# Policies for multiplexed quantum repeaters: theory and practical performance
analysis

Stav Haldar<EMAIL_ADDRESS>Hearne Institute for Theoretical Physics,
Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
Louisiana 70803, USA  Pratik J. Barge<EMAIL_ADDRESS>Hearne Institute for
Theoretical Physics, Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, Louisiana 70803, USA  Xiang Cheng
<EMAIL_ADDRESS>Fang Lu Mesoscopic Optics and Quantum Electronics
Laboratory, Department of Electrical and Computer Engineering, University of
California, Los Angeles, California 90095, USA  Kai-Chi Chang
<EMAIL_ADDRESS>Fang Lu Mesoscopic Optics and Quantum Electronics
Laboratory, Department of Electrical and Computer Engineering, University of
California, Los Angeles, California 90095, USA
Brian T. Kirby<EMAIL_ADDRESS>DEVCOM Army Research Laboratory,
Adelphi, MD 20783, USA  Tulane University, New Orleans, LA 70118, USA  Sumeet
Khatri<EMAIL_ADDRESS>Dahlem Center for Complex Quantum Systems,
Freie Universität Berlin, 14195 Berlin, Germany  Chee Wei Wong
<EMAIL_ADDRESS>Fang Lu Mesoscopic Optics and Quantum Electronics
Laboratory, Department of Electrical and Computer Engineering, University of
California, Los Angeles, California 90095, USA  Hwang Lee<EMAIL_ADDRESS>Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA

###### Abstract

Future quantum networks will have nodes equipped with multiple quantum
memories, providing the possibility to perform multiplexing and distillation
strategies in order to increase fidelities and reduce waiting times for end-
to-end entanglement distribution. In this paper, we introduce two policies
that adapt the well-known swap-as-soon-as-possible (swap-asap) policy to
multiplexed quantum repeater chains. Unlike the usual, fully local swap-asap
policy, these policies are “quasi-local”, making effective use of knowledge of
the states of the repeaters along the chain to optimize waiting times and end-
to-end fidelities. Our policies also make use of entanglement distillation. We
demonstrate via simulations one of our key findings, which is that these
policies can outperform the well-known and widely studied nested purification
and doubling swapping policy in practically relevant parameter regimes. Our
work also provides the tools to carefully examine the role of entanglement
distillation. We identify the parameter regimes in which performing
distillation makes sense and is useful. In these regimes, we also address the
question: “Should we distill before swapping, or vice versa?” We thus
formalize the trade-off between the advantages of adding distillation
capabilities to quantum networks against their technological and practical
challenges. Finally, to provide further practical guidance, we propose an
experimental implementation of a multiplexing-based linear network, and
experimentally demonstrate the key element, a high-dimensional biphoton
frequency comb (BFC). We then evaluate the anticipated performance of our
multiplexing-based policies in such a real-world network through simulation
results for two concrete memory platforms, namely rare-earth ions and diamond
vacancies.

##  I Introduction

The quantum internet [1, 2, 3] has the potential to revolutionize current
computation, communication, and sensing technologies by enabling exchange of
quantum data. Numerous significant applications have already been recognized,
encompassing areas such as secure quantum communication [4, 5], distributed
quantum computation [6, 7], and distributed quantum sensing [8, 9, 10], to
name a few. However, enormous hardware improvements are needed before a
practical quantum internet is realized [11, 12]. Currently, a significant
challenge lies in effectively distributing entanglement amongst network nodes,
particularly over significant distances. Recent, state-of-the-art experiments
have been performed for only a handful of nodes over short distances [13, 14].
This challenge stems from the delicate nature of quantum systems, leading to
issues like photon losses, imperfect measurements, and quantum memories with
short coherence times. Since comprehensive, fault-tolerant quantum error
correction has yet to be realized to overcome these obstacles, it becomes
important to explore how much we can overcome these limitations through other
means. This problem of designing effective protocols for entanglement
distribution using noisy, imperfect quantum hardware has been addressed in
multiple analytical [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33] and numerical [34, 35, 36, 37, 38, 39, 40, 41, 42, 43]
studies, among others; see also Refs. [44, 45, 46, 47, 48, 49, 50] for
reviews.

In the context of entanglement distribution in quantum repeater chains, the
“single-channel” approach has been commonly studied. In the single-channel
setting, every repeater is connected via a single channel to each of its two
neighbors, with a single quantum memory at the endpoint of each channel. These
channels are used to create elementary links of entangled qubit pairs between
the neighboring nodes. Then, using a particular policy for performing
probabilistic entanglement-swapping operations [51, 52], the elementary links
are transformed into a virtual link between the two end nodes of the repeater
chain, which can be further used for exchanging quantum information via the
quantum teleportation protocol [51]. Noise and loss along the channels, along
with quantum memories with short coherence times, result in long waiting times
and/or low fidelities of the end-to-end entangled links.

Figure 1: (a) Difference between the parallel and multiplexing approaches.
Unlike the parallel approach, cross-channel elementary link generation and
entanglement swapping is permitted in the multiplexing approach. (b)
Elementary links are created by multiple entanglement generation sources
associated with every channel that connects different pairs of quantum
memories. (c) Non-neighbouring nodes are connected by virtual links generated
by performing entanglement swapping operations between neighbouring active
quantum memories. (d) Linear chain of quantum repeaters with multiple quantum
memories capable of establishing multiple physical links, entanglement
distillation and entanglement swapping.

In this work, we look at the multiplexing approach [17] as a potential method
for reducing the overall waiting time for entanglement generation in linear
networks and increasing the fidelity of end-to-end entangled links; see Fig.
1. In a multiplexing protocol, every repeater contains multiple memories,
which are connected by multiple channels, as shown in Fig. 1(a). This allows
for simultaneous elementary link generation attempts. Furthermore, when
performing entanglement swapping, it is possible to go beyond the trivial
parallel approach and perform cross-channel entanglement swapping; see Fig.
1(a). There is also the possibility to perform entanglement
purification/distillation [53, 54, 55, 56].

Despite the significant benefits of approaches based on multiplexing and/or
distillation already studied and demonstrated in the literature [15, 16, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74], many open
questions remain in terms of efficient and optimal entanglement swapping and
distillation policies for linear quantum repeater chains. In particular, many
of the aforementioned works on multi-memory quantum repeaters with
multiplexing have made use of the “nested purification and doubling swapping”
protocol devised in the seminal works [15, 16] on quantum repeaters. We
therefore ask the following two questions.

  1. 1.

Is the nested purification and doubling swapping policy optimal, particularly
in the practically-relevant parameter regimes of low coherence times and low
elementary link success probabilities?

  2. 2.

What is the best way to combine entanglement swapping with entanglement
distillation? Should we distill first before swapping, or vice versa?

###  I.1 Overview of results

In this work, we start by extending the framework from Ref. [43] for modeling
quantum repeater chains in terms of Markov decision processes to the case of
multi-memory quantum repeater chains (see Sec. II). Then, we use this model to
address the two questions above.

Our main results and contributions comprise four parts.

  1. 1.

New policies for multiplexed quantum repeaters.

In Sec. III, we define our multiplexing-based policies. These policies are
generalizations of the so-called “swap-as-soon-as-possible” (swap-asap) policy
[29, 30] to the multiplexing setting. We refer to them as _strongest neighbor_
(SN) swap-asap, _farthest neighbor_ (FN) swap-asap, and _random_ swap-asap.
These policies are distinguished by how they pair memories within nodes for
the purposes of entanglement swapping. Intuitively, the SN swap-asap policy
prioritizes the creation of high-fidelity end-to-end links, and FN swap-asap
prioritizes lowering the waiting time for the creation of end-to-end links.
These policies are also “quasi-local”, requiring some knowledge of the network
state but not full, global knowledge.

In Sec. IV, we compare the performance of these policies and show that the SN,
FN, and random swap-asap policies can significantly reduce the average waiting
time compared to the parallel, non-multiplexed version of swap-asap, and at
the same time provide higher end-to-end link fidelities.

Notably, the SN and FN policies can outperform the widely used doubling
protocol for entanglement swapping in certain parameter regimes. More
specifically, in Sec. IV.1, we see improved performance of our policies in
more resource-constrained scenarios, such as high channel losses, high qubit
noise (small coherence time of quantum memories), and when the number of
multiplexing channels is small and/or the number of network nodes is large.
These parameter regimes are relevant for near-term quantum network
implementations.

  2. 2.

Policies with entanglement distillation.

Then, in Sec. V, we add distillation to our FN and SN swap-asap policies, and
we address the following questions: i) “When is distillation useful?” ii)
“Should we distill before swapping, or swap first and then distill?” We
demonstrate via simulations that these policy considerations have a strong
influence on the performance of the network. As such, with our study, we
provide the tools to weigh the technological and practical challenges of
adding distillation abilities to quantum networks against the advantages of
adding such an ability.

  3. 3.

The impact of classical communication costs.

In Sec. VI, we consider the classical communication (CC) costs associated with
our policies. By incorporating the time needed for CC into our model, we show
that our multiplexing-based FN and SN swap-asap policies outperform fully-
local policies that do not require any CC. We thus show that the benefit
gained by acquiring information about the network’s state is retained even
when the communication costs of acquiring such information is accounted for.
This result provides much needed evidence in support of exploring policies
that use knowledge of the network state, as promoted in previous literature
[42, 41, 43]. We also show that, even when CC costs are accounted for, the FN
and SN swap-asap policies can outperform the doubling policy in parameter
regimes corresponding to resource-constrained hardware settings.

  4. 4.

Design and analysis of real-world repeater chains.

In Sec. VII, we outline a proof-of-principle experimental realization of a
quantum network equipped with multiplexing policies using a high-dimensional
bi-photon frequency comb (BFC), and we map the relevant experimental
parameters to their counterpart parameters within our modeling framework.
Then, in Sec. VII.2, we assess the performance of our multiplexing-based
policies within these parameter regimes for two quantum memory platforms,
diamond vacancy and rare-earth metal ions.

##  II Linear chain quantum network with multiple channels

Throughout this work, we consider entanglement distribution in a linear chain
network of quantum repeaters. In this section, we present our theoretical
model for entanglement distribution in such networks; see also Fig. 1.

  * •

Nodes: The linear chain is made up of $n$ nodes. Every node contains multiple
quantum memories. Specifically, every node has $2n_{ch}$ quantum memories.
Given the linear nature of our networks, $n_{ch}\in\\{1,2,\dotsc\\}$
corresponds to the number of channels connecting the nearest-neighbor nodes.
(In Fig. 1, $n_{ch}=3$.) Every memory has a finite coherence time of
$m^{\star}\in\\{0,1,\dotsc\\}$ time steps, which specifies the maximum number
of (discrete) times steps that qubits can be stored in the memory.

  * •

Elementary and virtual links: Entanglement sources establish
physical/elementary links between nearest-neighbour quantum memories with
probability $p_{\ell}\in[0,1]$. A probabilistic Bell state measurement creates
a virtual link between distant quantum memories with success probability
$p_{sw}\in[0,1]$. These values are fundamentally limited by the technology
used to create the system, e.g., linear optics makes $p_{sw}\leq 0.5$, and
$p_{\ell}$ is determined by properties of the source, loss characteristics of
the sources, etc. These considerations are relevant for our proposed
experimental implementation in Sec. VII, and we provide details on how to
determine $p_{\ell}$ and $m^{\star}$ from physical properties of the system in
Sec. VII.1.

We keep track of the fidelity of both elementary and virtual links by
assigning every link an age. The age $m$ of a link starts from $m=m_{0}$, when
it is first created, with $m_{0}$ being a measure of its initial fidelity with
respect to a perfect Bell state. The age then increases in discrete steps,
such that $m\in\\{0,1,2,3...\\}$. Once the age of the link reaches
$m^{\star}$, the link is discarded. Throughout this work, we consider a
particular Pauli noise model for decoherence of the memories (see Appendix A),
such that the fidelity $f(m)$ of a link that has age $m$ is given by

$f(m)=\frac{1}{4}(1+3\mathrm{e}^{-\frac{m}{m^{\star}}}).$ (1)

The age of a link formed by a successful entanglement swapping operation is
determined by adding the ages of the two corresponding links: if $m_{1}$ and
$m_{2}$ are the ages of the two links, then if the entanglement swapping is
successful the age of the new link is $m^{\prime}=m_{1}+m_{2}$. This “addition
rule” for the age of virtual links is a direct consequence of the Pauli noise
model that we use, and a proof can be found in Ref. [43, Appendix C].

  * •

Entanglement distillation: Multiple low-fidelity links between any two nodes
can be distilled with probability $p_{ds}\in[0,1]$ to form a higher-fidelity
link. Throughout this work, we consider the BBPSSW protocol [53], and we
provide details on it in Appendix B.

Figure 2: (a) Illustrative example of the FN swap-asap policy. Links on both
sides of the $k^{\text{th}}$ node are ranked separately on the basis of their
lengths. The longest link is given a rank of 1, while shortest is ranked 3.
For entanglement swapping, we thus give first preference to links of rank 1
(marked in red), which would result in the longest possible link. Being a
probabilistic process, if the entanglement swapping fails, then next
preferences is given to links of rank 2, and so on. (b) In the case of the SN
swap-asap policy, link rankings are based on their ages. Preference is given
to links with lower ages.

Entanglement distribution protocols progress in a series of discrete time
steps, based on the Markov decision process model developed and used in Ref.
[43], which we refer to for further details on the model. In every time step,
the following events occur.

  1. 1.

Check the number of active links to the right of every node (except for the
right-most node). If there are inactive links, request the corresponding
elementary link.

  2. 2.

Check the number of active links to the left and right of every node, except
for the end nodes. If more than one link on either side is active, rank them
based on the policy, as described in detail in Sec. III below. If the doubling
policy is being used, pair the links for entanglement swapping only if they
are the same length. Perform the entanglement swapping operations in the
ranked order. Entanglement swapping is attempted only if the sum of the ages
of the two links is less than $m^{\star}$.

  3. 3.

If entanglement distillation is part of the policy, attempt to distill all
links between the same two nodes for a given number of rounds, as dictated by
the policy; see Sec. V for details. If the policy calls for distilling first,
then swapping, interchange steps 2 and 3.

  4. 4.

Increase the age of all active links by one time unit, accounting for
decoherence. If classical communication (CC) overheads are to be accounted
for, then add the CC time to the age of every active link; see Sec. VI for
details. Discard any link which has age greater than $m^{\star}$

##  III Policies for multiplexing

In this section, we introduce our two multiplexing policies for entanglement
distribution in a linear chain quantum network.

When multiple links are available between nodes of the network, there are many
possibilities for how to perform entanglement swapping. There is also the
possibility to perform entanglement distillation between nodes that share
several links. In linear quantum networks without multiplexing (i.e.,
$n_{ch}=1$ in Fig. 1), one of the simplest and best studied swapping protocols
is swap-as-soon-as-possible (swap-asap). As the name suggests, the swap-asap
policy dictates that entanglement swapping should be attempted as soon as two
memories in a given node become active, i.e., as soon as an entanglement
swapping policy becomes possible. swap-asap policies have been shown to be
better than fixed nesting or doubling policies for a large set of parameter
regimes in linear chain networks consisting of a single channel between nodes
[30, 25, 75]. The policies we present in this work are generalizations of the
swap-asap policy to the multiplexing setting.

The direct application of swap-asap in multiplexed linear chains is
unspecified due to the increased degrees of freedom. In particular, two
immediate questions arise:

  1. 1.

If the opportunity is available simultaneously, should we preferably perform
entanglement swapping on links that result in a virtual link between
_farthest_ nodes or the ones that result in a _strongest_ —highest
fidelity—virtual link? Based on this question, we define the following two
generalizations of swap-asap for multiplexed linear chains.

    1. (a)

The Farthest Neighbor (FN) swap-asap policy: This policy prioritizes the
creation of virtual links between faraway nodes. Accordingly, we rank the
links based on their lengths (farthest links are ranked first), and then
entanglement swapping is performed in that order. This policy is illustrated
with an example in Fig. 2.

    2. (b)

The Strongest Neighbor (SN) swap-asap policy: This policy prioritizes the
creation of virtual links with high fidelity. Accordingly, we rank the links
based on their ages (links with the lowest age are ranked first), and then
entanglement swapping is performed in that order.

For comparison, we also define the random swap-asap policy. In the random
swap-asap policy, links are randomly paired to perform entanglement swapping
operations, without any consideration of their length and/or age. We also
define the parallel swap-asap to be the non-multiplexed version of the usual
swap-asap policy.

  2. 2.

What should be the preference when it comes to the order of entanglement
distillation and swapping—should we distill first and then swap, or the other
way around? We call these two distinct options the distill-swap and swap-
distill policies. Both of these may be used in conjunction with the SN and FN
entanglement swapping policies defined above, leading to four distinct policy
combinations.

The motivation behind our definitions of the FN and SN swap-asap policies is
based on prior works [42, 41, 43], which have shown that policies that make
use of global knowledge of the state of the repeater chain offer an advantage
in terms of waiting time and fidelity. At the same time, such global knowledge
comes with considerable classical communication cost and hence is not
practical for networks with a large number of nodes and long link distances.
This limitation motivated the proposal of so-called quasi-local policies in
Ref. [43], which are policies that use global knowledge and coordination
between small groups of nodes, but still use local policies such as swap-asap
to connect such groups to each other. The SN and FN swap-asap policies being
defined here are also quasi-local policies: the nodes do not have knowledge of
the entire network, but they have information about other nodes they are
connected to. We stress here that access to such information comes with
classical communication (CC) costs, but such costs will be much smaller than
fully global policies, in which each node knows about the state of every other
network node at all times. More precisely, for fully global policies the CC
cost in each time step will be proportional to the communication time from one
end of the network to another. This would place considerable demands on the
coherence times of quantum memories used in the network. On the other extreme,
for fully-local policies all CC can be deferred to one round of end-to-end
communication at the end of the entanglement distribution protocol. Quasi-
local policies try to find a middle ground between these extremes.

Throughout the rest of this work, we demonstrate via simulations that our
quasi-local SN and FN swap-asap multiplexing policies can outperform (in terms
of waiting time and end-to-end fidelity) not only non-multiplexed or parallel
policies but also fully-local multiplexed policies in which decisions about
entanglement swapping are made without utilising any knowledge of the
network’s state,

like in the _random_ multiplexing policy. We also demonstrate via simulations
that our policies can outperform the well-known and widely-used doubling and
nested-purification-and-swapping protocol.

Before proceeding, we remark that very recently in Ref. [76], the idea of
using fidelity-based ranks to match links for entanglement swapping (similar
in spirit to our SN swap-asap policy) was proposed, but the focus was on
second-generation repeaters with error-correction capabilities. Our focus here
is on first-generation repeaters. See Refs. [77, 78] for the definitions of
first- and second-generation quantum repeaters.

##  IV Performance of multiplexing based policies

Having defined our SN and FN swap-asap policies in the previous section, we
now evaluate their performance. We consider two figures of merit: the average
waiting time and the average age of the youngest end-to-end link. In this
section, we measure the average waiting time in terms of the number of time
steps. This unitless, abstract quantity can be converted into physical times,
and also the ages converted to fidelity values, by incorporating the network
hardware and design parameters. Examples of such a translation are given in
Sec. IV and further details are provided in Sec. VII.1 and Appendix A.

Figure 3: Average waiting time and average age of the youngest end-to-end link
for a repeater chain with $n=7$ nodes, according to the multiplexed (SN and FN
swap-asap) and non-multiplexed (parallel swap-asap) policies. The multiplexed
versions can enormously reduce average waiting time compared to parallel swap-
asap, and at the same time increase end-to-end link fidelities. SN is the
better choice for fidelity enhancement, while FN reduces the waiting time more
significantly.

Intuitively, based on their definitions, the FN swap-asap policy aims to
create the longest possible links, with the aim of creating an end-to-end link
in the smallest possible number of time steps. Thus, the FN swap-asap policy
qualitatively prioritizes minimizing the average waiting time. On the other
hand, the SN swap-asap policy aims to create the strongest or highest-fidelity
links between the end nodes of the network. In order to confirm this intuitive
picture, and to ascertain the benefit of multiplexing with these two policies,
we benchmark the FN and SN policies against $n_{ch}$ copies of the usual swap-
asap run in parallel. In Fig. 3, we show that the multiplexed versions can
indeed, as expected, enormously reduce the average waiting time compared to
the parallel version of swap-asap, and at the same time provide higher end-to-
end fidelities.

In Fig. 3, and for all of the plots shown in this paper, the parameter values
that we choose are based on what is computationally feasible, but they are
also based on what we expect with actual hardware, as we outline in Sec.
VII.1. The choice of network parameters such as $p_{\ell},m^{\star}$ etc. in
specific examples is also motivated by the intent to show the most prominent
and overarching trends.

Figure 4: Average waiting time and average age of the youngest end-to-end link
for a repeater chain with $n=7$ nodes, according to the FN, SN, and random
swap-asap multiplexing policies. SN and FN both outperform the random policy,
indicating the benefit of using knowledge of the network state. SN is the
better choice for fidelity enhancement, while FN reduces the waiting time more
significantly.

The advantage of our SN and FN policies is not only based on the trivial
advantage one would expect by shifting from a parallel policy to a
multiplexing policy. Even amongst multiplexing policies, the SN and FN
policies can be advantageous. Indeed, the fact that we assign ranks to
different links and use this knowledge to perform entanglement swapping also
provides considerable advantage. To show this advantage, we compare SN and FN
multiplexing policies against the random swap-asap multiplexing policy defined
above. As expected, both SN and FN policies perform better than the random
policy, as we show in Fig. 4. We also see that the FN policy provides lower
average waiting times compared to the SN policy, and the SN policy provides a
higher-fidelity end-to-end link compared to the FN policy, verifying our
intuition.

Figure 5: Heat map of average waiting time (left) (denoted $\left<wt\right>$)
and ratio of the average age of youngest end-to-end link to the cutoff (right)
as a function of elementary link success probability and number of channels.
Contours of fixed values run along diagonals from the top-left to the bottom-
right, showing that lower link probabilities can be compensated for by
increasing the number of channels. Both the average waiting time and the
average age of the youngest end-to-end link reduce as the number of channels
increases, showing the usefulness of multiplexing. For the waiting time
results we use the FN policy, and for the average age results we use the SN
policy.

Next, we look at how increasing the number $n_{ch}$ of channels affects the
average waiting time of the repeater chain. Increasing $n_{ch}$ increases the
effective $p_{\ell}$ of the chain according to $p_{\ell}\to
1-(1-p_{\ell})^{n_{ch}}$; therefore, in Fig. 5, approximately constant values
of the waiting time run along the diagonal from the top-left to the bottom-
right, indicating that as the elementary link probability goes down, in order
to keep the average waiting time and end-to-end fidelity constant, we should
increase the number of channels. In other words, for a given average waiting
time or end-to-end fidelity requirement, low elementary link probabilities can
be compensated for by increasing the number of multiplexing channels.

Figure 6: Heat map of average waiting time (left) (denoted by
$\left<wt\right>$) and ratio of the average age of youngest end-to-end link to
the cutoff (right) as a function of link cutoff age and number of channels.
Contours of fixed values run along diagonals from the top-left to the bottom-
right, showing that lower cutoffs, which translates to memories with smaller
coherence times, can be compensated for by increasing the number of channels.
For waiting time results we use the FN policy and for average age results we
use SN policy.

In Fig. 6, we fix the elementary link and entanglement swapping success
probabilities while varying $n_{ch}$ and the coherence time $m^{\star}$. Here
we see a similar trend, a low value of the memory coherence time can again be
compensated for by increasing the number of channels for both figures of
merit.

Of course, FN, SN, and random are not the only three possible options for
multiplexing-based policies. For example, one could develop a ranking scheme
based on a combination of age and distance, or one could perform a search over
all possible pairings of links, looking for the optimal pairings in terms of
some figure of merit (such as the sum of lengths of all virtual links formed).
With the respect to such optimal ranking schemes, it is not clear how close
the SN and FN schemes are to being optimal; they are, however, simple and
intuitive policy choices that require no optimization in terms of pairing of
links for entanglement swapping, and hence are practical for near-term
implementation. We defer the discussion of other policy alternatives and
comments on the optimality of SN and FN swap-asap to Appendix D.

###  IV.1 Comparison to doubling

Let us now compare our policies with those in existing literature. In
particular, we compare our policies with one widely studied in previous works,
namely, the _doubling_ (or nesting) policy; see, e.g., Refs. [15, 16, 58, 59,
64, 79]. In this policy, which only applies to repeater chains with $2^{N}$
elementary links, the lengths of links are always doubled. As an example,
consider the following situation. Suppose two adjacent elementary links are
formed, an entanglement swap is performed successfully, and a link of length
two is obtained. Now, in contrast to the swap-asap policy, this length-two
link cannot be extended by swapping it with links of arbitrary lengths—links
of length two must be swapped with links of length two only—which means that
we must wait for another link of length two to be created before an
entanglement swap can be attempted. In the meantime, the first produced
virtual link keeps aging. We can thus see that this doubling policy is more
restricted compared to swap-asap. Accordingly, it was shown in Ref. [75] that
for low elementary link success probabilities and moderate swapping success
probabilities (such as 50%, as in the case of linear optics), the swap-asap
policy outperforms the doubling policy, although infinite coherence times for
memories was assumed in that work.

Figure 7: Average waiting time (top) and average age of the youngest end-to-
end link (bottom) for a repeater chain with $n=9$ nodes (eight elementary
links), according to the FN, SN, and random swap-asap policies, compared to
the SN doubling policy. For small values of $p_{\ell}$, both the FN and SN
swap-asap policies outperform the doubling policy in terms of waiting time.
The SN swap-asap policy outperforms doubling for the average age for all
values of $p_{\ell}$ up to $p_{\ell}=0.5$.

Now, within the multiplexing scenario, the doubling policy can be implemented
by ranking links based on age when more than one option of entanglement
swapping is available. Thus, we call this policy _Strongest-Neighbor doubling_
or SN doubling. Our aim now is to see if, within the multiplexing setting, and
with finite coherence times for the quantum memories, our FN and SN swap-asap
policies can outperform a doubling based multiplexing policy. This is what we
show in Fig. 7. In terms of the average waiting time, the FN swap-asap policy
outperforms the SN doubling policy for all values of link success probability
$p_{\ell}$ up to $p_{\ell}=0.5$, including very low values of $p_{\ell}$,
which is often most relevant in practice. It is important to mention here that
this result is not true for all parameter regimes; for high values of
$p_{\ell}$ and $m^{\star}$, the doubling policy can actually outperform our
swap-asap policies. Our results are consistent with the results in previous
works. Specifically, our results generalize those in Ref. [75] to the case of
finite coherence times.

At the same time, our results also show the importance of ranking the links in
an appropriate way, reflected in the fact that the _random_ swap-asap policy
and the SN swap-asap policies actually perform worse than the doubling policy
in terms of average waiting time. We see very similar results in terms of the
average age of the youngest end-to-end link. With respect to this figure of
merit, the SN swap-asap policy outperforms all other policies discussed here,
including the SN doubling policy. Below, in Sec. V, we also include
distillation in our swap-asap policies and the doubling policy to compare
their performance, since the latter has also been widely discussed in existing
literature; see, e.g., Refs. [15, 16, 58, 59, 64, 79]. The doubling policy is
also _quasi-local_ in the same sense as SN and FN swap-asap, because a node
needs to know the length of its links, i.e., it needs to know which nodes it
is connected to. This is an important consideration when accounting for
classical communication overheads of this policy, which we do in Sec. VI.

##  V Multiplexing policies with entanglement distillation

We now consider entanglement distillation with the probabilistic BBPSSW
protocol [53], which allows for distillation of two entangled links to one. We
provide a summary of the protocol in Appendix B. Briefly, if $f_{1}$ and
$f_{2}$ are the fidelities of the two links being distilled, then the success
probability $P_{\text{distill}}(f_{1},f_{2})$ of the protocol and the
resulting fidelity $F_{\text{distill}}(f_{1},f_{2})$ are given by

$\displaystyle P_{\text{distill}}(f_{1},f_{2})$
$\displaystyle=\frac{8}{9}f_{1}f_{2}-\frac{2}{9}(f_{1}+f_{2})+\frac{5}{9},$
(2) $\displaystyle F_{\text{distill}}(f_{1},f_{2})$
$\displaystyle=\frac{1-(f_{1}+f_{2})+10f_{1}f_{2}}{5-2(f_{1}+f_{2})+8f_{1}f_{2}}.$
(3)

The purpose of this section is to assess the usefulness of entanglement
distillation for quantum repeater chains with multiple memories per repeater.
In particular, in this section, we examine the extent to which entanglement
distillation can compensate for the loss of fidelity incurred due to
entanglement swapping and waiting (decoherence).

###  V.1 Initial considerations

Recall that entanglement swapping of two links of ages $m_{1}$ and $m_{2}$
results in a virtual link of age $m_{1}+m_{2}$, and waiting increases the ages
of all active links by one time unit. Similarly, as we show in Appendix B, the
age after distillation with the BBPSSW protocol is given by the following
formula:

$m^{\prime}=\left\lceil
m^{\star}\log\left(\frac{15-6(f(m_{1})+f(m_{2}))+24f(m_{1})f(m_{2})}{32f(m_{1})f(m_{2})-2(f(m_{1})+f(m_{2}))-1}\right)\right\rceil,$
(4)

where $m_{1}$ and $m_{2}$ are the ages of the two links being distilled, the
function $f(m)$ is defined in Eq. (1), and $\lceil x\rceil$ is the smallest
integer greater than or equal to $x$.

We begin by noting that entanglement distillation is useful and worthwhile
only when it increases the fidelity of the distilled link as compared to the
fidelities of the original two links. In particular, the fidelity after
distillation should be greater than the highest-fidelity link of the two being
distilled, i.e., $F_{\text{distill}}(f_{1},f_{2})>\max\\{f_{1},f_{2}\\}$. In
terms of ages, this translates to the following condition for entanglement
distillation with the BBPSSW protocol to be useful:
$m^{\prime}<\min\\{m_{1},m_{2}\\}$, where $m^{\prime}$ is the age after
distillation, as given by Eq. (4). If $m^{\prime}\geq\min\\{m_{1},m_{2}\\}$,
then we might as well use the youngest of the two links rather than attempt
entanglement distillation and risk its failure. In particular, if one of the
ages is 0, corresponding to a perfect Bell pair, then distillation should not
be performed.

Another important consideration is that both links being distilled should be
entangled. If one of the links is not entangled, then it is better to simply
discard the unentangled link and keep the entangled one. We elaborate on this
point in Appendix B. With this in mind, let us note that under the Pauli noise
model that we consider, the fidelity according to Eq. (1) is bounded from
below by $0.25$ (in the limit $m\to\infty$), and it is equal to approximately
$0.5259$ when $m=m^{\star}$. The entanglement threshold for the states that we
consider is $0.5$, meaning that the link is entangled if and only if its
fidelity strictly exceeds $0.5$, and thus the age at which the link is no
longer entangled is $m=\lceil m^{\star}\log(3)\rceil\approx\lceil
1.09m^{\star}\rceil$; see Appendix A for the details. This means that,
depending on the value of $m^{\star}$, it can be useful to distill links that
are older than $m^{\star}$ time steps. However, the probability of successful
distillation falls with increasing ages of the links, and hence using a very
old link in conjunction with a very young link increases the likelihood of
wasting the high-quality young link. Thus, while distillation might be useful
in this scenario, it is not necessarily worthwhile. Also, for the moderate
values of $m^{\star}$ considered in our simulations, we choose to discard
links at $m^{\star}$ time steps, instead of the strict limit of $\lceil
m^{\star}\log(3)\rceil$, even when distillation is considered.

###  V.2 Distillation policies

Now, when there are multiple (more than two) links that can be distilled, and
we consider the pairwise BBPSSW protocol, the question of what is the optimal
distillation policy naturally arises, i.e., what is the optimal pairing of
links when performing entanglement distillation of more than two links. This
question has been addressed in existing literature; see, e.g., Refs. [16, 79].
Here, we consider the following policies.

  * •

Banded policy: Referred to as “scheme A” in Ref. [16], this involves
distillation of entangled pairs that have the same age. Assuming that $n_{ch}$
is a power of two, and assuming that all initial $n_{ch}$ pairs have the same
fidelity, they are all put into pairs and a first round of $\frac{n_{ch}}{2}$
distillation attempts is made. The resulting links are all of the same
fidelity and are paired again such that, in the next round, $\frac{n_{ch}}{4}$
distillation attempts are made. This procedure continues for
$\log_{2}(n_{ch})$ rounds, at the end of which we have one link remaining.
Assuming all of the initial links have fidelity $f_{0}$, the fidelity after
$r$ rounds is given by the recurrence relation
$f_{r}=F_{\text{distill}}(f_{r-1},f_{r-1})$. It has been shown in Ref. [16]
that, as long as the initial links are all entangled ($f_{0}>\frac{1}{2}$),
this policy can achieve arbitrarily high fidelities with an increasing value
of $n_{ch}$.

  * •

Pumping policy: Referred to as “scheme C” in Ref. [16], this policy proceeds
as follows. Two links of fidelity $f_{0}$ are distilled, then a fresh link of
fidelity $f_{0}$ is used to distill the previously distilled link, and so on.
This policy is tailored to a practical situation in which fresh links are used
to distill the surviving links of previous time steps, and thus the links
being distilled do not necessarily have the same fidelity, as in the banded
policy. If the initial fidelity satisfies $f_{0}>\frac{1}{2}$, the fidelity
$f_{r}$ after $r\in\\{1,2,\dotsc\\}$ rounds of the pumping policy is given by
the recurrence relation $f_{r}=F_{\text{distill}}(f_{r-1},f_{0})$. In Appendix
B.1, we provide a closed-form, analytical solution to this recurrence
relation, and using that solution we show that in the limit of arbitrarily
many rounds of pumping, the fidelity saturates as follows:

$\lim_{r\to\infty}f_{r}=\frac{6f_{0}-3+\sqrt{7-26f_{0}+28f_{0}^{2}}}{8f_{0}-2}.$
(5)

The limiting fidelity on the right-hand side of this equation is strictly less
than one whenever $f_{0}<1$. This proves that, in contrast to the banded
policy, with the pumping policy it is not possible to distill links to
arbitrarily high fidelity, even with an infinite number of rounds.

  * •

Distill as soon as possible: This is a sequential policy, which we introduce
in this work as a hybrid of the banded and pumping policies, and very similar
in spirit to the greedy policy defined in Ref. [79]. We sort the links in
increasing order of their ages, and then pair them in increasing order. After
one round of pairwise distillation, successfully distilled links are again
sorted age-wise and the distillation is attempted again, until only one
distilled link or no active link remains. This approach therefore uses the
benefits of the banded policy, by pairing links of very similar ages, and at
the same time, by realising the restrictions introduced by decoherence and
finite number of channels, by pumping fresh links into the queue of available
links as soon as possible. We therefore refer to this policy as “distill-as-
soon-as-possible” (distill-asap), and this is the policy that we consider for
our simulations in Sec. V.3 below.

Now, how should we combine entanglement distillation with entanglement
swapping? Should we distill first then swap—a policy we call distill-swap—or
the other way around, which we refer to as swap-distill? In Appendix B.2 and
Appendix C, we provide analytical calculations that compare the two policies
for a three-node chain. For the rest of this section, we compare the policies
with full-fledged simulations of long quantum repeater chains.

###  V.3 Distill-swap vs. swap-distill: simulation results

Having developed some intuition about the performance of the distillation
based policies through analytical results, let us now consider the fully
dynamic setting with long quantum repeater chains. Here, the policy that we
use is the distill-asap policy defined in Sec. V.2 above. We stress again that
the aim here is not to show optimality of our approach. The pairing of links
for distillation is an optimization problem in its own right and not the focus
of this work.

We now present simulation results for a five-node chain with five channels. A
somewhat large value of the cutoff is chosen ($m^{\star}=24$) to show some
important scaling behaviour with increasing age $m_{0}$ of the fresh (newly
created) links. Fig. 8 shows the average waiting time and average age of end-
to-end links as a function of the ages of the elementary links $m_{0}$ when
they are fresh, i.e., newly created. For both swap-distill and distill-swap,
we have fixed the entanglement swapping policy to FN swap-asap.

Figure 8: Average waiting time (top) and average age of end-to-end links
(bottom) as a function of the age $m_{0}$ of newly-created elementary links.
The entanglement swapping policy is fixed to FN swap-asap. It is clear that
swap-distill is the policy of choice in terms of reducing the waiting time
when fresh elementary links have high fidelity (low age); otherwise, distill-
swap performs better.

It is clear that swap-distill is the policy of choice in terms of reducing the
waiting time when fresh elementary links have high fidelity (low age);
otherwise, distill-swap performs better. At the same time, distill-swap always
performs better in terms of the age of the end-to-end link. This result
reinforces our intuition, developed from the previous analytical results. We
also verify that waiting time rapidly increases beyond a threshold value of
$m_{0}$ for both swap-distill and distill-swap, the threshold being smaller
for swap-distill. Beyond the $m_{0}$ values reported in Fig. 8, end-to-end
entanglement generation time increases so much in the simulations that we are
unable to find any reasonable estimate of an average value. Thus, for all
practical purposes, the waiting time is essentially infinite. This is also
indicated by the average ages approaching $m^{\star}$.

Some estimates of the thresholds up to which the two distillation policies are
effective can be made using some rather simple arguments. If the fresh
elementary links have age $m_{0}$, the age of the distilled link saturates to
a minimum value $m^{d}_{min}(m_{0})$ which is a function of $m_{0}$. In order
to get an end-to-end link via entanglement swapping of such distilled links,
the following condition must be satisfied:

$(n-1)\cdot m^{d}_{min}(m_{0})\leq m^{\star}.$ (6)

For the configuration in Fig. 8, the above condition translates to
$m^{d}_{min}\lessapprox 4$, which in terms of $m_{0}$ is $m_{0}\leq 5$, and
this is the value around which we see that the average waiting time diverges.

In the case of swap-distill, a typical trajectory to get an end-to-end link
would be to first perform $(n-2)$ swaps to get links of age $(n-1)\cdot
m_{0}$, which are then distilled to links of age $m^{d}_{min}((n-1)\cdot
m_{0})$. The threshold condition thus takes the following form:

$m^{d}_{min}((n-1)\cdot m_{0}))\leq m^{\star},$ (7)

but since distillation is ineffective when both links involved are above the
cutoff age, this condition is equivalent to

$(n-1)\cdot m_{0}\leq m^{\star}.$ (8)

This also explains why the threshold is lower for swap-distill, since
$m^{d}_{min}(m_{0})\leq m_{0}$. Again, for the configuration in Fig. 8, this
translates to $m_{0}\lessapprox 4$, which agrees well with the simulation
results. Furthermore, the threshold in the case of the swap-distill policy is
identical to a policy without distillation.

Figure 9: Average waiting time improvement factor for distillation with the
swap-distill and distill-swap policies (with FN swap-asap entanglement
swapping). When $p_{\ell}$ is low, distill-swap performs better, and when
$p_{\ell}$ is high swap-distill performs better. These are general trends and
in concurrence with the Fig. 10 (center), where for low $p_{\ell}$ distill-
swap is better but as $p_{\ell}$ increases swap-distill catches up.
Furthermore, for large $p_{\ell}$, distillation is either not very useful, and
can even be detrimental.

We further elaborate on our answer to the question of whether to distill first
or to swap first in Fig. 9. We plot the waiting time improvement factor for
distillation, which we define as the ratio of the average waiting time without
distillation to the average waiting time with distillation. Apart from the
quality of fresh links, as shown in Fig, 8, the choice between the two
distillation policies also depends on the other network parameters. As an
illustration, we show that for the parameters ($n,n_{ch},m^{\star},p_{\ell}$)
chosen in Fig. 9, when $p_{\ell}$ is low, distill-swap performs better, and
when $p_{\ell}$ is high swap-distill performs better. This can be understood
in light of analytical results in Appendix C and Fig. 8. In the regime of low
$p_{\ell}$, links are on average older and we have seen in the previous
discussions in this section that distill-swap performs better in this case
since long entanglement swaps can only be possible once the ages of the
participating links have been reasonably reduced; also, distilling after
entanglement swapping is highly improbable to be successful because the ages
are going to only increase due to the swapping. On the other hand, when
$p_{\ell}$ is high, swap-distill performs better, because links are on average
younger and much more conducive to successful entanglement swapping. Further
we see that for large $p_{\ell}$ distillation is either not very useful or
even detrimental. Therefore, the decision _to distill or not to distill_ also
should be taken after considering the network parameters. We explore this
point further in Fig. 10.

Figure 10: Average waiting time improvement factor for distillation with the
swap-distill and distill-swap policies (with FN swap-asap entanglement
swapping). Distillation is not very useful, in fact mostly detrimental, when
(top) fresh elementary links are perfect ($m_{0}=0$, $100\%$ fidelity), but
when fresh elementary links are imperfect (center) ($m_{0}=1$, approx. $90\%$
fidelity), a dramatic reduction in waiting time can be obtained by performing
distillation. (Bottom) Improvement factor for average age of youngest end-to-
end link using distillation over non-distillation based (FN) policy. swap-
distill provides negligible advantage but distill-swap provides substantial
advantage.

The discussion above leads us to the next question pertaining to the
usefulness of distillation-based policies. We are yet to show whether there is
any advantage in performing distillation. Distillation is a probabilistic
process, and even when it succeeds it leads to a reduction in the number of
active links. Hence, it is not manifest that a policy without distillation,
such SN or FN swap-asap alone, cannot perform better than its distillation-
based counterpart. Let us look at two cases: when fresh elementary links are
perfect, i.e., $m_{0}=0$, and when they are imperfect, say $m_{0}=1$ for
concreteness. In Fig. 10 we look at a seven-node chain with $n_{ch}=12$
channels, $p_{sw}=0.5$, and $m^{\star}=8$ ($m_{0}=1$ corresponds to roughly
$90\%$ fidelity in this case). We choose FN swap-asap as our swapping policy.
We show the average waiting time improvement factor for distillation-based
policies, defined as the ratio of average waiting time without distillation
and with distillation as a function of the elementary link success probability
$p_{\ell}$. An improvement factor of greater than 1 indicates an advantage in
using distillation and vice versa. We see in Fig. 10 (top) that distillation
is mostly detrimental when freshly produced elementary links are perfect (100%
fidelity). Both swap-distill and distill-swap increase the average waiting
time, for all values of the elementary link probability. Further, the
disadvantage is higher in the case of distill-swap. This can understood by the
following observation. The primary role of distillation in the case when
elementary links are perfect is to reduce the age of long or older links
obtained as a consequence of waiting for other links and/or via entanglement
swaps. The longest links tend to be the oldest ones in the network, due to the
$m_{1}+m_{2}$ age-update rule for entanglement swapping. If we distill first,
then initially, since most links are perfect, distillation is never invoked
and after entanglement swaps, when distillation is needed, we have to wait an
extra time step before distillation is attempted. This extra time step reduces
the success probability of distillation of already old links, following Eq.
(2). This is also the reason why swap-distill has some positive improvement in
some parameter regimes. Although not shown in Fig. 10, there exists a small
parameter regime of low $m^{\star},p_{\ell}$ and $n_{ch}$ in which swap-
distill leads to a small positive improvement (3-5%). The choice of parameters
in the figure is motivated by the intent to show the most prominent and
overarching trends. On the other hand, When elementary links are imperfect,
distillation is essential in increasing the scale, fidelity and throughput of
the network, indicated by the significant reduction in waiting time seen in
Fig. 10 (bottom). When the elementary link probability is low, distill-swap
performs much better, and when it is is high, swap-distill catches up in terms
of improvement. This is in agreement with the results shown in Fig. 8 and 9.
Although, the trends of improvement factor variation with increasing
elementary link probability change between Fig. 9 and 10. In terms of the end-
to-end link fidelity swap-distill provides negligible advantage but distill-
swap provides substantial advantage as shown in Fig. 10 (bottom).

####  V.3.1 Comparison with the SN doubling policy

Figure 11: Comparison between the SN doubling and FN and SN swap-asap policies
with distillation according to distill-swap. (Top) FN swap-asap provides
smaller average waiting times than SN doubling, similar to the results in Sec.
III. (Bottom) The SN swap-asap policy is able to outperform the SN doubling
policy when considering the average age of the youngest end-to-end link.

Let us now provide a comparison with the widely studied nested purification
and doubling swapping policy. For this comparison, we fix the distillation
order according to distill-swap.

In Fig. 11 (top) we first consider the average waiting time. We see that, FN
swap-asap outperforms doubling, similar to the results in Sec. III in terms of
average waiting time. Furthermore, when considering at average age of the
youngest end-to-end link, we see in Fig. 11 (bottom) that the SN swap-asap
policy is able to outperform the SN doubling policy by providing higher end-
to-end fidelity. These trends are similar to those obtained in Fig. 7 in Sec.
III, and further establish the superiority of swap-asap based multiplexing
policies over the doubling approach in generally resource-constrained
settings, and more specifically when finite coherence times for quantum
memories are considered.

###  V.4 Main messages for distillation

In this section, we have addressed the following questions.

  * •

_To distill or not to distill?_

Yes, when fresh elementary links are not perfect. In this case, distillation
leads to a significant improvement in both the average waiting time and the
average age of the end-to-end link.

No, when fresh elementary links are perfect. In this case, distillation offers
either either negligible advantage or is in fact detrimental for most
parameter regimes.

More generally, the more resource constrained the scenario, i.e., in terms of
network’s hardware parameters, the more useful distillation is.

  * •

_To distill first or to swap first?_

If links are generally low fidelity either due to low $p_{\ell}$ or low
initial fidelity, distilling first is more advantageous, in the opposite case
it is better to swap first.

More generally, in resource constrained scenarios, i.e., in terms of network’s
hardware parameters, it is more useful to distill first and then swap.

##  VI Classical communication cost for multiplexing based swap-asap policies

In a realistic near-term entanglement distribution setting with imperfect
memories, considering classical communication (CC) costs is imperative. Since
entanglement distribution at the elementary link level is not deterministic,
it must be heralded via classical signals. This heralding sets a natural time
scale for entanglement distribution through a network. The heralding time is
equal to the classical communication time between two adjacent nodes. For the
purposes of our simulation, this heralding time becomes the duration of one
time step. Since entanglement swapping is also often probabilistic, two
distinct classes of entanglement distribution approaches can be envisaged:

  * •

A _local_ approach, where all elementary link generation and entanglement
swapping attempts are made agnostic of the success or failure of prior swaps.
This would then mean that all elementary links (once generated successfully)
are retained up to the cutoff time. When an entanglement swap has failed,
resulting in both participating links to become inactive, the surrounding
nodes must act as if the link is still active until it has reached its
expected cutoff. All the classical communication about successful swaps is
then done at the end of the protocol. Such an approach is fully local and
therefore all policy decisions are taken by individual nodes, without any
collaboration or exchange of information. The benefit is that no classical
communication overheads exist except for heralding of elementary links, and
therefore every time step has duration equal to the heralding time only. The
cost is instead paid by having to wait for a link to reach its cutoff time,
even though it might have become inactive long before that.

  * •

A _global_ approach, on the other hand, allows end-to-end classical
communication amongst all nodes in every time step, such that the duration of
every time step is equal to $(n-1)\times\Delta t$, where $\Delta t$ is the CC
time across an elementary link. The waiting time is subsequently largely
dominated by the CC overheads, putting a huge burden on the coherence time
requirements of the memories. The benefit, of course, is that all nodes can
now communicate with each other and take decisions collaboratively and in an
adaptive fashion.

In previous works such as Refs. [42, 43], global policies were shown to
optimize the average waiting time and fidelity; however, classical
communication costs were not included. Furthermore, in Ref. [43], a “quasi-
local” approach to entanglement distribution policies was proposed, in which
there is knowledge of the network state up to some length scales only, and not
globally. In this case, an advantage over fully local policies could still be
obtained. We know investigate whether the advantages in waiting time and
fidelity gained by global and quasi-local knowledge and collaboration can be
retained when CC overheads are taken into account. The multiplexing based FN
and SN swap-asap policies that we have introduced in this work are quasi-
local, and they have classical communication (CC) overheads. Indeed, whenever
an entanglement swapping decision is made, all the memories at the node are
assumed to have knowledge of which nodes/memories they are connected to. This
information is needed to decide the order of entanglement swaps based on age
(SN) or length (FN) of the links. Thus, every node has some “quasi-local”
knowledge of the network state but not the “global” network state. In the
previous sections, the CC overheads of the FN and SN swap-asap were not taken
into account when determining the waiting time. Let us now take the CC
overheads into account. We limit ourselves in this section to considering the
FN and SN swap-asap policies without distillation, and hence the CC costs
discussed will only be the ones associated with entanglement swapping.

Below is a summary of how we have taken classical communication into account
in our model.

  * •

In order to include the CC overheads, we approximate the CC time by adding
$t_{cc}$ time steps to every network evolution time step, where $t_{cc}$ is
determined by the length of the longest link involved in any entanglement swap
occurring in that step. This allows enough time for the classical
communication for all entanglement swaps attempted in that time step, and at
the end of such a CC-accounted time step, all nodes are aware of which nodes
they are connected to and the ages of those links (which can also be
classically communicated). This is an upper bound to the CC cost, since in
practice nodes could perform the next required action as soon as they receive
the CC tagged for or relevant to them.

  * •

In order to benchmark our quasi-local policies fairly against fully-local
policies, some modifications need to be made to the way the network evolves
under the fully-local policies. These modifications are necessary, because
when entanglement swaps are probabilistic, if links need to be restarted after
a failed Bell state measurement, but before their actual memory cutoff, some
CC will be needed. Therefore, if a policy has to be fully-local and free of
any CC overheads, nodes must be agnostic to entanglement swapping failures.
All nodes, once they have a heralded entangled link, must retain them until
their anticipated memory cutoff age $m^{\star}$. Once this restriction is
made, the only available information to nodes that can be used to decide the
rank of states is the perceived or local ages of its memories. In other words,
every link in the network now has two ages: one is its real age, as determined
by probabilistic swaps in the evolution “in real-time”; the second is its
_perceived_ age, which is the age that the nodes that hold the link assume
without any knowledge of the outcomes of entanglement swaps. When the
perceived ages are used to take policy decision in the network, there is no CC
overhead during the evolution. We call such a policy _local age-based_
(ranking) multiplexed swap-asap policy. For such a policy, the CC cost is give
by only one round of end-to-end communication time, in the last time step.

Figure 12: Average waiting time (top) and average age of an end-to-end link
(bottom) as a function of the elementary link success probability when
classical communication times are included. We compare the quasi-local FN
swap-asap policy with the fully-local age-based swap-asap policy. The FN swap-
asap policy provides smaller values for both figures of merit even when CC
costs are included. Similar trends can also be seen for the SN swap-asap
policy.

Having discussed our models for classical communication, before presenting
simulation results, let us make a brief remark. In a full simulation, such as
those which can be done via quantum network simulators such as NetSquid [35],
QuISP [38], SeQUeNCe [36], etc., time stamps are assigned to each quantum
operation and tasks are queued, such that classical communication costs are
intrinsically kept track of. Our simulations focus on optimization of
policies, similar in spirit to _rule sets_ used by many of the above mentioned
simulators; see also Refs. [80, 81]. The task that we undertake here is
highlighting that CC costs are relevant to the performance of policies or rule
sets for entanglement distribution in a realistic setting.

Let us first look at the average waiting time and average age of the end-to-
end link for the FN swap-asap policy when CC costs are included, as a function
of the elementary link success probability. As mentioned above, we benchmark
this policy against the local age-based swap-asap policy. From Fig. 12, it is
clear that substantial advantage in terms of both figures of merit is retained
even when CC times are taken into consideration.

Figure 13: (Top) Average waiting time as a function of the number of nodes
when classical communication times are included. We compare the quasi-local FN
swap-asap policy with the fully-local age-based swap-asap policy. The FN swap-
asap policy provides smaller waiting times, even when CC costs are included.
More surprisingly the advantage grows with an increasing number of nodes. This
provides evidence of the scalability of the advantage gained by using
knowledge of the network state. (Bottom) Number distribution of lengths of
entanglement swaps in the FN swap-asap policy. We observe an exponential
suppression of entanglement swaps of the order of the length of the chain,
which is an important reason for the retained advantage of this policy, even
with classical communication included.

A more surprising observation is the fact that even as the number of nodes
increases, the advantage of FN swap-asap over the local age-based ranking
policy remains strong—in fact, the advantage increases with an increasing
number of nodes. In Fig. 13 (top), we show the average waiting times for the
local age-based ranking and the FN swap-asap policies as a function of the
number of nodes. Intuitively, we might have guessed that as the number of
nodes increases, so too would the CC cost, because longer and longer swaps
would be required to distribute entanglement between the end points of the
chain, making quasi-local policies worse off in terms of average waiting
times. It is therefore surprising that not only a large improvement using the
FN and SN swap-asap policies is obtained for long chains, but further that the
improvement increases with the growing scale (number of nodes) of the network.

Nonetheless, we can explain this non-intuitive result as follows. First, we
remind the reader that although fully-local policies have the advantage of no
CC overheads, they suffer in terms of average waiting time because due to the
agnostic entanglement swapping. Waiting to restart links that are already dead
is detrimental to the average waiting time. Second, a quasi-local policy, as
the name suggests, is not fully global, and hence CC costs in a given time-
step are upper-bounded by the length of the longest entanglement swapping
operation. We define the length of an entanglement swapping operation as the
length (in terms of number of nodes) of the longer of two links involved in
the swap. In a typical trajectory of a network, from completely disconnected
to an end-to-end connected link, for the FN swap-asap policy, the number of
entanglement swapping operations of length equal to the number of nodes is
roughly $O(1)$, i.e., roughly constant with respect to the number of nodes.
This is seen in Fig. 13 (bottom). The average number distribution of the
length of entanglement swaps shows an exponentially suppressed tail (linear
fall on the logarithmic scale). Therefore, time steps in a typical MDP
trajectory that contribute the highest to the CC overhead are rare. This
justifies and supports the use of quasi-local policies over global policies
for large networks, because in the latter, the CC cost is equal for each time-
step and equal to the full network size, while for quasi-local policies the CC
cost can be considerably less.

###  VI.1 Comparison amongst quasi-local policies

Figure 14: Comparison of average waiting times for different quasi-local
policies when classical communication costs are included. In the parameter
regime selected for this example, reflecting a relatively resource-constrained
scenario, SN swap-asap outperforms both FN swap-asap and SN doubling for the
low elementary link success probability ($p_{\ell}$) regime. This could be
attributed to preference given to shorter swaps by SN and hence the avoidance
of large CC costs. On the other hand, when $p_{\ell}$ is larger, SN doubling
performs the best, followed by SN swap-asap and then by FN swap-asap.
Therefore, in this regime it is better to perform moderately longer swaps to
minimize the waiting time.

We now compare the performance of different quasi-local policies when
classical communication (CC) costs are included. We saw in Fig. 7 that the FN
swap-asap policy provided the smallest waiting time amongst all quasi-local
policies and thus also outperformed the doubling policy. Further, we observed
that SN swap-asap performed worse than both SN doubling and FN swap-asap in
terms of the average waiting time for end-to-end connection. With CC costs
included, new interesting trends emerge in a very similar resource-constrained
scenario, as shown in Fig. 14. First, in the low $p_{\ell}$ regime, we see
that it is SN swap-asap that provides the smallest waiting time, followed by
FN swap-asap, and then very closely followed by SN doubling. This result can
be attributed to the preference given to the shortest swaps (youngest links
are generally the shortest ones) by SN swap-asap compared to both the FN swap-
asap and SN doubling policies. Consequently, SN swap-asap is able to avoid any
large CC costs. In a scenario of low $m^{\star}$, as considered in this
example, when $p_{\ell}$ is also low, most links are already very close to
their cutoff time when an entanglement swap can be attempted; thus, it is
essential to avoid any large CC costs, thus making SN swap-asap the policy of
choice. On the other hand, when $p_{\ell}$ is larger, SN doubling performs the
best, followed by SN swap-asap, and then by FN swap-asap. This result can be
attributed to the fact that, in this regime, it is better to perform
moderately longer swaps to minimize the waiting time. As mentioned above, we
have not considered distillation-based policies here, since our model does not
account for CC costs associated with distillation.

##  VII Proposed Experimental Implementation

Figure 15:  (a) Proposed implementation of multiplexing policy using a high-
dimensional biphoton frequency comb. (b) Measured frequency correlation matrix
of the biphoton frequency comb. (c) Polarization entanglement measurements of
a biphoton frequency comb. (d) Schmidt number and Hilbert-space dimensionality
scaling for different dimensions in a biphoton frequency comb.

Based on the multiplexed model introduced in Fig. 1, in this section we
propose a proof-of-principle experimental realization of a quantum network
with multiplexing using a high-dimensional biphoton frequency comb (BFC) in
Fig. 15. First, the BFC state is prepared by passing the spontaneous
parametric down-conversion (SPDC) photons generated from a type-II
periodically-poled KTiOPO4 (ppKTP) waveguide through a fiber Fabry–Pérot
cavity (FFPC) [82, 83]. Due to the type-II phase-matching condition, the
signal and idler photons can be efficiently separated by a polarization beam-
splitter (PBS), enabling deterministic BFC generation without post-selection.
The signal and idler photons are then temporally overlapped at a fiber
beamsplitter with orthogonal polarizations, by using a tunable delay line (DL)
and polarization controllers [82]. This configuration generates polarization
entangled BFC state

$|\psi\rangle=\sum_{m=-N}^{N}|\omega_{p}/2+m\Delta\Omega\rangle_{1}|\omega_{p}/2-m\Delta\Omega\rangle_{2}\\\
\otimes(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2}),$ (9)

where $2N+1$ is the number of cavity lines passed by an overall bandwidth-
limiting filter (the number of frequency modes), $\Delta\Omega$ is the free
spectral range of the FFPC, and $\omega_{p}$ is the pump frequency [83]. This
scalable frequency-polarization hyperentangled BFC state [82] facilitates the
multiplexing network policy by transmitting polarization qubits via different
frequency channels to quantum network nodes, where entanglement swapping is
performed in polarization basis. Specifically, within the context of our
theoretical model, $n_{ch}=2N+1$. As shown in Fig. 15(a), one output channel
of the beamsplitter is directed to Bob, and the other is connected to a
commercial dense wavelength-division multiplexing (DWDM) module for de-
multiplexing and sent to quantum network nodes, consisting of multiple quantum
memories. Each frequency channel is assigned to couple to one quantum memory
at the quantum network node. The entanglement swapping inside a node is
assisted by an auxiliary SPDC source, where the signal photons are sent to
Bell-state measurement and idler photons are sent to another memory to
establish a fiber link to the next node.

Fig. 15(b) shows the measured frequency correlation matrix of a BFC, using a
FFPC of 50 GHz free spectral range (FSR) and finesse of 10. We measured 9
correlated frequency-bin pairs between signal and idler photons with a type-II
SPDC source of $\approx$ 250 GHz phase-matching bandwidth. Fig. 15(c) shows
the measured high-quality polarization fringes for the BFC state before de-
multiplexing, which shows a raw fringe visibility up to 94.1 $\pm$ 0.5 $\%$
and violates the Clauser-Horne-Shimony-Holt (CHSH) inequality by 19 standard
deviations with score $S=2.666\pm 0.034$. This polarization state yields a
state fidelity of $93.3\pm 0.1\%$ with respect to a Bell state
$\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2})$.
Such high-fidelity polarization entanglement facilitates the entanglement
swapping between quantum memories.

We perform the Schmidt mode decomposition of the measured joint spectral
intensity [82], and summarize the extracted Schmidt mode number ($K$), and the
corresponding Hilbert space dimension ($K^{2}$). When reconstructing the
density matrix, we observe that the Schmidt mode number and Hilbert-space
dimension scales with the dimension $d$ of the biphoton frequency comb, as
shown in Fig. 15(d). The Schmidt mode number is $2.98$ with $d=3$, and
$K=7.03$ when we increase the dimension to $d=9$. The deviation from ideal
scaling in Fig. 15(d) mainly comes from the sinc function of the SPDC
spectrum, which can be circumvented by using a flat-top broadband SPDC source.
The Hilbert-space dimension can be further increased by using a SPDC source
with broader spectrum bandwidth and FFPC with smaller FSR. We also note that
by passing only the signal photons through a FFPC, the idler photons will
still exhibit a comb-like behavior [84, 85]. Such a scheme can provide higher
photon flux due to less filtering loss, and with potentially higher secure key
rates for quantum key distribution [86, 87, 88]. Although we have already
experimentally implemented the multiplexed source required for the proposed
experimental implementation, integration of the scheme with quantum memories,
implementing entanglement swapping, and distillation of links has not been
done yet. At the same time, there has been great progress in each of these key
steps for the construction of a long-distance quantum network.

###  VII.1 Calculation of elementary link probability, time step, and memory
cutoff

In this section, we show how to translate physical parameters into the values
of $p_{\ell}$, $m^{\star}$, and $\Delta t$ within our theoretical framework

##### Elementary link success probability $p_{\ell}$.

In general, we have that

$p_{\ell}=\eta_{r}\eta_{\ell}\eta^{2},$ (10)

where $\eta_{\ell}=\exp(-\frac{\ell}{12\text{ km}})$ is the photon-loss
contribution, with $\ell$ being the length of the elementary link. For
$\ell=40\text{ km}$, we have $\eta_{\ell}\approx 0.036$. The factor of
$\frac{1}{12\text{ km}}$ in the exponent corresponds to the 3.63 dB loss in
the 10 km channel reported in the experiment in Ref. [85]. In addition,
$\eta=0.69$ is the Debye–Waller factor, the ideal efficiency of the memory
(fraction of photons captured in the memory out of all the incident photons)
[89], for a rare-Earth metal based quantum memory [90], and $\eta_{r}=0.79$
(approximately 1 dB) is the loss from other residual factors, which could
include detector efficiencies, optical component loss, etc. [82]. Altogether,
we have that $p_{\ell}=\eta_{r}\eta_{\ell}\eta^{2}\approx 0.0134$.

##### Heralding time step $\Delta t$.

This is given by the elementary link distance $\ell$ as

$\Delta t=\frac{n\ell}{c}\,\text{seconds},$ (11)

where $n$ is the refractive index of the communication medium, $\ell$ is
length of the elementary link, and $c$ is the speed of light in vacuum.

##### Memory cutoff $m^{\star}$.

Using the memory coherence time of $T_{2}\approx 1\,\rm{ms}$, as for a rare-
Earth metal based quantum memory [90], we can find the memory cutoff
$m^{\star}$ as

$m^{\star}=\frac{T_{2}}{\Delta t}.$ (12)

For $\ell=40~{}\text{km}$, we have $m^{\star}=5$.

In Table 1, we calculate our model parameters for various values of the
elementary link distance.

$\ell$ (km) | $p_{\ell}$ |  $\Delta t$ (ms) | $m^{\star}$
---|---|---|---
25.0 | 0.05 | 0.125 | 8
15.0 | 0.11 | 0.075 | 13
5.0 | 0.25 | 0.025 | 40
Table 1: Parameters $p_{\ell}$, $\Delta t$, and $m^{\star}$ of our theoretical
model, calculated for varying elementary link distances $\ell$ based on the
physical aspects of the quantum channels and memories.

###  VII.2 Figures of merit and repeater chain design for existing memory
platforms

Let us now address the following question. Given a particular number $n_{ch}$
of channels for multiplexing, and given other physical parameters
corresponding to channel losses, quantum memory coherence times, etc., what is
the optimal number of repeater nodes for a linear quantum repeater chain
spanning a certain distance, and what would be the rate and fidelity of end-
to-end entanglement distribution for this optimal setting? Intuitively, we do
expect such an optimal number of nodes to exist, because as the number of
nodes increases, even though the individual links become smaller and thus
$p_{\ell}$ and $m^{\star}$ both increase, the non-deterministic nature of
entanglement swapping will at some point begin to adversely affect the waiting
time as the number of nodes increases.

To answer this question, let us set the end-to-end distance to be 100 km. We
also use the FN swap-asap policy for illustrative purposes, and we also
include the classical communication overheads, but we leave out distillation-
based policies. We also let the entanglement swapping success probability be
$p_{sw}=0.5$ and the number of channels for multiplexing be $n_{ch}=9$. The
other physical parameters are chosen to reflect the proposed experimental
implementation above. For the quantum memories, we choose two state-of-the-art
quantum memory platforms for our analysis, a rare-Earth metal based memory
($\rm{Pr}^{3+}$ ions) and a diamond vacancy based memory.

Figure 16: Average waiting time (top), including classical communication
overheads, and average fidelity of the end-to-end link (bottom) for
entanglement distribution along a 100 km repeater chain with rare-Earth
($\text{Pr}^{3+}$ ion) (left) and diamond vacancy (right) based memories using
the FN swap-asap policy. Even though the $T_{2}$ coherence times and
elementary link probabilities are relatively low, multiplexed policies provide
reasonable communication rates and end-to-end link fidelities. Using five to
six repeaters between the end nodes seems to be optimal for both memories in
terms of waiting times.

We plot the average waiting time and fidelity in Fig. 16. We tabulate the
values for the fidelity and entanglement distribution rate for the two memory
platforms in the optimal setting, along with the hardware parameters, in Table
2. The optimal number of nodes in terms of the waiting time seems to be around
five to seven for both memory platforms, which have $T_{2}$ times on the order
of a few milliseconds. From the minimum value of the waiting time, we
anticipate entanglement distribution rates ranging from a few tens of Hertz to
a few hundred Hertz, with an average end-to-end link fidelity of 70-75%. At
the same time, for the end-to-end fidelity, the results of the two platforms
do not have the same trends. For the diamond vacancy based memory, the
fidelity seems to not have its minimum at around the same point as the waiting
time, indicating a direct trade-off between the two figures of merit. On the
other hand, for the rare-Earth metal based memory, the fidelity decreases with
an increasing number of nodes.

Memory | $T_{2}$ | $\eta$ | $f_{0}$ | $p_{\ell}$ | $m^{\star}$ | $m_{0}$ | $R$ | $f$
---|---|---|---|---|---|---|---|---
Rare-Earth | 1.2 ms | 0.69 | 0.98 | 0.046 | 9 | 0 | 200 Hz | 0.72
Diamond | 13 ms | 0.20 | 0.89 | 0.008 | 156 | 24 | 50 Hz | 0.74
Table 2: Hardware parameters for two quantum memory platforms (rare-earth ion
and diamond vacancy), and their respective optimal entanglement distribution
rates ($R$) and end-to-end fidelities ($f$), for a 100 km repeater chain,
based on the data in Fig. 16. $T_{2}$ is the coherence time of the memory,
$\eta$ is the absorption efficiency (Debye–Waller factor), and $f_{0}$ is the
fidelity of a fresh elementary link. Also shown are the parameters $p_{\ell}$,
$m^{\star}$, and $m_{0}$ of our theoretical model.

##  VIII Conclusions and directions for future work

In this paper, we have looked at practical multiplexing-based policies for
long-distance entanglement distribution using quantum repeaters with multiple
memories. We have proposed two paradigmatic policies, namely farthest
neighbour (FN) swap-asap and strongest neighbour (SN) swap-asap, adapting
swap-as-soon-as-possible policies for multiplexing-based linear networks. We
have benchmarked these policies against a non-multiplexed, or parallel swap-
asap policy, a policy that exploits multiplexing but does not use knowledge of
the network state, and the widely studied doubling policy. We have shown via
simulations that the FN and SN swap-asap policies can yield a considerable
advantage over the alternative policies considered, both in terms of reducing
the average waiting time increasing the fidelity of the end-to-end link. These
advantages occur in the most relevant parameter regimes for near-term quantum
networks, which correspond to the most resource-constrained settings.

We then considered policies with entanglement distillation, and we proposed a
new policy that we call distill-asap. This policy combines the benefits of
existing distillation policies, like the banded, greedy and pumping
approaches, and thus is able to outperform the well-studied doubling policy
nested with distillation. We also provide answers to two important policy
questions related to distillation: When is distillation useful, and when it is
useful, should we distill then swap, or the other way around? In this
direction, the next question of immediate interest for future work is: “How
much to distill?” Indeed, in this work, we only considered entanglement
distillation policies that take $N$ links and distill them to one, but one
could instead distill $N$ links to some $K$ links greater than one, and such
protocols have been the study of recent work [91].

An important consideration for practical and scalable implementation of
entanglement distribution policies is the classical communication (CC) cost
associated with non-deterministic entanglement swapping. Here, we have
considered these overheads and shown that not only do multiplexed policies
retain their advantage over non-multiplexed policies when CC costs are
included, but this advantage can also increase with an increasing number of
nodes. This is a surpising and counter-intuitive result, and shows that
policies that use some knowledge of the network state (but not necessarily
full, global knowledge) can enhance network performance even when CC costs
associated with such knowledge are accounted for. This is a very important
conclusion from the point of view determining useful policies beyond the
fully-local ones, especially for large quantum networks. Our work provides
much needed evidence in support of the widely believed conjecture that
policies that utilize knowledge of the network state can outperform policies
that do not use such information.

Finally, to make the numerical and analytical analyses of this work more
concrete, we performed an experimental demonstration of multiplexing using a
high-dimensional biphoton frequency comb (BFC), which would form the backbone
of our proposed experimental implementation of a linear-chain quantum network
with multiplexing capabilities. We then assessed the anticipated performance
of such a network over 100 km for two concrete memory platforms, namely rare-
earth ion and diamond vacancy based quantum memories, when using our
multiplexing-based policies.

A universal principle that is evident from the results of this work is that
_policies must be tailored to the hardware parameters of the network._ This
motivates us to further explore the question of policies, given that no single
entanglement distribution policy can provide an advantage in all network
scenarios.

###### Acknowledgements.

The authors thank Thomas Searles and Sanjaya Lohani for helpful discussions.
This work was supported by the Army Research Office Multidisciplinary
University Research Initiative (ARO MURI) through the grant number
W911NF2120214. SH also acknowledges support from the RCS program of Louisiana
Boards of Regents through the grant LEQSF(2023-25)-RD-A-04, NSF grant
PHY-2110273, funds from Xairos Inc., and funds from the Hearne Institute for
Theoretical Physics. PB and HL also acknowledge the support of the U.S. Air
Force Office of Scientific Research as well as the US-Israel Binational
Science Foundation. SK acknowledges financial support from the German BMBF
(Hybrid).

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##  Appendix A Noise model and decoherence

We consider the following Pauli channel noise model for qubit decoherence [92,
93]:

$\mathcal{N}_{m_{1}^{\star},m_{2}^{\star}}(\rho)=p_{I}\rho+p_{X}X\rho
X+p_{Y}Y\rho Y+p_{Z}Z\rho Z,$ (13)

where $X,Y,Z$ are the single-qubit Pauli operators, defined as

$X\coloneqq\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad
Y\coloneqq\begin{pmatrix}0&-\mathrm{i}\\\ \mathrm{i}&0\end{pmatrix},\quad
Z\coloneqq\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (14)

and the probabilities $p_{I},p_{X},p_{Y},p_{Z}$ are defined as

$\displaystyle p_{I}$
$\displaystyle=\frac{1+\mathrm{e}^{-\frac{1}{m_{2}^{\star}}}}{2}-\frac{1-\mathrm{e}^{-\frac{1}{m_{1}^{\star}}}}{4},$
(15) $\displaystyle p_{X}$
$\displaystyle=\frac{1-\mathrm{e}^{-\frac{1}{m_{1}^{\star}}}}{4},$ (16)
$\displaystyle p_{Y}$
$\displaystyle=\frac{1-\mathrm{e}^{-\frac{1}{m_{1}^{\star}}}}{4},$ (17)
$\displaystyle p_{Z}$
$\displaystyle=\frac{1-\mathrm{e}^{-\frac{1}{m_{2}^{\star}}}}{2}-\frac{1-\mathrm{e}^{-\frac{1}{m_{1}^{\star}}}}{4},$
(18)

where $m_{1}^{\star},m_{2}^{\star}\in\\{1,2,\dotsc\\}$. This channel is the
Pauli-twirled version of the concatenated amplitude damping and dephasing
channels.

Let us also define the two-qubit Bell states as follows:

$\displaystyle|\Phi^{\pm}\rangle$
$\displaystyle\coloneqq\frac{1}{\sqrt{2}}(|0,0\rangle\pm|1,1\rangle),\quad\Phi^{\pm}\coloneqq|\Phi^{\pm}\rangle\\!\langle\Phi^{\pm}|,$
(19) $\displaystyle|\Psi^{\pm}\rangle$
$\displaystyle\coloneqq\frac{1}{\sqrt{2}}(|0,1\rangle\pm|1,0\rangle),\quad\Psi^{\pm}\coloneqq|\Psi^{\pm}\rangle\\!\langle\Psi^{\pm}|.$
(20)

##### Decoherence of an entangled qubit pair.

Now, let us suppose that the initial state of an entangled qubit pair is the
perfect maximally-entangled Bell state $\Phi^{+}$. Then, after
$m\in\\{1,2,3,\dotsc\\}$ time steps, it is straightforward to show that the
decohered entangled state is equal to

$\displaystyle(\mathcal{N}_{m_{1}^{\star},m_{2}^{\star}}\otimes\mathcal{N}_{m_{1}^{\star},m_{2}^{\star}})^{\circ
m}(\Phi^{+})$
$\displaystyle\quad=\frac{1}{4}\left((1+\mathrm{e}^{-2m/m_{1}^{\star}}+2\mathrm{e}^{-2m/m_{2}^{\star}})\Phi^{+}\right.$
$\displaystyle\qquad\left.+(1+\mathrm{e}^{-2m/m_{1}^{\star}}-2\mathrm{e}^{-2m/m_{2}^{\star}})\Phi^{-}\right.$
$\displaystyle\qquad\left.+(1-\mathrm{e}^{-2m/m_{1}^{\star}})\Psi^{+}\right.$
$\displaystyle\qquad\left.+(1-\mathrm{e}^{-2m/m_{1}^{\star}})\Psi^{-}\right).$
(21)

For a proof, we refer to Ref. [43, Appendix E]. This implies that the fidelity
of the state after $m$ time steps is
$\frac{1}{4}(1+\mathrm{e}^{-2m/m_{1}^{\star}}+2\mathrm{e}^{-2m/m_{2}^{\star}})$.

In order to connect with the results presented in the main text, for a given
value of $m^{\star}$ as presented in the main text, let us take

$m_{1}^{\star}=2m^{\star},\quad m_{2}^{\star}=2m^{\star}.$ (22)

In other words, we consider equal dephasing and amplitude damping. The
fidelity after $m$ time steps is then equal to

$f(m)=\frac{1}{4}(1+3\mathrm{e}^{-\frac{m}{m^{\star}}}).$ (23)

Note that we have chosen the value of $m_{2}^{\star}$ in Eq. (22) such that,
at the cutoff time $m^{\star}$, the fidelity of the entangled qubit pair is
$f(m^{\star})=\frac{1}{4}(1+3/e)\approx 0.5259$. We emphasize that our results
can be applied to other choices of $m_{1}^{\star}$ and $m_{2}^{\star}$—in
particular, choices that could take dephasing as the dominant source of noise
[94] or in order to make connections to prior works [58, 20, 23, 41, 30].

We can invert the fidelity function in Eq. (23), such that for a given
fidelity $F$, the corresponding age of the qubits is given by

$f^{-1}(F)=\left\lceil-m^{\star}\log((4F-1)/3)\right\rceil,$ (24)

for all $F\in(f(m^{\star}),1)$, where we take the ceiling $\lceil\cdot\rceil$
because we want an integer for the age.

We also remark that the decohered entangled state in Eq. (21) is a Bell-
diagonal state of the form

$(a+b)\Phi^{+}+(a-b)\Phi^{-}+c\Psi^{+}+c\Psi^{-}.$ (25)

As such, its entanglement can be characterized entirely by the quantity $a+b$,
namely, the fidelity of the state with respect to $\Phi^{+}$. In particular,
the state is entangled if and only if this fidelity is strictly greater than
$\frac{1}{2}$; see, e.g., [28, Chapter 2]. Therefore, under the assumptions in
Eq. (22), such that the fidelity is given by Eq. (23), we find that the state
in Eq. (21) is entangled for all time steps
$m\in\\{0,1,2,\dotsc,m^{\star}\\}$. Furthermore, inserting $F=\frac{1}{2}$ in
Eq. (24), we find that the age at which entanglement is lost is $\lceil
m^{\star}\log(3)\rceil$.

##  Appendix B Entanglement distillation with the BBPSSW protocol

In this work, the entanglement distillation protocol we consider is the BBPSSW
protocol introduced in Ref. [53]. This protocol, involving the distillation of
two entangled pairs to one, consists of a CNOT gate applied locally to each
pair of qubits, followed by measuring the local target qubits of the CNOTs in
the Pauli-$Z$ basis. If $f_{1}$ and $f_{2}$ are the fidelities of the two
entangled qubit pairs, then the success probability of the distillation
procedure is

$P_{\text{distill}}(f_{1},f_{2})=\frac{8}{9}f_{1}f_{2}-\frac{2}{9}(f_{1}+f_{2})+\frac{5}{9},$
(26)

and the fidelity of the resulting state is

$\displaystyle F_{\text{distill}}(f_{1},f_{2})$
$\displaystyle=\frac{1}{P_{\text{distill}}(f_{1},f_{2})}\left(\frac{10}{9}f_{1}f_{2}-\frac{1}{9}(f_{1}+f_{2})+\frac{1}{9}\right)$
(27)
$\displaystyle=\frac{1-(f_{1}+f_{2})+10f_{1}f_{2}}{5-2(f_{1}+f_{2})+8f_{1}f_{2}}.$
(28)

With this expression for the fidelity after entanglement distillation, if we
let $f_{1}=f(m_{1})$ and $f_{2}=f(m_{2})$, with $m_{1}$ and $m_{2}$ the ages
of the two entangled states, then we can calculate the updated age as
$m^{\prime}=f^{-1}(F_{\text{distill}}(f(m_{1}),f(m_{2})))$, with the
expression for $f^{-1}$ given in Eq. (24). In particular, the updated age
$m^{\prime}$ is given by:

$m^{\prime}=\left\lceil
m^{\star}\log\left(\frac{15-6(f(m_{1})+f(m_{2}))+24f(m_{1})f(m_{2})}{32f(m_{1})f(m_{2})-2(f(m_{1})+f(m_{2}))-1}\right)\right\rceil.$
(29)

Figure 17: Usefulness of distillation with the BBPSSW protocol [53]. The dark
blue shaded region indicates the values of the initial fidelities, $f_{1}$ and
$f_{2}$, for which the fidelity after distillation exceeds the fidelity of the
best of the two links, i.e., when
$F_{\text{distill}}(f_{1},f_{2})\geq\max\\{f_{1},f_{2}\\}$.

Now, distillation should be considered useful only when it can improve upon
the fidelity of the best of the two links. In other words, in order for
distillation to be useful, we should have
$F_{\text{distill}}(f_{1},f_{2})>\max\\{f_{1},f_{2}\\}$. In Fig. 17, we plot
the $(f_{1},f_{2})$ region defined by this inequality.

###  B.1 The pumping policy

Let us recall that the fidelity of the pumping policy after
$r\in\\{1,2,\dotsc\\}$ rounds is defined by the recurrence relation
$f_{r}=F_{\text{distill}}(f_{r-1},f_{0})$, where $f_{0}$ is the initial
fidelity. Using Eq. (28), we can write this recurrence relation as

$f_{r}=\frac{a_{1}f_{r-1}+a_{2}}{a_{3}f_{r-1}+a_{4}},\quad a_{1}\coloneqq
10f_{0}-1,\quad a_{2}\coloneqq 1-f_{0},\quad a_{3}\coloneqq 8f_{0}-2,\quad
a_{4}\coloneqq 5-2f_{0}.$ (30)

In this section, we provide a closed-form, analytical solution to this
recurrence relation.

######  Lemma 1.

Define the matrix $A\coloneqq\begin{pmatrix}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{pmatrix}$, where $a_{1},a_{2},a_{3},a_{4}$ are defined in Eq.
(30). For $r\in\\{1,2,\dotsc\\}$, consider the vector

$\vec{v}_{r}=\begin{pmatrix}v_{r,0}\\\ v_{r,1}\end{pmatrix}\coloneqq
A^{r}\vec{v}_{0},\quad\vec{v}_{0}=\begin{pmatrix}f_{0}\\\ 1\end{pmatrix}.$
(31)

It holds that $f_{r}=\frac{v_{r,0}}{v_{r,1}}$.

###### Proof.

We prove this by induction on $r$. For the base case of $r=1$, we have

$\begin{pmatrix}v_{1,0}\\\ v_{1,1}\end{pmatrix}=\begin{pmatrix}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{pmatrix}\begin{pmatrix}f_{0}\\\
1\end{pmatrix}=\begin{pmatrix}a_{1}f_{0}+a_{2}\\\
a_{3}f_{0}+a_{4}\end{pmatrix},$ (32)

which is consistent with Eq. (30). Now, the induction step: assuming that the
result holds for $r\in\\{1,2,\dotsc\\}$, using Eq. (30) we obtain

$\displaystyle f_{r+1}$
$\displaystyle=\frac{a_{1}f_{r}+a_{2}}{a_{3}f_{r}+a_{4}}$ (33)
$\displaystyle=\frac{a_{1}\frac{v_{r,0}}{v_{r,1}}+a_{2}}{a_{3}\frac{v_{r,0}}{v_{r,1}}+a_{4}}$
(34)
$\displaystyle=\frac{a_{1}v_{r,0}+a_{2}v_{r,1}}{a_{3}v_{r,0}+a_{4}v_{r,1}}.$
(35)

From this, we see that

$\displaystyle\begin{pmatrix}a_{1}v_{r,0}+a_{2}v_{r,1}\\\
a_{3}v_{r,0}+a_{4}v_{r,1}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{pmatrix}\begin{pmatrix}v_{r,0}\\\ v_{r,1}\end{pmatrix}$ (36)
$\displaystyle=A\cdot A^{r}\vec{v}_{0}$ (37)
$\displaystyle=A^{r+1}\vec{v}_{0}$ (38) $\displaystyle=\vec{v}_{r+1},$ (39)

which means that $f_{r+1}=\frac{v_{r+1,0}}{v_{r+1,1}}$. This completes the
proof. $\blacksquare$

######  Theorem 1.

The fidelity after $r\in\\{1,2,\dotsc\\}$ rounds of the pumping policy is
given by $f_{r}=\frac{\alpha_{r}}{\beta_{r}}$,where

$\displaystyle\alpha_{r}$
$\displaystyle=-(1-4f_{0}+6f_{0}^{2})\left(\left(2+4f_{0}-\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}-\left(2+4f_{0}+\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}\right)$
$\displaystyle\qquad+f_{0}\sqrt{7-26f_{0}+28f_{0}^{2}}\left(\left(2+4f_{0}-\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}+\left(2+4f_{0}+\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}\right),$
(40) $\displaystyle\beta_{r}$
$\displaystyle=-(3-8f_{0}+8f_{0}^{2})\left(\left(2+4f_{0}-\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}-\left(2+4f_{0}+\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}\right)$
$\displaystyle\qquad+\sqrt{7-26f_{0}+28f_{0}^{2}}\left(\left(2+4f_{0}-\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}+\left(2+4f_{0}+\sqrt{7-26f_{0}+28f_{0}^{2}}\right)^{r}\right).$
(41)

###### Proof.

We start with the eigenvalues and eigenvectors of the matrix
$A=\begin{pmatrix}a_{1}&a_{2}\\\ a_{3}&a_{4}\end{pmatrix}$. The eigenvalues
$\lambda_{\pm}$ and corresponding eigenvectors $\vec{x}_{\pm}$ are readily
found to be

$\displaystyle\lambda_{\pm}$
$\displaystyle=2+4f_{0}\pm\sqrt{7-26f_{0}+28f_{0}^{2}},$ (42)
$\displaystyle\vec{x}_{\pm}$ $\displaystyle=\begin{pmatrix}\omega_{\pm}\\\
1\end{pmatrix},\quad\omega_{\pm}=\frac{-3+6f_{0}\pm\sqrt{7-26f_{0}+28f_{0}^{2}}}{-2+8f_{0}}.$
(43)

In particular, we have that $A\vec{x}_{\pm}=\lambda_{\pm}\vec{x}_{\pm}$. Then,
the initial vector $\vec{v}_{0}=\begin{pmatrix}f_{0}\\\ 1\end{pmatrix}$ can be
expressed in terms of the eigenvectors $\vec{x}_{\pm}$ as

$\vec{v}_{0}=\begin{pmatrix}f_{0}\\\
1\end{pmatrix}=c_{1}\vec{x}_{+}+c_{2}\vec{x}_{-},\quad
c_{1}=\frac{f_{0}-\omega_{-}}{\omega_{+}+\omega_{-}},\quad c_{2}=1-c_{1}.$
(44)

We therefore find that

$\displaystyle\vec{v}_{r}$ $\displaystyle=A^{r}\vec{v}_{0}$ (45)
$\displaystyle=c_{1}A^{r}\vec{x}_{+}+c_{2}A^{r}\vec{x}_{-}$ (46)
$\displaystyle=c_{1}\lambda_{+}^{r}\vec{x}_{+}+c_{2}\lambda_{-}^{r}\vec{x}_{-}$
(47) $\displaystyle=c_{1}\lambda_{+}^{r}\begin{pmatrix}\omega_{+}\\\
1\end{pmatrix}+c_{2}\lambda_{-}^{r}\begin{pmatrix}\omega_{-}\\\
1\end{pmatrix},$ (48)

which implies (via Lemma 1) that

$f_{r}=\frac{c_{1}\lambda_{+}^{r}\omega_{+}+c_{2}\lambda_{-}^{r}\omega_{-}}{c_{1}\lambda_{+}^{r}+c_{2}\lambda_{-}^{r}}=\frac{\lambda_{-}^{r}\omega_{-}+c_{1}(\lambda_{+}^{r}\omega_{+}-\lambda_{-}^{r}\omega_{-})}{\lambda_{-}^{r}+c_{1}(\lambda_{+}^{r}-\lambda_{-}^{r})}.$
(49)

After substitution and much simplification, we obtain the desired expressions
in Eq. (40) and Eq. (41). $\blacksquare$

######  Corollary 2.

For the sequence $\\{f_{r}\\}_{r}$ of fidelities defined by the pumping
policy, it holds that

$\lim_{r\to\infty}f_{r}=\frac{-3+6f_{0}+\sqrt{7-26f_{0}+28f_{0}^{2}}}{-2+8f_{0}}.$
(50)

###### Proof.

We can write the expression for $f_{r}$ in Eq. (49) as

$f_{r}=\frac{\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^{r}\omega_{-}+c_{1}\left(\omega_{+}-\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^{r}\omega_{-}\right)}{\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^{r}+c_{1}\left(1-\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^{r}\right)}.$
(51)

Now, it is straightforward to see that $\lambda_{\pm}\geq 0$ for all
$f_{0}\in[\frac{1}{2},1]$, and furthermore that $\lambda_{+}>\lambda_{-}$.
Therefore, $\frac{\lambda_{-}}{\lambda_{+}}\in[0,1)$, which means that
$\lim_{r\to\infty}\left(\frac{\lambda_{-}}{\lambda_{+}}\right)^{r}=0$. We
therefore find that $\lim_{r\to\infty}f_{r}=\omega_{+}$, and because
$\omega_{+}$ (defined in Eq. (43)) is precisely the expression on the right-
hand side of Eq. (50), we have the desired result. $\blacksquare$

###  B.2 Banding vs. pumping

To gain an understanding of the difference between the banding and pumping
policies of entanglement distillation, we consider here a three-node chain
with $n_{ch}$ channels between the nodes. In this setting, by the swap-distill
policy, we mean that all $n_{ch}$ links are active at the same time,
entanglement swapping is performed on all of them and we assume that they all
succeed, and then all of the end-to-end links are distilled into one end-to-
end link. By the distill-swap policy, we mean that all $n_{ch}$ on both sides
are distilled into one link, and then entanglement swapping is performed on
them and assumed to be successful.

Let us first consider the banding approach to distillation, as defined above.
Once all $n_{ch}$ links are put in pairs, a first round of $\frac{n_{ch}}{2}$
distillation attempts are made; the resulting links are all of the same
fidelity, and they are paired again. In the next round, $\frac{n_{ch}}{4}$
distillation attempts are made, and the process continues until we have one
link remaining. Thus a total number of $\log_{2}(n)$ rounds are needed
(supposing that $n_{ch}$ is a power of two).

Figure 18: Comparison of the swap-distill (top) and distill-swap (bottom)
policies for a three-node chain with $n_{ch}$ links connecting the neighboring
nodes, as described in Sec. B.2. Here we use the banded policy to perform
distillation. The swap-distill policy is not useful beyond initial age
$m^{\star}/2$, whereas distill-swap can extend this threshold to $m^{\star}$.

Results for the banding approach are shown in Fig. 18. We see that in the case
of swap-distill, the cut-off of $m^{\star}$ is breached by the end-to-end link
if the initial links are older than $\approx 0.5m^{\star}$. This is so because
distillation is not useful for links of age greater than the cut-off
$m^{\star}$, and swapping adds the ages of links. In the case of distill-swap,
the threshold is higher and equal to $m^{\star}$, because distillation occurs
first. More importantly, we note that in both cases these thresholds are
independent of the number of distillation channels used, indicated by the
intersection of the different curves. On the other hand, for links of age
below the threshold, using an increasing number of links, an arbitrarily high
fidelity link can be obtained.

Figure 19: Comparison of the swap-distill (top) and distill-swap (bottom)
policies for a three-node chain with $n_{ch}$ links connecting the neighboring
nodes, as described in Sec. B.2. Now, instead of the banding policy, we use
the pumping policy for distillation. As with banded purification, swap-distill
is not useful beyond an initial age of $m^{\star}/2$, whereas distill-swap can
extend this threshold, but now not quite up to $\approx m^{\star}$. This is a
consequence of the fact that the pumping policy has a strict limit on the
fidelity that can be achieved with an increasing number of channels, as shown
in Eq. (5).

All of the analysis above was done considering a idealised situation in which
links of the same age are available always for distillation. This is of course
not necessarily true in practice. In practical settings, links will have to be
used as and when they are produced. Therefore, it is important to look also at
the previously mentioned pumping policy.

Results for the pumping policy are shown in Fig. 19. Although the trends
remain very similar to Fig. 18, one important difference is noticeable
immediately. When the pumping policy is used for distillation, the final age
of the distilled link always saturates to a value approaching the limit in Eq.
(5) as the number of channels increases. This can be seen by the convergence
of the red (triangle) and blue (circle) curves for $n_{ch}=10$ and
$n_{ch}=50$, respectively. This is in contrast to the observations in Fig. 18,
in which we observe that increasing the number of channels progressively
decreases the final age.

##  Appendix C Distill-swap vs swap-distill policy: analytical results

Here we analytically compare the performance of the distill-swap and swap-
distill policies in the case of three nodes. To simplify the analysis, we
restrict ourselves to the cases with two and four channels, and consider all
elementary links to be active simultaneously with the same fidelity. The
general result for an arbitrary number of channels is difficult to calculate
analytically given the large number of possible trajectories that can lead to
the successful generation of at least one end-to-end link. Seeing in Appendix
B.2 that the banded policy performs better than pumping, in this section we
look exclusively at the banded policy. The performance of the policies is
evaluated by the success probability of establishing at least one end-to-end
entangled link and by the average fidelity of an end-to-end link.

###  C.1 Two-channel case

Let $f_{0}$ be the initial fidelity of both the active links on each side
given in terms of the initial age $m_{0}$ according to Eq. (23) as
$f_{0}=f(m_{0})=\frac{1}{4}(1+3\mathrm{e}^{-m_{0}/m^{\star}})$. Let us first
consider the distill-swap policy. The probability of distillation being
successful simultaneously on both sides followed by a successful entanglement
swapping of the distilled links is given by

$p_{ds}=p_{sw}p_{d}^{2}=p_{sw}((8f_{0}^{2}-4f_{0}+5)/9)^{2},$ (52)

where $p_{d}=(8f_{0}^{2}-4f_{0}+5)/9$ is the success probability of one
distillation attempt, as given by Eq. (26). The fidelity and ages of the
successfully distilled links on both sides is given by

$\displaystyle f_{d}$
$\displaystyle=(1-2f_{0}+10f_{0}^{2})/(5-4f_{0}+8f_{0}^{2}),$ (53)
$\displaystyle m_{d}$
$\displaystyle=\lceil-m^{\star}\log((4f_{d}-1)/3))\rceil.$ (54)

Finally, the age of virtual link produced by swapping is the sum of the ages
of the elementary links consumed, i.e., $2m_{d}$. Therefore, the fidelity of
the end-to-end link is given by:

$f_{ds}=\frac{1}{4}(1+3\mathrm{e}^{-2m_{d}/m^{\star}}).$ (55)

Next, we consider the swap-distill policy. For this policy, two cases may
arise depending on the number of successful entanglement swaps. Let
$p_{sd}^{(1)}$, $f_{sd}^{(1)}$ and $p_{sd}^{(2)}$, $f_{sd}^{(2)}$ be the
success probability and fidelity, respectively, of the two cases. The two
cases are based on whether one or both of the entanglement swaps succeed.
Then, the total probability of producing at least one end-to-end link is the
sum of two cases and is given by

$p_{sd}=p_{sd}^{(1)}+p_{sd}^{(2)},$ (56)

and the weighted (expected) fidelity of the final distilled virtual link
produced is given by

$f_{sd}=\frac{p_{sd}^{(1)}f_{sd}^{(1)}+p_{sd}^{(2)}f_{sd}^{(2)}}{p_{sd}^{(1)}+p_{sd}^{(2)}}.$
(57)

Let us now determine analytical expressions for
$p_{sd}^{(1)},f_{sd}^{(1)},p_{sd}^{(2)},f_{sd}^{(2)}$.

  1. 1.

Both entanglement swaps succeed. In this case, the probability that both
entanglement swaps succeed, followed by a successful distillation attempt, is
given by

$p^{(2)}_{sd}=p_{sw}^{2}(8f_{s}^{2}-4f_{s}+5)/9),$ (58)

where the “$(2)$” in the superscript of $p_{sd}^{(2)}$ indicates the number of
successful swaps and $f_{s}$ is the fidelity of the virtual links produced
after successful swapping, given by

$f_{s}=\frac{1}{4}(1+3\mathrm{e}^{-\frac{2m_{0}}{m^{\star}}}).$ (59)

The resulting fidelity, after distillation, is then

$f_{sd}^{(2)}=F_{\text{distill}}(f_{s},f_{s})=\frac{1-2f_{s}+10f_{s}^{2}}{5-4f_{s}+8f_{s}^{2}}.$
(60)

  2. 2.

Only one entanglement swap is successful. This can happen in two ways,
therefore the probability of successfully producing one end-to-end link in
this case is simply given by

$p_{sd}^{(1)}=2p_{sw}(1-p_{sw}),$ (61)

because no distillation can be performed in this case, which also means that

$f_{sd}^{(1)}=f_{s},$ (62)

with $f_{s}$ as defined in Eq. (59).

###  C.2 Four-channel case

Now, for the four-channel case, let us start with the distill-swap policy.
Recall that under the banding policy, at most three rounds of distillation can
be performed with four links. Based on this, there are three trajectories can
lead to one end-to-end link, and the corresponding success probabilities and
fidelities are derived below. We combine them to obtain the total probability
and weighted (expected) fidelity of an end-to-end link, in a manner analogous
to (56) and (57). Throughout these derivations, we let $f_{0}$ be the initial
fidelity of all links, we let $p_{d}\equiv P_{\text{distill}}(f_{0},f_{0})$
and $f_{d}\equiv P_{\text{distill}}(f_{0},f_{0})$ be the success probability
and fidelity, respectively, after the first round of distillation. Similarly,
we let $p_{d_{2}}\equiv P_{\text{distill}}(f_{d},f_{d})$ and $f_{d_{2}}\equiv
F_{\text{distill}}(f_{d},f_{d})$ be the success probability and fidelity,
respectively, after the second round of distillation. The corresponding ages
of the links after each round are denoted by $m_{d}\equiv f^{-1}(f_{d})$ and
$m_{d_{2}}\equiv f^{-1}(f_{d_{2}})$, where we recall the function
$f(m)=\frac{1}{4}(1+3\exp(-m/m^{\star}))$ and its inverse
$f^{-1}(F)=\lceil-m^{\star}\log((4F-1)/3)\rceil$.

  1. 1.

Only one pair of links is distilled successfully on each side, and the
distilled link is successfully swapped. There are four possible ways this can
occur, which means that the success probability and resulting fidelity in this
case are given by

$\displaystyle p^{(1,1)}_{ds}$
$\displaystyle=p_{sw}(4p_{d}^{2}(1-p_{d})^{2}),$ (63) $\displaystyle
f^{(1,1)}_{ds}$
$\displaystyle=\frac{1}{4}\Bigg{(}1+3\exp\bigg{(}-\frac{2m_{d}}{m^{\star}}\bigg{)}\Bigg{)},$
(64)

where the “$(1,1)$” in the superscript denotes that only one distillation
attempt succeeded on each side.

  2. 2.

All three distillation attempts succeed on one side and only one distillation
attempt succeeds on the other side, followed by a successful entanglement
swapping. (Note that the case of only two distillation successes on either
side does not lead to an active link, and thus entanglement swapping to
generate an end-to-end link is not possible.) This can happen in four ways -
two ways to choose which side all three distillation will succeed and when all
attempts succeed on one side, either the first or the second attempt could
fail on the other side. Denoting the corresponding success probability and
fidelity by $p_{ds}^{(3,1)}$ and $f_{ds}^{(3,1)}$, respectively, we have

$\displaystyle p^{(3,1)}_{ds}$
$\displaystyle=p_{sw}(4p_{d}^{3}(1-p_{d})p_{d_{2}}),$ (65) $\displaystyle
f^{(3,1)}_{ds}$
$\displaystyle=\frac{1}{4}\Bigg{(}1+3\exp\bigg{(}-\frac{m_{d}+m_{d_{2}}}{m^{\star}}\bigg{)}\Bigg{)}.$
(66)

  3. 3.

All three distillation attempts succeed on both sides, followed by successful
entanglement swapping. In this case, denoting the success probability and
fidelity by $p_{ds}^{(3,3)}$ and $f_{ds}^{(3,3)}$, respectively, we have

$\displaystyle p^{(3,3)}_{ds}$ $\displaystyle=p_{sw}(p_{d}^{2}p_{d_{2}})^{2},$
(67) $\displaystyle f^{(3,3)}_{ds}$
$\displaystyle=\frac{1}{4}\Bigg{(}1+3\exp\bigg{(}-\frac{2m_{d_{2}}}{m^{\star}}\bigg{)}\Bigg{)}.$
(68)

Therefore, the total success probability of producing at least one end-to-end
link is given by