longtable Ginwidth=,height=,keepaspectratio [a,b]Sofie Martins

# Towards the super Yang-Mills spectrum at large $N_{c}$

Michele Della Morte  Benjamin Jäger  Justus Tobias Tsang  Felix P. G. Ziegler

(2023-07-19)

###### Abstract

We examine one-flavour $SU(N_{c})$ gauge theories, where $N_{c}$ denotes the
number of colors, with one fermion in the antisymmetric representation as a
candidate to approximate $\mathcal{N}=1$ super Yang Mills due to their
equivalence in the large-$N_{c}$ limit. Summarising results on spectral
evaluations of $N_{c}=3$, we will report on the progress of dynamical
calculations for $N_{c}>3$. We discuss cut-off effects and challenges in
configuration generation.

20(15.0,1.70) CERN-TH-2023-242

##  1 Motivation

We are interested in studying the large-$N_{c}$ limit of a one-flavour theory
with one fermion in the two-index antisymmetric representation, following the
proposal of [1] to use this theory as a proxy to predict the low-lying mesonic
spectrum of a supersymmetric theory. In order to test phenomenological
predictions made in [2] and [3], we have performed a lattice study for the
case $N_{c}=3$ published in [4]. We now wish to, on the one hand, produce this
result for larger numbers of colours and, on the other hand, estimate the
impact of cut-off effects on the result we have obtained so far. This is a
progress report on this project.

##  2 $N_{c}=3$ summarized

We are using the Symanzik-improved gauge action and one flavour of clover-
improved Wilson fermions in the two-index antisymmetric representation at
$c_{\mathrm{sw}}=1$ with OpenQCD [5]. Due to having just a single flavour, we
need the RHMC [6].

Evaluating the spectrum of this one-flavour theory yields the low-lying
mesonic states that are the supersymmetric analogon to the lowest-energy
states of the glueball spectrum. We can obtain masses for these gluodynamic
states by evaluating the connected and disconnected contributions to the
correlation function and extracting the mass.

However, constructing the theory like this, we assume massive gluinos. In
order to predict the actual realisation of massless gluinos, we need to take
the chiral limit. As demonstrated in [7], the chiral point is characterized by
vanishing topological susceptibility. Extrapolating to the point of vanishing
topological susceptibility is equivalent to taking the mass in the connected
contributions to zero. As in [4], we denote the particle with this mass
$m_{\pi}^{\mathrm{fake}}$ as fake pion. Taking the mass of the fake pion to
zero is equivalent to taking the mass of the gluino constituents to zero.

Our $N_{c}=3$ study focused on testing the phenomenological predictions [2, 3]
for the ratio between pseudoscalar and scalar masses
$m_{\mathrm{P}}/m_{\mathrm{S}}$. This has the advantage that we can estimate
the size of $1/N_{c}$ effects by using $N_{c}=3$ without a number of technical
difficulties; for example, there is no necessity to set the scale.

The phenomenological prediction we aim to reproduce has been derived by [2]
and [3] with different methods. Here, [2] states the ratio between the scalar
and pseudoscalar meson to be

$\dfrac{m_{\mathrm{P}}}{m_{\mathrm{S}}}=1-\dfrac{22}{9N_{c}}-\dfrac{4}{9}\beta\lesssim
0.185$ (1)

up to unknown $\mathcal{O}(1/N_{c}^{2})$ effects, whereas [3] arrives at a
consistent result of

$\dfrac{m_{\mathrm{P}}}{m_{\mathrm{S}}}=\dfrac{N_{c}-2}{N_{c}}\,.$ (2)

Note that $\beta$ in (1) denotes a real, positive constant at
$\mathcal{O}(1/N_{c})$.

Our lattice study published in [4] finds

$\dfrac{m_{\mathrm{P}}}{m_{\mathrm{S}}}=0.356(54)\,.$ (3)

The predictions from [2] and [3] are trivially compatible in the large-$N_{c}$
limit but differ in the way they approach this limit. We now aim to test this
result using a larger number of colours. This approach reduces the
$\mathcal{O}(1/N_{c}^{2})$ effects in the estimate (1) and can therefore give
an indication which of these two predictions, which are based on different
sets of assumptions, describe the $\mathcal{O}(1/N_{c})$ effects more
accurately.

In addition, we are tuning ensembles without clover-improvement in $N_{c}=3$
to gauge how much the estimate at $N_{c}=3$ from [4] is affected by cut-off
effects.

##  3 Larger $N_{c}$

For testing this prediction for $SU(N_{c})$-theories with $N_{c}>3$, we use
HiRep [8, 9] using again Symanzik improvement in the gauge action and one
flavour of clover-improved Wilson fermions at $c_{\mathrm{sw}}=1$ in the two-
index antisymmetric representation for $N_{c}=4,5,6$. We have explored the
parameter space for these three $N_{c}$ to find approximate fake pion masses
in the range of $am_{\pi}^{\mathrm{fake}}\approx 0.2$ \- $1.0$ and lattice
spacings around $0.1~{}\mathrm{fm}$.

###  3.1 Matching physical parameters

For a reasonable estimate for the coupling $\beta$ to yield lattice spacings
of approximately $0.1~{}\mathrm{fm}$, we are following the suggestion from
t’Hooft in [10], to keep the t’Hooft coupling $\lambda=g^{2}N_{c}$ constant.
Since

$\beta=\dfrac{N_{c}}{g^{2}}\Rightarrow\lambda=\dfrac{N_{c}^{2}}{\beta}\,,$ (4)

we can naively rescale

$\beta\sim N_{c}^{2}\,,$ (5)

yielding the parameters displayed in table 1.

In practice, the rescaling described in [10] features the renormalised
coupling, and we initially expected the need to adjust $\beta$ due to
significant deviations from the target physical parameters. Surprisingly, we
found good agreement in the resulting physical parameter space, yielding the
same lattice spacings as for $N_{c}=3$ by just using the naive scaling.
Subsequently, it seems that the naive scaling suggested by t’Hooft works well
in this setting, even if it is only a tree-level estimate.

Unfortunately, when naively scaling the $\beta(N_{c}=3)=4.5$ to $N_{c}=4,5$
and $6$, we observed strong topological freezing in all simulations within the
interesting physical parameter space. In the following, we detail ensembles to
explore the parameter space to stay within target physical parameters but
avoid topological freezing.

$N_{c}$ | $\beta_{\mathrm{naive}}$ | $\beta\approx$
---|---|---
3 | 4.50 | 4.5
4 | 8.00 | 7.6
5 | 12.50 | 12.1
6 | 18.00 | –
Table 1: Naive estimates to rescale $\beta$ in order to simulate at the right
t’Hooft coupling, following [10] and an approximate value for $\beta$ at which
the freezing is sufficiently reduced.

###  3.2 Parameter dependence of topological freezing

Figure 1: Severity of topological freezing depending on the parameter space.
The left and middle image show pure gauge simulations over 500 trajectories
depending on the spatial lattice extent at fixed $N_{c}=4$ (middle) and
depending on the number of colours at fixed $L/a=24$ (left). The figure to the
right shows the development of the topological charge for dynamic ensembles
after thermalization for a fixed $\kappa=0.1450$ depending on the spatial
lattice extent $L/a$. Note that the data for pure gauge was already published
in [11].

For periodic boundary conditions, the parameter dependence of the number of
topological sectors explored depending on volume and quark mass is derived in
[12]. For one-flavour theories, they derive that the probability of finding a
gauge configuration in a given topological sector is normally distributed
around $Q=0$ as

$\sigma=\mathrm{Var}(Q)=\langle Q^{2}\rangle=V\Sigma m$ (6)

where $V$ is the volume $T\times L^{3}$, $m$ the quark mass and $\Sigma$
describes a low energy constant related to the quark condensate in the chiral
limit. Correspondingly, the probability of movement between different
topological sectors is increasing with $Vm$, so we can compensate a decrease
in $m$ with a corresponding increase in $V$. The topological susceptibility
will, in any case, go to zero in the chiral limit. The strength of the
freezing may also depend on $N_{c}$ and become worse when increasing $N_{c}$.
For this case we need to consider additional options to avoid autocorrelations
of our ensembles.

Another stronger predictor for topological freezing is the lattice spacing,
which we control over $\beta$. Since $\beta$ is inversely proportional to the
lattice spacing, a slight change in $\beta$ can have a significant effect on
the lattice spacing; the autocorrelation time of the topological charge
behaves as $\tau_{Q}\sim a^{-5}$ for lattice spacings of $\lesssim
0.1~{}\mathrm{fm}$ [13]. Relaxing $\beta$ is a way to reduce topological
freezing if this is sufficient for the precision one wants to achieve in the
continuum limit.

###  3.3 Overview of simulation parameters

$N_{c}$ | $\beta$ | $\kappa$ | $L/a$ | $T/a$ | BCs | start | $Q$
---|---|---|---|---|---|---|---
3 | 4.50 | 0.1410 | 24 | 64 | periodic | hot | suff.
3 | 4.50 | 0.1400 | 32 | 64 | periodic | hot | suff.
4 | 8.00 | 0.1470 | 12 | 48 | periodic | hot | (-1)
4 | 8.00 | 0.1450 | 24 | 48 | periodic | hot | (-20,-19,-18,-17)
5 | 12.50 | 0.1520 | 16 | 48 | periodic | cold | (0)
4 | 7.60 | 0.1520 | 16 | 48 | periodic | cold | suff.
5 | 12.00 | 0.1530 | 16 | 48 | periodic | cold | (0,1,2)
4 | 7.80 | 0.1440 | 16 | 96 | open | cold | suff.
4 | 7.80 | 0.1440 | 16 | 192 | open | cold | suff.
Table 2: Selected ensembles for $N_{c}>3$ to explore parameter space and
technical study setup and two representative ensembles from the $N_{c}=3$
study [4]. The column _BCs_ denotes the choice of boundary conditions, and the
column _start_ the chosen start configuration using either a cold or a hot
gauge configuration. All simulations were performed at $c_{\mathrm{sw}}=1.0$
with trajectory length $\tau=2.0$. The column $Q$ indicates the topological
sectors sampled after thermalisation, with _suff._ indicating that there is no
problem with topological freezing and that sufficiently many sectors were
sampled.

We see the effects described above in our ensembles. Table 2 shows a small
selection of ensembles $N_{c}>3$ and two representative ensembles at $N_{c}=3$
from [4] to illustrate the steps necessary to generate ergodic ensembles.

For $N_{c}>3$, all ensembles are at approximately $0.1~{}\mathrm{fm}$, and all
ensembles are frozen in a single topological sector if the lattice is small.
For these simulations, $\beta$ is scaled naively. Increasing the lattice size
can reduce the freezing, but using a hot start ensemble we may still only
explore topological sectors far from zero, while a cold start allows
thermalization near $Q=0$.

Figure 2: The topological charge moves after adjusting $\beta$ to $7.6$. We do
see some signs of freezing again for light fake pion masses. Promisingly,
$\kappa=0.1520$ corresponds to $am_{\pi}\approx 0.3$ and is therefore light
enough for production.

We also generated ensembles with a slightly adjusted $\beta$. As described in
[13], one can see that adjusting the spacing has the most significant effect
on increasing the number of topological sectors explored. For the precision
requirements of this study, it is sufficient to avoid lattices that are too
fine for this technical reason and perform the continuum extrapolation from
spacings that are not too strongly subjected to freezing. We found in our
simulations that decreasing $\beta$ from $8.0$ to $7.6$ completely solves the
problem of topological freezing at sufficiently small fake pion masses and
lattice spacings of around $0.1~{}\mathrm{fm}$ at $N_{c}=4$; see figure 2.

Figure 3: Finite-size effects for open boundary conditions. We see good
agreement between periodic boundary conditions and open boundary conditions
with T/a=192. The agreement is less good for T/a=96, however the effects are
at subpercent level.

In case of topological freezing at light quark masses, for example, at
$N_{c}=5$ or $6$ with fine lattice spacing, we have explored the option to use
open boundary conditions in addition to large lattices. While open boundary
conditions explore more topological sectors, we have to increase the temporal
extent by a factor of two to measure sufficiently far away from the boundary.
Our preliminary evaluations of the plaquette and the fake-pion mass show
finite-size effects contributing at the per cent level. See figure 3 for
comparing the plaquette between simulations with open and periodic boundary
conditions.

##  4 Cut-off effects

$N_{c}$ | $\beta$ | $\kappa$ | $L/a$ | $T/a$ | $am_{\pi}^{\mathrm{fake}}$ | $t^{\ast}$
---|---|---|---|---|---|---
3 | 4.555 | 0.1420 | 24 | 48 | 0.676(7) | 5.85(5)
3 | 4.555 | 0.1450 | 24 | 48 | 0.508(7) | 6.37(5)
3 | 4.555 | 0.1478 | 24 | 48 | 0.34(1) | 6.90(5)
Table 3: Overview of parameter choice for understanding cut-off effects. All
ensembles are with periodic boundary conditions, hot start, $\tau=2.0$ and
$c_{\mathrm{sw}}=0.0$. Results for $am_{\pi}^{\mathrm{fake}}$ and $t^{\ast}$
are preliminary. Figure 4: Finite-size effects in scale setting for $N_{c}=3$,
clover-improved data from [4]. $t^{\ast}$ is defined as the reference scale
for scale setting using the Wilson flow [14], using the symmetric definition
of $E$. We show preliminary results from ensembles at $c_{\mathrm{sw}}=0$ with
rescaled baseline $\beta=4.555$.

The deviations from the prediction found in the case of $N_{c}=3$ might not
only be due to effects of order $1/N_{c}^{2}$ but can also arise due to cut-
off effects. To understand the order of magnitude of cut-off effects, we are
generating ensembles for $N_{c}=3$ without clover-improvement that are tuned
to achieve the same physical parameters as their clover-improved counterparts.
We list planned ensembles with preliminary fake pion masses and $t^{\ast}$ in
table 3.

##  5 Computational cost

For $N_{c}=3$, the two-index antisymmetric representation coincides with the
fundamental representation. This one-flavour QCD theory is supported well by
different lattice community software libraries. Larger-$N_{c}$ with Wilson
fermions are supported by HiRep and Grid [15], with some GPU-support in Grid
and upcoming GPU-support for HiRep.

We compared benchmarks between HiRep and Grid for CPUs on LUMI-C and found
that HiRep significantly outperformed Grid for Wilson fermions. However, the
growing computational expense is unavoidable due to larger requirements for
memory and compute. One can estimate that the necessary computational time
scales at least with $N_{c}^{2}$ due to the matrix multiplication in the Dirac
operator. However, larger memory requirements degrade this scaling.

##  6 Conclusion

This is a progress report on the project. Moving from $N_{c}=3$ to higher
$N_{c}$ presents several challenges. These challenges are a substantial growth
of computational expense of the dynamical lattice calculations, severe
topological freezing at sufficiently low masses and high coupling, and
substantial finite-volume effects. We found that the right recipe for
conducting a study of this kind at a relatively fine lattice spacing is to
offset a decrease in quark mass with a proportional increase of the lattice
volume, consider a possible increase in lattice spacing for ensembles that
suffer from topological freezing, use GPUs to offset the significant
computational and memory overhead, simulate at large lattices of, at the very
least, $L/a=20$ to control finite-size effects in scale setting, start from a
unit configuration to thermalize around $Q=0$, and, if needed, use open
boundary conditions to avoid freezing additionally.

##  7 Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreements №813942 and 894103. F.P.G.Z. acknowledges support from UKRI Future
Leader Fellowship MR/T019956/1. We thank the EuroHPC Joint Undertaking for
providing computational resources on the Discoverer supercomputer under
proposal ID EHPC-REG-2023R01-102. Early simulations with Grid were performed
on LUMI-C and LUMI-G hosted by CSC (Finland) and the LUMI consortium using
resources provided by the University of Southern Denmark and the LUMI-G pilot
program. Dynamic simulations with HiRep on GPUs were possible using GPU nodes
provided by the UCloud interactive HPC system managed by the eScience Center
at the University of Southern Denmark. We would like to thank Steffen Ulrik
Jensen for his contributions to the exploration of the parameter space during
his master thesis work.

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A game of life with dormancy

Daniel Henrik Nevermann1, Claudius Gros1, Jay T. Lennon2,

1 Institute for Theoretical Physics, Goethe University Frankfurt, Germany

2 Department of Biology, Indiana University, Bloomington, IN 47405, USA

* corresponding author<EMAIL_ADDRESS>
## Abstract

The factors contributing to the persistence and stability of life are
fundamental for understanding complex living systems. Organisms are commonly
challenged by harsh and fluctuating environments that are suboptimal for
growth and reproduction, which can lead to extinction. Species often contend
with unfavorable and noisy conditions by entering a reversible state of
reduced metabolic activity, a phenomenon known as dormancy. Here, we develop
Spore Life, a model to investigate the effects of dormancy on population
dynamics. It is based on Conway’s Game of Life, a deterministic cellular
automaton where simple rules govern the metabolic state of an individual based
on the metabolic state of its neighbors. For individuals that would otherwise
die, Spore Life provides a refuge in the form of an inactive state. These
dormant individuals (spores) can resuscitate when local conditions improve.
The model includes a parameter $\alpha\in[0,1]$ that controls the survival
probability of spores, interpolating between Game of Life ($\alpha=0$) and
Spore Life ($\alpha=1$), while capturing stochastic dynamics in the
intermediate regime ($0<\alpha<1$ ). In addition to identifying the emergence
of unique periodic configurations, we find that spore survival increases the
average number of active individuals and buffers populations from extinction.
Contrary to expectations, the stabilization of the population is not the
result of a large and long-lived seed bank. Instead, the demographic patterns
in Spore Life only require a small number of resuscitation events. Our
approach, yields novel insight into what is minimally required for the
emergence of complex behaviors associated with dormancy and the seed banks
that they generate.

## Introduction

In nature, organisms are often challenged by conditions that are suboptimal
for growth and reproduction. For life to persist, organisms must contend with
a range of external forces. Scarcity of resources, fluctuating abiotic
variables, and the patchy distribution of suitable habitat are just a few of
the many exogenous factors that can reduce organismal fitness. In addition,
individual performance is affected by endogenous factors arising from
demographic rates and species interactions. Together, these factors pose a
risk for local extinction. Populations can escape this fate through behavioral
modifications, phenotypic plasticity, migration within a landscape, and
evolutionary adaptation [1, 2].

One process that is important in promoting the persistence of populations is
dormancy, which occurs when an individual enters a reversible state of reduced
metabolic activity [3]. Dormant individuals enjoy protection against
unfavorable conditions and have the capacity to resume growth when conditions
improve. The accumulation of inactive individuals results in a ”seed bank”,
which can buffer population dynamics leading to increased geometric mean
fitness and a reduced probability of stochastic extinction in variable
environments [4, 5].

Dormancy is achieved among species in different ways. In some cases, dormancy
requires hundreds of interacting genes that are integrated into a tightly
regulated developmental program [6]. Organisms adopting such a strategy often
rely on the interpretation of environmental cues to transition between
metabolic states in a responsive manner [7]. In other cases, dormancy is
achieved through simpler means. For example, organisms can stochastically fall
into an inactive state or randomly awaken independent of environmental
conditions consistent with a bet-hedging strategy [8, 9]. These fundamental
differences in dormancy affect the size and longevity of a seed bank. While
individuals belonging to some species only remain dormant for a short period
of time [10], individuals belonging to other species can persist indefinitely
[11]. Such differences in shallow vs. deep dormancy should have implications
for population stability and the persistence of life.

Dormancy has independently arisen many times across the tree of life [5]. In
this way, it is an example of convergent evolution [12], suggesting that
dormancy may be a solution to one of life’s major problems, i.e., surviving in
noisy and unpredictable environments. Despite its prevalence among diverse
lineages and ecosystems, there is no standard way for modeling dormancy. While
dynamical approaches have been developed [13, 14, 15] they typically do not
capture the fact that dormancy is an individual process that can lead to
complex emergent behaviors and patterns. Furthermore, modeling efforts have
not identified the minimal requirements for achieving the benefits of
dormancy, which could shed light on the origins and evolution of an important
life-history strategy.

One way to model living systems is through a bottom-up approach where rules
are encoded into individual agents that are then observed as the consequences
unfold over time. Perhaps the best examples come from cellular automata where
self-replicating behaviors are governed by the agent’s state (e.g., $alive$
vs. $dead$) as well as those of its neighbors on a two-dimensional lattice
with a random set of initial conditions. Despite their simplicity at the agent
level, cellular automata, most notably Conway’s Game of Life [16], can give
rise to periodic patterns [17], chaotic dynamics [18], self-organized
criticality [19], and other life-like complexities [20]. While dispersal in
space has been explored in cellular automata [21, 22], few studies have
investigated how dispersal in time (i.e., dormancy) affects the dynamics of
such systems [23, 24].

Here, we develop a cellular automaton called Spore Life, which includes
minimal modifications to the original Game of Life. We quantify the
distribution of metabolic states along with the probability of extinction with
updated rules that capture features of dormancy. In particular, we explored
shallow vs. deep dormancy by manipulating a spore survivorship parameter,
which also introduces stochasticity into the model. By characterizing the
lifetime of individuals in active ($A$) and inactive ($I$) states, along with
spore effects on demographic rates (i.e., births and deaths), we are able to
recreate and better understand what is minimally required for dormancy and its
contribution to population persistence.

## Methods

### Game of Life

We developed a cellular automaton that includes dormancy. The model is based
on Conway’s Game of Life (GoL), which takes place on a two-dimensional lattice
of sites $S$. In the original model, each site can be in dead state ($D$) or
occupied by an alive individual in a metabolically active state ($A$) such
that $S\in\\{\mathrm{active},\mathrm{dead}\\}\equiv\\{A,D\\}$. Initial
conditions ($t=0$) are created by seeding the $\ell\times\ell$ grid with
probability $q$ of each site being occupied by an active individual. In
subsequent time steps, the grid is updated in a density-dependent manner.
Specifically, the following rules are applied to the occupant of a focal site
based on the number of active individuals ($\mathcal{N}_{A}$) in neighboring
sites (Fig. 1):

  1. 1.

Active individual with $<$2 active neighbors dies ($A\rightarrow D$)

  2. 2.

Active individual with 2-3 active neighbors persists ($A\rightarrow A$)

  3. 3.

Active individual with $>$4 active neighbors dies ($A\rightarrow D$)

  4. 4.

Dead individual with 3 active neighbors is reborn ($D\rightarrow A$)

### Spore Life

We created a model called Spore Life by introducing dormancy into the cellular
automaton. This required the addition of an inactive state ($I$), allowing
individuals to be in one of three metabolic states:
$S\in\\{\mathrm{active},\mathrm{inactive},\mathrm{dead}\\}\equiv\\{A,I,D\\}$.
As in the original Game of Life, initial conditions are created by seeding the
$\ell\times\ell$ grid with a probability $q$ of each site being occupied by an
active individual. In subsequent time steps, the grid is updated according to
a slightly modified set of rules (Fig. 1):

  1. 1.

Active individual with 0 active neighbors dies ($A\rightarrow D$)

  2. 2.

Active individual with 1 active neighbor becomes a dormant spore
($A\rightarrow I$)

  3. 3.

Active individual with 2-3 active neighbors lives ($A\rightarrow A$)

  4. 4.

Active individual with $>$4 active neighbors dies ($A\rightarrow D$)

  5. 5.

Inactive individual (i.e., spore) with 2-3 active neighbors resuscitates
($I\rightarrow A$)

  6. 6.

Dead individual with 3 active neighbors is reborn ($D\rightarrow A$)

Fig 1: Transition rules and unique configurations In (a) we present rule
tables for the Game of Life (top) and Spore Life (middle). Individuals can be
in a dead ($D$), inactive ($I$), or active ($A$) state. Transitions among
these states are governed by the number of active neighbors
${\mathcal{N}}_{A}\in\\{1,2,\dots,8\\}$. Dormant individuals (i.e., spores) in
an inactive state $I$ die at every time step with probability $1-\alpha$
(bottom). In (b) a selection of repeating configurations $P_{n}$ with period
$n$ that arise in the deterministic limit ($\alpha=1$) of Spore Life. Active
individuals are shown in orange and inactive individuals in blue.

### Spore survivorship

To explore how variation in spore survivorship influences population dynamics,
we introduced a parameter $\alpha$, which determines the fate of an inactive
individual (Fig. 1 a). We specify that an inactive individual dies with
probability $(1-\alpha)$. As such, when $\alpha=1$, an inactive individual has
the potential to remain as a dormant spore for the remainder of the
simulation. At the opposite limit, when $\alpha=0$, all inactive individuals
deterministically transition into dead individuals before the next time step.
Under these conditions, the middle row of the Spore Life rule table is
identical to the top row of the original Game of Life rule table (Fig. 1 a).
When $\alpha\in(0,1)$, transitions are stochastic such that some inactive
individuals die while others resuscitate, reflecting variation in the degree
to which organisms can persist in a shallow vs. deep state of dormancy.

### Dormancy and population dynamics

Once implemented, we used Spore Life to characterize the effects of dormancy
on key population-level phenomena. In addition to identifying the emergence of
unique periodic configurations, we described how dormancy influences the
abundance and distribution of metabolic states ($A$, $I$, $D$) on different
sized grids. We also measured extinction probabilities of the population.
While controlling for starting densities across a broad range ($n$ = 1,000) of
initial condition, we quantified the number of time steps that occurred prior
to extinction ($T_{\mathrm{ext}}$), which we operationally defined based on
the constancy of $N_{A}$ after 100 time steps.

### Dormancy and demographic processes

To gain mechanistic insight into how dormancy affected population-level
phenomenon, we characterized two important demographic processes. First, we
quantified the lifetime of inactive individuals for different levels of
$\alpha$. This involved calculating the number of time steps before an
inactive spore either died or was resuscitated. We then compared these
estimates to the lifetime of an active individual, which involved measuring
the number of time steps before it either died or transitioned into an
inactive state $I$. Second, we examined how $\alpha$ contributed to the
population by comparing sources of births (i.e., $D$ $\to$ $A$ vs. $I$ $\to$
$A$) and sources of deaths ($A$ $\to$ $D$ vs. $A$ $\to$ $I$) as a function of
$\alpha$.

An interactive web page is available to explore the dynamics of Spore Life:
https://itp.uni-frankfurt.de/spore-life/

## Results

### Spore Life creates unique periodic configurations

In the Game of Life ($\alpha=0$) repeating configurations such as blinkers and
gliders exist [25]. This also holds in the opposite limit ($\alpha=1$) when
Spore Life becomes deterministic. We identified a number of configuration that
uniquely arise from the dormancy rule set. While our investigation of
configurations was neither systematic nor exhaustive, we documented several
patterns that repeat with periods of 2, 3, 4, 14, and 62 time steps (Fig. 1 b
and Fig. S2).

### Dormancy stabilizes population dynamics

When starting from a random distribution of metabolic states, we find that
Spore Life is much more stable than the Game of Life (Fig. 2). Without
dormancy ($\alpha=0$), the number of active individuals ($N_{A}$) rapidly
drops off leading to extinction in less than 100 time steps for a
$\ell\times\ell=30\times 30$ grid. Here, extinction occurs when there are no
active or dormant individuals on the lattice, or when the population is
comprised of isolated static or periodic configurations with densities of
$\sim 3.3\%$ active individuals. In contrast, when dormant spores were
generated due to local underpopulation, as illustrated in Fig. 1 a,
populations persist over prolonged time scales. In the deterministic limit,
$\alpha=1$, populations persisted for at least 1,000 time steps with much
higher average population sizes. A simulation comparing the dynamics of Spore
Life and the Game of Life can be viewed here. In between, for
$\alpha\in(0,1)$, there was a reduction in the average $N_{A}$ relative to
$\alpha=1$, suggesting an increased risk of local extinction (Fig. S1)

Fig 2: Population dynamics with dormancy Abundance of active individuals
($N_{A}$) over time ($t$) for different levels of spore survivorship
($\alpha$). For the Game of Life ($\alpha=0$), the abundance of active
individuals drops off rapidly to a small value $N_{A}/\ell^{2}\approx 0.033$,
which is due to a limited number of surviving elementary configurations,
mostly static or of period one. For a finite but low level of spore
survivorship ($\alpha=0.3$), $N_{A}$ is somewhat higher, but populations die
out as a result of stochastic fluctuations. When spore survivorship is higher
($\alpha=0.7$ and $\alpha=1$), there is a corresponding increase in $N_{A}$.
The grid size is $\ell\times\ell=30\times 30$ with a starting density
$N_{A}/\ell^{2}=0.37$

### Diminished effect of dormancy in small populations

We identify a finite-size effect of dormancy on population dynamics. When
spore survivorship ($\alpha$) is low, the stabilizing effect of dormancy on
the average density of active individuals ($\rho_{A}=\langle
N_{A}\rangle/\ell^{2}$) is weaker on smaller grids (e.g. $\ell=20$ and
$\ell=30$) (Fig. 3 a). Under such conditions, populations ($\langle
N_{A}\rangle$) are smaller and experience larger fluctuations (Fig. S1),
increasing the probability of extinction because locally static or periodic
configurations remain isolated for all time. On a larger grid (e.g.
$\ell=300$), however, the average density of active individuals increases near
linearly with $\alpha$ (Fig. 3 b). Under these conditions, where
$\alpha\approx 1$, the population is well-mixed and individuals successively
influence each other across all distances via the neighborhood-based update
rules (Fig. 1 a).

### Dormancy alters the distribution of metabolic states

Spore survivorship alters the distribution of metabolic states in the
population. Over the full range of $\alpha$, dormancy leads to an increase of
$\rho_{A}$ from $\approx 3\%$ for $\alpha=0$, to $\approx 30\%$ at $\alpha=1$
(Fig. 3 b). At the same time, the average density $\rho_{D}$ of dead
individuals decreases by $\approx 35\%$. In contrast, the average density of
inactive individuals $\rho_{I}$, which can be viewed as the ”seed bank”,
remains low ($\rho_{I}<0.1$), even in Spore Life where $\alpha=1$ (Fig. 3 b).

Fig 3: Finite-size effect of dormancy In (a), the average density of active
individuals $\rho_{A}=\langle N_{A}\rangle/\ell^{2}$ as a function of spore
survivorship ($\alpha$) on different sized grids. For a larger grid
($\ell=300$), the density of active individuals increases near linearly with
$\alpha$. The depression of $\rho_{A}$ on smaller grids ($\ell=20,30$)
represents a finite-size effect where small populations go extinct owing to
stochastic fluctuations. Compare Fig. 4 for further details. In (b), the
average density of active ($\rho_{A}$), inactive ($\rho_{I}$), and dead
($\rho_{D}$) individuals is shown for $\ell=300$.

### Dormancy reduces extinction probability

Motivated by the temporal dynamics presented in Fig. 2, we sought to
characterize the relationship between dormancy and the extinction probability.
Considering a $30\times 30$ grid, we found that extinction times
($T_{\mathrm{ext}}$) for a total of 1,000 simulations were distributed
according to an exponential law $\sim c\,\mathrm{e}^{-T_{\mathrm{ext}}/T}$
(Fig. 4 a). We find $c\;[\times 10^{2}]=1.2/0.38/0.17$ and $T=0.92/0.95/0.95$
for $\alpha=0.2/0.3/0.35$, respectively. While the decay constants ($T$) were
essentially unaffected by spore survivorship ($\alpha$), dormancy had a strong
effect on the scaling factors ($c$), which is evident from the different
intercepts in Fig. 4 a. The average time to extinction
$\langle{T_{\mathrm{ext}}}\rangle$ increases exponentially with increasing
spore survivorship. In the absence of dormancy ($\alpha=0$),
$\langle{T_{\mathrm{ext}}}\rangle$ was $\leq 500$ time steps, but even with
moderate levels of spore survivorship ($\alpha=0.4$),
$\langle{T_{\mathrm{ext}}}\rangle$ already increased to $\geq 25,000$ time
steps (Fig. 4 b). Numerically, extinction times ($T_{\mathrm{ext}}$) increase
with grid size.

Fig 4: Extinction times increase with dormancy In Spore Life, we consider a
population to be extinct if there are no active or dormant individuals, or if
the pattern of active individuals consists only of isolated periodic or static
configurations. From this, we define extinction time $T_{\mathrm{ext}}$ as the
number of time steps starting from the random initial distribution of active
individuals ($t=0$) until an extinct state is reached. For a
$\ell\times\ell=30\times 30$ grid, we averaged over $10^{3}$ initial
conditions, finding that the distribution of extinction times $T_{\rm ext}$ is
exponentially distributed (a). On average, extinction times rise sharply with
increasing spore survivorship $\alpha$ (b).

### Spore survivorship minimally extends spore lifetimes

To better understand how spore survivorship ($\alpha$) influences population
dynamics in Spore Life, we characterized the distribution of spore lifetimes.
Once created, an inactive individual’s lifetime $T_{I}$ is equal to the number
of time steps until a mortality event (when $\alpha<1$) or a resuscitation
event occurs. The probability that an isolated spore remains present after $t$
steps is

$\alpha^{t}=\mathrm{e}^{t\log(\alpha)}=\mathrm{e}^{-t/T_{\alpha}},\qquad
T_{\alpha}=\frac{-1}{\log(\alpha)}\,,$ (1)

indicating that the lifetime of an inactive individual is exponentially
distributed.

Owing to a sharp initial drop off (Fig. 5 a), mean lifetimes are substantially
shorter than the respective timescales $T$ obtained through fits (Fig. 5 b).
Regardless of $\alpha$, the average lifetime of a spore ($T_{I}$) is $<1.5$
time steps. Meanwhile, the average lifetime of an active individual ($T_{A}$)
monotonically decreases with increasing $\alpha$ to about one time step for
$\alpha\to 1$ (Fig. 5 b). Although we did not numerically explore the exact
behavior, it is worth noting that the lifetime of active individuals rapidly
increases as $\alpha$ approaches 0. See Supporting Information for a
discussion regarding the small values of $\alpha$.

From the exponential distribution, we conclude that the natural decay of
spores $\sim(1-\alpha)$ is the dominant cause for the loss of inactive
individuals when $\alpha$ is not too close to one. By examining the tails of
(Fig. 5 a), which can be approximated by an exponential $\sim\exp(-t/T)$, we
find $T=0.77/2.14/4.56$ for $\alpha=0.3/0.7/1.0$, whereas (1) would predict
$T_{\alpha}=0.82/2.80/\infty$, which is in turn reasonably close for
$\alpha=0.3$ and $\alpha=0.7$. On the other hand, inactive individuals can
only change to active individuals when $\alpha=1$, a process for which the
typical timescale is observed to be $T=4.56$.

Fig 5: Distribution of spore lifetimes (a) The time steps ($T$) that an
inactive individual remains in a dormant state ($D$) before dying or
resuscitating into an active state ($A)$ decays exponentially. (b) The average
life times of inactive individuals $\langle T_{I}\rangle$ and active
individuals $\langle T_{A}\rangle$ as a function of $\alpha$.

### Spore survival alters demographic processes

Spore survivorship ($\alpha$) substantially affects the birth rates and death
rates of active individuals (Fig. 6). For small $\alpha$, $\approx 20\%$ of
the population dies and is reborn at every time step. As $\alpha\to 1$, $\geq
50\%$ of the population dies and is reborn at every step. This is due to a
higher likelihood of mortality from overcrowding (Fig. 6 b) with increasing
population density (Fig. 3). Somewhat counterintuitively, the probability of
an active individual transitioning into the inactive state ($I$) increases
substantially over the same range of $\alpha$.

Regarding birth processes, at low- to intermediate-levels of $\alpha$, the
vast majority of births ($\geq 90\%$) occur when active individuals are
produced in an empty (dead) site that was surrounded by 3 active neighbors. As
$\alpha\to 1$, $\approx 13\%$ of new births are associated with the transition
from an inactive state ($I$) to an active ($A$) state. As illustrated in Fig.
1 a, these resuscitations happen when a spore is surrounded by either 2 or 3
active neighbors ($\mathcal{N}_{A}\in\\{2,3\\}$). Our analysis suggests that
the contributions of these two resuscitation pathways to the population birth
rate are roughly equivalent (Fig. 6 a). Regardless of whether an individual is
in a dead $D$ or inactive $I$ state, when $\mathcal{N}_{A}=3$, an active
individual will be born. This means that any birth-mediated effect of dormancy
can be attributed to the metabolic transitions that occur when
$\mathcal{N}_{A}=2$, which never accounts for more than $\approx 8\%$ of the
birth rate (Fig. 6 a). Instead, at least $88\%$ of all new births are
associated with $D\to A$ transitions, even when $\alpha=1$.

In a similar way, we analyzed how different metabolic transitions contributed
to the death rate of a population as a function of spore survival ($\alpha$).
The largest source of mortality ($\approx 80\%$) is due to overpopulation,
which occurs when an individual is surrounded by $\geq 4$ active neighbors
(Fig. 6 b). In contrast, underpopulation is the smallest source of mortality
($\approx 2\%$), which occurs when an active individual is surrounded by 0
active neighbors. Last, $\geq 18\%$ of the death rate was due to the loss of
active individuals that transitioned into an inactive state $I$ when they were
surrounded by 1 active individual.

Fig 6: Influence of dormancy on demographic processes The gray lines
correspond to (a) the birth rate and (b) the death rate of active individuals
($A$) as a function of spore survivorship ($\alpha$). The birth rate is
defined as the number of transitions to the active state ($D\to A$ or $I\to
A$) per time step normalized by the average number of active individuals
$\langle N_{A}\rangle$, that is
$\mathrm{Birth}\>\mathrm{Rate}=(\\#\text{transitions to active})/\langle
N_{A}\rangle$. The death rate is defined as the number of transitions out of
the active state ($A\to D$ or $A\to I$) per time step normalized by $\langle
N_{A}\rangle$. For the birth rate, the primary source of new individuals comes
from $D\to A$ transitions. Resuscitation events ($I\to A$) from individuals
surrounded by either 2 or 3 active neighbors contribute equally, but rather
minimally to the overall birth rate, especially at lower $\alpha$. For the
death rate, the most dominant cause of death is overcrowding ($A\to D$ when
$\geq 3$ active neighbors), while the contribution of underpopulation ($A\to
D$ for 0 active neighbors) is relatively small. Dormancy ($A\to I$) plays an
increasingly important role with increasing spore survivorship ($\alpha$).

## Discussion

Critical population-level phenomena commonly associated with dormancy emerged
via minimal modifications to a simple cellular automaton. We demonstrate that
dormancy confers stability, defined as population persistence over time (Figs.
2, S1). To address the effects of shallow vs. deep dormancy, we included a
spore survivorship parameter ($\alpha$) that also introduces stochasticity
into the model. As predicted, extinction times increase with spore
survivorship (Fig. 4). Unexpectedly, however, the benefits of dormancy do not
require the accumulation of inactive individuals into a large or long-lived
seed bank. Instead, dormancy promotes population persistence through slight
increases in spore lifetimes (Fig. 5), modest contributions to the metabolic
processes regulating demographic rates (Fig. 6), and feedbacks created by the
emergence of unique periodic configurations (Fig. 1 b). Our findings reveal
that small changes to a relatively simple cellular automaton can yield complex
behaviors that are consistent with theoretical expectations of how dormancy
stabilizes populations in fluctuating environments. In the following sections,
we discuss these findings in the context of what is known about dormancy from
other computational models and how our results may guide future research
efforts, including empirical studies relating to resiliency of host and
environmental health.

### Scale-dependence of dormancy

Dormancy is governed by a range of distinct processes that may operate across
spatial and temporal scales [3]. Some drivers of dormancy occur seasonally and
integrate over large spatial extents (e.g., photoperiod) [26]. Dormancy can
also be triggered by conditions that arise at the local scale, including
density-dependent fluctuations in the abundance of individuals belonging to a
cohort that experience intraspecific competition for a limiting resource [27].
In Spore Life, dormancy dynamics are entirely controlled by the metabolic
transitions of individuals and their nearest neighbors, and not because of
external forces. These localized shifts in metabolism and resulting
intraspecific interactions then play out over the entire landscape and
determine the emergent dynamics of the population.

One scale-dependent phenomenon observed in our study relates to the effect of
dormancy in habitats (i.e., grids) of different sizes. In the Game of Life,
which lacks dormancy ($\alpha=0$), the proportion of active individuals
($\rho_{A}$) rapidly declines over time (Fig. 2). Consistent with an Allee
effect [28], a critical point is reached where local neighborhoods are so
depleted of active individuals ($N_{A}$) that new individuals cannot be born
($D\to A$), and the population effectively goes extinct. In Spore Life, where
$\alpha>0$, the opposite is true, in particular when $\alpha$ is large
($\alpha\to 1$). The proportion of active individuals is sustained at a much
higher level (Fig. 2) owing to the dormancy refuge ($A\to I$) and
resuscitation events ($I\to A$). In the two deterministic limits
($\alpha=0/1$), the effects of dormancy are consistent across simulations
independent of grid size. However, at intermediate levels of spore
survivorship, the effects of dormancy are more variable. Specifically, on
small grids with low- to intermediate-levels of $\alpha$, there is a higher
probability for elementary configuration to become fixed, which leads to low
$N_{A}$ and subsequent extinction. In contrast, populations persist longer
with low- to intermediate-levels of $\alpha$ on larger grids. This finite-size
effect may reflect specific features that are unique to our cellular
automaton. Alternatively, the model may be capturing a more general phenomenon
– albeit one that is not well-documented – where smaller populations, perhaps
in more fragmented habitats, are less likely to be rescued by dormancy.

### Shallow vs. deep dormancy

Dormancy is a process that leads to the formation of a seed bank, a
subpopulation of metabolically inactive individuals. There are many seed-bank
attributes that can affect the ecology and evolution of a population [5]. One
important attribute is seed bank size. A large seed bank comprised of many
inactive individuals provides more opportunities for resuscitating individuals
to buffer a population against extinction. Another important attribute is seed
bank turnover, which is influenced by the amount of time that dormant
individuals can survive while in a metabolically inactive state. Longer-lived
dormant individuals should provide more insurance to a population in a
fluctuating environment, provided they retain their ability to resuscitate. At
the same time, prolonged dormancy can be maladaptive because as it delays
growth and reproduction [29]. These general hypotheses led us to introduce the
parameter $\alpha$ into our model to explore how spore survivorship
contributes to the outcomes of dormancy. We found that the effects of dormancy
on the population were strongly affected by $\alpha$, namely that the
abundance of active individuals ($N_{A}$) increases with $\alpha$, with the
probability of extinction decreasing with $\alpha$.

Contrary to our expectations, spore survivorship ($\alpha$) did not have a
strong effect on seed bank size (i.e., $\rho_{I}$), or on the lifetime of a
spore ($T_{I}$). Instead, the effects of dormancy were achieved by individuals
that spent only a short time in an inactive state ($I$). The average lifetime
of a spore was quite low, on the order of only a single time step (or
generation). Meanwhile, the contribution of resuscitation ($I\to A$) to the
overall birth rate of active individuals was low ($<12\%$) compared to births
not directly associated with dormancy ($D\to A$). On the larger grids (e.g.
$300\times 300$), the density of inactive individuals ($\rho_{I}$) never
comprised more than 8% of the grid or 22% of the viable (i.e., active +
inactive) individuals. Seed bank size, defined as $N_{I}$:$N_{A}$, increased
with $\alpha$, but was generally low (0.02 - 0.27). While turnover increased
with $\alpha$, resuscitation with $\mathcal{N}_{A}=2$ living neighbors never
accounted for $>11\%$ of all births.

One potential explanation for population persistence in Spore Life (Fig. 2) is
sustained chaotic transients. In complex systems, the introduction of noise
can lead to super-persistent transient chaos [30]. Specifically, lattice-wide
noise could be generated in simulations with stochastic spore survivorship
($\alpha\in(0,1)$). However, this explanation is not consistent with observed
behavior in the deterministic limit of Spore Life ($\alpha=1$) where we see
clear stabilization of populations induced by dormancy.

Another possibility is that dormancy can stabilize population even when
inactive individuals are short-lived and seed banks are relatively small. Such
ideas are not well-developed in theoretical treatments of dormancy, although
biologists have noted ”cryptic” forms of metabolic stasis in some groups of
taxa [31, 32]. Thus, our findings suggest there may be unexplored regimes
where dormancy may be important for stabilizing populations, but this requires
the simultaneous development and integration of theory and empirical
approaches.

### How ”easy” is dormancy?

Dormancy has evolved independently many times throughout the tree of life. It
would be impossible, therefore, to create a single model representing all
features and attributes of dormancy as a diverse life-history strategy. As
such, our goal was to develop a compact model with the fewest number of rules
and assumptions. We find that Spore Life is successful in recapitulating a
range of canonical behaviors that are associated with dormancy, in particular
the dampening of population dynamics in a fluctuating environment [33].

That being said, there are other seed-bank attributes that could be
incorporated into future versions of Spore Life. In nature, dormant and active
individuals are typically not well-mixed [5]. For example, much of an actively
growing plant is found aboveground while dormant seeds, after being dispersed
from their natal sites, often reside belowground in the soil. This population
structure could be represented in the metabolic-transition rules of Spore Life
(Fig. 1 a). Also, there are ways that coarser-grained heterogeneity could be
encoded onto a grid, or even a 3D spatial lattice of a cellular automaton [34,
35], that would allow for tests of seed bank dynamics in a more complex
universe. One might expect that the spatial decoupling of dormant and active
individuals would reduce the reactivity of the system and give rise to larger
and longer-lived seed banks.

Furthermore, in the current version of Spore Life, there is only a single
condition when dormancy acts as a refuge (Fig. 1), which occurs due to
underpopulation when an active individual is surrounded by a single active
individual. However, initiation of dormancy could also be triggered by
crowding [36], for example, when an active individual is surrounded by $\geq
4$ active individuals (see Supporting Information). Such modifications would
affect the rule table in a way that would increase the size of the seed bank.

Last, while we incorporated stochasticity into spore survivorship, there are
opportunities to explore how mixtures of random and deterministic transitions
between metabolic states affect population dynamics [37]. Such efforts would
allow for exploration and synergy with other applications of dormancy-related
research, including those focused on evolutionary behaviors such as adaptive
dynamics [38]; interacting particle systems in statistical physics [39]; the
treatment of diseases, such as chronic infections [40] and cancer [41]; and
the response of multi-species communities to climate-change drivers [42].
Ultimately, our approach and finding have the potential to shed light on the
origins and diversification of a widespread trait characterized by complex
spatial and temporal dynamics, together with the associated metabolic feedback
loop.

## Acknowledgments

We acknowledge support from the US National Science Foundation (DEB 1934554
and DBI-2022049 to JTL), the US Army Research Office Grant (W911NF2210014 to
JTL), the National Aeronautics and Space Administration (80NSSC20K0618 to
JTL), and the Alexander von Humboldt Foundation (to JTL). We also acknowledge
Pat Wall for discussion during the early stages of this project, along with
Bulcsú Sándor and Kevin Webster for helpful comments. Code for the model and
analyses is available on GitHub: https://github.com/ISCOTTYI/spore-life

## Supporting Information

### Population fluctuations are Gaussian

In Fig. 2 and 3 we demonstrate how population dynamics change in response to
spore survival probability ($\alpha$). Increasing $\alpha$ leads to larger
average population sizes containing larger numbers of active individuals
($N_{A}$). This pattern is accompanied by decreased fluctuations in population
size, which are Gaussian distributed (Fig. S1).

Fig S1: Distribution of the population size For different spore survival
probabilities $\alpha$, we constructed the distribution of active population
densities, $N_{A}/\ell^{2}$, where we considered $\ell=300$ (top) and
$\ell=30$ (bottom). Shown are least-square Gaussian fits (lines) to the data
(circles), where in the smaller grid ($\ell=30$) we find the means
$\mu=0.13/0.23/0.29$ and standard deviations $\sigma=0.041/0.032/0.024$ for
$\alpha=0.3/0.7/1.0$, respectively. In the larger grid ($\ell=300$), means are
unchanged and standard deviations are smaller by a factor 10. The insets show
examples for the respective time series. For $\ell=30$ and $\alpha=0.3$ the
distribution overlaps with unstable densities which cause fluctuation induced
extinction (dashed line).

The means of the fitted Gaussians $\langle N_{A}\rangle$ increase near
linearly with the spore survivability $\alpha$, as demonstrated in Fig. 3 b.
We further find that the standard deviation decreases roughly linearly with
$\alpha>0$, where the treatment at small $\alpha\sim 0$ is numerically
demanding (see below). For smaller grids (e.g., $\ell=30$) fluctuations are
much broader than in larger grids (e.g., $\ell=300$), where the observed
standard deviations are scaled by a factor of 10. Importantly, for small
$\alpha$ a high standard deviation may cause fluctuation induced extinction
events, when the distribution overlaps with densities of actives cells that
are unstable towards extinction (Fig. S1, dashed line).

### Lifetimes of active and inactive cells close to GoL

In Fig. 5 b we did present the average lifetimes of active and inactive cells,
$\langle T_{A}\rangle$ and $\langle T_{I}\rangle$ respectively. Here we expand
upon the behavior close to the transition to the deterministic GoL ($\alpha$
small). For inactive cells, it is clear that the average lifetime $\langle
T_{I}\rangle=0$ at $\alpha=0$ for $\alpha\to 0$, as spores die in the GoL
limit (Fig. 1). The average lifetimes of active individuals, $\langle
T_{A}\rangle$, are however not well-defined when $\alpha\sim 0$. In this
parameter regime, the average extinction time, $\langle
T_{\mathrm{ext}}\rangle$, is low (Fig. 4 b). According to our definition, an
extinct system may contain active (and inactive) individuals in stable
configurations or blinkers, thus active individuals in an extinct state may
either stay active for eternity (or rather for the duration of the
simulation), or periodically die after a single time step. Taking the average
after population extinction therefore seems unreasonable. As a consequence, as
extinction times are short for small $\alpha$, there is no window for which a
stable transient state may be defined. It is hence not possible to measure, or
to define, the average lifetime of active cells in the GoL limit.

### Overpopulation induced dormancy

In unfavorable environmental conditions an individual may enter a dormant
state with reduced metabolic activity, which therefore increases the endurance
of life. External factors challenging organisms can be of very different
nature. So far, we took the neighborhood of a living cell as a predictor for
the quality of the environment (Fig. 1), where we assumed that harsh
conditions, and thus an impetus for dormancy, are present when the number of
active neighbors was low. Nevertheless, external factors challenging the
survival of individuals could arise also in the case of overpopulation, e.g.
when resources or habitat become scarce as a result of high population
densities. Overpopulation is hence an alternative route for dormancy to arise.
As a robustness check we implemented a modified version of Spore Life compared
to Fig. 1, where a further transition to the inactive state ($I$) is included
for active individuals with $\mathcal{N}_{A}=4$ living neighbors, viz for mild
overpopulation. While we find that absolute figures change in this modified
model, the aforementioned phenomenology remains valid.

Fig S2: Patterns Additional to Fig. 1 b we present periodic patterns $P_{n}$
of periods $n=2,3,4$ and $14$ found the deterministic Spore Life ($\alpha=1$).
See https://itp.uni-frankfurt.de/spore-life/ to interactively explore more
periodic patterns.

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# Overcoming the rate-distance limit of device-independent quantum key
distribution

Yuan-Mei Xie  Bing-Hong Li  Yu-Shuo Lu  Xiao-Yu Cao  Wen-Bo Liu  Hua-Lei Yin
<EMAIL_ADDRESS>Zeng-Bing Chen<EMAIL_ADDRESS>National Laboratory of
Solid State Microstructures, School of Physics, and Collaborative Innovation
Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

###### Abstract

Device-independent quantum key distribution (DIQKD) exploits the violation of
a Bell inequality to extract secure key even if the users’ devices are
untrusted. Currently, all DIQKD protocols suffer from the secret key capacity
bound, i.e., the secret key rate scales linearly with the transmittance of two
users. Here we propose a heralded DIQKD scheme based on entangled coherent
states to improve entangling rates whereby long-distance entanglement is
created by single-photon-type interference. The secret key rate of our scheme
can significantly outperform the traditional two-photon-type Bell-state
measurement scheme and, importantly, surpass the above capacity bound. Our
protocol therefore is an important step towards a realization of DIQKD and can
be a promising candidate scheme for entanglement swapping in future quantum
internet.

Quantum key distribution (QKD) Bennett and Brassard (IEEE Press, New York,
1984); Ekert (1991), based on intrinsic properties of quantum systems, can
extract a secret key from correlations against an all powerful adversary Lo
and Chau (1999); Shor and Preskill (2000); Mayers (2001). The QKD scheme has
been successfully implemented over long distances Yin et al. (2016); Boaron et
al. (2018); Fang et al. (2020); Chen et al. (2020); Yin et al. (2020).
Nevertheless, the implementation of these QKD protocols depends on perfect
characteristics of the underlying source or detection devices, which cannot
always be satisfied in practice. This dependence leads to various quantum
hacking strategies Zhao et al. (2008); Lydersen et al. (2010); Gerhardt et al.
(2011). By exploiting the violation of a Bell inequality, device-independent
QKD (DIQKD) Acín et al. (2007); Masanes et al. (2011); Vazirani and Vidick
(2014); Arnon-Friedman et al. (2018); Tan et al. (2020) can achieve secure key
distribution without detailed device characterization. This will close the gap
between theoretical analyses and practical realizations of QKD, enabling the
systems more reliable against such attacks. The unprecedented level of
security of DIQKD makes it the ultimate goal of researchers in the field of
quantum communications despite current experimental challenges Murta et al.
(2019).

Protocols for DIQKD rely on a Bell test, typically the Clauser-Horne-Shimony-
Holt (CHSH) test Clauser et al. (1969), which requires a pair of entangled
states shared between two distant systems. In traditional setups Acín et al.
(2007), distant entanglement is usually generated by photons distributed via
an optical fiber. Unfortunately, photon losses impair the entangled state
detection efficiency. Even by using novel techniques such as heralded qubit
amplifier Gisin et al. (2010) or local Bell test Lim et al. (2013), the secure
key rate of DIQKD is still extremely limited at practical distances. One
solution to the above limitation is heralded entangling scheme Bell (1980);
Żukowski et al. (1993), in which photon losses do not influence the entangled
state detection efficiency or fidelity. However, previous heralded schemes
suffer from low entangling rate, leading to a low key rate for long distance.
Besides, for the point-to-point lossy quantum channel without quantum
repeaters, the secret-key capacity Takeoka et al. (2014); Pirandola et al.
(2017) is restrained by the Pirandola-Laurenza-Ottaviani-Bianchi (PLOB) bound
Pirandola et al. (2017). To increase the key rate and transmission distance of
DIQKD, in this paper we propose a different heralded scheme utilizing single-
photon-type interference Wang et al. (2018); Ma et al. (2018); Cui et al.
(2019); Liu et al. (2019); Wang et al. (2019); Zhong et al. (2019); Lucamarini
et al. (2018); Minder et al. (2019), which was first proposed in the twin-
field QKD Lucamarini et al. (2018).

In previous heralded entanglement schemes Hofmann et al. (2012); Bernien et
al. (2013); Hensen et al. (2015), the two distant parties, Alice and Bob, each
generate a pair of qubits in the spin–photon entanglement
$\left|\phi^{+}\right\rangle=(\left|+z\right\rangle\left|+z\right\rangle+\left|-z\right\rangle\left|-z\right\rangle)/\sqrt{2}$,
where $\left|\pm z\right\rangle$ represent the eigenstates of Pauli operator
$\sigma_{z}$. They send one photon respectively to the central station
‘Charlie’, who could even be untrusted. Subsequently, Charlie performs a Bell-
state measurement (BSM) on the two received photons, causing a two-photon
interference followed by a coincidence count in Charlie’s detectors. Detection
of the photons heralds the projection of the spin qubits onto an entangled
state. By applying these schemes to DIQKD, the key rate is linearly dependent
on channel transmittance $\eta$.

In our heralded scheme, the generation of entanglement between distant systems
is based on entangled coherent states (ECSs) Wineland (2013). First, Alice and
Bob each prepare an entangled atom-light Schrödinger cat state, where cat
state refers to a superposition of two coherent states with opposite phases
and arbitrary amplitudes. Then they send their optical states to the beam
splitter (BS) of Charlie to generate a ECS by a single-photon interference,
leading to successful entanglement between Alice’s and Bob’s atoms Yin and
Chen (2019). In this way, the key rate of DIQKD will scale with $\sqrt{\eta}$,
making it feasible to overcome the PLOB bound. Although there still exist
difficulties generating entangled atom–light cat states with high rate, recent
works have shown great experimental progress. An entangled artificial atom-cat
state in a cavity has been created with high-fidelity state measurement
Vlastakis et al. (2015). Furthermore, deterministic realization of
entanglement state with atom–light Schrödinger cat states has been
successfully demonstrated Hacker et al. (2019). With this backdrop, we could
expect ECS to become a promising alternative to Bell state for intercity-
distance entanglement swapping in the near future.

When applying our heralded scheme to DIQKD, based on time-reversed idea Biham
et al. (1996), Alice and Bob can alternatively measure the atom spins they
have kept before Charlie performs the ECS measurement Hensen et al. (2015). In
this case, the requirement of quantum memories can be removed and repetition
rate can be increased. The scheme is detailed below and shown in Fig.1:

Figure 1: The heralded DIQKD schemes. (a) Alice and Bob each prepare a pair of
qubits in the spin-photon entangled state
$\left|\phi^{+}\right\rangle=(\left|+z\right\rangle\left|+z\right\rangle+\left|-z\right\rangle\left|-z\right\rangle)/\sqrt{2}$
and send one photon respectively to the central station Charlie. Entanglement
between the distant spins is achieved through a BSM performed by Charlie. (b)
Alice and Bob each prepare an entangled atom–light Schrödinger cat state
$\left|\Phi^{+}\right\rangle=(\left|+z\right\rangle\left|\alpha\right\rangle+\left|-z\right\rangle\left|-\alpha\right\rangle)/\sqrt{2}$
and send optical pulse respectively to the central station Charlie.
Entanglement between distant spins is realized through an ECS measurement
performed by Charlie.

Step 1.——Alice and Bob each generate an entangled atom–light Schrödinger cat
state
$\left|\Phi^{+}\right\rangle=(\left|+z\right\rangle\left|\alpha\right\rangle+\left|-z\right\rangle\left|-\alpha\right\rangle)/\sqrt{2}$,
where $\left|\pm\alpha\right\rangle$ are the coherent states with opposite
phases. Then Alice chooses to measure spin on her atom among three possible
directions $A_{0}$, $A_{1}$, and $A_{2}$, while Bob between two possible
directions $B_{1}$ and $B_{2}$, where
$A_{0}=B_{1}=\sigma_{z},~{}B_{2}=\sigma_{x},~{}A_{1}=(\sigma_{z}+\sigma_{x})/\sqrt{2},~{}A_{2}=(\sigma_{z}-\sigma_{x})/\sqrt{2}$.
All measurements have binary outcomes labeled by
$a_{i},~{}b_{j}\in\\{+1,~{}-1\\}$.

Step 2.——Alice and Bob send the optical pulses to interfere on the BS of an
intermediate station Charlie. Then Charlie performs an ECS measurement on the
two received pulses and broadcasts her detection results to Alice and Bob.
Define a successful detection as the case where one and only one of two
detectors clicks, denoted by D1 click or D2 click. When Charlie announces that
D1/D2 detector clicks, Alice and Bob keep their corresponding outcome
$\\{a_{i},~{}b_{j}\\}$. In addition, Bob flips his outcome $b_{1}$ if his
basis selection is $Z$ and in the meantime Charlie’s announcement is D2 click.

Step 3.——Alice and Bob announce their choices $A_{i}$ and $B_{j}$, over an
authenticated classical channel. The raw key is extracted from the pair
$\\{A_{0},~{}B_{1}\\}$. The quantum bit error rate (QBER) in Z basis is
defined as $e_{zz}=P(a_{0}\neq b_{1})$. The measurements
$A_{1},~{}A_{2},~{}B_{1}$ and $B_{2}$ are used on a subset of the particles to
estimate the CHSH value $S=\langle A_{1}B_{1}\rangle-\langle
A_{1}B_{2}\rangle+\langle A_{2}B_{1}\rangle+\langle A_{2}B_{2}\rangle$, where
the $\langle a_{i}b_{j}\rangle$ is defined as $P(a_{i}=b_{j})-P(a_{i}\neq
b_{j})$.

The key idea of heralded schemes is to record an additional signal announced
by Charlie, which indicates whether the entangled state was successfully
shared between Alice and Bob. When Charlie announces a successful detection
signal, the corresponding trial operated by Alice and Bob will be valid. That
is, failed entanglement distribution events can be excluded upfront from being
used in the trial. It needs to be pointed out that our secret key rate
analysis is under the assumption of collective attacks in the asymptotic
regime. By using the entropy accumulation theorem Arnon-Friedman et al.
(2018), our protocol can be proven secure against the general attacks in the
framework of universal composability. We let the probability of both Alice and
Bob choosing Z basis $p_{zz}\approx 1$ in the asymptotic limit. Therefore, the
key rate of our protocol is Acín et al. (2007)

$R=Q_{zz}\left[1-H_{2}(e_{zz})-H_{2}\left(\frac{1+\sqrt{(S/2)^{2}-1}}{2}\right)\right],$
(1)

where $Q_{zz}$ denotes the total gain of the $Z$ basis and
$H_{2}(x)=-x\log{(x)}-(1-x)\log{(1-x)}$ is the binary Shannon entropy
function.

We use the compressed subscript notation $x_{a,a^{\prime}}$
($x_{b,b^{\prime}}$) to indicate $x_{a},~{}x_{a^{\prime}}$
($x_{b},~{}x_{b^{\prime}}$), where the first label refers to Alice’s (Bob’s)
atom and the second to pulse sent by Alice (Bob). By combining the initial
states $\left|\Phi^{+}\right\rangle_{aa^{\prime}}$ and
$\left|\Phi^{+}\right\rangle_{bb^{\prime}}$, we obtain

$\displaystyle\left|\Phi^{+}\right\rangle_{aa^{\prime}}$
$\displaystyle\left|\Phi^{+}\right\rangle_{bb^{\prime}}=$ (2)
$\displaystyle\frac{1}{2\sqrt{2}}\Big{(}$
$\displaystyle\sqrt{N_{+}}\left|\phi^{+}\right\rangle_{ab}\left|\phi(\alpha)^{+}\right\rangle_{a^{\prime}b^{\prime}}+\sqrt{N_{-}}\left|\phi^{-}\right\rangle_{ab}\left|\phi(\alpha)^{-}\right\rangle_{a^{\prime}b^{\prime}}$
$\displaystyle+$
$\displaystyle\sqrt{N_{+}}\left|\psi^{+}\right\rangle_{ab}\left|\psi(\alpha)^{+}\right\rangle_{a^{\prime}b^{\prime}}+\sqrt{N_{-}}\left|\psi^{-}\right\rangle_{ab}\left|\psi(\alpha)^{-}\right\rangle_{a^{\prime}b^{\prime}}\Big{)},$

where the forms of four ECSs Jeong and Kim (2002) are
$\left|\phi(\alpha)^{\pm}\right\rangle=\frac{1}{\sqrt{N_{\pm}}}\big{(}\left|\alpha\right\rangle\left|\alpha\right\rangle\pm\left|-\alpha\right\rangle\left|-\alpha\right\rangle\big{)}$
and
$\left|\psi(\alpha)^{\pm}\right\rangle=\frac{1}{\sqrt{N_{\pm}}}\big{(}\left|\alpha\right\rangle\left|-\alpha\right\rangle\pm\left|-\alpha\right\rangle\left|\alpha\right\rangle\big{)}$.
$N_{\pm}=2(1\pm e^{-4\mu})$ are the normalization factors, with intensity of
coherent state $\mu=|\alpha|^{2}$. The four Bell states are
$\left|\phi^{\pm}\right\rangle=\frac{1}{\sqrt{2}}\big{(}\left|+z\right\rangle\left|+z\right\rangle\pm\left|-z\right\rangle\left|-z\right\rangle\big{)}$
and
$\left|\psi^{\pm}\right\rangle=\frac{1}{\sqrt{2}}\big{(}\left|+z\right\rangle\left|-z\right\rangle\pm\left|-z\right\rangle\left|+z\right\rangle\big{)}$.
Let two inputs of BS as $a$ and $b$ modes while the two outputs modes are
$c=(a+b)/\sqrt{2}$ and $d=(a-b)/\sqrt{2}$, detected by D1 and D2 detectors,
respectively. After passing through the lossless symmetric BS, the above four
ECSs are evolved into Yin and Chen (2019)

$\displaystyle\left|\phi(\alpha)^{+}\right\rangle_{a^{\prime}b^{\prime}}\stackrel{{\scriptstyle
BS}}{{\longrightarrow}}|even\rangle_{c}\left|0\right\rangle_{d},$ (3)
$\displaystyle\left|\phi(\alpha)^{-}\right\rangle_{a^{\prime}b^{\prime}}\stackrel{{\scriptstyle
BS}}{{\longrightarrow}}|odd\rangle_{c}\left|0\right\rangle_{d},$
$\displaystyle\left|\psi(\alpha)^{+}\right\rangle_{a^{\prime}b^{\prime}}\stackrel{{\scriptstyle
BS}}{{\longrightarrow}}\left|0\right\rangle_{c}|even\rangle_{d},$
$\displaystyle\left|\psi(\alpha)^{-}\right\rangle_{a^{\prime}b^{\prime}}\stackrel{{\scriptstyle
BS}}{{\longrightarrow}}\left|0\right\rangle_{c}|odd\rangle_{d},$

where $|even\rangle_{k}$ ($|odd\rangle_{k}$) means that the output mode $k$
contains even (odd) number of photons. For the case of lossless channel and
ideal photon-number-resolving detector, we can unambiguously discriminate the
four ECSs using photon-number parity measurement. However, since practical
detectors are threshold detectors, only the case with or without detector
clicks can be discriminate. For the states
$\left|\phi(\alpha)^{+}\right\rangle$ and
$\left|\psi(\alpha)^{+}\right\rangle$, the zero-photon component is close to
unit when we choose small intensity $\mu$, leading to the low click
probabilities. Therefore, we make only D1 (D2) click represent that the result
of the ECS measurement is the state $\left|\phi(\alpha)^{-}\right\rangle$
($\left|\psi(\alpha)^{-}\right\rangle$), which indicates the bell state
$\left|\phi^{-}\right\rangle$ ($\left|\psi-\right\rangle$) has been
successfully shared between Alice and Bob. As a result, the states
$\left|\phi(\alpha)^{+}\right\rangle$ and
$\left|\psi(\alpha)^{+}\right\rangle$ will always be mismeasured as quantum
states $\left|\phi(\alpha)^{-}\right\rangle$ and
$\left|\psi(\alpha)^{-}\right\rangle$, respectively. This introduces an
intrinsic error, making the CHSH value less than $2\sqrt{2}$ even in the case
of ideal threshold detectors and lossless channel.

Figure 2: Key rates of DIQKD based on Bell states and ECSs, with the inherent
loss of fiber $\beta=0.2~{}\mbox{dB/km}$, the efficiency of threshold detector
$\eta_{d}=0.8$ and the dark count rate $p_{d}=10^{-7}$. The dashed line
represents optimized intensity of coherent state $\mu$. For the heralded DIQKD
scheme with Bell states, the total gain of the $Z$ basis is
$Q_{zz}=(1-p_{d})^{2}\big{(}\frac{\eta^{2}}{2}+(4\eta-3\eta^{2})p_{d}+4(1-\eta)^{2}p_{d}^{2}\big{)}$,
the CHSH value is
$S=2\sqrt{2}\eta^{2}(1-p_{d})/[8\eta(1-2p_{d})p_{d}+8p_{d}^{2}+\eta^{2}(1-6p_{d}+8p_{d}^{2})]$,
and the QBER of the $Z$ basis is
$e_{zz}=\frac{1}{2}\big{\\{}1-\eta^{2}(1-2p_{d})/[8\eta(1-2p_{d})p_{d}+8p_{d}^{2}+\eta^{2}(1-6p_{d}+8p_{d}^{2})]\big{\\}}$.

It is convenient to define
$\chi_{\theta}:=\cos{\theta}\sigma_{z}+\sin{\theta}\sigma_{x}$. Thus we can
rewrite $A_{0}=B_{1}=\chi_{0}$, $A_{1}=\chi_{\frac{\pi}{4}}$,
$A_{2}=\chi_{-\frac{\pi}{4}}$, and $B_{2}=\chi_{\frac{\pi}{2}}$. The entangled
atom–light cat states along the direction $\chi_{\theta}$ can be expressed as

$\displaystyle\left|\Phi^{+}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}\left|+z\right\rangle\left|\alpha\right\rangle+\left|-z\right\rangle\left|-\alpha\right\rangle\Big{)}$
(4)
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}M^{+}\left|+\xi\right\rangle^{\theta}\left|\xi^{+}(\alpha)\right\rangle^{\theta}+M^{-}\left|-\xi\right\rangle^{\theta}\left|\xi^{-}(\alpha)\right\rangle^{\theta}\Big{)},$

where
$\left|+\xi\right\rangle^{\theta}=\cos{\frac{\theta}{2}}\left|+z\right\rangle+\sin{\frac{\theta}{2}}\left|-z\right\rangle$
and
$\left|-\xi\right\rangle^{\theta}=\sin{\frac{\theta}{2}}\left|+z\right\rangle-\cos{\frac{\theta}{2}}\left|-z\right\rangle$
represent the atom spin eigenstates in $\chi_{\theta}$ basis, corresponding to
outcomes $+1$ and $-1$, respectively.
$\left|\xi^{+}(\alpha)\right\rangle^{\theta}=\frac{1}{M^{+}}\big{(}\cos{\frac{\theta}{2}}\left|\alpha\right\rangle+\sin{\frac{\theta}{2}}\left|-\alpha\right\rangle\big{)}$
and
$\left|\xi^{-}(\alpha)\right\rangle^{\theta}=\frac{1}{M^{-}}\big{(}\sin{\frac{\theta}{2}}\left|\alpha\right\rangle-\cos{\frac{\theta}{2}}\left|-\alpha\right\rangle\big{)}$
are corresponding optical modes sent to Charlie, where
$M^{\pm}=\sqrt{1\pm\sin{\theta}\,e^{-2\mu}}$ are the respective normalization
factors.

The Eq.(4) means that when Alice (Bob) measures atom along $\chi_{\theta_{i}}$
($\chi_{\theta_{j}}$) and gets outcome $a_{i}$ ($b_{j}$), the joint quantum
states sent to Charlie are projected into
$\left|\xi^{a_{i}}(\alpha)\right\rangle^{\theta_{i}}$
($|\xi^{b_{j}}(\alpha)\rangle^{\theta_{j}}$) correspondingly. Considering the
case of ideal threshold detectors and lossless channel, we can get the total
gain of the $Z$ basis $Q_{zz}=1-e^{-2\mu}$ and CHSH value
$S=\sqrt{2}(1+e^{-2\mu})$, in violation of the CHSH-Bell inequality $S\leq 2$
when $\mu\leq 0.4407$.

For the practical case with threshold detectors and a lossy channel which is
assumed symmetrical for Alice and Bob, we have
$Q_{zz}=(1-p_{d})[1-(1-2p_{d})e^{-2\mu\eta}]$, the CHSH value
$S=2\sqrt{2}\big{\\{}\sinh(2\mu)-\cosh[2\mu(1-\eta)]+(1-p_{d})e^{-2\mu(1-\eta)}\big{\\}}/[e^{2\mu}-(1-2p_{d})e^{2\mu(1-\eta)}]$,
and the QBER of the $Z$ basis
$e_{zz}=p_{d}e^{2\mu(1-\eta)}/[e^{2\mu}-(1-2p_{d})e^{2\mu(1-\eta)}]$, where
$p_{d}$ is the dark count rate, $\eta=\eta_{d}\times 10^{-\frac{\beta
L/2}{10}}$, $\eta_{d}$ is the total efficiency of detector, $\beta$ denotes
the intrinsic loss coefficient of fiber channel and $L$ represents the
distance between Alice and Bob. Note that the factor $L/2$ in $\eta$ is due to
Charlie being placed midway between Alice and Bob.

We simulate the performance of the heralded DIQKD protocol utilizing different
light sources under practical experimental environments. Specifically, we
numerically optimize the secret key rate of ECS protocol over the free
parameter $\mu$. The simulation results are shown in Fig. 2. We can see that
the ECS protocol can exceed the Bell-state protocol when
$100~{}\mbox{km}<L<450$ km, and its key rate achieves 4 orders of magnitude
higher than the one using a Bell state at $400$ km. However, the maximum
transmission distance of ECS protocol is less than Bell-state protocol. This
zero key rate at long distance of ECS protocol is attributed to the intrinsic
error mentioned above, leading to CHSH value $S$ dropping dramatically when
losses are too severe.

In the following, we investigate the performances of our heralded DIQKD
protocol with ECSs considering misalignment errors. Define $\delta_{0}$ as the
fixed phase drift difference between the two pluses from Alice and Bob owing
to propagation in fibers, causing the optical misalignment error
$e_{d}=\frac{1-\cos{\delta_{0}}}{2}$. As a consequence, after going through
the channel and BS, the four joint coherent states would be involved into

$\displaystyle\left|\alpha\right\rangle\left|\alpha\right\rangle\xrightarrow[BS]{channel}|\sqrt{2(1-e_{d})}\,\alpha\rangle|\sqrt{2e_{d}}\,\alpha\rangle,$
(5)
$\displaystyle\left|-\alpha\right\rangle\left|-\alpha\right\rangle\xrightarrow[BS]{channel}|-\sqrt{2(1-e_{d})}\,\alpha\rangle|-\sqrt{2e_{d}}\,\alpha\rangle,$
$\displaystyle\left|\alpha\right\rangle\left|-\alpha\right\rangle\xrightarrow[BS]{channel}|\sqrt{2e_{d}}\,\alpha\rangle|\sqrt{2(1-e_{d})}\,\alpha\rangle,$
$\displaystyle\left|-\alpha\right\rangle\left|\alpha\right\rangle\xrightarrow[BS]{channel}|-\sqrt{2e_{d}}\,\alpha\rangle|-\sqrt{2(1-e_{d})}\,\alpha\rangle.$

By applying Eq.(5) to the input state
$\left|\xi^{a_{i}}(\alpha)\right\rangle^{\theta_{i}}|\xi^{b_{j}}(\alpha)\rangle^{\theta_{j}}$,
we find $Q_{zz}=(1-p_{d})[e^{-2\mu\eta e_{d}}+e^{2\mu(-\eta+\eta
e_{d})}-2(1-p_{d})e^{-2\mu\eta}]$ and $S=\sqrt{2}w_{ed}/[e^{2\mu(1-\eta
e_{d})}+e^{2\mu(1-\eta+\eta e_{d})}-2(1-p_{d})e^{2\mu(1-\eta)}]$, where
$w_{ed}=2\sinh[2\mu(1-\eta e_{d})]-2\cosh[2\mu(1-\eta+\eta
e_{d})]+2(1-p_{d})e^{-2\mu(1-\eta)}$. The QBER of the $Z$ basis under
misalignment errors is $e_{zz}=[e^{2\mu(1-\eta_{d}+\eta
e_{d})}-(1-p_{d})e^{2\mu(1-\eta)}]/[e^{2\mu(1-\eta e_{d})}+e^{2\mu(1-\eta+\eta
e_{d})}-2(1-p_{d})e^{2\mu(1-\eta)}]$.

Figure 3: Key rates of DIQKD based on ECSs under different optical
misalignment errors, with the inherent loss of fiber
$\beta=0.2~{}\mbox{dB/km}$, the efficiency and dark count rate of threshold
single-photon detector $\eta_{d}=0.8$ and $p_{d}=10^{-7}$. The dashed line
represents the CHSH value S under the misalignment error $e_{d}=0.01$, the
black line represents the PLOB bound
$R_{\mbox{PLOB}}=-\log_{2}{(1-\eta_{AB})}$, where $\eta_{AB}=\eta_{d}\times
10^{-\beta L/10}$ is the efficiency between Alice and Bob.

We draw the variation of the key generation rates with different optical
misalignment errors $e_{d}$ and the variation of CHSH value S under the
misalignment error $e_{d}=0.01$, as shown in Fig. 3. Even if the misalignment
error is as large as $7\%$, the key rate can still largely exceed the absolute
PLOB bound, which indicates the robustness of our improved method under great
noise. Here we note that our protocol is not a point-to-point scheme. Thus it
can overcome the PLOB bound without using quantum repeaters.

In summary, we introduce a novel heralded DIQKD scheme that uses ECSs to
establish entanglement between distant users. Compared with former heralded
protocols, the key rate of our protocol is increased by approximately $2-4$
orders of magnitude at intercity distances. Additionally, our protocol is able
to exceed the PLOB bound when $L>150$ km with practical settings such as dark
counts, detector efficiency and misalignment errors. These improvements make a
proof-of-principle experimental demonstration of DIQKD more feasible. Future
work can focus on developing new techniques with the ECS source to reduce the
intrinsic error rate, so that the secure transmission distance can be further
increased. Because the ECS measurement devices are identical to the BSM
devices, our process is compatible with the previous infrastructure. We could
therefore envision an unconditional secure global quantum network combined
with ECS scheme at short and medium distances and traditional Bell-state
scheme at long distance.

##  I Acknowledgments

We gratefully acknowledge support from the National Natural Science Foundation
of China (61801420), the Key Research and Development Program of Guangdong
Province (2020B0303040001)and the Fundamental Research Funds for the Central
Universities.

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