# Parameter regimes for surpassing the PLOB bound with error-corrected qudit
repeaters

Daniel Miller<EMAIL_ADDRESS>Timo Holz  Hermann Kampermann  Dagmar Bruß
Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf,
D-40225 Düsseldorf, Germany

(August 27, 2024)

###### Abstract

A potential quantum internet would open up the possibility of realizing
numerous new applications, including provably secure communication. Since
losses of photons limit long-distance, direct quantum communication and
widespread quantum networks, quantum repeaters are needed. The so-called PLOB-
repeaterless bound [Pirandola et _al._ , Nat. Commun. 8, 15043 (2017)] is a
fundamental limit on the quantum capacity of direct quantum communication.
Here, we analytically derive the quantum-repeater gain for error-corrected,
one-way quantum repeaters based on higher-dimensional qudits for two different
physical encodings: Fock and multimode qudits. We identify parameter regimes
in which such quantum repeaters can surpass the PLOB-repeaterless bound and
systematically analyze how typical parameters manifest themselves in the
quantum-repeater gain. This benchmarking provides a guideline for the
implementation of error-corrected qudit repeaters.

###### Contents

  1. 1 Introduction
  2. 2 Identification of genuine quantum repeaters
    1. 2.1 The PLOB-repeaterless bound
    2. 2.2 Bosonic qudits
    3. 2.3 Abstract description of qudits
    4. 2.4 Error-corrected qudit repeaters and the quantum-repeater gain
  3. 3 Parameter regimes for genuine quantum repeaters
    1. 3.1 Noise model
      1. 3.1.1 Approximation of pure-loss channels with generalized Pauli-channels
    2. 3.2 Error statistics for error-corrected, one-way qudit repeaters
      1. 3.2.1 Error statistics for multimode qudits
      2. 3.2.2 Error statistics for Fock qudits
    3. 3.3 Optimizing the quantum-repeater gain
    4. 3.4 Influence of operational errors on the quantum-repeater gain
    5. 3.5 Estimate of resources
  4. 4 Conclusion and Outlook

##  1 Introduction

The prospect of an eventual world-spanning quantum internet motivates
tremendous interest and investments [1, 2, 3]. A quantum internet offers—among
an increasing number of other applications [3, 4, 5, 6, 7]—the possibility of
quantum key distribution (QKD), a cryptographic procedure whose security is
not based on computational hardness assumptions but on the laws of quantum
mechanics [8, 9, 10]. Although state-of-the-art experiments can perform fiber-
based direct-transmission QKD across a few hundred kilometers [11], they face
fundamental limitations [12, 13, 14]. The so-called PLOB-repeaterless bound
(named after Pirandola, Laurenza, Ottaviani and Banchi) states that the
quantum capacity of a fiber directly connecting two parties is exponentially
suppressed in their distance [14]. As the quantum capacity is closely related
to the amount of transmissible quantum information, direct transmission
channels are not well suited for long distance quantum connections. To
overcome these limitations, quantum repeaters have been proposed [15, 16, 17,
18, 19, 20]. They shorten the distance of direct transmissions by introducing
intermediate repeater stations such that losses and errors can be tackled
using entanglement heralding, quantum memories, entanglement distillation, or
quantum error-correcting codes (QECCs) [20]. Recent investigations have shown
that quantum repeaters based on currently available technology have the
potential to surpass the PLOB-repeaterless bound, even with a single
intermediate repeater station [21, 22, 23, 24, 25, 26]. Laboratory experiments
have been reported which prove that it is in principle possible to surpass the
PLOB-repeaterless bound over distances of tens and hundreds of kilometers [27,
28, 29].

For world-spanning quantum communication, error-corrected, one-way [17, 18,
19] (also known as third generation [20]) quantum repeaters are promising
candidates. Since the implementation of such quantum repeaters will be
demanding and expensive, it is crucial to identify under which circumstances
they can be superior to direct quantum communication. In Refs. [30, 31], this
is done for qubit based quantum repeaters employing static linear optics.
Here, we address this problem in the case of error-corrected, one-way quantum
repeaters based on qudits (discrete variable quantum systems of dimension
$D\geq 2$), as such higher-dimensional qudits offer the advantage that more
noise can be tolerated before entanglement is lost [32]. (See Refs. [33, 34,
35, 36] for previous investigations in quantum repeaters based on qudits.) To
conclude that a quantum repeater can overcome the PLOB-repeaterless bound, it
is instrumental to find a lower bound on the achievable quantum capacity of
quantum repeaters. In previous approaches [21, 22, 23, 25, 26, 24], this
figure of merit usually was given by the secret key rate achievable with a
specific protocol; see also Refs. [37, 38, 39, 40, 41] for earlier
investigations in secret key rates of quantum repeaters. In this paper, we use
a different approach by exploiting that the quantum capacity of an error-
corrected quantum repeater can be lower bounded by $\log_{2}(D)-H(P)$, where
$H(P)$ is the Shannon entropy of the error probability distribution $P$ of the
state distributed by the repeater [14, 42]. There are numerous parameters
influencing the performance of quantum repeaters, e.g., total distance, number
of intermediate repeater stations, various error rates and choice of a QECC.
In Ref. [35], we derived an expression of the error probability distribution
$P$ in terms of these parameters. Here, we identify and discuss parameter
regimes in which error-corrected, one-way quantum repeaters based on qudits
can beat the PLOB-repeaterless bound.

This paper is structured as follows. In Sec. 2, we explain our method to
assess the quantum capacity of quantum repeaters. In Sec. 3, we identify
parameter regions where error-corrected qudit repeaters can surpass the PLOB-
repeaterless bound. Finally, in Sec. 4 we conclude and give an outlook on
possible future work.

##  2 Identification of genuine quantum repeaters

Consider two remote parties called Alice and Bob. A quantum channel
$\mathcal{E}$ from Alice to Bob is a completely positive, trace-preserving map
from the space of density operators on Alice’s Hilbert space to that of Bob.
The (two-way) _quantum capacity_ , $\mathcal{C}(\mathcal{E})$, quantifies how
much quantum information Alice can transmit asymptotically to Bob through
$\mathcal{E}$ using adaptive local operations and classical communications.
See Ref. [14] for the formal definition. We call such a quantum channel a
_genuine quantum repeater_ if it has a quantum capacity that is larger than
that of any direct transmission. In Sec. 2.1, we explain how this
characterization can be assessed via the multimode PLOB-repeaterless bound. In
Sec. 2.2, we propose two physical encodings of qudits into photons and provide
error models for the transmission of these bosonic qudits. In Sec. 2.3, we
recall a more abstract description of qudits which we will use throughout this
paper. In Sec. 2.4, we present the here-considered protocol for an error-
corrected qudit repeater and define its _quantum-repeater gain_. If this
figure of merit is positive, a genuine quantum repeater is identified.

###  2.1 The PLOB-repeaterless bound

Consider a photonic mode with a bosonic creation operator $b^{\dagger}$. The
_pure-loss channel_
$\mathcal{E}^{(\eta)}_{\mathrm{loss}}:b^{\dagger}\mapsto\sqrt{\eta}\,b^{\dagger}+\sqrt{1-\eta}\,b_{\mathrm{E}}^{\dagger},$
mixes such a mode with a vacuum mode via a beam splitter with transmissivity
$\eta$, where $b_{\mathrm{E}}^{\dagger}$ is the creation operator of an
environmental bosonic mode initialized in the zero-photon state
$\ket{0}_{\mathrm{E}}$ [43, 44]. The _PLOB-repeaterless bound_ states that the
quantum capacity of every quantum channel $\mathcal{E}$ is limited by that of
$\mathcal{E}^{(\eta)}_{\mathrm{loss}}$,

$\displaystyle\mathcal{C}(\mathcal{E})\leq\mathcal{C}(\mathcal{E}^{(\eta)}_{\mathrm{loss}})=-\log_{2}(1-\eta),$
(1)

provided there exists a decomposition of the form
$\mathcal{E}=\mathcal{E}_{\mathrm{B}}\circ\mathcal{E}^{(\eta)}_{\mathrm{loss}}\circ\mathcal{E}_{\mathrm{A}}$
for some quantum channels $\mathcal{E}_{\mathrm{A}}$ and
$\mathcal{E}_{\mathrm{B}}$ [14]. As this type of decomposition is typical for
direct quantum communication scenarios, this bound is fundamental. Moreover,
such decompositions are known for all Gaussian channels [44, 14]. In order to
overcome this limit on direct quantum communication, one could employ $M>1$
bosonic modes in parallel and perform measurements at repeater stations in
between Alice and Bob. Since the quantum capacity of the pure-loss channel is
additive, the quantum capacity $\mathcal{C}(\mathcal{E}_{\mathrm{rep}})$ of a
genuine quantum repeater of this form has to beat the _multimode PLOB-
repeaterless bound_ ,

$\displaystyle\mathcal{C}(\mathcal{E}_{\mathrm{direct}})\leq-M\times\log_{2}(1-\eta).$
(2)

This is the ultimate limit of direct quantum communication using $M$ bosonic
modes through free space or an optical fiber with transmissivity $\eta$. At
high loss $\eta\approx 0$, the multimode PLOB-repeaterless bound scales
linearly in the transmissivity according to $-M\times\log_{2}(1-\eta)\approx
1.44M\eta$, i.e., it is exponentially suppressed in the distance [14].

###  2.2 Bosonic qudits

In order to investigate under which circumstances a quantum repeater has the
potential to overcome the PLOB-repeaterless bound, it is necessary to specify
the physical encoding of quantum information into bosonic modes. We consider
two types of bosonic qudits: one-mode Fock qudits and multimode qudits. Here,
we specify how the pure-loss channel acts on these encodings.

The pure-loss channel transforms a pure Fock number state
$\ket{k}=\frac{1}{\sqrt{k!}}(b^{\dagger})^{k}\ket{0}$ into

$\displaystyle\mathcal{E}^{(\eta)}_{\mathrm{loss}}\left(\ket{k}\bra{k}\right)=\sum_{j=0}^{k}{k\choose
j}\eta^{j}(1-\eta)^{k-j}\,\ket{j}\bra{j}.$ (3)

Let $\mathcal{E}^{(\eta)}_{\mathrm{Fock};D}$ denote the $D$-dimensional
restriction of the pure-loss channel to inputs with $k\leq D-1$ photons. This
channel possesses a decomposition
$\mathcal{E}^{(\eta)}_{\mathrm{Fock};D}=\mathcal{E}_{\mathrm{CV\rightarrow
DV}}\circ\mathcal{E}^{(\eta)}_{\mathrm{loss}}\circ\mathcal{E}_{\mathrm{DV\rightarrow
CV}}$, where $\mathcal{E}_{\mathrm{DV\rightarrow CV}}$ is the inclusion map
from $\mathbb{C}^{D}$ to the single mode Fock space, i.e.,
$\mathcal{E}_{\mathrm{DV\rightarrow CV}}$ sends a computational basis state to
the state with the corresponding photon number, and similarly for
$\mathcal{E}_{\mathrm{CV\rightarrow DV}}$, cf. Ref. [14]. Thus, the PLOB-
repeaterless bound yields

$\displaystyle\mathcal{C}\left(\mathcal{E}^{(\eta)}_{\mathrm{Fock},D}\right)\leq\mathcal{C}(\mathcal{E}^{(\eta)}_{\mathrm{loss}})=-\log_{2}(1-\eta),$
(4)

where equality is reached in the limit $D\rightarrow\infty$. However, if $D$
is finite, the inequality is strict, as not the full potential of the pure-
loss channel is exploited.

Instead of encoding a $D$-dimensional qudit into the Fock basis, one could
also use time-bin encoding [45, 46, 47, 48], temporal modes (TM) [49, 50] or
modes of orbital angular momentum (OAM) [51, 52, 53]. For any of these
implementations—to which we will from now on refer to as _multimode encoding_
—the computational basis states are given by a single photon in one of $D$
modes, i.e.,

$\displaystyle\ket{m}:=b^{\dagger}_{m}\ket{\mathrm{vac}},$ (5)

where $b^{\dagger}_{m}$ is the creation operator of a bosonic mode labeled by
$m\in\\{0,\ldots,D-1\\}$ and $\ket{\mathrm{vac}}=\ket{0,\ldots,0}$ is the
vacuum state of all $D$ modes. Sending a multimode qudit in state $\rho$
through a beam splitter (with zero environmental photons) gives rise to the
$D$-dimensional erasure channel,

$\displaystyle\mathcal{E}^{(\eta)}_{\mathrm{erase};D}:\rho\longmapsto\eta\,\rho+(1-\eta)\,\ket{\mathrm{vac}}\bra{\mathrm{vac}}.$
(6)

Its quantum capacity is known to be
$\mathcal{C}\left(\mathcal{E}^{(\eta)}_{\mathrm{erase};D}\right)=\eta\log_{2}(D)$
[14]. This is smaller than
$\mathcal{C}\left((\mathcal{E}^{(\eta)}_{\mathrm{loss}})^{\otimes
D}\right)=-D\times\log_{2}(1-\eta)$ because
$\mathcal{E}^{(\eta)}_{\mathrm{erase};D}$ is only defined on the domain of
one-photon excitations, while for $D$ pure-loss channels arbitrary
superpositions of $D$-mode Fock states are valid inputs.

###  2.3 Abstract description of qudits

Formally, a qudit is a quantum system with a Hilbert space of dimension $D\geq
2$. We label its computational basis states $\ket{j}$ by elements $j$ in
$\mathbb{Z}/D\mathbb{Z}=\\{0,1,\ldots,D-1\\}$, the ring of integers modulo
$D$. For example,

$\displaystyle\ket{+}:=\frac{1}{\sqrt{D}}\sum_{j\in\mathbb{Z}/D\mathbb{Z}}\ket{j},$
(7)

is the equally weighted superposition of all computational basis states. Up to
a global phase, the generalized Pauli-operators of a qudit are products of

$\displaystyle X:=\sum_{k\in\mathbb{Z}/D\mathbb{Z}}\ket{k+1}\bra{k}\hskip
20.00003pt\text{and}\hskip
10.00002ptZ:=\sum_{k\in\mathbb{Z}/D\mathbb{Z}}\omega^{k}\ket{k}\bra{k},$ (8)

where $\omega:=\mathrm{e}^{2\pi i/D}$, i.e., the generalized Pauli-operators
are of the form

$\displaystyle
X^{r}Z^{s}=\sum_{k\in\mathbb{Z}/D\mathbb{Z}}\omega^{ks}\ket{k+r}\bra{k},$ (9)

where $r,s\in\mathbb{Z}/D\mathbb{Z}$. They constitute a basis of the vector
space of complex $D\times D$ matrices [55, 56]. The generalized Pauli-error
channel, given an error probability distribution
${P=(p_{r,s})_{r,s\in\mathbb{Z}/D\mathbb{Z}}}$ with $\sum_{r,s}p_{r,s}=1$, is
defined as the quantum channel,

$\displaystyle\mathcal{E}_{P}:\rho\longmapsto\sum_{r,s\in\mathbb{Z}/D\mathbb{Z}}p_{r,s}(X^{r}Z^{s})\rho(X^{r}Z^{s})^{\dagger},$
(10)

with Kraus operators $\sqrt{p_{r,s}}X^{r}Z^{s}$. It corresponds to the random
application of a Pauli-operator $X^{r}Z^{s}$ to the state $\rho$ with
probability $p_{r,s}$. The depolarizing channel,

$\displaystyle\mathcal{E}_{\mathrm{depol};D}^{(1-f)}:\rho\longmapsto(1-f)\rho+f\frac{\mathbbm{1}}{D},$
(11)

is an example of a generalized Pauli-error channel where the trivial error
$X^{0}Z^{0}=\mathbbm{1}$ occurs with probability $p_{0,0}=1-f+f/D^{2}$ and any
other error occurs with probability $f/D^{2}$ [35]. The controlled-$Z$ gate,

$\displaystyle\textsc{{cz}}:=\sum_{k\in\mathbb{Z}/D\mathbb{Z}}\ket{k}\bra{k}\otimes
Z^{k}$ (12)

is a two-qudit Clifford gate which can be used to produce the maximally-
entangled state,

$\displaystyle\ket{\Phi}:=\textsc{{cz}}\ket{+}^{\otimes
2}=\frac{1}{D}\sum_{j,k\in\mathbb{Z}/D\mathbb{Z}}\omega^{jk}\ket{j}\otimes\ket{k},$
(13)

from two copies of the $\ket{+}$ state.

###  2.4 Error-corrected qudit repeaters and the quantum-repeater gain

Assume that Alice and Bob make use of the one-way quantum repeater protocol
described in the caption of Fig. 1.

a) Quantum repeater lineb) Physical encodingMultimode
(MM)Fock$\ket{3}={b_{3}^{\dagger}}\ket{\mathrm{vac}}$.$\ket{3}=\frac{1}{\sqrt{3!}}{(b^{\dagger})^{3}}\ket{\mathrm{vac}}$...
Figure 1: a) An error-corrected, one-way qudit quantum repeater line with
$N-1$ intermediate repeater stations [39, 40, 41, 35]. Alice produces the two-
qudit state $\ket{\Phi}=\textsc{{cz}}\ket{+}^{\otimes 2}$, stores one qudit
into a quantum memory (QM), and sends the other qudit to the first
intermediate repeater station. At every repeater station, the incoming qudit
is entangled via a cz gate with a new qudit prepared in the $\ket{+}$ state.
Then, the previous qudit is measured in the $X$ basis and the other qudit is
sent to the next repeater station. After $N$ transmissions, Bob receives the
last qudit, entangles it with his own $\ket{+}$ state, measures it, and stores
the remaining qudit in his own QM. As a result, Alice and Bob have stored an
entangled qudit pair in their QMs. The protocol takes place on a logical level
where each logical qudit consists of $n$ physical qudits. b) Visualization of
two different encoding methods into photons.

In the ideal case, they obtain a maximally entangled state
$B^{\dagger}\ket{\Phi}$ in their quantum memories (QMs) which only differs
from $\ket{\Phi}$ of Eq. (13) by the application of the byproduct operator
$B=X^{c_{\mathrm{even}}}Z^{c_{\mathrm{odd}}}$ to Bob’s qudits, where the
number of elementary links $N$ is even. The exponents

$\displaystyle c_{\mathrm{even}}:=\sum_{i=1}^{N/2}(-1)^{i}c_{2i}\hskip
20.00003pt\text{and}\hskip
10.00002ptc_{\mathrm{odd}}:=\sum_{i=1}^{N/2}(-1)^{i+1}c_{N+1-2i},$ (14)

are computed from the measurement outcomes $c_{i}$ at the $i$-th repeater
station, i.e., $c_{\mathrm{even}}$ depends on the measurement outcomes of
even-numbered repeater stations and likewise for $c_{\mathrm{odd}}$. See Ref.
[35] for more details.

To overcome the limit of direct quantum communication, the quantum repeater
employs logical qudits which are encoded using an $\llbracket
n,1,d\rrbracket_{D}$ QECC such that each qudit is replaced by $n$ physical
qudits of dimension $D$. The code distance $d$ determines the number $t$ of
correctable errors by the QECC according to $t=\lfloor(d-1)/2\rfloor$, see
Ref. [57] for its formal definition. Note that QECCs do not exist for all code
parameters, e.g., all QECCs fulfill the quantum singleton bound $2d-1\leq n$.
However, if $D$ is a prime number and $d\leq(D-1)/2$, an explicit construction
of $\llbracket 2d-1,1,d\rrbracket_{D}$ QECCs saturating the quantum singleton
bound is known in the form of quantum polynomial codes [58, 59, 60, 61]. We
focus on this encoding because quantum polynomial codes can give an advantage
over other QECCs for quantum repeaters [33, 34].

At every repeater station, the (logical) $X$ measurement is performed as
follows. All physical qudits are measured individually in the eigenbasis of
the $X$ operator. If the number of erroneous measurement results is smaller
than or equal to $t$, the error pattern can be corrected successfully.
Otherwise, the correction attempt leads to a logical error. For the considered
QECCs, it is infeasible to determine the success probability for this
strategy. Thus, we bound the capacity of the repeater by using a random
logical error (instead of the actual logical error) whenever the number of
physical errors is larger than $t$. Note that this simplification does not
overestimate the capacity of the quantum repeater. We will make this statement
more precise in Eq. (17). For this strategy, it turns out [35] that all
repeater stations (except for the first) have the same probability
$p^{\mathrm{rep}}_{\mathrm{succ}}$ of a successful correction. For the first
repeater station, where fewer error sources contribute, we denote this
probability by $p^{1.\mathrm{rep}}_{\mathrm{succ}}$. To simplify the error
analysis, we furthermore assume that Bob performs a (perfect) round of
stabilizer measurements. As quantum polynomial codes belong to the class of
Calderbank-Shor-Steane codes [61], Bob can independently correct $X$ and $Z$
errors. Again, if the number of $X$ ($Z$) errors exceeds $t$, Bob guesses a
recovery operation. Otherwise, with probability
$p_{\mathrm{succ}}^{\mathrm{Bob},X}$ ($p_{\mathrm{succ}}^{\mathrm{Bob},Z}$),
he can successfully reveal the error pattern and applies the appropriate
recovery operation. In the final step of the entanglement swapping protocol,
Bob has to apply the byproduct operator
$B=X^{c_{\mathrm{even}}}Z^{c_{\mathrm{odd}}}$. In doing so, errors on the
measurement results of the even-numbered repeater stations propagate to $X$
errors on Bob’s qudit which gives rise to an $X$ dephasing channel,
$\rho\mapsto
p_{\mathrm{succ}}^{\mathrm{rep}}\rho+\frac{1-p_{\mathrm{succ}}^{\mathrm{rep}}}{D}\sum_{r\in\mathbb{Z}/D\mathbb{Z}}X^{r}\rho
X^{-r}$. Likewise, wrong measurement results at odd-numbered repeater stations
induce $Z$ errors on Bob. In conclusion, the overall error statistics are of
the form
$P=(p^{\mathrm{fin}}_{r,s})_{r,s\in\mathbb{Z}/D\mathbb{Z}}=(p^{\mathrm{fin},X}_{r}p^{\mathrm{fin},Z}_{s})_{r,s\in\mathbb{Z}/D\mathbb{Z}}$,
where

$\displaystyle p^{\mathrm{fin},X}_{0}$
$\displaystyle=\frac{1}{D}\left[1+(D-1)(p^{\mathrm{rep}}_{\mathrm{succ}})^{\frac{N}{2}}p^{\mathrm{Bob},X}_{\mathrm{succ}}\right]\text{
and}$ (15) $\displaystyle p^{\mathrm{fin},X}_{r\neq 0}$
$\displaystyle=\frac{1}{D}\left[1-(p^{\mathrm{rep}}_{\mathrm{succ}})^{\frac{N}{2}}p^{\mathrm{Bob},X}_{\mathrm{succ}}\right]$

are the probabilities of $X$ errors on Bob’s qudit, and

$\displaystyle p^{\mathrm{fin},Z}_{0}$
$\displaystyle=\frac{1}{D}\left[1+(D-1)p^{\mathrm{1.rep}}_{\mathrm{succ}}(p^{\mathrm{rep}}_{\mathrm{succ}})^{\frac{N}{2}-1}p^{\mathrm{Bob},Z}_{\mathrm{succ}}\right]\text{
and}$ (16) $\displaystyle p^{\mathrm{fin},Z}_{s\neq 0}$
$\displaystyle=\frac{1}{D}\left[1-p^{\mathrm{1.rep}}_{\mathrm{succ}}(p^{\mathrm{rep}}_{\mathrm{succ}})^{\frac{N}{2}-1}p^{\mathrm{Bob},Z}_{\mathrm{succ}}\right]$

are the probabilities of $Z$ errors. We will explain in Sec. 3.2 how these
success probabilities depend on the physical error rates.

The erroneous state distributed by the quantum repeater is
$\rho=\mathcal{E}_{P}(\Phi)$, where $\Phi=\ket{\Phi}\bra{\Phi}$ is the
projector onto the maximally entangled state defined in Eq. (13), and
$\mathcal{E}_{P}$ is the generalized Pauli-error channel, recall Eq. (10),
corresponding to the error distribution $P$ acting on Bob’s qudit. The quantum
capacity of a generalized Pauli-error channel $\mathcal{C}(\mathcal{E}_{P})$
can be lower bounded by

$\displaystyle\mathcal{C}(\mathcal{E}_{\mathrm{rep}})=\mathcal{C}(\mathcal{E}_{P})\geq\max\\{0,\,\log_{2}(D)-H(P)\\}=:{B}^{\downarrow}_{\mathrm{rep}},$
(17)

see suppl. of Ref. [14]. As mentioned before, guessing logical errors leads to
an overestimation of $H(P)$, thus, even for the implementable strategy,
${B}^{\downarrow}_{\mathrm{rep}}$ is still a lower bound on the capacity of
the quantum repeater. Note that ${B}^{\downarrow}_{\mathrm{rep}}$ is also a
lower bound on the distillable entanglement of $\mathcal{E}_{P}(\Phi)$ which
is achievable with a distillation protocol given in Ref. [42].

Since a logical qudit is encoded into $n$ physical qudits, the number of
photonic modes used to connect two repeater stations is $M=n$ and $M=nD$ for
Fock and MM encoding, respectively. Thus, the multimode PLOB-repeaterless
upper bound, recall Eq. (2), is given by

$\displaystyle{B}^{\uparrow}_{\mathrm{PLOB}}:=\begin{cases}-n\times\log_{2}(1-\eta),&\text{Fock
encoding}\\\ -nD\times\log_{2}(1-\eta),&\text{MM encoding}\hskip
10.00002pt,\end{cases}$ (18)

where $\eta$ is the transmissivity of a pure-loss channel corresponding to the
total distance $L$ from Alice to Bob. Every direct transmission channel
employing the same number of photonic modes, as the quantum repeater has a
quantum capacity smaller than ${B}^{\uparrow}_{\mathrm{PLOB}}$. The quantum
repeater, on the other hand, has a quantum capacity larger than
${B}^{\downarrow}_{\mathrm{rep}}$. Hence, the quantum repeater is genuine if
(but not necessarily only if) the _quantum-repeater gain_ ,

$\displaystyle\Delta:={B}^{\downarrow}_{\mathrm{rep}}-{B}^{\uparrow}_{\mathrm{PLOB}},$
(19)

is positive.

##  3 Parameter regimes for genuine quantum repeaters

As we have argued in Sec. 2.4, error-corrected, one-way qudit repeaters have a
quantum capacity of at least
${B}^{\downarrow}_{\mathrm{rep}}=\log_{2}(D)-H(P)$, where the error
probability distribution
$P=(p^{\mathrm{fin}}_{r,s})_{r,s\in\mathbb{Z}/D\mathbb{Z}}$ depends on various
parameters of the quantum repeater such as the qudit dimension $D$, the
distance $L$ between Alice and Bob, the number $N-1$ of repeater stations, and
the parameters of the $\llbracket n,1,d\rrbracket_{D}$ QECC. In Sec. 3.1, we
introduce our noise model and derive a worst-case approximation of the pure-
loss channel for Fock qudits. In Sec. 3.2, we provide the analytical
dependence of $P$ on all these parameters for both Fock and MM encoding.
Afterwards, in Secs. 3.3$-$3.5, we investigate parameter regimes for genuine
quantum repeaters.

###  3.1 Noise model

In a realistic scenario, Alice and Bob have to deal with errors. Each
transmission channel from one repeater station to the next is modeled as a
pure-loss channel $\mathcal{E}^{(\eta_{0})}_{\mathrm{loss}}$ with
transmissivity

$\displaystyle\eta_{0}=10^{-\frac{\alpha L_{0}}{10}},$ (20)

where $L_{0}$ is the distance between the repeater stations and $\alpha=0.2$
dB/km is the attenuation of optical fibers at 1550 nm wavelength [62]. We
derive an approximation of pure-loss channels with generalized Pauli-error
channels in Sec. 3.1.1 as preparation for an error analysis using the
framework of Ref. [35].

Besides transmission losses, we also include measurement errors
($f_{\mathrm{M}}$), modeled by a depolarizing channel
$\mathcal{E}^{(1-f_{\mathrm{M}})}_{\mathrm{depol};D}$, recall Eq. (11), before
each measurement. Unless stated otherwise, we use the value
$f_{\mathrm{M}}=0.01$, as the best single-photon detector efficiencies of
about $95\%$ match this order of magnitude [63]. To model gate errors
($f_{\mathrm{G}}$), we furthermore assume that all cz gates are followed by
two depolarizing channels (one on each qudit) with an optimistic error
parameter $f_{\mathrm{G}}=10^{-3}$. Deterministic photon-photon gates for
polarization qubits based on light-matter interactions have been demonstrated
with an average gate fidelity of $76.2\pm 3.6\%$ [64]. Two-mode gates for Fock
qudits could in principle be realized using Kerr-interactions [65], however,
high-fidelity phase gates have only been reported for a single photonic mode
[66]. In Sec. 3.4, we will examine how the quantum-repeater gain $\Delta$
depends on the operational error rates $f_{\mathrm{M}}$ and $f_{\mathrm{G}}$.

Finally, storage errors ($f_{\mathrm{S}}$) affecting Alice’s physical qudits
in the QMs are modeled by depolarizing channels,
$\mathcal{E}^{(1-f_{\mathrm{S}})}_{\mathrm{depol};D}$, with

$\displaystyle 1-f_{S}=10^{-\frac{\gamma L/c}{10}},$ (21)

where $\gamma$ is the decaying rate of Alice’s QM and $c=200\mathrm{km/ms}$ is
the speed of light in a fiber with a refractive index of 1.5. Optical fiber
loop QMs, with $\gamma_{\mathrm{fiber}}=\alpha c=40\mathrm{dB/ms}$, are not
useful, as the stored qudits decay with the same rate as the flying qudits.
However, matter-based QMs have been demonstrated: Cold atomic ensembles
provide QMs with $\gamma_{\mathrm{atom}}=5\mathrm{dB/ms}$ ($50\%$ efficiency
in $0.6\mathrm{ms}$) [67]. Promising candidates are based on nitrogen vacancy
centers in diamond which range from $\gamma_{\mathrm{NV}}=4\times
10^{-3}\mathrm{dB/ms}$ (coherence time $T_{2}\approx 1\mathrm{s}$)[68, 69] to
$\gamma_{\mathrm{NV}}=7\times 10^{-5}\mathrm{dB/ms}$ ($T_{2}\approx
60\mathrm{s}$)[70]. Using trapped ions, decaying rates of
$\gamma_{\mathrm{ion}}=7\times 10^{-6}\mathrm{dB/ms}$ are possible [71]. Since
these proof-of-principle QMs do not take storage-and-retrieval efficiencies
into account, we use a more realistic value of $\gamma=10^{-2}\mathrm{dB/ms}$
($T_{2}\approx 100\mathrm{ms}$ [72]) for our analysis.

####  3.1.1 Approximation of pure-loss channels with generalized Pauli-
channels

For our error analysis it is crucial that all error channels are modeled as
Pauli error channels. Recall from Eq. (6) that the pure-loss channel,
$\mathcal{E}_{\mathrm{loss}}^{(\eta_{0})}$, acts on MM qudits as an erasure
channel. If we do not use the information of an occurred erasure, i.e., we
replace the flag $\ket{\mathrm{vac}}\bra{\mathrm{vac}}$ by the completely
mixed state $\mathbbm{1}/D$, we can replace
$\mathcal{E}_{\mathrm{loss}}^{(\eta_{0})}$ by
$\mathcal{E}_{\mathrm{depol};D}^{(\eta_{0})}$. Note that this simplification
cannot overestimate the performance of the quantum repeater.

If, on the other hand, the qudits are encoded in the Fock basis, the pure-loss
channel introduces errors on the number of photons. Losing exactly $r$ photons
can be regarded as an application of the error operator $E=X^{-r}$, recall Eq.
(8). For a given input state $\ket{k}$, the probability for this to happen is
given by

$\displaystyle\mathrm{Pr}\left(E=X^{-r}\ \big{|}\
\rho_{\mathrm{in}}=\ket{k}\bra{k}\right)=\begin{cases}{k\choose
r}\eta_{0}^{k-r}(1-\eta_{0})^{r},&r\leq k\\\ 0,&r>k\end{cases},$ (22)

where $k,r\in\\{0,\ldots,D-1\\}$, as we have seen in Eq. (3). To upper bound
the $X$ error probabilities, we set

$\displaystyle
p^{\mathrm{appr}}_{-r}:=\max_{k\in\\{0,\ldots,D-1\\}}\mathrm{Pr}\left(E=X^{-r}\
\big{|}\ \rho_{\mathrm{in}}=\ket{k}\bra{k}\right)$ (23)

for all $r\neq 0$. If
$p^{\mathrm{appr}}_{0}:=1-\sum_{r=1}^{D-1}p^{\mathrm{appr}}_{-r}$ is positive,
we can model the pure-loss channel on Fock state qudits of a single bosonic
mode by the generalized Pauli-error channel,

$\displaystyle\mathcal{E}^{(\eta_{0})}_{\mathrm{appr};D}:\rho\longmapsto\sum_{r=0}^{D-1}p^{\mathrm{appr}}_{-r}X^{-r}\rho(X^{-r})^{\dagger},$
(24)

as a worst-case approximation. In the error analysis for Fock-state encoding,
we thus replace each pure-loss
channel,$\mathcal{E}_{\mathrm{loss}}^{(\eta_{0})}$, between any two repeater
stations by $\mathcal{E}^{(\eta_{0})}_{\mathrm{appr};D}$. Note that this
further decreases ${B}^{\downarrow}_{\mathrm{rep}}$; thus, $\Delta>0$ will
still indicate a genuine quantum repeater. We will only consider repeater
lines for which the repeater stations are spaced close enough such that
$p^{\mathrm{appr}}_{0}\geq 0$, i.e., $\eta_{0}\approx 1$. To compute the error
probabilities in Eq. (23), we have to find the input state $\ket{k}$ with the
highest probability to lose $r$ photons. By differentiating the analytical
continuation of Eq. (22), we obtain

$\displaystyle 0$ $\displaystyle=\frac{\partial}{\partial k}\left[{k\choose
r}\eta_{0}^{k-r}(1-\eta_{0})^{r}\right]={k\choose
r}\eta_{0}^{k-r}(1-\eta_{0})^{r}\left(\ln(\eta_{0})+H_{k}-H_{k-r}\right),$
(25)

where

$\displaystyle
H_{k}:=\sum_{j=1}^{k}\frac{1}{j}=\ln(k)+\gamma_{\mathrm{EM}}+\frac{1}{2k}-\frac{1}{12k^{2}}+\frac{1}{120k^{4}}-\ldots$
(26)

is the $k$th Harmonic number and $\gamma_{\text{EM}}\approx 0.577$ is the
Euler-Mascheroni constant. For large $k$, the approximation
$H_{k}-H_{k-r}\approx\ln(k)-\ln(k-r)$ yields $k\approx{r}/({1-\eta_{0}})$ as
the solution of Eq. (25). This is indeed a maximum because the sign of the
derivative given in Eq. (25) changes at $k\approx r/(1-\eta_{0})$ from plus to
minus, when increasing $k$. This follows from the positivity and the strictly
monotonic decrease of the derivative of the analytical continuation of
$H_{k}$. Let $\mathrm{rd}(x)$ denote the integer that is closest to
$x\in\mathbb{R}$. Because of $\eta_{0}\approx 1$, the approximation is so good
that the integer
$\tilde{k}(r,\eta_{0}):=\min\\{\mathrm{rd}\left(r/({1-\eta_{0}})\right),D-1\\}$
is the number of input photons having the highest probability (among inputs of
up to $D-1$ photons) to lose exactly $r$ photons. This yields

$\displaystyle p^{\mathrm{appr}}_{-r}={\tilde{k}(r,\eta_{0})\choose
r}\eta_{0}^{\tilde{k}(r,\eta_{0})-r}(1-\eta_{0})^{r}$ (27)

as the solution of Eq. (23).

###  3.2 Error statistics for error-corrected, one-way qudit repeaters

In order to evaluate the quantum-repeater gain defined in Eq. (19), one has to
know the error probability distribution $P$ of the entangled state distributed
by the quantum repeater. In Sec. 3.2.1 and Sec. 3.2.2, we derive for MM and
Fock qudits, respectively, the expression of the success probabilities
$p_{\mathrm{succ}}^{\mathrm{1.rep}}$, $p_{\mathrm{succ}}^{\mathrm{rep}}$,
$p_{\mathrm{succ}}^{\mathrm{Bob},X}$, and $p_{\mathrm{succ}}^{\mathrm{Bob},Z}$
(recall the paragraph above Eq. (15) for their definitions) from which $P$
follows via Eqs. (15) and (16).

####  3.2.1 Error statistics for multimode qudits

Here, we review our previous results [35] which hold for error-corrected qudit
repeater lines where the qudits are encoded into multiple bosonic modes, e.g.,
time-bin, TM, and OAM qudits. Recall that we replace the pure-loss channel
with transmissivity $\eta_{0}$ by a depolarizing channel with transmission
error rate $f_{\mathrm{T}}=1-\eta_{0}$. Depolarizing channels can be regarded
as sources of discrete $X$ and $Z$ errors which propagate through the repeater
line. It turns out that there are six sources from which a $Z^{e_{i}}$ error
at the $X$ measurement of qudit $i$ can originate: two transmission, three
gate, and one measurement error channel [39, 40, 41, 35]. Such an error will
change the physical measurement result $c_{i}$ into
$c_{i}+e_{i}\in\mathbb{Z}/D\mathbb{Z}$ with probability
$p^{\mathrm{rep}}_{e_{i}}$, where

$\displaystyle p^{\mathrm{rep}}_{{e_{i}}\neq 0}$
$\displaystyle=\frac{1}{D}\left[1-(1-f_{\mathrm{T}})^{2}(1-f_{\mathrm{G}})^{3}(1-f_{\mathrm{M}})\right]$
(28)

and $p^{\mathrm{rep}}_{0}=1-(D-1)p^{\mathrm{rep}}_{e_{i}\neq 0}$. For the
first repeater station, fewer error sources contribute:
$p^{1.\mathrm{rep}}_{e_{i}\neq
0}=\frac{1}{D}\left[1-(1-f_{\mathrm{T}})(1-f_{\mathrm{G}})^{2}(1-f_{\mathrm{M}})\right]$
and $p^{\mathrm{1.rep}}_{0}=1-(D-1)p^{\mathrm{1.rep}}_{e_{i}\neq 0}$.
Similarly, the probability of an $X^{e_{i}}$ and $Z^{e_{i}}$ error on Bob’s
qudit right before the stabilizer measurement is given by
$p^{\mathrm{Bob},X}_{e_{i}}$ and $p^{\mathrm{Bob},Z}_{e_{i}}$, respectively,
where $p^{\mathrm{Bob},X}_{e_{i}\neq
0}=\frac{1}{D}\left[1-(1-f_{\mathrm{G}})^{2}(1-f_{\mathrm{S}})\right],$ and
$p^{\mathrm{Bob},Z}_{e_{i}\neq
0}=\frac{1}{D}\left[1-(1-f_{\mathrm{T}})(1-f_{\mathrm{G}})^{3}(1-f_{\mathrm{S}})\right],$
and $p^{\mathrm{Bob},X}_{0}$, $p^{\mathrm{Bob},Z}_{0}$ again follow from
normalization. See Ref. [35] for more details.

In the following, let $p_{e_{i}}$ be either $p^{\mathrm{rep}}_{e_{i}}$,
$p^{1.\mathrm{rep}}_{e_{i}}$, $p^{\mathrm{Bob},X}_{e_{i}}$, or
$p^{\mathrm{Bob},Z}_{e_{i}}$, and likewise for $p_{\mathrm{succ}}$. In either
situation, $n$ individual measurement results are employed for the error
correction attempt based on the $\llbracket n,1,d\rrbracket_{D}$ QECC. Since
the error probability of a single error $e_{i}\in\mathbb{Z}/D\mathbb{Z}$ is
given by $p_{e_{i}}$, the probability of an error pattern
$\textbf{e}=(e_{1},\ldots,e_{n})$ at the respective error correction attempt
is given by

$\displaystyle
p_{\textbf{e}}=\prod_{i=0}^{n}p_{e_{i}}=p_{0}^{n-\mathrm{wt}(\textbf{e})}p_{e\neq
0}^{\mathrm{wt}(\textbf{e})},$ (29)

where the Hamming weight $\mathrm{wt}(\textbf{e})$ denotes the number of
nonzero entries of e. As the QECC can correct up to $\lfloor(d-1)/2\rfloor$
errors, the probability of a correctable error pattern is given by

$\displaystyle p_{\mathrm{succ}}$
$\displaystyle=\sum_{k=0}^{\left\lfloor\frac{d-1}{2}\right\rfloor}(D-1)^{k}{n\choose
k}p_{0}^{n-k}p_{e\neq 0}^{k}.$ (30)

If an error pattern occurs which cannot be corrected, a logical error is
guessed with probability $p_{\mathrm{guess}}={(1-p_{\mathrm{succ}})/D}$.
Combining the respective success probabilities according to Eqs. (15) and (16)
yields the final error distribution on the distributed state.

####  3.2.2 Error statistics for Fock qudits

Here, we adapt the error analysis of Ref. [35] to error-corrected qudit
repeater lines with physical qudits encoded in the Fock basis of a single
bosonic mode. In this case, the propagation of errors is more complicated, as
the error probabilities of $\mathcal{E}^{(\eta_{0})}_{\mathrm{appr};D}$ all
differ from each other. A logical cz gate for the $\llbracket
n,1,d\rrbracket_{D}$ quantum polynomial code is transversal in the sense that
there are invertible elements $s_{1},\ldots,s_{n}\in\mathbb{Z}/D\mathbb{Z}$
such that $\bigotimes_{i=1}^{n}\textsc{{cz}}^{s_{i}}(a_{i},b_{i})$ acts as a
cz gate between two logical qudits $a$ and $b$, where
$\textsc{{cz}}(a_{i},b_{i})$ denotes a physical cz gate from the $i$th
physical qudit of the logical qudit $a$ to the $i$th qudit of $b$. For MM
qudits, in the previous section, this is not important because, at a
depolarizing channel, every nontrivial error occurs with the same probability.
Here, however, an $X^{s_{i}^{-1}e_{i}}$ error, occurring during the
transmission to physical qudit $i$, will induce a $Z^{e_{i}}$ error to qudit
$i$ of the next logical qudit for all $e_{i}\in\mathbb{Z}/D\mathbb{Z}$.
Employing the error tracking tools of Ref. [35] and taking all relevant error
sources into account, we obtain that the probability for an error $e_{i}$ on
the measurement result of qudit $i$ at every repeater station but the first is
given by

$\displaystyle p^{\mathrm{rep}}_{e_{i}}$
$\displaystyle=p^{\mathrm{appr}}_{s_{i}^{-1}e_{i}}(1-f_{\mathrm{G}})^{3}(1-f_{\mathrm{M}})+\frac{1}{D}\left[1-(1-f_{\mathrm{G}})^{3}(1-f_{\mathrm{M}})\right],$
(31)

recall Eq. (27) for the definition of $p^{\mathrm{appr}}_{-r}$. For the first
repeater station,

$\displaystyle p^{1.\mathrm{rep}}_{e_{i}\neq 0}$
$\displaystyle=\frac{1}{D}\left[1-(1-f_{\mathrm{G}})^{2}(1-f_{\mathrm{M}})\right]$
(32)

and $p^{1.\mathrm{rep}}_{0}$ follows from normalization; note that
transmission errors do not contribute, as they are of $X$ type for the channel
$\mathcal{E}^{(\eta_{0})}_{\mathrm{appr};D}$. Similarly,
$p^{\mathrm{Bob},X}_{e_{i}}$ and $p^{\mathrm{Bob},Z}_{e_{i}}$ are given by

$\displaystyle p^{\mathrm{Bob},X}_{e_{i}\neq 0}$
$\displaystyle=\frac{1}{D}\left[1-(1-f_{\mathrm{G}})^{2}(1-f_{\mathrm{S}})\right],$
(33) $\displaystyle\text{and}\hskip 10.00002ptp^{\mathrm{Bob},Z}_{e_{i}}$
$\displaystyle=p^{\mathrm{appr}}_{s_{i}^{-1}e_{i}}(1-f_{\mathrm{G}})^{3}(1-f_{\mathrm{S}})+\frac{1}{D}\left[1-(1-f_{\mathrm{G}})^{3}(1-f_{\mathrm{S}})\right].$

This time, let $p_{e_{i}}$ be either $p^{\mathrm{rep}}_{e_{i}}$ or
$p^{\mathrm{Bob},Z}_{e_{i}}$ (for $p^{1.\mathrm{rep}}_{e_{i}}$ and
$p^{\mathrm{Bob},X}_{e_{i}}$ we can continue as in Sec. 3.2.1), and likewise
for $p_{\mathrm{succ}}$. Again, the probability of an error pattern
$\textbf{e}=(e_{1},\ldots,e_{n})$ is given by
$p_{\textbf{e}}=\prod_{i=1}^{n}p_{e_{i}}$, but here we cannot simplify this
expression using the Hamming weight because the nontrivial error probabilities
do not coincide. The probability that a correctable error pattern occurs is
given by the sum over all probabilities $p_{\mathbf{e}}$ where
$\mathrm{wt}(\mathbf{e})\leq\lfloor(d-1)/2\rfloor$. Since the substitution
$e_{i}^{\prime}:=s_{i}^{-1}e_{i}$ does not change the Hamming weight, this sum
does not depend on the $s_{i}$ and can be expressed as

$\displaystyle p_{\mathrm{succ}}$
$\displaystyle=\sum_{k=0}^{\left\lfloor\frac{d-1}{2}\right\rfloor}{n\choose
k}p_{0}^{n-k}\left(\sum_{\textbf{r}\in\\{1,\ldots,D-1\\}^{k}}p_{\textbf{r}}\right),$
(34)

where $p_{\textbf{r}}:=p_{r_{1}}p_{r_{2}}\ldots p_{r_{k}}$. By combining terms
with equal probability in the inner sum over all combinations of nontrivial
error patterns $\textbf{r}=(r_{1},\ldots,r_{k})$, we find

$\displaystyle\sum_{\textbf{r}\in\\{1,\ldots,D-1\\}^{k}}p_{\textbf{r}}$
$\displaystyle=\sum_{\ell_{1}+\ldots+\ell_{D-1}=k}{k\choose\ell_{1},\ldots,\ell_{D-1}}p_{1}^{\ell_{1}}\ldots
p_{D-1}^{\ell_{D-1}},$ (35)

where for $\ell_{1}+\ldots+\ell_{D-1}=k$ the multinomial coefficient is
defined as

$\displaystyle{k\choose\ell_{1},\ldots,\ell_{D-1}}$
$\displaystyle=\frac{k!}{\ell_{1}!\ldots\ell_{D-1}!}.$ (36)

Note that $s_{1}=\ldots=s_{n}=1$ can be assumed for the evaluation of Eq.
(35). Because $p_{e_{i}}\neq p_{{e^{\prime}}_{i}}$ for
$e_{i}\neq{e^{\prime}}_{i}$, no further simplification can be made through
combining coinciding terms. As before, the final error distribution follows
from the corresponding success probabilities.

###  3.3 Optimizing the quantum-repeater gain

In order to identify genuine quantum repeaters, we want to find parameter
regions where the quantum-repeater gain,
$\Delta={B}^{\downarrow}_{\mathrm{rep}}-{B}^{\uparrow}_{\mathrm{PLOB}}$, takes
values which are significantly larger than zero. The first parameter we focus
on is the repeater spacing $L_{0}$. In Fig. 2,

$L_{0}$ in km

Figure 2: Quantum-repeater gain
$\Delta={B}^{\downarrow}_{\mathrm{rep}}-{B}^{\uparrow}_{\mathrm{PLOB}}$ in
terms of the spacing $L_{0}$ between adjacent repeater stations for a quantum
repeater line of total length $L=200$km. The qudits are encoded with a
$\llbracket D,1,(D+1)/2\rrbracket_{D}$ QECC where the qudit dimension $D$ is
color coded. Dashed and solid curves correspond to MM and Fock qudits,
respectively. Note that $\Delta$ is negative for $D=13$ Fock qudits because it
cannot be ensured that sufficiently many errors can be corrected. We use the
error model of Sec. 3.1 with $\alpha=0.2\mathrm{dB/km}$,
$f_{\mathrm{M}}=10^{-2}$, $f_{\mathrm{G}}=10^{-3}$, and
$\gamma=10^{-2}\mathrm{dB/ms}$, i.e., $f_{\mathrm{S}}\approx 2.3\times
10^{-3}$.

$\Delta$ is plotted as a function of $L_{0}$ for a quantum repeater line of
total length $L=200\mathrm{km}$, a distance large enough that $\Delta$ takes
positive values while the $M$-mode PLOB-repeaterless bound,
${B}^{\uparrow}_{\mathrm{PLOB}}\approx 1.44M\times 10^{-4}$, still has a
recognizable influence. The selected QECCs have parameters $\llbracket
D,1,(D+1)/2\rrbracket_{D}$, i.e., they saturate the quantum singleton bound.
Note that this also is the largest possible code distance for which quantum
polynomial codes are available at a given dimension $D$, where $D$ is a prime
number. Thus, the considered QECCs are the best available for a given qudit
dimension. The prime dimensions $D\in\\{13,17,29,37\\}$ are selected such that
the code distance $d=(D+1)/2$ of the QECC is odd, as this ensures that the
number of correctable single qudit errors is $t=(d-1)/2$ (and not
$t=(d-2)/2$). For large $D$, the error correction capabilities perform
sufficiently good, such that $H(P)$, the Shannon entropy of the error
probability distribution, is almost zero, i.e., Alice and Bob have almost
perfect knowledge about the state of their qudits. In this saturated regime,
where $\Delta$ does not significantly change with respect to $L_{0}$ over some
orders of magnitude, the height of the plateau,
$\max_{L_{0}}\Delta=\log_{2}(D)-H(P)-{B}^{\uparrow}_{\mathrm{PLOB}}\approx\log_{2}(D)-{B}^{\uparrow}_{\mathrm{PLOB}}$,
is, by Eq. (18), larger for Fock qudits than for MM qudits because logical
Fock qudits employ only $M=D$ modes while MM qudits require $M=D^{2}$ modes.
Thus, for short distances $L_{0}$, the repeaterless quantum capacity is larger
for MM qudits. Indeed, the gap between the Fock and the MM plateau is given by

$\displaystyle{B}^{\uparrow}_{\mathrm{PLOB}}(\mathrm{MM})-{B}^{\uparrow}_{\mathrm{PLOB}}(\mathrm{Fock})\approx
1.44\times 10^{-4}\times(D^{2}-D)$
$\displaystyle\approx\begin{cases}0.2&\text{ for }D=37\\\ 0.1&\text{ for
}D=29\ \ .\end{cases}$ (37)

As $L_{0}$ further increases, transmission losses start to significantly
deteriorate the error correction procedure, causing a sudden drop of $\Delta$.
The largest possible repeater spacing for which the PLOB-repeaterless bound is
surpassed is on the order of $L_{0}\sim 1$km for MM qudits and $L_{0}\sim
0.01-0.1$km for Fock qudits, respectively. For Fock qudits, the quantum-
repeater gain is more vulnerable to transmission losses because of our worst-
case approximation of the corresponding pure-loss channel, recall Eq. (24). On
the other hand, for repeater lines with a very small repeater spacing $L_{0}$,
operational errors (gate and measurement errors) start to play a critical
role, as more repeater stations increase the number of error sources, until
eventually the lower bound on the repeater’s quantum capacity,
${B}^{\downarrow}_{\mathrm{rep}}$, vanishes and $\Delta$ coincides with
$-{B}^{\uparrow}_{\mathrm{PLOB}}$. Since we assume depolarizing noise for
operational errors in both encodings, repeaters based on MM and Fock qudits
qualitatively show the same behavior for small $L_{0}$. For small $D$, the
code distance $d$ is too small, both transmission losses and operational
errors deteriorate the error correction procedure, which prohibits the
formation of a plateau where $\Delta(L_{0})$ is constant. We stress that the
repeater spacing $L_{0}$ can be raised tremendously if the intended quantum-
repeater gain $\Delta$ is sub-optimal, e.g.,
${B}^{\downarrow}_{\mathrm{rep}}=0.9\times
B^{\downarrow\mathrm{max}}_{\mathrm{rep}}$.

In Fig. 3,

Multimode qudits Fock qudits

$\Delta$

$L$ in km $L$ in km

Figure 3: Quantum-repeater gain
$\Delta={B}^{\downarrow}_{\mathrm{rep}}-{B}^{\uparrow}_{\mathrm{PLOB}}$ and
corresponding repeater spacing $L_{0}$ (white lines) for $D$-dimensional
qudits based on MM (left) and Fock (right) encoding, The qudits are encoded
with a $\llbracket D,1,(D+1)/2\rrbracket_{D}$ QECC where the qudit dimension
varies in steps of 4 from $D=5$ to $D=93$ and $D=33$ for MM and Fock encoding,
respectively. Note that the computational complexity of Eq. (34) limits $D\leq
33$ for Fock qudits. The total distance between Alice and Bob varies from
$L=0\mathrm{km}$ to $L=500\mathrm{km}$ and the repeater spacing $L_{0}=L/N$ is
adjusted such that ${B}^{\downarrow}_{\mathrm{rep}}=0.9\times
B^{\downarrow\mathrm{max}}_{\mathrm{rep}}$. We use the error model of Sec. 3.1
with $\alpha=0.2\mathrm{dB/km}$, $f_{\mathrm{M}}=10^{-2}$,
$f_{\mathrm{G}}=10^{-3}$, and $\gamma=10^{-2}\mathrm{dB/ms}$.

we display the quantum-repeater gain $\Delta$ (color coded) for quantum
repeater lines of varying total length $L$ (abscissa) and qudit dimension $D$
(ordinate). We vary $D$ in steps of 4 such that the considered $\llbracket
D,1,(D+1)/2\rrbracket_{D}$ QECCs have an odd code distance $d$, in between, we
interpolate. The corresponding values of $L_{0}$ are included in Fig. 3 via
white contour lines. For MM and Fock qudits, respectively, the repeater
spacing is on the order of $L_{0}\sim 1\mathrm{km}$ and $L_{0}\sim
0.01\mathrm{km}$, respectively. If the total length $L$ of the repeater line
is shortened, transmission losses become less important and operational errors
begin to dominate. Thus, to reach ${B}^{\downarrow}_{\mathrm{rep}}=0.9\times
B^{\downarrow\mathrm{max}}_{\mathrm{rep}}$, the spacing $L_{0}$ between two
adjacent repeater stations is increased, as this decreases the number of
operational error sources. The spacing $L_{0}$ also increases with $D$ because
QECCs with a higher code distance $d=(D+1)/2$ can correct more errors.

We find three distinct regions in Fig. 3, each with a typical signature: (i)
For small $L$, the PLOB-repeaterless bound cannot be surpassed, i.e.,
$\Delta<0$. (ii) For large $L$ and a large code distance $d=(D+1)/2$ we
observe $\Delta>0$. (iii) For large $L$ and small $d$ we find $\Delta\approx
0$. We now discuss the signatures of these three regions:

  * (i)

[$\Delta<0$] At the brown region on the left, the $M$-mode PLOB-repeaterless
bound is much larger than $\log_{2}(D)\geq{B}^{\downarrow}_{\mathrm{rep}}$.
Asymptotically, it is even unbounded,

$\displaystyle{B}^{\uparrow}_{\mathrm{PLOB}}=-M\times\log_{2}(1-\eta)\overset{L\rightarrow
0}{\longrightarrow}\infty.$ (38)

As logical $\llbracket D,1,(D+1)/2\rrbracket_{D}$ MM qudits are encoded into
$M=D^{2}$ modes, this region extends to longer total lengths $L$ of the
repeater line if the qudit dimension $D$ is larger, e.g.,
$L(\Delta<0,D=13)\lesssim 100\mathrm{km}$ and $L(\Delta<0,D=85)\lesssim
150\mathrm{km}$. For Fock qudits, which only require $M=D$ modes, this effects
is barely noticeable and $L(\Delta<0)\lesssim 50\mathrm{km}$ for all $D$.

  * (ii)

[$\Delta>0$] At the purple region on the upper right, quantum repeaters can
surpass the PLOB-repeaterless bound because
${B}^{\uparrow}_{\mathrm{PLOB}}\approx 0$ while
${B}^{\downarrow}_{\mathrm{rep}}>0$. The quantum-repeater gain
$\Delta={B}^{\downarrow}_{\mathrm{rep}}-{B}^{\uparrow}_{\mathrm{PLOB}}\approx\log_{2}(D)-H(P)$
increases (for a fixed $L$) in $D$ if the distance $d=(D+1)/2$ of the QECC is
large enough, as this causes $H(P)\approx 0$. Genuine MM quantum repeaters are
possible for $D\geq 13$, whereby the minimal repeater length increases with
the number of modes $M=D^{2}$, as already discussed for region (i). For
quantum repeaters with Fock encoding, the PLOB-repeaterless bound can be
outperformed for $D\geq 17$ and $L>50\mathrm{km}$. For $D=13$, we observe a
small quantum-repeater gain $\Delta\in(0.1,0.5)$ for quantum repeater lines
with a total length $L$ between $60\mathrm{km}$ and $110\mathrm{km}$.

  * (iii)

[$\Delta\approx 0$] At the white region on the lower right, the total length
$L$ is too large and the code distance $d$ is too small such that
${B}^{\uparrow}_{\mathrm{PLOB}}\approx 0$ and
${B}^{\downarrow}_{\mathrm{rep}}=0$, respectively. Recall that we consider
$\llbracket D,1,(D+1)/2\rrbracket_{D}$ quantum polynomial codes, as they have
the highest code distance for a given dimension, as well as transversal cz
gates and transversal $X$ measurements. For $D\leq 9$, we find
${B}^{\downarrow}_{\mathrm{rep}}=0$ which implies $\Delta\approx 0$.

Let us summarize what can be learned from Figs. 2 and 3. Using higher-
dimensional qudits, it is possible to reach a larger quantum-repeater gain
because the ideal quantum capacity of the quantum repeater is given by
$\log_{2}(D)$. In many cases, this optimum can be reached by an appropriate
choice of $L_{0}$. Since the code distance of the best known QECCs also grows
with the qudits’ dimension, a side effect of higher-dimensional qudits is the
possibility to increase the distance $L_{0}$ between two neighboring repeater
stations. For MM qudits, $L_{0}$ is two orders of magnitude larger than for
Fock qudits. We attribute this to our worst-case Pauli-approximation of the
pure-loss channel for Fock qudits which leads to a lower bound on $L_{0}$ for
genuine quantum repeaters. We stress that this bound is not tight. Within our
error model, intermediate repeater stations of a genuine quantum repeater
cannot be separated by a distance exceeding $L_{0}\sim 1$km. Even if all error
parameters except for fiber attenuation are set to zero, $L_{0}$ does not
improve by an order of magnitude. Finally, within our error model, Fock and MM
quantum repeaters can surpass the PLOB-repeaterless bound for
$L>50\mathrm{km}$ and $L>100-150\mathrm{km}$, respectively. MM repeaters
require a larger total length because the PLOB-repeaterless bound is higher
for a larger number of modes.

###  3.4 Influence of operational errors on the quantum-repeater gain

As it is easier to implement a $\llbracket D,1,(D+1)/2\rrbracket_{D}$ QECC for
smaller qudit dimension $D$, it is desirable to lower the demands on error
correction. One way to achieve this is the improvement of operational error
rates. As we have shown in Sec. 3.2, operational errors mainly originate from
intermediate repeater stations, where gate and measurement errors enter via
$(1-f_{\text{G}})^{3}$ and $(1-f_{\text{M}})$, respectively, recall Eqs. (28)
and (31). Thus, gate errors affect the quantum-repeater gain three times as
large as measurement errors do, but they otherwise lead to the same
qualitative behavior of $\Delta$. Hence, we restrict the investigation of the
influence of operational errors to the measurement error rate $f_{\text{M}}$
and fix $f_{\text{G}}=10^{-3}$, as before. Figure 4

$f_{\mathrm{M}}$

Figure 4: The quantum-repeater gain $\Delta$ in terms of the measurement error
rate $f_{\mathrm{M}}$. The other error parameters are fixed to
$\alpha=0.2\mathrm{dB/km}$, $f_{\mathrm{G}}=10^{-3}$, and
$\gamma=0.01\mathrm{dB/ms}$, i.e., $f_{\mathrm{S}}\approx 2.3\times 10^{-3}$.
The qudits are encoded with a $\llbracket D,1,(D+1)/2\rrbracket_{D}$ QECC. The
repeater line has a total length of $L=200$km and the repeater spacing
$L_{0}=L/N$ is adjusted such that ${B}^{\downarrow}_{\mathrm{rep}}=0.9\times
B^{\downarrow\mathrm{max}}_{\mathrm{rep}}$.

shows the quantum-repeater gain as a function of the measurement error rate
$f_{\text{M}}$. We observe a similar pattern for all curves: At low error
rates $f_{\text{M}}<10^{-3}$ the quantum-repeater gain $\Delta$ is almost
constant. As $f_{\text{M}}$ increases, the smaller the dimension $D$, the
sooner the corresponding quantum-repeater gain drops to zero, as fewer errors
can be corrected by the QECC. In terms of quantum-repeater gain and in direct
comparison, the MM encoding is more tolerant towards measurement errors than
the Fock encoded repeater line. As expected, lower operational error rates
allow genuine quantum repeaters with smaller dimension $D$, as fewer errors
need to be corrected. In this range of $f_{\mathrm{M}}$, the smallest
dimension for which genuine quantum repeaters are possible is $D=11$ with MM
qudits.

###  3.5 Estimate of resources

For a resource-efficient use of quantum repeaters it is crucial to identify
cost-saving candidates. Naturally, the costs of developing and maintaining a
single repeater station will increase with $D$, as a $\llbracket
D,1,(D+1)/2\rrbracket_{D}$ QECC is employed. However, higher-dimensional
qudits have the advantage of better error correction capabilities, thus, the
number of necessary repeater stations is lower. Table 1

| $D$ | 13 | 17 | 29 | 73
---|---|---|---|---|---
MM | $N_{\mathrm{min}}$ | 371 | 248 | 163 | 119
| $L(N_{\mathrm{min}})$ | 120km | 130km | 140km | 170km
Fock | $N_{\mathrm{min}}$ | - | 2070 | 1705 | ?
| $L(N_{\mathrm{min}})$ |  | 56km | 60km |
Table 1: The minimal number $N_{\mathrm{min}}$ of repeater stations for which
the PLOB-repeaterless bound can be just surpassed by a quantum repeater line
of a total length $L(N_{\mathrm{min}})$, see also Fig. 5. We use the error
model of Sec. 3.1 with $\alpha=0.2\mathrm{dB/km}$, $f_{\mathrm{M}}=10^{-2}$,
$f_{\mathrm{G}}=10^{-3}$, and $\gamma=10^{-2}\mathrm{dB/ms}$. For $D=13$ Fock
qudits, the PLOB-repeaterless bound is not surpassed, and for $D=73$ Fock
qudits, the corresponding minimum cannot be evaluated due to the computational
complexity of Eq. (34).

provides the minimal requirement on the number of repeater stations of a
genuine error-corrected qudit repeater.

A relevant figure of merit to compare different quantum repeater lines is the
minimum number of repeater stations per unit length $N_{\mathrm{min}}/L$,
which we plot in Fig. 5

$L$ in km

Figure 5: The minimal number of repeater stations per km for which the PLOB-
repeaterless bound can be just surpassed by a quantum repeater line of total
length $L$ with MM and Fock qudits encoded by a $\llbracket
D,1,(D+1)/2\rrbracket_{D}$ QECC. The error parameters are
$\alpha=0.2\mathrm{dB/km}$, $f_{\mathrm{M}}=10^{-2}$,
$f_{\mathrm{G}}=10^{-3}$, and $\gamma=10^{-2}\mathrm{dB/ms}$.

as a function of the total length $L$ for various encodings. All curves
qualitatively show the same behavior: For very small $L$, the PLOB-
repeaterless bound is not surpassed, as direct quantum communication is still
possible. Eventually, with increasing $L$, the PLOB-repeaterless bound has
dropped to a quantum capacity which can be surpassed by
${B}^{\downarrow}_{\mathrm{rep}}$. At this minimal total length
$L_{\mathrm{min}}$, the curves in Fig. 5 begin. For Fock qudits,
$L_{\mathrm{min}}$ is smaller than for MM qudits because fewer modes are used,
consistent with previous observations above. If $L$ is slightly above
$L_{\mathrm{min}}$, the PLOB-repeaterless bound
${B}^{\uparrow}_{\mathrm{PLOB}}$ quickly approaches zero. Thus, the lower
bound on the quantum capacity of the repeater,
${B}^{\downarrow}_{\mathrm{rep}}=\log_{2}(D)-H(P)$, is allowed to decrease as
well, which leads to the possibility of setting up the quantum repeater with
fewer repeater stations per unit length. This explains the initial drop of the
curves for $L\gtrsim L_{\mathrm{min}}$. At some point, the global minima from
Tab. 1 are reached. For larger $L$, the regime
${B}^{\uparrow}_{\mathrm{PLOB}}\approx 0$ is entered. Since we consider
$0\lesssim\Delta$, this implies $H(P)\lesssim\log_{2}(D)$. That is, the number
of repeater stations is adjusted such that the Shannon entropy of the error
distribution on the distributed state is kept slightly below $\log_{2}(D)$.
If, in this regime, $L$ is increased, the amount of error correction overhead
has to be adjusted accordingly. Therefore, the minimal number of repeater
stations per unit length increases with $L$. The (log-log) slope of the
corresponding curves in this intermediate region decreases with $D$ because
QECCs with a larger code distances can more readily cope with the additional
transmission losses. For $D=73$ MM qudits, the code distance $d=37$ of the
QECC is large enough such that the curve just barely grows. Eventually, so
many storage errors of Alice’s quantum memory have accumulated that the
pseudothreshold111The (code capacity) pseudothreshold of a QECC is the error
rate at which the physical error rate is equal to the logical error rate, in
Eq. (30) with $f_{\mathrm{S}}=p_{e\neq 0}$. For $\llbracket
29,1,15\rrbracket_{29}$ and $\llbracket 73,1,37\rrbracket_{73}$ QECCs, our
calculations show that this pseudothreshold is approximately $20\%$. By Eq.
(21), $f_{\mathrm{S}}=0.2$ is reached for $L\approx 20,000$km since
$\gamma=10^{-2}\mathrm{dB/ms}$. of the respective QECC is reached. In that
region, storage errors strongly influence $H(P)$ until the condition
$H(P)\lesssim\log_{2}(D)$ cannot be fulfilled for any choice of $N/L$. This
explains the sudden growth of the curves in Fig. 5 for large values of $L$ and
is clearly visible in Fig. 6.

$\frac{N}{L[\mathrm{km}]}$

Figure 6: The Shannon entropy $H(P)$ of the error distribution of a state
distributed by a $\llbracket 29,1,15\rrbracket_{29}$ error-corrected MM
quantum repeater for different total lengths $L$ as a function of the inverse
repeater spacing $1/L_{0}=N/L$. Perfect error correction means $H(P)=0$. For
$L\leq 20,000$km, one can reach $H(P)\lesssim\log_{2}(D)$ by an adjustment of
$N/L$. Since the global minimum of $H(P)$ grows in $L$, the value of $N/L$
where $\log_{2}(D)$ intersects $H(P)$ exponentially increases. For
$L=25,000$km, ${B}^{\downarrow}_{\mathrm{rep}}=\max\\{\log_{2}(D)-H(P),0\\}$
is zero for any choice of $N/L$, i.e., the quantum-repeater gain $\Delta$ is
negative. As in Fig. 5, we use $\alpha=0.2\mathrm{dB/km}$,
$f_{\mathrm{M}}=10^{-2}$, $f_{\mathrm{G}}=10^{-3}$, and
$\gamma=10^{-2}\mathrm{dB/ms}$.

In conclusion of this subsection, we observe that for our error model and
existing QECCs, genuine error-corrected qudit repeater lines require at least
about $10^{2}$ and $10^{3}$ repeater stations for MM and Fock qudits,
respectively. For a total length $L$ between $10^{2}$km and $10^{4}$km, the
PLOB-repeaterless bound can be surpassed while the minimum number of repeater
stations per unit length, $N_{\mathrm{min}}/L$ , gradually increases in $L$.
This increase is less pronounced for quantum repeaters with better error-
correcting capabilities, i.e., for a higher qudit dimension $D$.

##  4 Conclusion and Outlook

In this paper, we have analyzed the applicability of error-corrected quantum
repeaters based on higher-dimensional qudits as long-term candidates of a
quantum communication infrastructure. To our knowledge, this work shows for
the first time how the PLOB-repeaterless bound can be surpassed using error-
corrected qudit repeaters. By making explicit how the PLOB-repeaterless bound
relates to the encoding of abstract qudits into photonic modes, we obtain a
bound on the capacity of the quantum repeater using the Shannon entropy of the
error distribution of the final state. We defined the quantum repeater-gain as
a figure of merit and used it to identify genuine quantum repeaters by a
systematic analysis of its dependency on a variety of parameters.

We derived an analytical solution of the quantum-repeater gain for error-
corrected, one-way qudit repeaters based on two different types of physical
encoding: Fock encoding, where each qudit is encoded into a single photonic
mode; and multimode encoding, where each computational basis state of a qudit
has its own mode. While Fock encoding is interesting from a theoretical
perspective, as it allows to surpass the PLOB-repeaterless bound over shorter
distances by harnessing higher photon numbers of the photonic mode, multimode
qudits pose the more realistic way of implementing error-corrected qudit
repeaters, as they are more readily available in the form of e.g., time-bin
qudits, temporal modes, and modes of orbital angular momentum. We found that
genuine quantum repeaters are feasible if the distance $L_{0}$ between
adjacent repeater stations is on the order of 1km for multimode encoding,
independent of its total length. For Fock qudits, we can only prove that
$L_{0}\sim 10\mathrm{m}$ is sufficient however, we expect that this is due to
our worst-case approximation of the pure-loss channel and that Fock repeaters
can also surpass the PLOB-repeaterless bound with $L_{0}\sim
100\mathrm{m}-1\mathrm{km}$.

We have shown that an improvement of operational error rates makes it possible
to decrease the necessary number of physical qudits per logical qudit, as well
as the qudit dimension. For realistic error rates, the smallest qudit
dimension with which a genuine quantum repeater could be realized within our
error model is $D=13$ and $D=11$ for Fock and MM qudits, respectively.
Although theoretical proposals for the generation of Fock states with an
arbitrary photon number exist [73], high-quality Fock states have
experimentally only been realized up to four photons, i.e.,
$D_{\mathrm{Fock}}\leq 5$, and no significant improvement was made over the
last 10-15 years [74, 75]. For MM qudits, on the other hand, the state-of-the-
art continuously progresses: Qudits based on temporal modes, orbital angular
momentum and time bin can be realized up to $D_{\mathrm{TM}}\leq 7$ [50],
$D_{\mathrm{OAM}}\lesssim 100$ [54], and $D_{\text{time-bin}}\lesssim 10^{5}$
[47, 48], respectively. An experimental challenge that has to be overcome to
realize the here-considered protocol is the development of high-fidelity, two-
photon controlled-phase gates, as well as the preparation of multipartite
entangled photons, in particular in the logical $\ket{+}$ state of a quantum
polynomial code.

While near-term candidates such as the so-called single photon scheme based on
a nitrogen vacancy architecture [23] and twin-field quantum key distribution
[24, 25, 26] are within experimental reach, they are spatially restricted to
tens and hundreds of kilometers, respectively. As we showed here, error-
corrected qudit repeaters, on the other hand, have the potential to overcome
the PLOB-repeaterless bound over length scales on the same order of magnitude
as the Trans-Siberian railroad, i.e., $10^{4}$km. These length scales are
sufficient to connect any two points on earth.

Here, we have focused on subspace quantum error-correcting codes (QECCs) [57],
in particular, quantum polynomial codes [58, 59, 60, 61]. However, recently it
was shown that subsystem QECCs can have an advantage in fighting leakage
errors in ion trap quantum computers [76]. It would be interesting to find out
whether subsystem codes, such as Bacon-Shor codes [77, 78, 79], subsystem
surface codes [80], 2D compass codes [81], and optimal generalized Bacon-Shor
codes [82], also have an advantage in coping with photon losses in an error-
corrected quantum repeater protocol. Finally, note that the Gottesman-Kitaev-
Preskill (GKP) code which encodes each physical qudit into a single bosonic
mode is naturally suited in the case of leakage errors [83]. It would be
valuable to extend our analysis for Fock qudits and multimode qudits to GKP
qudits as well, similar to Ref. [84].

The authors thank Mohsen Razavi and Federico Grasselli for helpful discussions
and Eric Sabo for feedback on the manuscript. The authors acknowledge support
from the Federal Ministry of Education and Research (BMBF, Project Q.Link.X).

## References

  * [1] M. Riedel, D. Binosi, R. Thew, and T. Calarco, _The European quantum technologies flagship programme_ , Quantum Sci. Technol. 2, 030501 (2017).
  * [2] A. Acín, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. Riedel, P. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. Wilhelm, _The quantum technologies roadmap: a European community view_ , New J. Phys. 20, 080201 (2018).
  * [3] S. Wehner, D. Elkouss, and R. Hanson, _Quantum internet: A vision for the road ahead_ , Science 362, 6412 (2018).
  * [4] V. Giovannetti, S. Lloyd, and L. Maccone, _Quantum-enhanced positioning and clock synchronization_ , Nature 412, 417 (2001).
  * [5] M. Christandl and S. Wehner, _Quantum Anonymous Transmissions_ , ASIACRYPT 2005, 217-235 (2005).
  * [6] D. Gottesman, T. Jennewein, and S. Croke, _Longer-Baseline Telescopes Using Quantum Repeaters_ , Phys. Rev. Lett. 109, 070503 (2012).
  * [7] E. Khabiboulline, J. Borregaard, K. De Greve, and M. Lukin, _Optical Interferometry with Quantum Networks_ , Phys. Rev. Lett. 123, 070504 (2019).
  * [8] C. Bennett, G. Brassard, _Public Key Distribution and Coin Tossing_ , Theor. Comput. Sci. 560, 7 (2014).
  * [9] A. Ekert, _Quantum cryptography based on Bell’s theorem_ , Phys. Rev. Lett. 67, 661 (1991).
  * [10] D. Bruß, _Optimal Eavesdropping in Quantum Cryptography with Six States_ , Phys. Rev. Lett. 81, 3018 (1998).
  * [11] A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M. Li, D. Nolan, A. Martin, and H. Zbinden, _Secure Quantum Key Distribution over 421 km of Optical Fiber_ , Phys. Rev. Lett. 121, 190502, (2018).
  * [12] M. Takeoka, S. Guha, and M. Wilde, _Fundamental rate-loss tradeoff for optical quantum key distribution_ , Nat. Comm. 5, 5235 (2014).
  * [13] M. Christandl and A. Müller-Hermes, _Relative Entropy Bounds on Quantum, Private and Repeater Capacities_ , Commun. Math. Phys. 353, 821 (2017).
  * [14] S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, _Fundamental limits of repeaterless quantum communications_ , Nat. Commun. 8, 15043 (2017).
  * [15] H. Briegel, W. Dür, J. Cirac, and P. Zoller, _Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication_ , Phys. Rev. Lett. 81, 5932 (1998).
  * [16] P. van Loock, T. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. Munro, and Y. Yamamoto, _Hybrid Quantum Repeater Using Bright Coherent Light_ , Phys. Rev. Lett. 96, 240501 (2006).
  * [17] L. Jiang, J. Taylor, K. Nemoto, W. Munro, R. Van Meter, and M. Lukin, _Quantum repeater with encoding_ , Phys. Rev. A 79, 032325 (2009).
  * [18] A. Fowler, D. Wang, C. Hill, T. Ladd, R. Van Meter, and L. Hollenberg, _Surface Code Quantum Communication_ , Phys. Rev. Lett. 104, 180503 (2010).
  * [19] S. Muralidharan, J. Kim, N. Lütkenhaus, M Lukin, and L. Jiang, _Ultrafast and Fault-Tolerant Quantum Communication across Long Distances_ , Phys. Rev. Lett. 112, 250501 (2014).
  * [20] S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. Lukin and L. Jiang, _Optimal architectures for long distance quantum communication_ , Sci Rep. 6, 20463 (2016).
  * [21] D. Luong, L. Jiang, J. Kim, and N. Lütkenhaus, _Overcoming lossy channel bounds using a single quantum repeater node_ , Appl. Phys. B 112: 96 (2016).
  * [22] F. Rozpędek., K. Goodenough, J. Ribeiro, N. Kalb, V. Caprara Vivoli, A. Reiserer, R. Hanson, S. Wehner, and D. Elkouss, _Parameter regimes for a single sequential quantum repeater_ , Quantum Sci. Technol. 3, 034002 (2018).
  * [23] F. Rozpędek, R. Yehia, K. Goodenough, M. Ruf, P. Humphreys, R. Hanson, S. Wehner, and D. Elkouss, _Near-term quantum-repeater experiments with nitrogen-vacancy centers: Overcoming the limitations of direct transmission_ , Phys. Rev. A 99, 052330 (2019).
  * [24] M. Lucamarini, Z. Yuan, J. Dynes, and A. Shields, _Overcoming the rate-distance limit of quantum key distribution without quantum repeaters_ , Nature 557, 400 (2018).
  * [25] M. Curty, K. Azuma, and H. Lo, _Simple security proof of twin-field type quantum key distribution protocol_ , npj Quantum Inf. 5, 64 (2019).
  * [26] F. Grasselli and M. Curty, _Practical decoy-state method for twin-field quantum key distribution_ , New J. Phys. 21, 073001 (2019).
  * [27] M. Minder, M. Pittaluga, G. Roberts, M. Lucamarini, J. Dynes, Z. Yuan, and A. Shields, _Experimental quantum key distribution beyond the repeaterless secret key capacity_ , Nat. Photonics 13, 334 (2019).
  * [28] S. Wang, D. He, Z. Yin, F. Lu, C. Cui, W. Chen, Z. Zhou, G. Guo, and Z. Han, _Beating the Fundamental Rate-Distance Limit in a Proof-of-Principle Quantum Key Distribution System_ , Phys. Rev. X 9, 021046 (2019).
  * [29] X. Zhong, J. Hu, M. Curty, L. Qian, and H. Lo, _Proof-of-Principle Experimental Demonstration of Twin-Field Type Quantum Key Distribution_ , Phys. Rev. Lett. 123, 100506 (2019).
  * [30] F. Ewert and P. van Loock, _Ultrafast fault-tolerant long-distance quantum communication with static linear optics_ , Phys. Rev. A 95, 012327 (2017).
  * [31] F. Schmidt and P. van Loock, _Efficiencies of logical Bell measurements on Calderbank-Shor-Steane codes with static linear optics_ , Phys. Rev. A 99, 062308 (2019).
  * [32] S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, _Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise_ , Phys. Rev. X 9, 041042 (2019).
  * [33] S. Muralidharan, C. Zou, L. Li, J. Wen, and L. Jiang, _Overcoming erasure errors with multilevel systems_ , New J. Phys. 19, 013026 (2017).
  * [34] S. Muralidharan, C. Zou, L. Li, and L. Jiang, _One-way quantum repeaters with quantum Reed-Solomon codes_ , Phys. Rev. A 97, 052316 (2018).
  * [35] D. Miller, T. Holz, H. Kampermann, and D. Bruß, _Propagation of generalized Pauli errors in qudit Clifford circuits_ , Phys. Rev. A 98, 052316 (2018).
  * [36] M. Bergmann and P. van Loock, _Hybrid quantum repeater for qudits_ , Phys. Rev. A 99, 032349 (2019).
  * [37] S. Abruzzo, S. Bratzik, N. Bernardes, H. Kampermann, P. van Loock, and D. Bruß, _Quantum repeaters and quantum key distribution: Analysis of secret-key rates_ , Phys. Rev. A 87, 052315 (2013).
  * [38] S. Bratzik, H. Kampermann, and D. Bruß, _Secret key rates for an encoded quantum repeater_ , Phys. Rev. A 89, 032335 (2014).
  * [39] M. Epping, H. Kampermann, and D. Bruß, _On the error analysis of quantum repeaters with encoding_ , Appl. Phys. B 122: 54 (2016).
  * [40] M. Epping, H. Kampermann, and D. Bruß, _Large-scale quantum networks based on graphs_ , New J. Phys. 18, 53036 (2016).
  * [41] M. Epping, H. Kampermann, and D. Bruß, _Robust entanglement distribution via quantum network coding_ , New J. Phys. 18, 103052 (2016).
  * [42] K. Vollbrecht and M. Wolf, _Efficient distillation beyond qubits_ , Phys. Rev. A 67, 012303 (2003).
  * [43] S. Braunstein and P. van Loock, _Quantum information with continuous variables_ , Rev. Mod. Phys. 77, 513 (2005).
  * [44] C. Weedbrook, S. Pirandola, R. García-Patrón, N. Cerf, T. Ralph, J. Shapiro, and S. Lloyd, _Gaussian quantum information_ , Rev. Mod. Phys. 84, 621 (2012).
  * [45] J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, _Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication_ , Phys. Rev. Lett. 82, 2594 (1999).
  * [46] H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, _Creating high dimensional time-bin entanglement using mode-locked lasers_ , Quant. Inf. Comp. 2, 425 (2002).
  * [47] T. Zhong, H. Zhou, R. Horansky, C. Lee, V. Verma, A. Lita, A. Restelli, J. Bienfang, R. Mirin, T. Gerrits, S. Nam, F. Marsili, M. Shaw, Z. Zhang, L. Wang, D. Englund, G. Wornell, J. Shapiro, and F. Wong, _Photon-efficient quantum key distribution using time-energy entanglement with high-dimensional encoding_ , New J. Phys. 17, 022002 (2015).
  * [48] N. Montaut, O. Magaña-Loaiza, T. Bartley, V. Verma, S. Nam, R. Mirin, C. Silberhorn, and T. Gerrits, _Compressive characterization of telecom photon pairs in the spatial and spectral degrees of freedom_ , Optica 5, 1418 (2018).
  * [49] B. Brecht, D. Reddy, C. Silberhorn and M. Raymer, _Photon Temporal Modes: A Complete Framework for Quantum Information Science_ , Phys. Rev. X 5, 041017 (2015).
  * [50] V. Ansari, J. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, _Tomography and Purification of the Temporal-Mode Structure of Quantum Light_ , Phys. Rev. Lett. 120, 213601 (2018).
  * [51] L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, _Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes_ , Phys. Rev. A 45, 8185 (1992).
  * [52] G. Calvo, A. Picón, and E. Bagan, _Quantum field theory of photons with orbital angular momentum_ , Phys. Rev. A 73, 013805 (2006).
  * [53] W. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, _Quantum orbital angular momentum of elliptically symmetric light_ , Phys. Rev. A 87, (2013).
  * [54] M. Krenn, M. Malik, M. Erhard, and A. Zeilinger, _Orbital angular momentum of photons and the entanglement of Laguerre–Gaussian modes_ , Phil. Trans. R. Soc. A 375: 20150442 (2017).
  * [55] E. Knill, _Group Representations, Error Bases and Quantum Codes_ , Technical Report LAUR-96-2807, Los Alamos National Laboratory, arXiv:9608049 [quant-ph] (1996).
  * [56] D. Gottesman, _Fault-Tolerant Quantum Computation with Higher-Dimensional Systems_ , Chaos Solitons Fractals 10, 1749-1758 (1999).
  * [57] D. Lidar and T. Brun, _Quantum Error Correction_ , Cambridge University Press (2013).
  * [58] R. Cleve, D. Gottesman and H. Lo, _How to share a quantum secret_ , Phys. Rev. Lett. 83, 648 (1999).
  * [59] D. Aharonov and M. Ben-Or, _Fault-Tolerant Quantum Computation with Constant Error Rate_ , SIAM J. Comput. 38(4), 1207 (2008).
  * [60] A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, _Nonbinary stabilizer codes over finite fields_ , IEEE Trans. Inf. Theory, 52(11), 4892 (2006).
  * [61] A. Cross, _Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes_ , MIT 1721.1/44407 (2008).
  * [62] N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, _Quantum cryptography_ , Rev. Mod. Phys. 74, 145 (2002).
  * [63] R. Hadfield, _Single-photon detectors for optical quantum information applications_ , Nat. Photonics 3, 696 (2009).
  * [64] B. Hacker, S. Welte, G. Rempe, and S. Ritter, _A photon–photon quantum gate based on a single atom in an optical resonator_ Nature 536, 193 (2016).
  * [65] G. Alber, A. Delgado, N. Gisin, and I. Jex, _Efficient bipartite quantum state purification in arbitrary dimensional Hilbert spaces_ , J. Phys. A: Math. Gen. 34 8821 (2001).
  * [66] R. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. Albert, L. Frunzio, L. Jiang, and R. Schoelkopf, _Cavity State Manipulation Using Photon-Number Selective Phase Gates_ , Phys. Rev. Lett. 115, 137002 (2015).
  * [67] Y. Cho, G. Campbell, J. Everett, J. Bernu, D. Higginbottom, M. Cao, J. Geng, N. Robins, P. Lam, and B. Buchler, _Highly efficient optical quantum memory with long coherence time in cold atoms_ , Optica 3, 100 (2016).
  * [68] P. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Yao, S. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. Twitchen, J. Cirac, and M. Lukin, _Room-Temperature Quantum Bit Memory Exceeding One Second_ , Sci Rep. 336, 6086 1283 (2012).
  * [69] M. Abobeih, J. Cramer, M. Bakker, N. Kalb, M. Markham, D. Twitchen, and T. Taminiau, _One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment_ , Nat. Comm. 9, 2552 (2018).
  * [70] C. Bradley, J. Randall, M. Abobeih, R. Berrevoets, M. Degen, M. Bakker, M. Markham, D. Twitchen, T. Taminiau, _A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute_ , Phys. Rev. X 9, 031045 (2019).
  * [71] Y. Wang, M. Um, J. Zhang, S. An, M. Lyu, J. Zhang, L. Duan, D. Yum, and K. Kim, _Single-qubit quantum memory exceeding ten-minute coherence time_ , Nat. Photonics 11, 646 (2017).
  * [72] M. Körber, O. Morin, S. Langenfeld, A. Neuzner, S. Ritter, and G. Rempe, _Decoherence-protected memory for a single-photon qubit_ , Nat. Photonics 12, 18 (2018).
  * [73] K. Brown, K. Dani, D. Stamper-Kurn, and K. Whaley, _Deterministic optical Fock-state generation_ , Phys. Rev. A 67, 043818 (2003).
  * [74] E. Waks, E. Diamanti, and Y. Yamamoto, _Generation of photon number states_ , New J. Phys. 8, 4 (2006).
  * [75] J. Tiedau, T. Bartley, G. Harder, A. Lita, S. Nam, T. Gerrits, and C. Silberhorn, _On the scalability of parametric down-conversion for generating higher-order Fock states_ , Phys. Rev. A 100, 041802(R) (2019).
  * [76] N. Brown, M. Newman, and K. Brown, _Handling Leakage with Subsystem Codes_ , New J. Phys. 21 073055 (2019).
  * [77] P. Shor, _Scheme for reducing decoherence in quantum computer memory_ , Phys. Rev. A 52, R2493(R) (1995).
  * [78] D. Bacon, _Operator quantum error-correcting subsystems for self-correcting quantum memories_ , Phys. Rev. A 73, 012340 (2006).
  * [79] P. Aliferis and A. Cross, _Subsystem Fault Tolerance with the Bacon-Shor Code_ , Phys. Rev. Lett. 98, 220502 (2007).
  * [80] S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, _Subsystem surface codes with three-qubit check operators_ , Quant. Inf. Comp. 13, 963 (2013).
  * [81] M. Li, D. Miller, M. Newman, Y. Wu, and K. Brown, _2D Compass Codes_ , Phys. Rev. X 9, 021041 (2019).
  * [82] T. Yoder, _Optimal quantum subsystem codes in 2-dimensions_ , Phys. Rev. A 99, 052333 (2019).
  * [83] D. Gottesman, A. Kitaev, and J. Preskill, _Encoding a qubit in an oscillator_ , Phys. Rev. A 64, 012310 (2001).
  * [84] C. Vuillot, H. Asasi, Y. Wang, L. Pryadko, and B. Terhal, _Quantum error correction with the toric Gottesman-Kitaev-Preskill code_ , Phys. Rev. A 99, 032344 (2019).