}^{2t})$ |  |  |
|  $(D_{2}^{2t},D_{4}),(D_{2},D_{4}^{2t})$ |  |  |
|  $(D_{5}^{2t},D_{6}),(D_{5},D_{6}^{2t})$ |  |  |
|  $(D_{7}^{2t},D_{8}),(D_{7},D_{8}^{2t})$ |  |  |
|  $(D_{5}^{2t},D_{7}),(D_{5},D_{7}^{2t})$ |  |  |
|  $(D_{6}^{2t},D_{8}),(D_{6},D_{8}^{2t})$ |  |  |
$2t$ |  $(D_{1}^{2t},D_{1}),(D_{2}^{2t},D_{2})$ |  $|\phi_{0}^{++}\rangle_{AB^{\prime}}$ |  none |
|  $(D_{3}^{2t},D_{3}),(D_{4}^{2t},D_{4})$ |  |  |
|  $(D_{5}^{2t},D_{5}),(D_{6}^{2t},D_{6})$ |  |  |
|  $(D_{7}^{2t},D_{7}),(D_{8}^{2t},D_{8})$ |  |  |
|  $(D_{1}^{2t},D_{4}),(D_{1},D_{4}^{2t})$ |  |  |
|  $(D_{2}^{2t},D_{3}),(D_{2},D_{3}^{2t})$ |  |  |
|  $(D_{5}^{2t},D_{8}),(D_{5},D_{8}^{2t})$ |  |  |
|  $(D_{6}^{2t},D_{7}),(D_{6},D_{7}^{2t})$ |  |  |

After flipping the polarization and spatial mode of photon $B^{\prime}$ of
state $|\phi_{1}^{++}\rangle_{AB^{\prime}}$, it is transformed into

$\displaystyle\begin{split}|\phi_{2}^{++}\rangle_{AB^{\prime}}=&(\alpha^{\prime}|HH\rangle+\beta^{\prime}|VV\rangle)\otimes(\gamma^{\prime}|a_{1}b_{1^{\prime}}\rangle\\\
&+\delta^{\prime}|a_{2}b_{2^{\prime}}\rangle).\end{split}$ (29)

It is obvious that Eq. (29) has a similar form to a partially hyperentangled
Bell state
$(\alpha|HH\rangle+\beta|VV\rangle)\otimes(\gamma|a_{1}b_{1}\rangle+\delta|a_{2}b_{2}\rangle)$
as described in Ref. hyper-ECP1 . Then, applying the method in Ref. hyper-ECP1
, Alice and Bob can further extract the maximal hyperentangled Bell state
$|\phi_{0}^{++}\rangle_{AB^{\prime}}$ from
$|\phi_{2}^{++}\rangle_{AB^{\prime}}$ with a success probability of
$4|\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}|^{2}$.
Therefore, this improves the total success probability of our hyper-ECP from
$P_{1}$ to

$\displaystyle\begin{split}P_{2}&=P_{1}+4|\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}|^{2}(|\alpha|^{4}+|\beta|^{4})(|\gamma|^{4}+|\delta|^{4})\\\
&=4|\alpha\beta\gamma\delta|^{2}+\frac{4|\alpha\beta\gamma\delta|^{4}}{(|\alpha|^{4}+|\beta|^{4})(|\gamma|^{4}+|\delta|^{4})}.\end{split}$
(30)

Figure 2: The total success probability of the presented hyper-Bell state
concentration protocol (a) without and (b) with recycling state
$|\phi_{1}^{++}\rangle_{AB^{\prime}}$.

Figure 2 plots the success probabilities of $P_{1}$ and $P_{2}$ as a function
of $|\alpha|^{2}$ and $|\gamma|^{2}$, where
$|\alpha|^{2}=1-|\beta|^{2}\in(0,0.5]$ and
$|\gamma|^{2}=1-|\delta|^{2}\in(0,0.5]$ are taken. As shown in Fig. 2,
$P_{1}=0.25$ without recycling $|\phi_{1}^{++}\rangle_{AB^{\prime}}$, and it
is equal to the result shown in Ref. hyper-ECP1 . However, after concentrating
state $|\phi_{1}^{++}\rangle_{AB^{\prime}}$ again, as shown in Fig. 2, $P_{1}$
can be increased to $P_{2}=0.3125$ in principle.

###  II.2 Hyper-ECP for partially hyperentangled GHZ states with unknown
parameters

Suppose two maximally hyperentangled GHZ states from $S_{1}$ and $S_{2}$ are
given by

$\displaystyle\begin{split}|\psi\rangle_{ABC}=&\frac{1}{2}(|HHH\rangle+|VVV\rangle)\\\
&\otimes(|a_{1}b_{1}c_{1}\rangle+|a_{2}b_{2}c_{2}\rangle),\\\
|\psi\rangle_{A^{\prime}B^{\prime}C^{\prime}}=&\frac{1}{2}(|HHH\rangle+|VVV\rangle)\\\
&\otimes(|a_{1^{\prime}}b_{1^{\prime}}c_{1^{\prime}}\rangle+|a_{2^{\prime}}b_{2^{\prime}}c_{2^{\prime}}\rangle).\end{split}$
(31)

The $|a_{k}\rangle$ ($|a_{k^{\prime}}\rangle$), $|b_{k}\rangle$
($|b_{k^{\prime}}\rangle$), and $|c_{k}\rangle$ ($|c_{k^{\prime}}\rangle$),
$k=1,2$, are spatial modes of photons $A$ ($A^{\prime}$), $B$ ($B^{\prime}$),
and $C$ ($C^{\prime}$), respectively. As shown in Fig. 3, after photon $B$ in
the spatial mode $|b_{1}\rangle$ ($|b_{2}\rangle$) and photon $A^{\prime}$ in
the spatial mode $|a_{1^{\prime}}\rangle$ ($|a_{2^{\prime}}\rangle$) pass
through BS, photons $A$, $B$, and $C$ are distributed to three distant parties
Alice, Bob, and Charlie, respectively. Due to the operations of BSs and noisy
channels, the six-photon system composed of photons $A$, $B$, $C$,
$A^{\prime}$, $B^{\prime}$, and $C^{\prime}$ is given by

$\displaystyle\begin{split}|\Psi_{0}\rangle=&\frac{1}{\sqrt{2}}(\alpha^{2}|HHHHHH\rangle+\beta^{2}|VVVVVV\rangle)\\\
&\otimes[\gamma^{2}|a_{1}b_{1^{\prime}}c_{1}\rangle(|b_{1}b_{1}\rangle-|a_{1^{\prime}}a_{1^{\prime}}\rangle)|c_{1^{\prime}}\rangle\\\
&+\delta^{2}|a_{2}b_{2^{\prime}}c_{2}\rangle(|b_{2}b_{2}\rangle-|a_{2^{\prime}}a_{2^{\prime}}\rangle)|c_{2^{\prime}}\rangle]\\\
&+\frac{\alpha\beta}{2}(|HVHVHV\rangle+|VHVHVH\rangle)\\\
&\otimes[\gamma^{2}|a_{1}b_{1^{\prime}}c_{2}\rangle(|b_{1}b_{1}\rangle-|a_{1^{\prime}}a_{1^{\prime}}\rangle-|b_{1}a_{1^{\prime}}\rangle\\\
&+|a_{1^{\prime}}b_{1}\rangle)|c_{1^{\prime}}\rangle+\delta^{2}|a_{2}b_{2^{\prime}}c_{2}\rangle(|b_{2}b_{2}\rangle\\\
&-|a_{2^{\prime}}a_{2^{\prime}}\rangle-|b_{2}a_{2^{\prime}}\rangle+|a_{2^{\prime}}b_{2}\rangle)|c_{2^{\prime}}\rangle]\\\
&+\frac{\gamma\delta}{2}(\alpha^{2}|HHHHHH\rangle+\beta^{2}|VVVVVV\rangle\\\
&+\alpha\beta|HVHVHV\rangle+\beta\alpha|VHVHVH\rangle)\\\
&\otimes[|a_{1}b_{2^{\prime}}c_{1}\rangle(|b_{1}b_{2}\rangle-|a_{1^{\prime}}a_{2^{\prime}}\rangle-|b_{1}a_{2^{\prime}}\rangle\\\
&+|a_{1^{\prime}}b_{2}\rangle)|c_{1^{\prime}}\rangle+|a_{2}b_{1^{\prime}}c_{2}\rangle(|b_{2}b_{1}\rangle\\\
&-|a_{2^{\prime}}a_{1^{\prime}}\rangle-|b_{2}a_{1^{\prime}}\rangle+|a_{2^{\prime}}b_{1}\rangle)|c_{2^{\prime}}\rangle].\end{split}$
(32)

The architecture of distilling the maximally hyperentangled GHZ states from
$|\Psi_{0}\rangle$ is shown in Fig. 3, where Alice and Bob perform the same
operations as in Fig. 1. After Alice, Bob, and Charlie have completed
concentration operations, similar to that as described in Sec. II, the
relationship between the output states and the corresponding detectors
triggered is shown in Tab. 2. Here, these states in Tab. 2 are as follows:

$\displaystyle\begin{split}|\psi^{+\pm}_{0}\rangle_{AB^{\prime}C}=&\frac{1}{2}(|HHH\rangle+|VVV\rangle)\\\
&\otimes(|a_{1}b_{1^{\prime}}c_{1}\rangle\pm|a_{2}b_{2^{\prime}}c_{2}\rangle),\\\
|\psi^{\pm+}_{0}\rangle_{AB^{\prime}C}=&\frac{1}{2}(|HHH\rangle\pm|VVV\rangle)\\\
&\otimes(|a_{1}b_{1^{\prime}}c_{1}\rangle+|a_{2}b_{2^{\prime}}c_{2}\rangle).\\\
|\psi^{+\pm}_{1}\rangle_{AB^{\prime}C}=&(\alpha^{\prime}|HVH\rangle+\beta^{\prime}|VHV\rangle)\\\
&\otimes(\gamma^{\prime}|a_{1}b_{2^{\prime}}c_{1}\rangle\pm\delta^{\prime}|a_{2}b_{1^{\prime}}c_{2}\rangle),\\\
|\psi^{\pm+}_{1}\rangle_{AB^{\prime}C}=&(\alpha^{\prime}|HVH\rangle\pm\beta^{\prime}|VHV\rangle)\\\
&\otimes(\gamma^{\prime}|a_{1}b_{2^{\prime}}c_{1}\rangle+\delta^{\prime}|a_{2}b_{1^{\prime}}c_{2}\rangle).\end{split}$
(33)

Table 2: The relations between the detection signatures, the output states, and the classical feed-forward operations to complete the hyper-ECP for a hyperentangled GHZ state with unknown parameters. $|\psi_{0}^{\pm\pm}\rangle_{AB^{\prime}C}$ and $|\psi_{1}^{\pm\pm}\rangle_{AB^{\prime}C}$ are the desired and recyclable outcomes, respectively. Detectors | Detectors | Outcomes of | Feed-forward
---|---|---|---
of Charlie | of Alice and Bob |  $A$, $B^{\prime}$, and $C$ | on $B^{\prime}$
$D_{H1}$ | $D_{1},\,\,\,D_{2},\,\,\,D_{3},\,\,\,D_{4},\,\,\,D_{5},\,\,\,D_{6},\,\,\,D_{7},\,\,\,D_{8}$ | $|\psi_{1}^{++}\rangle_{AB^{\prime}C}$ | none
$D_{H2}$ | $(D_{2},D_{3}),\,(D_{1},D_{4}),\,(D_{6},D_{7}),\,(D_{5},D_{8})$ |  |
$D_{V1}$ | $(D_{1},D_{2}),\,(D_{3},D_{4}),\,(D_{5},D_{6}),\,(D_{7},D_{8})$ |  |
$D_{V2}$ | $(D_{1},D_{3}),\,(D_{2},D_{4}),\,(D_{5},D_{7}),\,(D_{6},D_{8})$ |  |
$D_{H2}$ | $D_{1},\,\,\,D_{2},\,\,\,D_{3},\,\,\,D_{4},\,\,\,D_{5},\,\,\,D_{6},\,\,\,D_{7},\,\,\,D_{8}$ | $|\psi_{1}^{+-}\rangle_{AB^{\prime}C}$ | $Z^{S}$
$D_{H1}$ | $(D_{2},D_{3}),\,(D_{1},D_{4}),\,(D_{6},D_{7}),\,(D_{5},D_{8})$ |  |
$D_{V2}$ | $(D_{1},D_{2}),\,(D_{3},D_{4}),\,(D_{5},D_{6}),\,(D_{7},D_{8})$ |  |
$D_{V1}$ | $(D_{1},D_{3}),\,(D_{2},D_{4}),\,(D_{5},D_{7}),\,(D_{6},D_{8})$ |  |
$D_{V1}$ | $D_{1},\,\,\,D_{2},\,\,\,D_{3},\,\,\,D_{4},\,\,\,D_{5},\,\,\,D_{6},\,\,\,D_{7},\,\,\,D_{8}$ | $|\psi_{1}^{-+}\rangle_{AB^{\prime}C}$ | $Z^{P}$
$D_{V2}$ | $(D_{2},D_{3}),\,(D_{1},D_{4}),\,(D_{6},D_{7}),\,(D_{5},D_{8})$ |  |
$D_{H1}$ | $(D_{1},D_{2}),\,(D_{3},D_{4}),\,(D_{5},D_{6}),\,(D_{7},D_{8})$ |  |
$D_{H2}$ | $(D_{1},D_{3}),\,(D_{2},D_{4}),\,(D_{5},D_{7}),\,(D_{6},D_{8})$ |  |
$D_{V2}$ | $D_{1},\,\,\,D_{2},\,\,\,D_{3},\,\,\,D_{4},\,\,\,D_{5},\,\,\,D_{6},\,\,\,D_{7},\,\,\,D_{8}$ | $|\psi_{1}^{--}\rangle_{AB^{\prime}C}$ | $Z^{S},Z^{P}$
$D_{V1}$ | $(D_{2},D_{3}),\,(D_{1},D_{4}),\,(D_{6},D_{7}),\,(D_{5},D_{8})$ |  |
$D_{H2}$ | $(D_{1},D_{2}),\,(D_{3},D_{4}),\,(D_{5},D_{6}),\,(D_{7},D_{8})$ |  |
$D_{H1}$ | $(D_{1},D_{3}),\,(D_{2},D_{4}),\,(D_{5},D_{7}),\,(D_{6},D_{8})$ |  |
$D_{H2}$ | $(D_{1},D_{8}),\,(D_{2},D_{7}),\,(D_{3},D_{6}),\,(D_{4},D_{5})$ | $|\psi_{0}^{++}\rangle_{AB^{\prime}C}$ | none
$D_{V1}$ | $(D_{1},D_{6}),\,(D_{2},D_{5}),\,(D_{3},D_{8}),\,(D_{4},D_{7})$ |  |
$D_{H1}$ | $(D_{1}^{2t},D_{1}),\,(D_{2}^{2t},D_{2}),\,(D_{3}^{2t},D_{3}),\,(D_{4}^{2t},D_{4}),\,(D_{1}^{2t},D_{4}),\,(D_{2}^{2t},D_{3}),\,(D_{1},D_{4}^{2t}),\,(D_{2},D_{3}^{2t})$ |  |
| $(D_{5}^{2t},D_{5}),\,(D_{6}^{2t},D_{6}),\,(D_{7}^{2t},D_{7}),\,(D_{8}^{2t},D_{8}),\,(D_{5}^{2t},D_{8}),\,(D_{6}^{2t},D_{7}),\,(D_{5},D_{8}^{2t}),\,(D_{6},D_{7}^{2t})$ |  |
$D_{V2}$ | $(D_{1}^{2t},D_{2}),\,(D_{3}^{2t},D_{4}),\,(D_{1}^{2t},D_{3}),\,(D_{2}^{2t},D_{4}),\,(D_{1},D_{2}^{2t}),\,(D_{3},D_{4}^{2t}),\,(D_{1},D_{3}^{2t}),\,(D_{2},D_{4}^{2t})$ |  |
| $(D_{5}^{2t},D_{6}),\,(D_{7}^{2t},D_{8}),\,(D_{5}^{2t},D_{7}),\,(D_{6}^{2t},D_{8}),\,(D_{5},D_{6}^{2t}),\,(D_{7},D_{8}^{2t}),\,(D_{5},D_{7}^{2t}),\,(D_{6},D_{8}^{2t})$ |  |
$D_{H1}$ | $(D_{1},D_{8}),\,(D_{2},D_{7}),\,(D_{3},D_{6}),\,(D_{4},D_{5})$ | $|\psi_{0}^{+-}\rangle_{AB^{\prime}C}$ | $Z^{S}$
$D_{V2}$ | $(D_{1},D_{6}),\,(D_{2},D_{5}),\,(D_{3},D_{8}),\,(D_{4},D_{7})$ |  |
$D_{H2}$ | $(D_{1}^{2t},D_{1}),\,(D_{2}^{2t},D_{2}),\,(D_{3}^{2t},D_{3}),\,(D_{4}^{2t},D_{4}),\,(D_{1}^{2t},D_{4}),\,(D_{2}^{2t},D_{3}),\,(D_{1},D_{4}^{2t}),\,(D_{2},D_{3}^{2t})$ |  |
| $(D_{5}^{2t},D_{5}),\,(D_{6}^{2t},D_{6}),\,(D_{7}^{2t},D_{7}),\,(D_{8}^{2t},D_{8}),\,(D_{5}^{2t},D_{8}),\,(D_{6}^{2t},D_{7}),\,(D_{5},D_{8}^{2t}),\,(D_{6},D_{7}^{2t})$ |  |
$D_{V1}$ | $(D_{1}^{2t},D_{2}),\,(D_{3}^{2t},D_{4}),\,(D_{1}^{2t},D_{3}),\,(D_{2}^{2t},D_{4}),\,(D_{1},D_{2}^{2t}),\,(D_{3},D_{4}^{2t}),\,(D_{1},D_{3}^{2t}),\,(D_{2},D_{4}^{2t})$ |  |
| $(D_{5}^{2t},D_{6}),\,(D_{7}^{2t},D_{8}),\,(D_{5}^{2t},D_{7}),\,(D_{6}^{2t},D_{8}),\,(D_{5},D_{6}^{2t}),\,(D_{7},D_{8}^{2t}),\,(D_{5},D_{7}^{2t}),\,(D_{6},D_{8}^{2t})$ |  |
$D_{V2}$ | $(D_{1},D_{8}),\,(D_{2},D_{7}),\,(D_{3},D_{6}),\,(D_{4},D_{5})$ | $|\psi_{0}^{-+}\rangle_{AB^{\prime}C}$ | $Z^{P}$
$D_{H1}$ | $(D_{1},D_{6}),\,(D_{2},D_{5}),\,(D_{3},D_{8}),\,(D_{4},D_{7})$ |  |
$D_{V1}$ | $(D_{1}^{2t},D_{1}),\,(D_{2}^{2t},D_{2}),\,(D_{3}^{2t},D_{3}),\,(D_{4}^{2t},D_{4}),\,(D_{1}^{2t},D_{4}),\,(D_{2}^{2t},D_{3}),\,(D_{1},D_{4}^{2t}),\,(D_{2},D_{3}^{2t})$ |  |
| $(D_{5}^{2t},D_{5}),\,(D_{6}^{2t},D_{6}),\,(D_{7}^{2t},D_{7}),\,(D_{8}^{2t},D_{8}),\,(D_{5}^{2t},D_{8}),\,(D_{6}^{2t},D_{7}),\,(D_{5},D_{8}^{2t}),\,(D_{6},D_{7}^{2t})$ |  |
$D_{H2}$ | $(D_{1}^{2t},D_{2}),\,(D_{3}^{2t},D_{4}),\,(D_{1}^{2t},D_{3}),\,(D_{2}^{2t},D_{4}),\,(D_{1},D_{2}^{2t}),\,(D_{3},D_{4}^{2t}),\,(D_{1},D_{3}^{2t}),\,(D_{2},D_{4}^{2t})$ |  |
| $(D_{5}^{2t},D_{6}),\,(D_{7}^{2t},D_{8}),\,(D_{5}^{2t},D_{7}),\,(D_{6}^{2t},D_{8}),\,(D_{5},D_{6}^{2t}),\,(D_{7},D_{8}^{2t}),\,(D_{5},D_{7}^{2t}),\,(D_{6},D_{8}^{2t})$ |  |
$D_{V1}$ | $(D_{1},D_{8}),\,(D_{2},D_{7}),\,(D_{3},D_{6}),\,(D_{4},D_{5})$ | $|\psi_{0}^{--}\rangle_{AB^{\prime}C}$ | $Z^{S},Z^{P}$
$D_{H2}$ | $(D_{1},D_{6}),\,(D_{2},D_{5}),\,(D_{3},D_{8}),\,(D_{4},D_{7})$ |  |
$D_{V2}$ | $(D_{1}^{2t},D_{1}),\,(D_{2}^{2t},D_{2}),\,(D_{3}^{2t},D_{3}),\,(D_{4}^{2t},D_{4}),\,(D_{1}^{2t},D_{4}),\,(D_{2}^{2t},D_{3}),\,(D_{1},D_{4}^{2t}),\,(D_{2},D_{3}^{2t})$ |  |
| $(D_{5}^{2t},D_{5}),\,(D_{6}^{2t},D_{6}),\,(D_{7}^{2t},D_{7}),\,(D_{8}^{2t},D_{8}),\,(D_{5}^{2t},D_{8}),\,(D_{6}^{2t},D_{7}),\,(D_{5},D_{8}^{2t}),\,(D_{6},D_{7}^{2t})$ |  |
$D_{H1}$ | $(D_{1}^{2t},D_{2}),\,(D_{3}^{2t},D_{4}),\,(D_{1}^{2t},D_{3}),\,(D_{2}^{2t},D_{4}),\,(D_{1},D_{2}^{2t}),\,(D_{3},D_{4}^{2t}),\,(D_{1},D_{3}^{2t}),\,(D_{2},D_{4}^{2t})$ |  |
| $(D_{5}^{2t},D_{6}),\,(D_{7}^{2t},D_{8}),\,(D_{5}^{2t},D_{7}),\,(D_{6}^{2t},D_{8}),\,(D_{5},D_{6}^{2t}),\,(D_{7},D_{8}^{2t}),\,(D_{5},D_{7}^{2t}),\,(D_{6},D_{8}^{2t})$ |  |

After performing the feed-forward operations on photon $B^{\prime}$ shown in
Tab. 2, Alice, Bob, and Charlie can obtain the maximally hyperentangled GHZ
state $|\psi^{++}_{0}\rangle_{AB^{\prime}C}$ with a success probability of
$4|\alpha\beta\gamma\delta|^{2}$ or a recyclable hyperentangled state
$|\psi^{++}_{1}\rangle_{AB^{\prime}C}$ with a probability of
$(|\alpha|^{4}+|\beta|^{4})(|\gamma|^{4}+|\delta|^{4})$. It is easy to verify
that the total success probability of the scheme shown in Fig. 3 is equal to
that of the one shown in Fig. 2 by recycling
$|\psi^{++}_{1}\rangle_{AB^{\prime}C}$. Additionally, as long as the arbitrary
single-photon pair $(D_{i},D_{j})$ $(i,j=1,2,\cdots,8)$ belonging to Alice and
Bob is triggered with a time interval of $t$ or $3t$, the concentration
process fails and is terminated, regardless of the detection signatures of
Charlie.

Figure 3: (Color online) Schematic diagram of the hyper-ECP for a
hyperentangled GHZ state with unknown parameters. $S_{1}$ and $S_{2}$ are
partial hyperentanglement sources for $|\psi\rangle_{ABC}$ and
$|\psi\rangle_{A^{\prime}B^{\prime}C^{\prime}}$, respectively.

##  III Discussion and Summary

Hyperentanglement concentration is one of the methods to decrease the
influence of channel noise in high-capacity long-distance quantum
communication. ECP or hyper-ECP is used to deal with the case where the pure
state remains after being subjected to noisy channels. In practice, when
affected by noise, the input state is not necessarily still pure but may be a
mixed state, in which case EPPs are required. In this article, we design a
hyper-ECP for unknown hyperentangled Bell states in both the polarization and
spatial DOFs that can be unambiguously heralded by the detection signatures.
Remarkably, this heralded hyper-ECP can be extended to concentrate the GHZ
state case.

In the constructions of previous hyper-ECPs using linear optics, PBSs are
employed to complete the parity-check measurement of polarization photon pairs
hyper-ECP1 ; hyper-ECP2 ; hyper-ECP3 ; hyper-ECP4 . Meanwhile, postselection
is necessary to explicitly identify the instances where each of the spatial
modes contains exactly one photon without destruction of the incident photon,
which leads to schemes that cannot be accomplished simply with linear optical
elements. We avoid postselections or sophisticated photon-number-resolving
detectors by introducing time-delay DOF as in Ref. BSA3 , and by the detection
signatures of the common single-photon detector pair $(D_{i},D_{j})$, the
instances $\\{|HH\rangle|a_{1}b_{1^{\prime}}\rangle$,
$|HH\rangle|a_{2}b_{2^{\prime}}\rangle$,
$|VV\rangle|a_{1}b_{1^{\prime}}\rangle$,
$|VV\rangle|a_{2}b_{2^{\prime}}\rangle\\}$,
$\\{|HV\rangle|a_{1}b_{2^{\prime}}\rangle$,
$|HV\rangle|a_{2}b_{1^{\prime}}\rangle$,
$|VH\rangle|a_{1}b_{2^{\prime}}\rangle$,
$|VH\rangle|a_{2}b_{1^{\prime}}\rangle\\}$, and the other terms can be
perfectly distinguished without destruction of the incident photon.
Additionally, the success probability of our linear optic architectures can be
increased from $\frac{1}{4}$ to $\frac{5}{16}$ by recycling partially
hyperentangled states. In previous works, only hyper-ECPs assisted by
nonlinear mediates allow the recyclable program to efficiently improve the
success probability hyper-ECP5 ; hyper-ECP6 ; hyper-ECP7 ; hyper-ECP8 . That
is attributed to the quantum anti-Zeno effect, which accelerates the dynamic
evolution towards the target state by frequent measurement Zeno1 . Although
the success probability of $\frac{5}{16}$ is still lower than that of
nonlinear hyper-ECPs, linear optic implementations of our schemes are readily
available in practice with current technology. In the overall process, we
assume that the performances of linear optical elements and single-photon
detectors are perfect. However, this does not work in a practical experiment,
and the schemes may be negatively affected by noise during application
Wang2018 . Quantum Zeno effect can enhance the measurement accuracy of
entangled probes to resist these disadvantages Zeno2 .

In summary, we present a practically enhanced hyper-ECP for the hyperentangled
Bell state in both the polarization and spatial mode DOFs with unknown
parameters using available linear optical elements and common single-photon
detectors. By introducing the time-delay DOF to unbalanced interferences, the
practicality of our protocol is efficiently enhanced as the scheme can be
perfectly heralded by the unique detection signatures, and postselection or
sophisticated photon-number-resolving detectors are not needed. Moreover, the
success probability of our linear optical hyper-ECP is higher than that of
Ref. hyper-ECP1  by recycling partially hyperentangled states. Note that
different from previous works, the hyperparallel parity-check measurements in
our scheme acting on polarization and spatial DOFs are simultaneously
accomplished in one step. We also extend the hyper-ECP for hyperentangled Bell
states to generic hyperentangled GHZ states. We will study nonpostselection
EPP in the future.

## ACKNOWLEDGEMENTS

This work is supported by the Fundamental Research Funds for the Central
Universities under Grant No. FRF-TP-19-011A3 and the National Natural Science
Foundation of China under Grant No. 11604012.

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# Towards Behavioral-aware Crowd Management System

Yixin Zhang  Carnegie Mellon University4616 Henry StreetPittsburghUSA
<EMAIL_ADDRESS>,  Tianyu Zhao  University of California, IrvineIrvine, CA
92697IrvineUSA<EMAIL_ADDRESS>and  Salma Elmalaki  University of California,
IrvineIrvine, CA 92697IrvineUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

Instances of casualties resulting from large crowds persist, highlighting the
existing limitations of current crowd management practices. One notable
drawback is the insufficient provision for disadvantaged individuals who may
require additional time to evacuate due to their slower running speed.
Moreover, the existing escape strategies may fall short of ensuring the safety
of all individuals during a crowd surge. To address these pressing concerns,
this paper proposes two crowd management methodologies. Firstly, we advocate
for the implementation of a fair evacuation strategy following a surge event,
which takes into account the diverse needs of all individuals, ensuring
inclusivity and mitigating potential risks. Secondly, we propose a
preventative approach involving the adjustment of attraction locations and
switching between stage performances in large-crowded events to minimize the
occurrence of surges and enhance crowd dispersion. To assess the effectiveness
of our proposals, we used high-fidelity crowd management simulators. Our
findings demonstrate the positive impact of the fair evacuation strategy on
safety measures and inclusivity, which increases fairness by $41.8\%$ on
average. Furthermore, the adjustment of attraction locations and stage
performances has shown a significant reduction in the incidence of surges by
$34\%$ on average, thereby enhancing overall crowd safety.

crowd management, human-in-the-loop, fairness, surge prevention

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: Under Review; ;

##  1\. Introduction

Crowd surge incidents, which occur when a large number of people enter or exit
a confined space, pose significant risks to public safety (Illiyas et al.,
2013). Recent stampede incidents in Itaewon (Sharma et al., 2023) and
Astroworld Festival (The New York Times, 2021) highlight the need for improved
crowd management methods. In Itaewon, trapped individuals struggled to move or
breathe, leading to $156$ deaths and $170$ injuries. At the Astroworld
Festival, $10$ deaths and multiple injuries from the crush near the stage.
Limited entry points, uneven terrain, and unexpected choke points increase the
risk of stampedes. Event organizers must implement strategies, such as careful
scheduling, venue planning, sufficient staffing, and clear communication to
ensure attendees’ safety and prevent property damage. These tragedies
emphasize the importance of effective crowd management at mass gathering
events.

“Smart Crowd Management and Control System” (CMS) is tasked with monitoring,
directing, and managing large groups of people—with an eye toward safety,
efficiency, and satisfaction. CMS requires a diverse range of knowledge,
including engineering, technology, and understanding of crowd behavior (Sharma
et al., 2018). The goal is to prevent crowd surge incidents through meticulous
planning and execution. Effective crowd management involves multiple stages:
pre-event planning, event monitoring and control, post-event feedback, and
improvement. This holistic approach ensures continuous enhancement in crowd
management strategies for future events. However, we believe that each of
these approaches has different effects on healthy individuals, people with
disabilities, children, pregnant women, and neurodivergent communities.
Therefore, while managing the crowd flow, CMS should not only aim to maximize
the efficiency, but also ensure fairness in distributing the adverse effect of
the CMS control actions among different individuals. This vision is attainable
with the advancement in sensor technologies to estimate the human state
through wearable devices, and the decision-making algorithms that provide
trade-offs between system performance, fairness, and privacy in multi-human
environment (Elmalaki, 2021; Taherisadr et al., 2023). This technological leap
in human sensing and decision-making algorithms should be exploited in CMS to
ensure inclusivity in crowd management algorithms.

In this paper, we advocate for implementing a fair evacuation strategy and
prevention approaches that account for the diverse needs of all individuals.
By embracing an inclusive approach, we can provide the necessary time and
assistance to disadvantaged individuals, helping them to evacuate safely and
efficiently. Through thoughtful planning and coordination, we can mitigate
potential risks and minimize casualties.

The contribution of this paper is in two aspects, (1) fair evacuation
strategy: we investigated the correlation between assigning escape exits and
the evacuation efficiency and fairness measured in time-to-escape,
particularly after a crowd surge incident; and (2) prevention approach: we
performed experiments for timely adjustments to the location of stages in big
gathering events to prevent crowd surge incidents. We utilize high-fidelity
CMS simulators, including Vadere (Kleinmeier et al., 2019) and NetLogo (Tisue
and Wilensky, 2004) to simulate various crowd scenarios.

##  2\. Background & Related Work

Crowd surges pose a significant risk when a large number of individuals
attempt to enter or exit a confined area, leading to increased pressure and
potential danger. In tightly packed crowds, people lose control of their
movements and face difficulties breathing. The lack of space for recovery
makes stumbling or falling particularly hazardous, putting individuals at risk
of suffocation and injuries from being crushed. The probability of a surge
occurring is closely related to crowd density. When there are 4-5 people per
square meter, the crowd remains relatively safe, with enough space for
individuals to make movement decisions. However, when the number exceeds 6 per
square meter, the limited available space forces tight packing and diminishes
individual control, significantly increasing the likelihood of a surge
(Abuarafah et al., 2012). A single stumble or jolt within the crowd can
trigger a chain reaction, creating voids that disrupt the crowd’s equilibrium.
Subsequently, more people stumble into these voids, setting off a domino
effect and generating additional voids. This interplay of forces can cause
abrupt collapses, intensifying pressure and chaos within the crowd,
potentially leading to injuries or fatalities if not managed properly (Aalami
and Kattan, 2020). Throughout history, large-scale stampedes have taken place
worldwide, leading to severe loss of life and property damage. On October 29,
2022, a Halloween event occurred in Seoul, South Korea, attracting tens of
thousands of costumed attendees to the Itaewon district. This marked the first
unrestricted Halloween celebration in over two years due to COVID-19
lockdowns. The massive crowd in the narrow streets, coupled with limited entry
and exit points, created a dangerous situation. Videos from that night show
trapped individuals struggling to move or breathe, fueling panic that spiraled
out of control. This catastrophe led to one of South Korea’s worst stampede
disasters, with 156 deaths and 170 crush injuries (Sharma et al., 2023).
Similarly, on April 30, 2021, a devastating crowd surge occurred at Mount
Meron, Israel, during an annual pilgrimage on the Jewish holiday of Lag
BaOmer. The event attracted around 100,000 attendees and resulted in 45 deaths
and approximately 150 injuries, making it Israel’s deadliest civilian
disaster. The tragedy unfolded as people exited a mountainside compound,
moving through a wet, sloping passageway towards a staircase. Witnesses
described attendees tripping and slipping, while others unknowingly continued,
causing trampling, crushing, and asphyxiation. Israeli media reported that
COVID-19 precautions created unanticipated choke points. Additionally,
bonfires were not lit simultaneously, leading to people attending multiple
lightings and increasing the number of attendees.

These tragic incidents highlight the importance of crowd management at mass
gathering events. The sheer number of people in a confined space can create a
dangerous situation that can quickly spiral out of control, resulting in
stampedes and crush injuries. Factors such as limited entry and exit points,
uneven terrain, and unexpected choke points can exacerbate the risk of a
stampede. Therefore, it is crucial for event organizers and authorities to
implement effective crowd management strategies to prevent such incidents.
This comprises actions such as appropriate scheduling of event timing,
meticulous event venue planning and design, sufficient staffing, and
unambiguous communication and signage. Effective crowd management not only
ensures the safety and well-being of attendees but also helps to prevent
damage to property and infrastructure. In light of the recent stampede
incidents, it is clear that crowd management should be given the utmost
importance in planning and executing mass gathering events.

Crowd management is a multifaceted field that necessitates knowledge of
engineering and technology, as well as comprehension of crowd behavior and
crowd flow management, encompassing psychological and sociological aspects
(Sharma et al., 2018). By meticulous planning and execution, the objective of
crowd management is to prevent crowd incidents (Martella et al., 2017).
Effective crowd management is a holistic process that includes several stages.
It begins with meticulous planning before the event, considering all potential
scenarios and preparing for them. During the event, the crowd needs to be
closely monitored and controlled to ensure everyone’s safety. After the event,
it’s important to gather feedback to understand what worked well and what
didn’t. Finally, these insights and lessons learned should be reported and
used to improve crowd management strategies for future events. This approach
ensures continuous improvement in managing crowds effectively (Sharma et al.,
2018).

In the pre-event planning stage, two primary technologies play a crucial role:
crowd modeling and simulation, and social and web data mining. Crowd modeling
and simulation enable the creation of virtual crowd scenarios, which serve as
testing grounds for various crowd management strategies and their
effectiveness. On the other hand, social and web data mining provides valuable
insights into crowd demographics, behaviors, and trends. These insights help
inform decision-making and enable the customization of crowd-management
strategies to suit specific audience profiles. By leveraging these
technologies in pre-event planning, crowd management can be approached with a
greater level of knowledge, strategy, and effectiveness(Sharma et al., 2018).

As for the in-event control period, the acquisition of crowd data during
monitoring, decision-making based on data analysis, and the implementation of
crowd control measures are three key steps for success(Sharma et al., 2018).
The primary goal of crowd control during the event is to detect instances of
mass panic and respond quickly to dangerous situations. Various existing
research proposed numerous methods for detecting crowd density to prevent
surge incidents or to enforce social distancing. For instance, infrared
thermal video sequences have been employed to monitor and estimate the density
of crowds in real-time during large-scale public events (Abuarafah et al.,
2012). In addition, given the widespread Wi-Fi availability, it has been used
to monitor crowd behavior and interaction (Zhou et al., 2020; Weppner et al.,
2016).

Post-event feedback is crucial for preventing future incidents, and in this
regard, social media data plays a pivotal role. The system for situational
awareness can be enhanced by integrating feedback from the crowd and
information related to the crisis that comes from social media. For example,
in a system known as the HADRian, social media data was scrutinized after the
Boston Marathon bombing in April 2013 to identify any unexploded or additional
bombs(Ulicny et al., 2013). Another example is Ushahidi, which is a versatile
data collection, management, and visualization tool that enables data
collection from multiple sources such as SMS, email, web, Twitter, and RSS,
and offers robust features for post management and triaging through filters
and workflows(Ushahidi, [n. d.]). Systems built on Ushahidi have been
implemented worldwide in numerous situations, for instance, to oversee
disaster relief efforts after the Haiti Earthquake in January 2010(Yuan et
al., 2013).

Moreover, the outbreak of COVID-19 needed new real-time approaches for crowd
monitoring and management systems for social distancing. Furthermore, Virtual
Reality (VR) technology has been applied to replicate the crucial parts of the
2010 Love Parade tragedy. Analyzing the emotional responses and stress levels
of participants helps decision-makers gain enhanced insights into crowd
management strategies for comparable occurrences (Zhao, 2016). Additionally,
some studies have devised crowd management approaches that consider balancing
crowd density and movement efficiency. One such study suggested a method that
combines Geographic Information Systems and Agent-Based Modeling to simulate
the movement of pilgrims during the Hajj days. The study simulated and
evaluated five different scheduling plans for pilgrim movement and the results
allow hajj authorities to make informed decisions about the most appropriate
scheduling plans in terms of safety and effectiveness(Yaagoubi et al., 2023).

##  3\. Evacuation Fairness Experiment

The importance of fairness in evacuating crowds lies in achieving a balanced
distribution of evacuees across routes, ensuring equitable waiting times for
different groups. Vulnerable groups, such as the elderly or pregnant women,
are often overlooked in standard evacuation plans due to their physical
limitations (United Nations, 2004). The motivation for this section is to
propose evacuation plans that ensure similar evacuation times for all
individuals, regardless of their physical condition.

###  3.1. Modeling and Simulation

Our experimental goal is to explore the design of evacuation routes for
different groups of people in various crowd-gathering locations, aiming to
achieve both high evacuation efficiency and fairness towards vulnerable
populations. In real-life scenarios, the different running speeds of
vulnerable groups and normal individuals can influence evacuation times,
potentially leading to hazards like pushing or tripping. To address this, we
propose a strategy where a dedicated evacuation exit is designated exclusively
for vulnerable groups, guided by mobile notifications or other means, while
other individuals can use the nearest exit. We hypothesize that this design
can reduce overall evacuation time, especially for vulnerable groups, ensuring
efficiency and fairness simultaneously. To validate our hypothesis, we
proposed $3$ strategies across multiple crowded event scenarios to assign the
evacuation gate:

  * •

Randomly gate assignment(RGA): Individuals evacuate by randomly selecting a
gate without any specific guidance.

  * •

Vulnerable people exclusive gate assignment(VEGA): Vulnerable individuals are
directed to a designated gate exclusively, while normal people are assigned
the closest gate.

  * •

Closest gate assignment(CGA): All individuals are assigned the closest gate
regardless of their physical state.

We used Vadere (Kleinmeier et al., 2019) to simulate various crowd event
setups which we call a map. Four evacuation exits were placed at the corners
of the map. On the map, there are $1363$ people, consisting of $340$
vulnerable people and $1023$ healthy normal people. The average slow running
speed of healthy and young individuals (aged 20-45 years) is $\approx$ 5.4
miles per hour or $\approx$ 2.4 meters per second (Lung et al., 2021). Hence,
we set the average speed of normal people to be $1.0-1.3$ meters per timestep.
Each timestep represents $0.48$ seconds. Vulnerable people, such as the
elderly move at a slower pace We exploited the Optimal Steps Model (OSM) in
Vadere which incorporates the psychological principle of “social distance”
into its mathematical framework, which means that agents strive to avoid
encroaching on others’ personal or intimate space and to prevent physical
contact (Kleinmeier et al., 2019).

(a)(b)

(c)(d)

Figure 1. The four scenarios of crowd distribution in Vadere, where blue dots
represent individuals, and orange blocks represent exit locations. (a): Center
crowd gathering; (b): Non-center crowd gathering; (c): Evenly crowd
dispersing; (d): Unevenly crowd dispersing

###  3.2. Evaluation

We simulated various scenarios for crowd distribution on the map illustrated
in Figure 1:

  * •

Scenario 1: Center crowd gathering: Serving as an exemplar for an event with a
setup focused on the center stage.

  * •

Scenario 2: Non-center crowd gathering: Serving as an exemplar for an event
with a setup focused on a non-center stage.

  * •

Scenario 3: Evenly crowd dispersing: Representing carnival event across the
entire map.

  * •

Scenario 4: Unevenly crowd dispersing: Representing carnival event with
varying area of crowd densities: $75\%$ in top-left, $10\%$ in top-right,
$10\%$ in bottom-left, and $5\%$ in bottom-right.

| RGA | VEGA | CGA
---|---|---|---
$(V,N,Ratio)$ | $all$ | $(G1,G2,G3,G4)$ | $(V,N,Ratio)$ | $all$ | $(G1,G2,G3,G4)$ | $(V,N,Ratio)$ | $all$ | $(G1,G2,G3,G4)$
$S_{1}$ |  (149.9, 80.9, 0.54) | 98.1 | (201, 187, 195, 195) | (154.3, 72.4, 0.47) | 92.8 | (198, 99, 88, 98) |  (125.7, 68.3, 0.54) | 82.6 |  (143, 144, 146, 150)
$S_{2}$ | (153.4, 83.8, 0.55) | 101.1 | (143, 193, 250, 200) |  (97.0, 83.1, 0.86) | 86.5 |  (140, 105, 115, 103) | (132.1, 70.9, 0.54) | 86.2 | (90, 153, 204, 158)
$S_{3}$ | (138.0, 70.5, 0.51) | 87.3 | (263, 254, 227, 257) | (137.4, 33.0, 0.24) | 59.0 | (262, 61, 63, 63) |  (64.7, 33.5, 0.52) | 41.3 |  (122, 116, 129, 112)
$S_{4}$ | (140.7, 69.5, 0.49) | 87.3 | (226, 233, 258, 250) |  (91.2, 58.3, 0.64) | 66.5 | (260, 93, 60, 90) | (79.0, 41.8, 0.53) | 51.1 |  (132, 122, 120, 116)
Table 1. Comparison of average evacuation times for vulnerable ($V$) and
normal people ($N$) and across all all people ($all$), fairness index
$Ratio=\frac{N}{V}$, and Gate Time $(G1,G2,G3,G4)$ using $3$ different
strategies under $4$ different scenarios ($S_{1},\dots S_{4}$). Time is
measured in the simulation step.

We measure the fairness of the time-to-evacuation across all individuals. We
used a metric fairness index (FI) which indicates the ratio of normal healthy
people’s time-to-evacuation to vulnerable people’s time-to-evacuation. This
means the higher the FI value, the higher the notion of fairness. Table 1
presents the results of average time-to-evacuation and FI using three
different strategies. Gate Time $(G1,G2,G3,G4)$ shows the last human exiting
time in each gate, providing insight into the gate utility. RGA shows the
worst time-to-evacuation for all people in all scenarios. In scenarios 1 and
3, CGA demonstrates the best average time-to-evacuation for all people. In
scenarios 2 and 4, VEGA achieves the highest FI of $86\%$ and $64\%$,
respectively, with a slightly slower average time-to-evacuation compared to
CGA for all individuals.

#### Takeaways:

Our preliminary experiment aims to investigate whether a specific evacuation
strategy for vulnerable groups, such as VEGA, enhances fairness while
maintaining overall efficiency. In crowd scenarios 1 and 3, the presence of a
vulnerable exclusive gate may not improve fairness or efficiency when people
initially gather around the center of each exit. Nonetheless, when a
considerable number of vulnerable individuals congregate near a single exit,
such as in scenarios 2 and 4, the VEGA strategy leads to a fairness
improvement of $41.8\%$ on average, compared to both RGA and CGA. In scenario
2, the VEGA strategy demonstrates a fairness increase of $56\%$ and $59\%$
compared to RGA and CGA, while in scenario 4, it shows a fairness improvement
of $31\%$ and $21\%$ compared to RGA and CGA. VEGA can decrease the time gap
between vulnerable groups and others, and the average time-to-evacuation
either reduces or remains relatively unchanged. In real-world scenarios, it is
anticipated that some individuals may not adhere to the recommended guidelines
set by the organizers or the automated crowd management system. Therefore, a
comprehensive study into the social psychological dynamics of the crowd
becomes imperative to better understand and address such situations.

##  4\. Preventative Strategy

Our second experiment is motivated by the tragic Astroworld Festival accident
in 2021 (The New York Times, 2021). The festival featured two stages: the main
stage for the main performance and a secondary stage with performances by
other artists throughout the day. During the concert night, after one
performance at the secondary stage, the audience began moving towards the
already crowded area near the main stage, resulting in a surge and crush near
the stage, as depicted in Figure 2.

Figure 2. The illustration of the 2021 Astroworld Festival accident.

Our prevention strategy suggests alleviating congestion during stage
performances by employing stage switching at designated time intervals. This
approach aims to decrease the likelihood of overcrowding and improve crowd
management throughout the event. To determine the optimal switching points
between stages, we propose three metrics; the Panic State and the Surge State
per individual, and the Crowded State for each subarea. We will explain how to
determine these metrics in Section 4.1. We developed a tool using NetLogo, an
agent-based programmable modeling environment (Tisue and Wilensky, 2004; Zia
and Ferscha, 2020), to configure various user-definable attributes, such as
position and walking speed.

Figure 3. Our proposed simulator tool interface using NetLogo. The top-left
and the bottom-right red rectangles in the simulation map represent two
stages. The upper yellow dot and lower blue dot symbolize the restroom and
bar, respectively. The blue area represents patches close to the left stage,
while the green area represents patches close to the right stage.

Map A

Map B

Map C

Figure 4. Three different stage setups.

###  4.1. Modeling and Simulation

We simulate a crowded scene with two stages in a two-dimensional square world
composed of $51\times 51$ patches. Each patch is represented by a xy-
coordinate point, with the origin $(0,0)$ located at the left bottom. The map
is divided into multiple subareas using a $5\times 5$ grid, with each subareas
containing $25$ patches. Each human, depicted as triangular shapes, resides on
these patches and utilizes information from the underlying patch and
neighboring patches and people to make decisions. The scene includes two
stages, a bar, and a restroom. The stages are positioned at the far left and
right sides of the scene, respectively, while the bar and restroom are located
at the top and bottom sides of the scene. Figure 3 provides a visual
representation of the map. In order to make the simulation scenes more
realistic, the simulation considers four factors: the speed of humans,
psychological factors, such as comfort zone and the preferred distance from
the stage performance, and hesitation time the frequency and duration of
visits to the bar or restroom, which can be changed in the interface,

The first is speed. The paper uses the ”random” method in NetLogo’s code to
randomly set the speed of half of the agents to 1 step per time-step, and the
speed of the other half of the agents to 2 steps per time-step.

The second is a comfort zone and the preferred distance. This paper takes into
account that everyone has a different level of comfort when it comes to the
distance from the stage. For example, some people like to watch performances
right by the stage, while others are satisfied with watching from a slightly
farther distance. Therefore, in the NetLogo code, we randomly set the comfort
distance of each agent between 1-10 meters.

Third, People in the scene randomly go to the bar and a restroom. The total
time spent and the frequency of going to the bar or restroom could be set in
the NetLogo interface. Our current setting is that 40% of the agents go to the
bar or restroom every 50 time-steps, and the duration of each trip is 50 time-
steps. Additionally, we have included adjustment bars for users to set the
frequency and duration of going to the bar or restroom in the NetLogo
interface according to their requirements.

The last one is hesitation time. This paper considers the variation in the
time agents take to decide whether to switch stages after a performance has
ended. Some agents may move to another stage immediately, while others may
stay around the original stage for some time before moving to another one. To
capture this, we assigned a random hesitation time between 1-20 timesteps to
each agent in the NetLogo code.

Our prevention strategy focuses on determining the best time to switch stages,
considering various parameters. To achieve this, we have developed a
simulation tool that can evaluate different scenarios and parameters, allowing
us to estimate the most suitable moment for switching the performance to a new
stage. We propose using a set of metrics to determine the status of
individuals and subareas during the event:

  * •

Panic state: Once an individual is blocked and stuck on the way to the
restroom or bar for some time more than a particular threshold (panic
threshold (PT)) the individual enters the panic status.

  * •

Surge state: Once an individual is blocked and stuck on the way to the stage
for some time more than a particular threshold (surge threshold (ST)), the
individual enters the surge state.

  * •

Crowded state: A subarea enters the crowded state when over $70\%$ of its
patches have people, and at least one person is in a panic or surge state.

  * •

Switch index (SI): The switch index (SI) refers to a threshold on the time for
which a subarea is continuously in the crowded state before the performance is
switched to the other stage.

When a subarea consistently stays in a crowded state, exceeding the threshold
of the switch index (SI), and its two neighboring subareas are in a crowded
state (not necessarily passed the threshold of the SI), it indicates a
critical surge situation. At this point, the currently performing stage will
receive instructions to stop, and the performance switches to another stage.

(a)(b)(c)

(d)(e)

Figure 5. Frequency of switching stage(F) and Average Panic/Surge value with
four different Switch Index (SI) (10, 20, 30, 40) using different parameters.
(a): PN500, BRF50, PT10, ST30; (b): PN750, BRF50, PT10, ST30; (c): PN500,
BRF30, PT10, ST30; (d): PN500, BRF50, PT10, ST40; (e): PN500, BRF50, PT20,
ST30.

###  4.2. Evaluation

We first explored the correlation between stage positions and the likelihood
of surge accidents. The Astroworld tragedy highlighted the significant impact
of stages’ close proximity. Figure 4 illustrates three maps with different
stage positions: Map A features stages placed directly facing each other, Map
B has one stage in the bottom-right corner and another in the middle on the
left side, while Map C positions two stages at the top-left and bottom-right,
respectively. In each map, the speed of half the people doubles the other
half. Each individual’s comfort distance is randomly set between $1-10$ units
patch-size, and their hesitation time is randomly assigned between $1-20$
time-steps. The Switch Index(SI) is set at $10$ time-steps. We assign $40\%$
for the people to go to the bar or restroom and each trip lasts for $50$ time-
steps. Additionally, we have four adjustable parameters: total number of
people (PN), frequency of bar/restroom visits (BRF), Panic Threshold (PT), and
Surge Threshold (ST). The default parameters are set as follows: PN $=500$,
BRF $=50$, PT $=10$, and ST $=30$.

We utilize several metrics to assess crowd behavior. We consider the Frequency
of stage switching (F) and Average Panic/Surge (APS). The Average Panic/Surge
(APS) metric reflects the average number of people in Panic/Surge Status per
timestep. These metrics collectively help us understand and evaluate the
effectiveness of our strategies and response measures. Our experiments showed
that in Map A, the Frequency of stage switching (F) is $0.010$ and the Average
Panic/Surge (APS) is $0.85$. In Map B, the F is $0.009$ and APS is $1.05$. In
Map C, the F is $0.007$ and APS is $0.62$. On average, Map C reduced F by
$26\%$ and APS by $34\%$. The results in Map C show that increasing the
distance between the two stages can reduce both the likelihood of crowd surge
occurrence and the stage switching frequency. Therefore, the subsequent
experiments will Map C to explore the effect of choosing different values of
SI on F and APS under different parameters including:

  * •

Total number of people (PN): Figure 4(a) is 500. In Figure 5(b), PN increases
to 750, showcasing a more crowded environment.

  * •

Frequency of bar/restroom use (BRF): Figure 4(a) is 50 while in Figure 5(c),
BRF decreases to 30, indicating limited availability of restroom and bar
facilities.

  * •

Panic threshold (PT) and Surge threshold (ST): Different values of ST and PT
represent varying audience compositions. Figure 4(a) has ST=10 and PT=10. In
Figure 5(d)(e), with ST increasing to 40, and with PT increasing to 20.

In all of these setups, we observed that the value of SI provides a tradeoff
between the frequency of switching stages (F) and the average of panic/surge
(APS). Indeed this tool can provide insights to the event organizers to
balance the frequency of switching the stages.

#### Takeaways:

Our experiments have provided insights into the correlation between stage
positions and crowd state. The analysis of SI and different parameters
emphasizes the significance of adapting the SI according to the crowd state
and the availability of facilities. In real situations, to estimate the
individual state (panic/surge state), people’s intention to move can be
detected using wearable sensors, such as accelerometers, gyroscopes, and heart
rate data from wearable devices. These sensors enable the estimation of gait
parameters, making them useful for determining individuals’ states.

##  5\. Future Work & Conclusion

Crowd evacuation and surge prevention represent critical research directions
due to their significant implications for public safety, urban planning, and
smart building systems. As urbanization and population density continue to
rise, ensuring efficient and safe evacuation procedures during emergencies or
crowded events becomes increasingly challenging. Our research focuses on
addressing crowd management challenges by exploring both evacuation strategies
and preventive methodologies. We emphasize the importance of balancing
fairness and efficiency in evacuation plans while considering psychological
factors influencing individual social distancing behavior during evacuations.
Our preliminary results showed that by utilizing simulation tools like Vadere,
we can design evacuation strategies that consider vulnerable people.
Additionally, we developed a NetLogo-based tool to simulate preventive
strategies based on the current crowd state. In the future, we aim to further
optimize our preventive approach by integrating post-event analysis and
privacy protection measures. These steps will ensure ethical data usage and
enhance the overall effectiveness of crowd management.

###### Acknowledgements.

This research was partially supported by NSF award # CNS-2105084.

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