# Entangling remote qubits using the single-photon protocol: an in-depth
theoretical and experimental study

S. L. N. Hermans  QuTech and Kavli Institute of Nanoscience, Delft University
of Technology, Delft, The Netherlands  Present address: Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena,
California, USA  M. Pompili  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, Delft, The Netherlands  Present address: Pritzker
School of Molecular Engineering, University of Chicago, Chicago, Illinois, USA
L. Dos Santos Martins  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, Delft, The Netherlands  Present address: Sorbonne
Université, LIP6, CNRS, Paris, France  A. R.-P. Montblanch  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, Delft, The
Netherlands  H. K. C. Beukers  QuTech and Kavli Institute of Nanoscience,
Delft University of Technology, Delft, The Netherlands  S. Baier  QuTech and
Kavli Institute of Nanoscience, Delft University of Technology, Delft, The
Netherlands  Present address: Institut für Experimentalphysik, Universität
Innsbruck, Innsbruck, Austria  J. Borregaard  QuTech and Kavli Institute of
Nanoscience, Delft University of Technology, Delft, The Netherlands  R. Hanson
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
Delft, The Netherlands

###### Abstract

The generation of entanglement between remote matter qubits has developed into
a key capability for fundamental investigations as well as for emerging
quantum technologies. In the single-photon, protocol entanglement is heralded
by generation of qubit-photon entangled states and subsequent detection of a
single photon behind a beam splitter. In this work we perform a detailed
theoretical and experimental investigation of this protocol and its various
sources of infidelity. We develop an extensive theoretical model and
subsequently tailor it to our experimental setting, based on nitrogen-vacancy
centers in diamond. Experimentally, we verify the model by generating remote
states for varying phase and amplitudes of the initial qubit superposition
states and varying optical phase difference of the photons arriving at the
beam splitter. We show that a static frequency offset between the optical
transitions of the qubits leads to an entangled state phase that depends on
the photon detection time. We find that the implementation of a Charge-
Resonance check on the nitrogen-vacancy center yields transform-limited
linewidths. Moreover, we measure the probability of double optical excitation,
a significant source of infidelity, as a function of the power of the
excitation pulse. Finally, we find that imperfect optical excitation can lead
to a detection-arm-dependent entangled state fidelity and rate. The conclusion
presented here are not specific to the nitrogen-vacancy centers used to carry
out the experiments, and are therefore readily applicable to other qubit
platforms.

##  I Introduction

Entanglement between different nodes will be an essential element of future
quantum networks. Entangled states will serve as a key ingredient for many
applications, such as secure communication, distributed quantum computation
and advanced quantum network protocols [1, 2, 3, 4, 5, 6, 7]. Remote
entanglement between distant nodes can be generated using different protocols.
One of these protocols, the single-photon protocol based on emitted photons
encoded in number states [8, 9], is especially suited to establish
entanglement between distant stationary qubits with high generation rates in
the presence of significant photon loss. Since a single photon has to travel
only half of the distance between the emitters, the total photon loss is
reduced compared to direct photon transmission or to two-photon entangling
protocols [10]. The single-photon protocol has been implemented on various
qubit platforms, such as electron and hole spins in quantum dots, nitrogen-
vacancy centers in diamond and atomic ensembles in rare-earth-ion doped
crystals [11, 12, 13, 14].

The single-photon protocol works as follows. Two remote qubits are each
prepared in the superposition state
$\sqrt{\alpha}\ket{0}+\sqrt{1-\alpha}\ket{1}$. State-selective excitation of
$\ket{0}$, the so-called bright state, creates a qubit-photon entangled state.
Interference of the photon states on a balanced beam splitter erases the
which-path information. Detection of a single photon projects the two remote
qubits into an entangled state
$\ket{\Psi}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$, up to a single qubit phase
correction. However, in the presence of photon loss we cannot discriminate
between the emission of a single photon and the case in which both qubits were
in the bright state, $\ket{00}$, and emitted each one photon but one photon
was lost. The latter events falsely herald entanglement and reduce the average
fidelity. In the high photon-loss regime, given the detection of one photon,
the probability that both qubits are in the bright state is given by the
initial population in $\ket{0}$, $\alpha$. Hence, the average heralded density
matrix is $(1-\alpha)\ket{\Psi}\bra{\Psi}+\alpha\ket{00}\bra{00}$, with a
fidelity of $F=1-\alpha$ with respect to the maximally entangled state.

Apart from this infidelity intrinsic to the protocol, other sources of error
can degrade the heralded state. In this paper we provide a detailed
theoretical and experimental study of error sources and characteristics
associated with the single-photon protocol.

The paper is structured as follows. In Section II we describe the single-
photon entanglement protocol step-by-step for a general experimental setting
and we develop a model describing the effect of experimental imperfections. We
introduce our experimental system, the nitrogen-vacancy (NV) center in
diamond, in Section III and we tailor the model to our system in Section IV.
Afterwards, we discuss the effect of various parameters on the heralded state;
the bright state population (Section V), the phase of the entangled state
(Section VI), photon indistinguishability (Section VII), double optical
excitation (Section VIII) and non-excited bright state population (Section
IX). Several of these experimental parameters are interrelated and in Section
X we discuss fidelity and rate optimization of the heralded state. We present
our conclusions in Section XI.

This work contributes to a better understanding of the effect of general
experimental imperfections and its platform-independent insights can be used
to improve entanglement generation on various systems, such as other solid
state defects and quantum dots [15, 16, 17, 18, 19, 20, 21, 22, 23].

##  II The single-photon entanglement protocol

Figure 1: Simplified level structure, generalized experimental layout and
overview of unitary operations. (a) L-scheme level structure for the single-
photon entanglement protocol. Two ground states define the qubit subspace
$\ket{0}$ and $\ket{1}$, and we can drive the transition between them.
Furthermore, $\ket{0}$ is optically connected to an excited state $\ket{e}$.
(b) In a generalized experimental layout, excitation pulses are sent to qubits
A and B and the emitted photons are led to a beam splitter. The output ports C
and D of the beam splitter are connected to two single photon detectors. (c)
Gate circuit diagram describing the single-photon entanglement protocol.
Stationary qubits A and B are initialized into the $\ket{0}_{i}$ state, where
the subscript $i$ denotes qubit A or B. Since there are no photons yet at the
start of the protocol, all photon modes are described by the vacuum state,
$\ket{0}_{\gamma,i}$. The same holds for the photon modes indicating lost
photons, $\ket{0}_{\gamma,ri}$. Throughout the protocol, several unitaries $U$
act on the qubits and the photon states, and finally a state between the
stationary qubits is heralded by a joint measurement of the photonic states.
In the main text, we describe the different unitaries and the resulting
heralded state.

In this section we provide a step-by-step description of the single-photon
protocol and derive the resulting two-qubit state. Figure 1a shows an example
of the energy levels used by the protocol, in this work we employ a L-scheme
for the optical excitation. We would like to emphasize that the protocol can
also be executed with a Lambda (or Raman) excitation scheme. In that case, the
optical excitation induces the qubit to flip [8]. Here we develop our model
based on the L-scheme, as depicted in Figure 1a. We label two levels $\ket{0}$
and $\ket{1}$ as our qubit subspace, and we can coherently drive the
transition between them to create any superposition state. Furthermore, the
$\ket{0}$ ground state is connected to an optically excited state $\ket{e}$,
allowing for state-selective excitation and qubit-photon entanglement. Figure
1b shows a general experimental layout, the two qubits can be individually
excited and the emitted photons are led to a beam splitter. The output ports
of the beam splitter are connected to two photon detectors.

Figure 2: Overview of the different time definitions used in the derivation.
(a) At $t=0$ the excitation pulse is generated at position $x=0$ with phase
$\phi_{l,i}$. (b) At time $t=t_{\text{exc},i}$, the excitation pulse arrives
at the emitter and transfers the population from the $\ket{0}$ ground to the
excited state $\ket{e}$. (c) At time $t=t_{\text{em},i}$, the population in
the excited state spontaneously decays and a photon is emitted. (d) Finally,
the emitted photon arrives at the detector at position $x=L_{1,i}+L_{2,i}$ and
is detected at time $t=t_{\text{em},i}+\frac{L_{2,i}}{c}=t^{\prime}$

Throughout this section, we describe various unitaries acting on the
individual qubits and photons, and we finally discuss a photon detection event
heralding the two-qubit state. Figure 1c summarizes the protocol in the form
of a gate circuit diagram. $\ket{0}_{A}$ and $\ket{0}_{B}$ represent the
(stationary) qubits at the start of the protocol. $\ket{0}_{\gamma,A/B}$
describe the photon modes and $\ket{0}_{\gamma,r,A/B}$ the photon modes of any
lost photons, both equal to the vacuum state at the start of the protocol as
there are no photons yet. Figure 2 provides an overview of the different
timing definitions and schematically shows the single-photon entanglement
protocol at different points in time.

We work in the rotating frame of both qubits and we start with qubits A and B
initialized in the $\ket{0}$ ground state

$\ket{\Psi_{0}}_{A}\otimes\ket{\Psi_{0}}_{B}=\ket{0}_{A}\otimes\ket{0}_{B}.$
(1)

Next we create a superposition state on each of the qubits using the unitary

$U_{1}^{\alpha_{i},\vartheta_{i}}:\ket{0}\rightarrow\sqrt{\alpha_{i}}\ket{0}+\sqrt{1-\alpha_{i}}e^{-i\vartheta_{i}}\ket{1}.$
(2)

In unitary $U_{1}^{\alpha_{i},\vartheta_{i}}$, $\alpha_{i}$ denotes the
population in $\ket{0}$, i.e. the bright state, and $\vartheta_{i}$ represents
the phase of the superposition state. Subscript $i$ denotes either of the two
emitters used in the protocol, A or B.

To create the qubit-photon entangled states, we expose the emitter to
excitation light. At time $t=0$, a short optical pulse is being created, see
Figure 2a. At time $t_{\text{exc},i}$, qubit $i$ will be state-selectively
excited by this optical excitation pulse (Figure 2b). In this derivation we
assume the optical excitation to be instantaneous. Subsequent decay to the
ground state will result in the emission of a photon (Figure 2c) and the
photonic state can be written as a superposition of different photon modes
$\zeta_{i}$ with different emission times $t_{\text{em},i}$

$\begin{split}\int_{t_{\text{exc},i}}^{\infty}c_{e,i}\zeta_{i}(x,t,t_{\text{exc},i},t_{\text{em},i})dt_{\text{em},i}&=\int
c_{e,i}\vec{\epsilon}_{i}\mathcal{E}_{i}(t_{\text{exc},i},t_{\text{em},i})e^{-i(\omega_{i}(t-t_{\text{em},i})-k_{i}(x-L_{1,i})+\phi_{l,i}+\pi/2+\omega_{i}(t_{\text{em},i}-t_{\text{exc},i}))}dt_{\text{em},i}\\\
&=\int
c_{e,i}\vec{\epsilon}_{i}\mathcal{E}_{i}(t_{\text{exc},i},t_{\text{em},i})e^{-i(\omega_{i}(t-t_{\text{exc},i})-k_{i}(x-L_{1,i})+\phi_{l,i}+\pi/2)}dt_{\text{em},i}\end{split}$
(3)

In Equation (3), $|c_{e,i}|^{2}$ is the probability to transfer the population
to the excited state $\ket{e}$, $\vec{\epsilon}_{i}$ denotes the polarization
of the emitted photon, $\omega_{i}$ is the transition frequency
$\ket{0}\rightarrow\ket{e}$ with the corresponding wave number $k_{i}$,
$\phi_{l,i}+\pi/2$ is the phase of the excitation laser imprinted on the
photon by means of the Rabi drive,
$\mathcal{E}_{i}(t_{\text{exc},i},t_{\text{em},i})$ is the temporal envelope
and $x$ and $t$ as the spatial and time coordinates respectively. For
spontaneous decay, this envelope can be modeled as

$\mathcal{E}_{i}(t_{\text{exc},i},t_{\text{em},i})=\frac{H(t_{\text{em},i}-t_{\text{exc},i})e^{-(t_{\text{exc},i}-t_{\text{em},i})/2\tau}}{\sqrt{\tau}},$
(4)

with $\tau$ the excited state lifetime and $H(t)$ the Heaviside function.

At a later point in the protocol we will describe the detection of a photon at
time $t_{\text{em},i}+\frac{L_{2,i}}{c}$ (see Figure 2d), therefore we can
already write the photon state as being emitted in that specific mode

$\begin{split}c_{e,i}\zeta_{i}(x,t,t_{\text{exc},i},t_{\text{em},i})=c_{e,i}\vec{\epsilon}_{i}\mathcal{E}_{i}(t_{\text{exc},i},t_{\text{em},i})e^{-i(\omega_{i}(t-t_{\text{exc},i})-k_{i}(x-L_{1,i})+\phi_{l,i}+\pi/2)}\\\
\end{split}$ (5)

Additionally, in practice, the excitation laser pulse has a finite duration
and therefore re-excitation of the qubit is possible in case the first decay
happens during the optical pulse (we ignore any higher-order emissions). For
this reason we define $t_{\text{exc},2,i}$ and $t_{\text{em},2,i}$ as the
times at which the re-excitation and second spontaneous decay occur, with the
condition $t_{\text{exc},2,i}>t_{\text{em},i}$. We write the joint photonic
mode as $\zeta_{ii}(t_{\text{exc},i},t_{\text{exc},2,i})$ and define the
double excitation probability as $|c_{ee,i}|^{2}$ [24, 25]. Again we assume
both photons to be emitted in specific modes.

Together with the probability to remain in the ground state $\ket{0}$,
$|c_{0,i}|^{2}$, the unitary describing the state-selective excitation and
emission can be written as

$U_{2}^{c_{i}}:\begin{cases}\begin{split}\ket{0}\ket{0}_{\gamma}\rightarrow\ket{0}\otimes(&c_{0,i}\ket{0}_{\gamma}+c_{e,i}\zeta_{i}\hat{a}^{\dagger}(t_{\text{em},i})\ket{0}_{\gamma}+\\\
&c_{ee,i}\zeta_{ii}\hat{a}^{\dagger}(t_{\text{em},2,i})\hat{a}^{\dagger}(t_{\text{em},i})\ket{0}_{\gamma}),\end{split}\\\
\ket{1}\ket{0}_{\gamma}\rightarrow\ket{1}\ket{0}_{\gamma},\end{cases}$ (6)

where $\hat{a}^{\dagger}$ is the photon creation operator. Note that we
exclude the possibility for off-resonant excitation of any transition related
to the $\ket{1}$ ground state.

The emitted photons are subject to losses, which we assume to be equal for all
the photons associated with one of the nodes. We use a beam splitter
transformation $U_{3}^{\eta_{i}}$ to model photon loss as

$\begin{split}U_{3}^{\eta_{i}}:\hat{a}^{\dagger}_{i}\ket{0}_{\gamma}\ket{0}_{\gamma,r}\rightarrow&\sqrt{\eta_{i}}\hat{a}^{\dagger}_{out,i}\ket{0}_{\gamma}\ket{0}_{\gamma,r}\\\
&+\sqrt{1-\eta_{i}}\hat{a}^{\dagger}_{r,i}\ket{0}_{\gamma}\ket{0}_{\gamma,r}.\end{split}$
(7)

$\hat{a}_{r,i}^{\dagger}$ creates a photon in the loss mode
$\ket{0}_{\gamma,r}$ and $\hat{a}_{out,i}^{\dagger}$ denotes the photon
arriving at the beam splitter. We omit the out subscript for brevity. In
$U_{3}^{\eta_{i}}$, $|\eta_{i}|^{2}$ represents the photon detection
probability. The (separable) state in front of the central beam splitter will
then be given by

$\begin{split}\ket{\Psi_{3}}_{AB}=&U_{3}^{\eta_{A}}U_{2}^{c_{A}}U_{1}^{\alpha_{A},\vartheta_{A}}\ket{0}_{A}\ket{0}_{\gamma,A}\ket{0}_{\gamma,rA}\\\
&\otimes
U_{3}^{\eta_{B}}U_{2}^{c_{B}}U_{1}^{\alpha_{B},\vartheta_{B}}\ket{0}_{B}\ket{0}_{\gamma,B}\ket{0}_{\gamma,rB}.\end{split}$
(8)

Let us now turn to the detection. We assume non-number resolving single photon
detectors. Due to photon loss, double excitation and the presence of noise
counts, different detection patterns will look identical. As a consequence, we
will herald an average state, averaged over the different possible detection
patterns. However, we can reject any experimental repetition in which two
photons are detected in different detectors. These events suggest either
double excitation or both qubits being in the bright state, deteriorating the
average state compared to the maximally entangled. Furthermore, assuming the
presence of modest photon loss, we consider at most two photons arriving at
the beam splitter. We treat all the detection patterns separately and later on
combine the result to obtain the average heralded density matrix. We assume
the loss after the central beam splitter to be symmetric, for this reason we
treat any loss after the beam splitter as an additional contribution to the
loss handled by $U_{3}^{\eta_{i}}$.

Single photon -  The first detection pattern we consider is the case where one
photon is emitted and one photon is detected. For a detection at time
$t^{\prime}=t_{\text{em},A}+\frac{L_{2,A}}{c}=t_{\text{em},B}+\frac{L_{2,B}}{c}$
in port C of the beam splitter, we model the detection by projecting the state
onto $\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})$, where
$\hat{a}_{C}(t^{\prime})$ is the photon annihilation operator acting at time
$t^{\prime}$. The effect of the beam splitter is mapping the state to a
superposition of a photon originating from A or B

$\begin{split}\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})=\frac{1}{\sqrt{2}}\bra{0}_{\gamma}\left(\hat{a}_{A}(t^{\prime})+\hat{a}_{B}(t^{\prime})\right).\end{split}$
(9)

Similarly we can define the projector for detecting a photon in port D as

$\begin{split}\bra{0}_{\gamma}\hat{a}_{D}(t^{\prime})=\frac{1}{\sqrt{2}}\bra{0}_{\gamma}\left(\hat{a}_{A}(t^{\prime})-\hat{a}_{B}(t^{\prime})\right).\end{split}$
(10)

For detecting a photon in port C, the corresponding (unnormalized) density
matrix is given by

$\rho_{1}=\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})\ket{\Psi_{3}}\bra{\Psi_{3}}_{AB}\hat{a}_{C}^{\dagger}(t^{\prime})\ket{0}_{\gamma}.$
(11)

Two photons -  Secondly, we deal with the case where two photons are emitted,
either both by the same node or by the two nodes separately, and both photons
arrive at the beam splitter. As mentioned above, two-photon events will be
rejected as they suggest both qubits to be in the bright state or double
optical excitation. Therefore, we reject events where two photons are being
detected in different ports of the beam splitter. When two photons leave the
beam splitter via the same port, we cannot discriminate a two-photon event
from a single photon event due to the non-number resolving detectors and the
first photon will herald a state. In this case, the first photon will be
detected in port C at time $t^{\prime}$ and the second photon will be
similarly absorbed at (unknown) time $t^{\prime\prime}$, also in port C. The
projector is then given by

$\begin{split}\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})\hat{a}_{C}(t^{\prime\prime})=\frac{1}{2}\bra{0}_{\gamma}(\hat{a}_{A}(t^{\prime})&+\hat{a}_{B}(t^{\prime}))(\hat{a}_{A}(t^{\prime\prime})+\hat{a}_{B}(t^{\prime\prime})),\end{split}$
(12)

and we obtain

$\rho_{2}=\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})\hat{a}_{C}(t^{\prime\prime})\ket{\Psi_{3}}\bra{\Psi_{3}}_{AB}\hat{a}_{C}^{\dagger}(t^{\prime\prime})\hat{a}_{C}^{\dagger}(t^{\prime})\ket{0}_{\gamma},$
(13)

again an unnormalized density matrix.

At least one lost photon -  When at least one photon is lost, detection of any
remaining photon will falsely herald entanglement. This cannot lead to
coherence between the qubit states and therefore we can project on each
detection pattern separately and sum over the resulting density matrices

$\begin{split}\rho_{\text{incoherent}}=&\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})\hat{a}_{rA}(t_{r})\ket{\Psi_{3}}\bra{\Psi_{3}}_{AB}\hat{a}_{rA}^{\dagger}(t_{r})\hat{a}_{C}^{\dagger}(t^{\prime})\ket{0}_{\gamma}\\\
&+\bra{0}_{\gamma}\hat{a}_{C}(t^{\prime})\hat{a}_{C}(t^{\prime\prime})\hat{a}_{rA}(t_{r})\ket{\Psi_{3}}\bra{\Psi_{3}}_{AB}\hat{a}_{rA}^{\dagger}(t_{r})\hat{a}_{C}^{\dagger}(t^{\prime\prime})\hat{a}_{C}^{\dagger}(t^{\prime})\ket{0}_{\gamma}\\\
&+...\\\ \end{split}$ (14)

Since the detection time of the lost photon $t_{r}$ and therefore its emission
time $t_{\text{em,i}}$ is unknown, we can integrate over all possible emission
times; $|\int
c_{e,i}\zeta_{i}(t_{\text{em,i}})dt_{\text{em,i}}|^{2}=|c_{e,i}|^{2}$.

Noise photon -  A detector dark count or stray light will additionally lead to
falsely heralded entangled states. We assume the contribution of such noise
counts small compared to actual signal photons and we ignore the (small)
probability of both a signal photon and noise photon arriving at the detector.
We distinguish two scenarios: no photon is emitted, or none of the emitted
photons arrived at the beam splitter. The first scenario leads to a separable
state, but the single qubit coherence is not lost, while the latter projects
the corresponding qubit state and its coherence is lost. We can deal with
these two scenarios separately. Regarding the first scenario, by projecting on
the vacuum state $\ket{0}_{\gamma}$, we get the density matrix

$\rho_{0}=\ket{\Psi_{0}}\bra{\Psi_{0}}_{A}\otimes\ket{\Psi_{0}}\bra{\Psi_{0}}_{B},$
(15)

with

$\begin{split}\ket{\Psi_{0}}_{i}=(\sqrt{\alpha_{i}}c_{0,i}\ket{0}+e^{-i\vartheta_{i}}\sqrt{1-\alpha_{i}}\ket{1}).\end{split}$
(16)

Alternatively, when none of the emitted photons arrive at the beam splitter,
we can trace over all the lost photons to obtain

$\begin{split}\rho_{\text{lost}}=&\bra{0}_{\gamma}\hat{a}_{rA}(t_{r})\ket{\Psi_{3}}\bra{\Psi_{3}}_{AB}\hat{a}_{rA}^{\dagger}(t_{r})\ket{0}_{\gamma}\\\
&+\bra{0}_{\gamma}\hat{a}_{rA}(t_{r})\hat{a}_{rA}(\tilde{t_{r}})\ket{\Psi_{3}}\bra{\Psi_{3}}_{AB}\hat{a}_{rA}^{\dagger}(\tilde{t_{r}})\hat{a}_{rA}^{\dagger}(t_{r})\\\
&+...\\\ \end{split}$ (17)

We add the two noise contributions and multiply by the probability for a noise
count being detected, $p_{d}$ and construct

$\rho_{\text{noise}}=p_{d}(\rho_{0}+\rho_{\text{lost}}).$ (18)

Having discussed all the detection patterns we can finally combine all the
contributions to extract the probability to get a detection event in port C

$\begin{split}p_{\text{click,C}}=&\Tr(\rho_{1})+\Tr(\rho_{2})+\Tr(\rho_{\text{incoherent}})+\Tr(\rho_{\text{noise}}).\\\
\end{split}$ (19)

We add all the density matrices and normalize using the detection probability
$p_{\text{click,C}}$ and we obtain an expression for the average density
matrix $\rho_{C}$ heralded by a detection in port C

$\begin{split}\rho_{C}=&\frac{1}{p_{\text{click,C}}}(\rho_{1}+\rho_{2}+\rho_{\text{incoherent}}+\rho_{\text{noise}}).\\\
\end{split}$ (20)

In Appendix A, we provide the full description of the density matrix.

In the introduction we discussed the expected fidelity in the high loss regime
and here we check that our model matches the intuitive result. We use a
shorthand notation for the detected photon modes defined in Equation (5) as
$\zeta_{i}(t^{\prime})$, with $t^{\prime}$ ($t^{\prime\prime}$) the detection
time of the first (second) photon. In the limit of high photon loss ($\eta\ll
1$), equal experimental settings ($\eta_{A}=\eta_{B}\equiv\eta$ and
$\alpha_{A}=\alpha_{B}\equiv\alpha$), perfect optical excitation pulses
$|c_{e,i}|^{2}=1$, perfectly indistinguishable photons with equal optical
phase upon arrival at the beam splitter
$\zeta_{A}(t^{\prime})=\zeta_{B}(t^{\prime})\equiv\zeta(t^{\prime})$ and
absence of noise counts $p_{d}$, we can simplify the results to

$\begin{split}p_{\text{click},\eta\ll
1,C}=&\Tr(\rho_{1})+\Tr(\rho_{\text{incoherent}})\\\
=&\alpha(1-\alpha)\eta|\zeta(t^{\prime})|^{2}+\alpha^{2}\eta|\zeta(t^{\prime})|^{2}\\\
=&\alpha\eta|\zeta(t^{\prime})|^{2},\\\ \end{split}$ (21)

where $\eta|\zeta(t^{\prime})|^{2}$ can be interpreted as the probability to
detect a photon in the detection window. The density matrix can be written as

$\displaystyle\begin{split}\rho_{\eta\ll
1,C}=&\frac{1}{p_{\text{click},\eta\ll
1,C}}(\rho_{1}+\rho_{\text{incoherent}}),\\\
=&\frac{\alpha(1-\alpha)\eta|\zeta(t^{\prime})|^{2}}{p_{\text{click},\eta\ll
1,C}}\ket{\Psi}\bra{\Psi}+\frac{\alpha^{2}\eta|\zeta(t^{\prime})|^{2}}{p_{\text{click},\eta\ll
1,C}}\ket{00}\bra{00},\\\
=&(1-\alpha)\ket{\Psi^{+}}\bra{\Psi^{+}}+\alpha\ket{00}\bra{00},\\\
\end{split}$ (22)

with $\ket{\Psi^{+}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$, the maximally
entangled Bell state, matching our intuitive prediction presented in Section
I. Similarly, a photon detection in port D will give the same result, albeit
with the entangled state
$\ket{\Psi^{-}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$.

##  III Experimental setup: NV centers

In this work we use nitrogen-vacancy centers (NV) in bulk diamond as our qubit
system. This defect consists of a substitutional nitrogen-atom with the
adjacent lattice site left vacant. In the negative charge state an additional
electron from the environment is trapped and a spin-1 system is formed. We use
two ground states, $\ket{0}=\ket{m_{s}=0}$ and $\ket{1}=\ket{m_{s}=-1}$ (or
$\ket{m_{s}=+1}$), as our qubit subspace. The $\ket{m_{s}=0}$ state is
connected to an optically excited state $\ket{e}=\ket{E_{x}}$ 111In one of the
experimental setups, node C, we use $\ket{E_{y}}$ as the excited state., and
the transition can be selectively addressed at cryogenic temperatures [26,
27]. Spontaneous decay from the excited state $\ket{e}$ to $\ket{0}$ happens
resonantly $\approx 3\%$ of the time, into the so-called zero-phonon line
(ZPL) [28]. In the remaining $\approx 97\%$, the decay happens off-resonantly
into the phonon-side band (PSB); in this case the emitted photon is
accompanied with the emission of a phonon. Tracing over the (undetected)
phonon state erases the spin-photon coherence, prohibiting PSB photons to be
used for entanglement generation. In the optical setup we separate the ZPL
from the PSB photons using a dichroic mirror [29].

In this work, we use the same experimental setups as used in the quantum
network of References [29] and [30]. We label the nodes Alice (A), Bob (B) and
Charlie (C), which are connected in such a way that we can generate
entanglement on two links; Alice-Bob (AB) and Bob-Charlie (BC). The exact
connections are given in [29]. By applying a DC voltage via on-chip
electrodes, we tune the optical transitions of nodes A and C to match the
frequency of node B using the DC Stark effect [31]. For each link the nodes
share the excitation laser and the short excitation pulses are generated using
an electro-optic modulator (EOM, Jenoptik) driven by an arbitrary waveform
generator (AWG, Zürich Instruments). To provide additional extinction of the
pulse, we make use of an acousto-optic modulator (AOM, Gooch&Housego). We use
microwave (MW) pulses to drive the transition between the qubit states and
with I- and Q-modulation we can generate any superposition. We actively
stabilize the optical phases acquired by the excitation pulses and photons
using a combination of heterodyne and homodyne phase detection methods and
feedback [29].

To perform a readout on one of the individual qubit, the qubit is rotated to
the required basis using a MW pulse and the NV center is read out by exposure
to light resonant with the $\ket{0}\rightarrow\ket{e}$ transition. Detection
of a photon in the PSB detection path marks a $\ket{0}$ state readout, while
absence of photons corresponds to the $\ket{1}$ state.

The optical transition frequencies of the NV center are sensitive to (laser-
induced) changes in the charge-environment [32, 33, 34, 35]. To mitigate this
effect we perform a Charge-Resonance (CR) check prior to every experimental
run [27]. During a CR check, we turn on the control lasers to ensure the
emitter is on resonance with the control lasers and in the correct charge
state. Only when a number of PSB photons above a pre-set threshold is
detected, an experimental repetition is started.

##  IV Tailoring the model for NV centers

In Section II we have considered the single-photon protocol in a general way.
Considering our experimental implementation using NV centers in bulk diamond,
we can make several approximations to simplify the results. Due to the small
fraction of resonantly emitted photons ($\approx 3\%$) and limited detection
efficiency ($<15\%$), we can assume $\eta\ll 1$. By using high-power laser
pulses we can assume all population to be transferred to the optically excited
state, $c_{0,i}=0$. We set a detection window which starts after the arrival
time of the optical pulse to mitigate counts due to imperfectly rejected laser
light reaching the detectors. As a consequence, during a double excitation
event the first photon will never be detected and we set
$\eta_{i}(1-\eta_{i})|\zeta_{ii}(t^{\prime},t_{r})|^{2}=0$. In this way, we
can define the parameter $p_{de}$ as the probability that a second photon is
emitted given a photon detection:
$|c_{ee,i}\zeta_{ii}(t_{r},t^{\prime})|^{2}=p_{de}$.

With these assumptions we can simplify Equation (19) and (20), and we obtain
for the probability to detect a photon in port C

$\begin{split}p_{\text{click,NV,C}}=&\Tr(\rho_{1})+\Tr(\rho_{\text{incoherent}})+\Tr(\rho_{\text{noise}}).\\\
\end{split}$ (23)

The heralded density matrix is then given by

$\begin{split}\rho_{\text{NV,C}}=&\frac{1}{p_{\text{click,NV,C}}}(\rho_{1}+\rho_{\text{incoherent}}+\rho_{\text{noise}})\\\
=&\frac{1}{p_{\text{click,NV,C}}}\begin{pmatrix}a_{00}&0&0&0\\\
0&a_{11}&a_{12}&0\\\ 0&a_{12}^{\ast}&a_{22}&0\\\ 0&0&0&a_{33}\\\
\end{pmatrix}\end{split}$ (24)

with the elements

$\displaystyle\begin{split}a_{00}=&\alpha_{A}\alpha_{B}(\frac{1}{2}\eta_{A}|c_{e,A}\zeta_{A}(t^{\prime})|^{2}+\frac{1}{2}\eta_{B}|c_{e,B}\zeta_{B}(t^{\prime})|^{2}\\\
&+\frac{1}{2}\eta_{A}p_{de}+\frac{1}{2}\eta_{B}p_{de}+p_{d}),\\\ \end{split}$
(25)
$\displaystyle\begin{split}a_{11}=\alpha_{A}(1-\alpha_{B})(\frac{1}{2}\eta_{A}|c_{e,A}\zeta_{A}(t^{\prime})|^{2}+\frac{1}{2}\eta_{A}p_{de}+p_{d}),\\\
\end{split}$ (26)
$\displaystyle\begin{split}a_{22}=(1-\alpha_{A})\alpha_{B}(\frac{1}{2}\eta_{B}|c_{e,B}\zeta_{B}(t^{\prime})|^{2}+\frac{1}{2}\eta_{B}p_{de}+p_{d}),\\\
\end{split}$ (27) $\displaystyle\begin{split}a_{12}=\frac{1}{2}Me^{-i\phi},\\\
\end{split}$ (28)
$\displaystyle\begin{split}a_{33}=p_{d}(1-\alpha_{A})(1-\alpha_{B}).\\\
\end{split}$ (29)

In the expression of $a_{12}$, $M$ is the magnitude of the coherence term

$\begin{split}M=&\sqrt{\alpha_{A}(1-\alpha_{A})\alpha_{B}(1-\alpha_{B})\eta_{A}\eta_{B}}\\\
&\times(\vec{\epsilon_{A}}\cdot\vec{\epsilon_{B}})c_{e,A}c_{e,B}\mathcal{E}_{A}(t_{\text{exc},A},t_{\text{em},A})\mathcal{E}_{B}(t_{\text{exc},B},t_{\text{em},B}),\\\
\end{split}$ (30)

and $\phi$ represents the phase of the entangled state

$\begin{split}\phi=&\vartheta_{B}-\vartheta_{A}-\omega_{A}(t_{\text{em},A}-t_{\text{exc},A})\\\
&+\omega_{B}(t_{\text{em},B}-t_{\text{exc},B})-\phi_{l,A}+\phi_{l,B}.\\\
\end{split}$ (31)

In our experimental implementation, the emitters share the excitation laser
and the optical paths are much smaller than the coherence length of the laser.
Thus, we can rewrite Equation (31). The phase of the laser is
$\phi_{l,A}=\phi_{l,B}\equiv\phi_{l}$ and the frequency of the laser
$\omega_{l,A}=\omega_{l,B}\equiv\omega_{l}$. We introduce a detuning
$\Delta_{i}$ between the laser and the optical transitions,
$\omega_{l}-\omega_{i}=\Delta_{i}$ of the emitters ($i=A,B$). Even though the
emitters share the same excitation pulse, the path lengths of the excitation
pulse and single photons do not have to be same and their difference is given
by $L_{1,A}+L_{2,A}-L_{1,B}-L_{2,B}=dL$. Hence, for a photon detection event
at time $t^{\prime}$, the time spent in the excited state can be different for
the two emitters, $t_{\text{em},A}-t_{\text{exc},A}\neq
t_{\text{em},B}-t_{\text{exc},B}$. We introduce the variable $t_{d,i}$ as the
time spent in the excited state, $t_{\text{em},i}-t_{\text{exc},i}$, see
Figure 3. Note that $t_{d,i}$ can also be viewed as the detection time with
respect to the arrival time of the excitation pulse at the beam splitter.
Using these definitions we can write $\phi$ as

$\begin{split}\phi=&\vartheta_{B}-\vartheta_{A}-\omega_{A}t_{d,A}+\omega_{B}t_{d,B}\\\
=&\vartheta_{B}-\vartheta_{A}-\omega_{l}(t_{d,A}-t_{d,B})+\Delta_{A}t_{d,A}-\Delta_{B}t_{d,B}\\\
=&\vartheta_{B}-\vartheta_{A}+\omega_{l}\frac{dL}{c}+\Delta_{A}t_{d,A}-\Delta_{B}t_{d,B}\end{split}$
(32)

where $\omega_{l}\frac{dL}{c}$ is the optical phase difference between the
paths, $c$ denotes the speed of light. As mentioned in Section III, we
stabilize this phase difference to a setpoint $\delta\varphi$. We can rewrite
$t_{d,B}$ as a function of $t_{d,A}$; $t_{d,B}=t_{d,A}+\frac{dL}{c}$ and
obtain

$\begin{split}\phi=&\vartheta_{B}-\vartheta_{A}+\delta\varphi+\Delta_{A}t_{d,A}-\Delta_{B}t_{d,A}-\Delta_{B}\frac{dL}{c}).\\\
\end{split}$ (33)

The last term in Equation (33) can be ignored when either the arrival time
difference $\frac{dL}{c}$ or the detuning $\Delta$ of one of the nodes is
small. Assuming $\frac{dL}{c}\ll t_{d,A}$, we can write the entangled state
phase as

$\phi=\vartheta_{B}-\vartheta_{A}+\delta\varphi+\Delta_{A}t_{d}-\Delta_{B}t_{d}.$
(34)

where $t_{d}$ is the photon detection time with respect to the (nearly equal)
arrival time of the optical pulses to the detectors.

Figure 3: Photon detection time. The photon detection happens at time
$t=t^{\prime}$. We introduce an extra variable, $t_{d,i}$ with $i=(A,B)$,
which is defined as the time spent in the excited state
$t_{\text{em,i}}-t_{\text{exc,i}}$. $t_{d,i}$ can also be viewed as the photon
detection time with respect to the arrival times of the optical pulses, which
occur at $t_{\text{exc,i}}+L_{2,i}/c$ (with $c$ the speed of light constant).
The arrival time difference between the optical pulses depends on the optical
path length difference $dL=L_{1,A}+L_{2,A}-L_{1,B}-L_{2,B}$, and is given by
$dL/c$.

In the remainder of the paper, we use this model to simulate the fidelity with
respect to the maximally entangled state and the success probability. The
simulations include a Monte Carlo simulation to incorporate the exponential
probability distribution of the photon detection times, as well as Gaussian
probability distributions of the noise of the phase stabilization setpoint and
frequency detunings [36].

##  V Bright state population

Figure 4: Effect of bright state population. Fidelity with respect to the
maximally entangled state (a) and probability to herald a state (b) as a
function of the bright state populations $\alpha$. The data (circles) is
measured on two links of the network of Reference [29], AB and BC. The x-axis
represents the bright state population of setup B in both cases. The bright
state population of node A is scaled to be
$\alpha_{A}=\frac{\eta_{B}}{\eta_{A}}\alpha_{B}$, while
$\alpha_{C}=\alpha_{B}$. The solid lines are given by our model, see Table B2
for the parameters. (c) In absence of errors other than the protocol error, we
calculate the entangled state fidelity for various settings of
$(\alpha_{A},\alpha_{B})$ for the case $\eta_{B}=2\eta_{A}$ with $\eta_{A}=$
4e-4. The black and white solid lines are isolines for the entangled state
fidelity and success probability respectively. The red solid line indicates
the optical settings to obtain the highest fidelity with respect to the
maximally entangled state for a fixed success probability. The red dashed line
represents the $\eta_{A}\alpha_{A}=\eta_{B}\alpha_{B}$ scenario.

In this section we study the effect of the bright state population $\alpha$.
First, we will vary $\alpha$ and discuss the effect on the fidelity of the
heralded state in the presence of noise. Second, we will discuss optimal
settings for the individual $\alpha_{i}$ when the detection efficiencies of
the nodes are not the same.

In a practical experimental setting, entanglement might be falsely heralded by
noise counts. The noise counts can originate from different sources, such as
dark counts of the detector, excitation light leaking into the detectors or
stray light. The effect on the average heralded state depends on the ratio
between noise and signal photons. In Figure 4a we plot the measured and
simulated fidelity with the maximally entangled state for various settings of
$\alpha$. For high values of $\alpha$ we observe the linear scaling of the
fidelity with $\alpha$, as suggested by the model for the high-photon loss
regime $\eta\ll 1$. For low values of $\alpha$, the fidelity deviates from the
linear behavior and for sufficiently low $\alpha$ we observe a sharp drop-off,
indicating a significant contribution of falsely heralding noise counts. We
note that off-resonant excitation of undesired transitions or errors in the
preparation of the superposition state (unitary $U_{1}^{\alpha,\vartheta}$)
can also lead to a sharp drop-off in fidelity for low values of $\alpha$. In
Figure 4b we plot the success probability $p_{\text{click}}$ as a function of
$\alpha$, displaying the expected linear behavior (see Equation (21)).

We now turn to the individual detection efficiency $\eta$. Generally, the
detection efficiency or loss parameter, $\eta$, is not the same for the two
nodes due to differences in the individual experimental setups or unequal
fiber loss in the paths from the nodes to the beam splitter. For this reason
we would like to find optimal settings for $\alpha_{A}$ and $\alpha_{B}$ to
establish remote entanglement with the highest fidelity for a fixed success
probability $p_{click}$. In other words, we want to optimize
$F(\alpha_{A},\alpha_{B})$ subject to
$\eta_{A}\alpha_{A}+\eta_{B}\alpha_{B}=p_{click}$ in the high-loss regime. For
simplicity, we assume no errors other than the protocol error (i.e. no noise
counts, no double excitation, perfectly indistinguishable photons), set an
entangled state phase of $\phi=0$ and integrate over all possible detection
times such that $\int|\zeta_{i}(t^{\prime})|^{2}dt^{\prime}=1$. To compute the
fidelity we calculate the overlap with the maximally entangled state

$\displaystyle\begin{split}F=&\bra{\Psi}\rho_{C}\ket{\Psi},\\\
=&\frac{1}{4p_{click,C}}(a_{11}+a_{22}+2|a_{12}|),\\\
=&\frac{1}{4p_{click,C}}(\alpha_{A}\eta_{A}(1-\alpha_{B})+\alpha_{B}\eta_{B}(1-\alpha_{A})\\\
&+2\sqrt{\alpha_{A}\alpha_{B}(1-\alpha_{A})(1-\alpha_{B})}),\\\ \end{split}$
(35)

and the success probability per attempt

$\begin{split}p_{click}=&a_{00}+a_{11}+a_{22},\\\
=&\frac{1}{2}(\eta_{A}\alpha_{A}+\eta_{B}\alpha_{B}).\\\ \end{split}$ (36)

To find the optimal settings, we use a Langrangian formalism. The optimal
settings are met when the gradients of the fidelity and success probability
are parallel, i.e. when the fidelity is maximal for a fixed success
probability. We can write this condition as

$\begin{split}(\nabla p_{click})_{\perp}\cdot\nabla F&=0\\\
-\eta_{B}\frac{\partial F}{\partial\alpha_{A}}+\eta_{A}\frac{\partial
F}{\partial\alpha_{B}}&=0.\\\ \end{split}$ (37)

We solve Equation (37) numerically for the case $\eta_{B}=2\eta_{A}$, with
$\eta_{A}=$ 4e-4. In Figure 4c we plot the entangled state fidelity for
different values of $\alpha_{A}$ and $\alpha_{B}$. The optimal settings are
represented by the red solid line. For high-fidelity states, the optimal
settings are close to $\alpha_{A}\eta_{A}=\alpha_{B}\eta_{B}$ (red dashed
line). We can interpret this result as balancing the probability of the photon
to originate from either setup. For low-fidelity states, the optimal settings
differ from balancing the detection probabilities. This can be explained with
a simple example; by setting $\alpha_{A}=0$ and $\alpha_{B}\neq 0$, the
detected photon will always originate from setup B and thus we will measure
perfect classical anti-correlations. However any quantum correlations are
completely washed out and we obtain a fidelity $F=0.5$, irrespective of the
value of $\alpha_{B}$. On the contrary, if we set both
$\alpha_{A},\alpha_{B}\neq 0$ we do get quantum correlations but the protocol
error can now push the fidelity below 0.5. Hence, for low-fidelity states the
optimal settings for $\alpha_{A}$ and $\alpha_{B}$ optimize classical anti-
correlation at the expense of quantum correlations.

##  VI Phase of the entangled state

Figure 5: Entangled state phase. (a) We determine the entangled state phase by
measuring along the $+X$ axis on setup B (red arrow), while sweeping the
measurement axis of setup C over the equator plane of the Bloch sphere (black
arrows). In this way we obtain oscillating correlations between the
measurement outcomes, as shown in (b), for states heralded by detector C
(turquoise circles) and D (purple circles). We jointly fit the data for the
two detectors to extract $\phi$, the phase offset with respect to a cosine.
(c) The entangled state phase $\phi$ as a function of the phase of the
microwave (MW) pulse we use to create the initial superposition state (see
Equation (2)). Here we fix $\vartheta_{C}=0$ and sweep $\vartheta_{B}$. Due to
a small difference of the stabilized path compared to the path of the
excitation pulses and the single photons, there is a nonzero phase offset
[29]. We fit the measured phase values with a line with slope 1 and subtract
the fitted offset. The solid line shows the fit with the subtracted offset.
(d) The entangled state phase as a function of the optical phase stabilization
setpoint. Again we account for the phase offset due to path difference and
show the fit with the subtracted offset (solid line). (e) The entangled state
phase as function of the detection time of the photon for different frequency
offsets $\Delta f$ between the setups. The photon detection times are binned
in bins of 4 ns, and the x value represents the middle of the bin. The time
scale is with respect to the highest intensity point of the optical pulse. The
solid lines are a joint fit to the data with the zero point crossing $x_{0}$
as the only free parameter. The fit gives $x0=(0.8\pm 0.3)$ ns. The error bars
of the data plotted in (C), (D) and (E) are smaller than the symbol size.

In Section IV we discussed the phase of the entangled state $\phi$ and derived
the expression for the phase in the case where the excitation laser is shared
between the emitters, Equation (34). Here we experimentally verify the effect
of the different parameters on the entangled state phase, using nodes B and C.

We measure the phase of the entangled state by sweeping the readout basis of
node C over the XY-plane of the Bloch sphere (black arrows in Figure 5a) while
we fix the readout basis of node B to be along the +X axis (red arrow). In
Figure 5b we plot the correlations of the measured readout outcomes as a
function of the readout basis of node C for states heralded by detecting a
photon in port C (turquoise circles) or port D of the beam splitter (purple
circles). To extract the phase we jointly fit the two curves and extract
$\phi$, the phase offset with respect to a cosine.

First we vary the phase of the microwave pulse that creates the superposition
state on setup B, i.e. we change $\vartheta_{B}$, the phase of the initial
superposition state. We fit the data with a line with slope 1 and subtract the
fitted offset. The data with the subtracted offset and the fit are are plotted
in Figure 5c and we observe the expected linear dependence.

The next parameter we vary is the setpoint for the optical phase stabilization
to change $\delta\varphi$. In Figure 5d we plot the measured entangled state
phase $\phi$ (again with a fitted offset subtracted) for different setpoints
of the stabilization (circles) together with the fit with a fixed slope 1
(solid line). We use the phase stabilization architecture of Reference [29].
In this architecture, the optical phase difference $\delta\varphi$ is governed
by a linear combination of the setpoints of the three individual
interferometers. This enables us to stabilize to values of $\delta\varphi$
further away from 0, but ultimately the non-linear sinusoidal phase signals
hinder effective stabilization. In the shaded regime of Figure 5d, the slopes
of the individual phase signals are below 0.9 and we assume this regime to be
insufficiently linear for effective stabilization.

The third parameter we modify is the frequency difference between the emitters
of each setups. We change the frequency of the excitation laser and shift the
resonance condition of one of the emitters to the new frequency using the DC
Stark effect (we apply a DC voltage via on-chip electrodes), while leaving the
other emitter at its original emission frequency (thus introducing a detuning
to the excitation laser). As derived in Equation (34), a frequency difference
results in a shift of the entangled state phase depending on the detection
time of the photon. This dependence of phase on the photon detection time has
also been observed in [37, 38]. In Figure 5e we plot the measured entangled
state phase as a function of the detection time of the photon (in bins of 4
ns) for various frequency offsets between the emitters. The time axis is with
respect to the highest intensity point of the excitation pulse. We perform a
combined fit of the data with fixed slopes given by our model. In a quantum
jump picture, the fitted crossing of the lines can be interpreted as the
average point in time where the excitation from $\ket{0}$ to $\ket{e}$ occurs,
and thus the starting point of the time spent in the excited state. For large
frequency detunings (comparable to the inverse of the pulse width), the
effective averaging over different excitation times could also lead to a
decrease of the average fidelity, however we expect this contribution to be
small for our pulse width (2 ns). The observed dependence of the entangled
state phase on the photon detection time is consistent with our model.

##  VII Photon distinguishability

Figure 6: Transform limited spectral linewidth. (a) After passing a Charge-
Resonance (CR) check, see Section III, we expose the emitter to a laser pulse
with a variable frequency and count the emitted photons. We fit the result
with a Lorentzian pulse shape (solid lines) and repeat the measurements for
different powers of the applied laser pulse. (b) Extracted linewidths (colored
circles) from panel (A) as function of optical power. We fit the curve given
by Equation (39) to extract the natural linewidth (solid gray line). The
dashed gray line shows the expected transform-limited linewidth. (c) Histogram
of phonon-side band photon (PSB) counts. We use the region between the dashed
lines to extract the excited state lifetime and compute the expected linewidth
in panel (B). (d) Fidelity with respect to the maximally entangled state as a
function of the detection time of the heralding photon. We measure the
entangled state fidelity for both the AB (red circles) and the BC (black
triangles) links of Reference [29]. The detection time is binned in bins of 1
ns and the x value represents the start of the bin. The x axis is the
detection time with respect to the highest intensity point of the optical
pulse. Using a Monte-Carlo simulation, we model the entangled state fidelity
for a frequency difference between the emitted photons given by a Gaussian
distribution, for different values of the full-width half-maximum of the
distribution.

The next aspect we study is the photon distinguishability. For the case of
indistinguishable photons, the which-path information is completely erased by
the beam splitter. In turn, any distinguishability of the photons will
therefore affect their interference and hence alter the average heralded
density matrix, see Section II. A difference in arrival time or temporal shape
of the photons will result in different probabilities to detect a photon
originating from either node at a certain point in time (Section V). We note
that different spatial modes of the photons will have a similar effect, but
since, in our case, the photons are interfered on a fiber-coupled beam
splitter this effect can be neglected.

Distinguishability in polarization or frequency will act on the off-diagonal
terms of the density matrix. A difference in polarization decreases the
magnitude of the off-diagonal term, see Equation (30). We work with a fiber-
based beam splitter consisting of polarization maintaining (PM) fibers. We
align the polarization to the slow or fast axis better than 20 dB, therefore
we can assume the polarization mismatch to be small. As discussed in the
previous section, a fixed frequency offset causes the entangled state phase to
be dependent on the detection time of the photon. In Figure 5e, we introduced
a fixed frequency difference throughout the entire duration of the experiment.
However, even if the average frequency offset is zero, if the frequency
difference varies in each experimental run, an entangled state with a
different phase will be heralded in each repetition. While averaging over many
repetitions would still yield a constant phase, the fidelity with respect to
the target state will be decreased, with lower fidelity for later detection
times.

In Section III we discussed the Charge-Resonance (CR) check as a way to ensure
the NV centers are on resonance with the excitation laser and to eliminate any
frequency difference between the emitters. Here we assess the performance of
the CR check by measuring the spectral linewidth of the NV after passing the
CR check. We turn on an additional laser with a variable frequency and count
the emitted photons. We perform this procedure many times before moving to a
different frequency setpoint of the additional laser. We scan over a range of
$\pm$ 30 MHz around the frequency of the excitation laser, see Figure 6a. We
fit the measured counts $N$ with a Lorentzian shape

$N=\frac{a}{\pi}\frac{2\gamma}{4f^{2}+\gamma^{2}}+A$ (38)

and extract $\gamma$, the full-width at half-maximum (FWHM). In Equation (38),
$a$ represents a scaling factor and $A$ an offset. Non-zero laser power
induces broadening of the linewidth [39], so we repeat this measurement for
different powers of the scanning laser. In Figure 6b we plot the extracted
linewidth for the different optical powers of the additional laser and fit the
measured linewidths as a function of the optical power $P$ with the expression

$\gamma=\sqrt{\gamma_{0}^{2}+b\cdot P}$ (39)

to find the natural linewidth $\gamma_{0}$. In Equation (39), $b$ is the
scaling factor relating the externally calibrated applied power to the optical
Rabi-frequency. We find $\gamma_{0}=(12.4\pm 0.8)$ MHz and $b=(690\pm 40)$
MHz2/nW.

We compare the observed natural linewidth with the expected linewidth
extracted from an excited-state lifetime measurement on the same NV center. We
apply an optical $\pi$-pulse to the NV center and record the detection times
of the photons, see Figure 6c. We fit the regime between the dashed lines, for
which the influence of the pulse and dark counts is negligible, with an
exponential decay and we find a lifetime $\tau=(12.43\pm 0.02)$ ns. The
corresponding lifetime limited linewidth is $\gamma_{0,l}=(12.81\pm 0.02)$
MHz. We thus conclude that implementing a CR check can yield transform-limited
linewidths within measurement accuracy and thus allows access to
(near-)perfectly coherent photons.

Having addressed the spectral properties of a single NV center, we now move to
the photon distinguishability of two emitters and its effect on the entangled
state fidelity. We measure the fidelity of generated entangled states between
setups A and B as well as between setups B and C. In Figure 6d, we show the
measured fidelity (binned) versus the detection time of the photon. We observe
a drop in fidelity for later detection times of the photon. Since the signal-
to-noise ratio is approximately constant over the entire detection window, the
observed drop in fidelity may be attributed to a varying frequency mismatch of
the emitters.

We use the Monte-Carlo simulation to predict how large the frequency mismatch
would have to be to explain the measured behavior. We pick a random frequency
difference from a Gaussian distribution, calculate the resulting entangled
state phase and compute the fidelity to the target state (with target phase).
By averaging over many repetitions and repeating for different widths of the
Gaussian distribution, we obtain the entangled state fidelity as a function of
detection time and FWHMs of the frequency distribution (solid lines in Figure
6d). The measured entanglement data appears to be consistent with a
fluctuating frequency mismatch with standard deviation of $\approx 13$ MHz
between the emitters. Interestingly, we observe a quantitatively similar
dependence for the link between Alice and Bob and for the link between Bob and
Charlie. Note that this $\approx 13$ MHz standard deviation of the frequency
mismatch is inconsistent with the observed lifetime limited linewidth and the
accuracy of the wavemeter to which the lasers are locked ($<$2 MHz). Future
work should focus on confirming the nature of this noise and its source.

##  VIII Double optical excitation

Figure 7: Double excitation probability. (a) Temporal shape of the excitation
pulse. The exact shape is determined by the different components we use to
generate the pulse. The region between the dashed lines is the detection
window in which we accept heralding photons. The inset displays the temporal
shape of the pulse on a linear scale. (b)(c) Photon detection probabilities
for off-resonant phonon-side band (PSB) photons (b) and resonant zero-phonon
line (ZPL) photons (c). The dashed curves show expected behavior. In (b) we
indicate the voltage applied to the AOM in the top axis. (d) Extracted double
excitation probability $p_{de}$ using Equation (43) and simulated curve
resulting from our model (gray line).

In Sections II and IV we have briefly discussed double optical excitation.
During the finite duration of the excitation pulse, the emitter can get re-
excited after emission of a first photon. In the high-photon loss regime, the
probability that both photons arrive at the beam splitter is negligible. Loss
of one of the two photons projects the qubits and result in a lowered fidelity
of the heralded state. In our experiment we start the detection window a few
nanoseconds after the highest intensity point of the pulse. We define double
excitation as the probability that one emitter has emitted two photons given a
photon detection event in the heralding window, $p_{de}=|\zeta(t_{r},t)|^{2}$.
Here we extract the double excitation probability for our specific optical
excitation pulse and measure its dependence on the power of the optical pulse.

The double excitation probability depends on the pulse duration with respect
to the lifetime of the excited state, but also on the exact pulse shape and
power of the pulse. As mentioned in Section III, we generate the optical pulse
using an AWG, EOM and AOM. The combination of the response times and output of
these three instruments determines the shape of the excitation pulse. In
Figure 7a we plot the optical pulse intensity and indicate the heralding
detection window.

We measure the double excitation probability for different powers of the
optical pulse for nodes B and C. We scale the intensity of the pulse by
changing the voltage sent to the AOM to maintain the same pulse shape for
different powers (as opposed to changing the amplitude of the EOM pulse). We
measure the emitted photons using two different detection paths, namely the
detection path of the resonant ZPL photons (one of the detectors after the
beam splitter in Figure 1a), and an additional detection path for the off-
resonant PSB photons [30]. We extract the optical rotation angles $\theta$ and
the detection efficiency $\eta_{p}$ ($\eta_{z})$ from the photon detection
probability in the PSB (ZPL) detector (Figure 7b), assuming that double
excitation is small. The optical rotation angle $\theta$ serves as a measure
of how much of the population of the $\ket{0}$ ground state is transferred to
the excited state $\ket{e}$, where 180o marks the point where all population
has been transferred. For each power, or rotation angle $\theta$, we measure
the number of coincidence events where both detectors (ZPL and PSB) detected a
photon, $N_{\text{coin}}$. The ZPL detection window is indicated in Figure 7a
and we set the detection window of the PSB photons to start and end well
before and after the pulse.

Since we start the heralding detection window of the ZPL after the optical
pulse has (approximately) ended, re-excitation is not possible during this
window, so we can assume that all measured coincidence events consist of a PSB
photon during the optical pulse and a ZPL photon in the detection window. To
compute the double excitation probability $p_{de}$ from the measured number of
coincidence events $N_{\text{coin}}$, we can thus reformulate $p_{de}$ as

$p_{de}=\frac{P_{2}}{P_{1}+P_{2}},$ (40)

where $P_{1}$ ($P_{2}$) is the probability of a single (two) photon emission,
given a ZPL photon detected in the window. However, single or two photon
emissions have different probabilities to be detected, and we can define the
corresponding detection probabilities as

$\displaystyle P_{2}^{\prime}=$ $\displaystyle\eta_{P}\eta_{Z}P_{2},$ (41)
$\displaystyle P_{1}^{\prime}=$ $\displaystyle\eta_{Z}P_{1}.$ (42)

Here $\eta_{Z}$ and $\eta_{P}$ are the detection efficiencies of the ZPL and
PSB detection paths respectively. Filling in the definitions of $P_{1}$ and
$P_{2}$ in Equation (40) gives

$\displaystyle\begin{split}p_{de}=&\frac{P_{2}^{\prime}}{\eta_{P}P_{1}^{\prime}+P_{2}^{\prime}},\\\
=&\frac{N_{\text{coin}}}{\eta_{P}N_{\text{all}}-(\eta_{P}-1)N_{\text{coin}}},\end{split}$
(43)

in which we have used $N_{\text{coin}}=nP_{2}^{\prime}$ and
$N_{\text{all}}=n(P_{1}^{\prime}+P_{2}^{\prime})$ for $n$ repetitions.

Figure 7d shows the extracted double excitation probability $p_{de}$ for
different rotation angles $\theta$ for the two emitters (turquoise and blue
circles), together with the simulated double excitation probability for our
exact pulse shape using the model from Reference [30]222The numerical model
and the exact pulse shape are included in the supporting software and data
package[36]. There is a clear qualitative agreement between the data and the
simulations. The quantitative difference between the data and simulations
could potentially be explained by measurement errors in the pulse shape
displayed in Figure 7a. We note that the simulations are very sensitive to the
exact shape of the pulse, and any measurement artifact such as reflections in
the optical path could broaden the measured pulse shape. Importantly, we find
from both the measured data and simulations that the double optical excitation
probability can be mitigated by choosing a smaller rotation angle $\theta$,
albeit at the cost of a lowered entanglement generation rate (which scales
with $\cos^{2}\frac{\theta}{2}$). In contrast, setting a $\theta>180^{o}$
results both in a lowered entanglement rate and fidelity, and therefore should
be avoided.

##  IX Non-excited bright state population

Figure 8: Fidelity and relative detection probability in case of non-excited
bright state population. In the top panels, we measure the entangled state
fidelities as a function of optical excitation rotation angle $\theta$ for
states heralded by a photon detection in detectors 1 (circles) and 2
(triangles), and the weighted average (squares) for different values of
$\alpha$ ( $\alpha=0.05,0.2,0.4$ for the purple, blue and green data points
respectively). In the bottom panels we plot the relative probability to detect
the heralding photon in each detector. We perform these measurements for two
different setpoints $\delta\varphi$ of the optical phase stabilization (left
and right panels). The x-axis indicates the rotation angle of node B. The
rotation angle of node C is approximately equal, but differs slightly due to
small differences in the delivered optical power to the nodes. The dashed and
solid lines are the results of the model (see main text). To include other
error sources than the protocol error, we scale the results of the model to
the measured average fidelities.

Up to now we have considered perfect optical pulses, such that $c_{0,i}=0$. In
this section, we study the case of $c_{0,i}\neq 0$. To isolate the effect of
$c_{0,i}\neq 0$, we make the following assumptions. We assume high photon loss
($\eta\ll 1$), no double excitation ($|c_{ee,i}|^{2}=0$), perfectly
overlapping polarization of the photons
($\vec{\epsilon_{A}}\cdot\vec{\epsilon_{B}}=1$), no frequency difference
between the emitters ($\omega_{A}=\omega_{B}$), no noise photons ($p_{d}=0$),
we consider the two setups to have equal bright state populations and photon
losses ($\alpha_{A}=\alpha_{B}\equiv\alpha,\eta_{A}=\eta_{B}\equiv\eta$) and
we write the phase of the entangled state as the optical phase difference in
front of the beam splitter $\phi=-\delta\varphi$. Furthermore, we define

$\displaystyle c_{0}=$ $\displaystyle\cos\frac{\theta}{2}$ (44) $\displaystyle
c_{e}=$ $\displaystyle\sin\frac{\theta}{2},$ (45)

where $\theta$ can be considered as the optical excitation rotation angle
between the ground and excitation state, as explained in the previous Section.

We rewrite Equations (19) and (20) using these assumptions. The probability
$p_{\text{click,C/D}}$ to detect a photon in either port C or D of the beam
splitter is now given by

$\begin{split}p_{\text{click,C/D}}=&\alpha\eta\sin^{2}\frac{\theta}{2}\pm\alpha^{2}\eta\cos\delta\varphi\sin^{2}\frac{\theta}{2}\cos^{2}\frac{\theta}{2},\\\
=&\alpha\eta\sin^{2}\frac{\theta}{2}(1\pm\alpha\cos\delta\varphi\cos^{2}\frac{\theta}{2}),\end{split}$
(46)

where the +(-) sign correspond to a photon detection in port C (port D). Using
the corresponding density matrices $\rho_{C,D}$, we compute the fidelity with
the maximally entangled state with a phase $\phi_{T}$ as

$\begin{split}F_{C/D}=&\bra{\Psi}\rho_{C,D}\ket{\Psi}\\\
=&\frac{1}{4p_{\text{click,C/D}}}(a_{11}+a_{22}+a_{12}e^{-i\phi_{T}}+a_{12}^{\ast}e^{i\phi_{T}})\\\
=&\frac{(1-\alpha)}{2(1\pm\alpha\cos\delta\varphi\cos^{2}\frac{\theta}{2})}(1+\cos(\delta\varphi-\phi_{T}))\end{split}$
(47)

From Equations (46) and (47) it is apparent that the fidelity of the heralded
entangled state depends on which detector detects the single photon, the
optical phase difference $\delta\varphi$ and the optical excitation rotation
angle $\theta$. Surprisingly, for $\delta\varphi=0$ and $\theta\rightarrow 0$
the fidelity of the entangled state heralded by one of the detectors
approaches 1, albeit with a small probability to occur. On the contrary, for
$\delta\varphi=90^{o}$ no difference in entangled state fidelities and their
probabilities to be heralded is expected. This result can be interpreted as
the interference of the different photonic states associated with the
$\ket{00}_{AB}$ qubit states. Dependent on $\delta\varphi$, constructive or
destructive interference of the
$\hat{a}_{A}^{\dagger}\ket{0}_{\gamma}\ket{00}_{AB}$ state with
$\hat{a}_{B}^{\dagger}\ket{0}_{\gamma}\ket{00}_{AB}$ causes different
heralding probabilities and consequently different average heralded state
fidelities. The fact that the heralded fidelity can approach 1 for particular
detector and settings is explained by the reduction of the protocol error (the
error resulting from both qubits being in the bright state): a low excitation
probability makes the probability that two photons were emitted small and
destructive interference ensures that if one photon was emitted it is directed
towards the other detector.

Figure 9: Entangled state optimization. (a) For a fixed fidelity with respect
to the maximally entangled state, we extract the maximal success probability
$p_{\text{click}}$ for different temporal widths of the excitation pulse, by
varying $\alpha$ and fixing the optical rotation angle $\theta=180^{o}$
(circles) or by varying $\alpha$ and $\theta$ (triangles).Note that for the
smallest pulse width (green line) both optimization nearly overlap. (b) (c)
For the optimization where both $\alpha$ and $\theta$ are varied (circles in
panel (a)), the extracted optimal settings for the bright state population
$\alpha$ (b) and the optical excitation rotation angle $\theta$ (c) are shown.

We compare these theoretical results with our experimental data in Figure 8.
We stabilize the interferometer to different setpoints to obtain
$\delta\varphi=0.3\pm 0.9^{o}$ and $\delta\varphi=91.3\pm 1.2^{o}$ (left and
right panels). For different values of $\alpha$ ($\alpha=[0.05,0.2,0.4]$,
purple, blue and green data points respectively) we generate entanglement
while varying the optical rotation angle $\theta$. We record the fidelity (top
panels) for heralding signals detected by the two different detectors
(triangles and circles) and the mean fidelity (square data points), and the
relative probability to detect a photon on each detector (bottom panels). In
the same figure we plot the theoretical model scaled to the mean measured
fidelity to incorporate additional errors (solid, dashed and dotted lines).
Our measured data is in excellent agreement with the theoretical model, we
observe the effect of the lowered excitation power on the fidelity heralded by
the different detectors for the case $\delta\varphi\approx 0^{o}$ and the
absence of this effect for $\delta\varphi\approx 90^{o}$. Note that the (mean)
fidelity for small rotation angles $\theta$ is additionally improved by the
reduction of the double excitation errors due to the lowered optical power,
see Figure 7d. However the data displayed here has a too high measurement
uncertainty to resolve this effect (we expect a maximum improvement of the
fidelity $F$ by $\approx 0.01$ for the measured rotation angles).

##  X State optimization

Let us now turn again to the optimization of the entanglement generation. The
effects discussed in the previous two sections, double optical excitation and
non-excited bright state population, are both related to the optical power of
the excitation pulse. Together with the bright state population (as discussed
in Section V), we have two degrees of freedom in optimizing the entangled
state and we would like to find the optimal settings to produce a state with
fixed fidelity with the highest success probability. In this Section, we
assume symmetric settings of the experimental setups
($\alpha_{A}=\alpha_{B},\eta_{A}=\eta_{B},\theta_{A}=\theta_{B}$).
Furthermore, we assume no noise counts and perfectly indistinguishable
photons.

In Figure 9a, we plot the maximal success probability versus fidelity of the
(numerically simulated) heralded state for different temporal widths of the
excitation pulse (estimated as a Gaussian shaped pulse), in units of the
excited state lifetime $\tau$. The triangular data points represent the case
where $\alpha$ is varied, but the optical power of the excitation pulse is set
such that it transfers all population from the ground to the excited state
(rotation angle $\theta=180^{o}$). The circular data points show the case
where both $\alpha$ and $\theta$ are varied. Figure 9b and 9c show the
settings for the bright state population $\alpha$ and optical excitation angle
$\theta$ that optimize the success probability for fixed fidelity.

We can interpret the strategy for the optimized settings as follows. When the
intrinsic protocol error ($\alpha$) is the dominating source of infidelity, it
is most efficient to choose a smaller $\alpha$ to increase the entangled state
fidelity. Furthermore, the numerical optimization reproduces our conclusion
from Section VIII; reducing slightly the optical power of the excitation pulse
mitigates the effect of double excitation at only a small cost in success
probability. On the contrary, when the double excitation contribution becomes
the dominating error source, lowering the optical power allows the target
fidelity to be met while with fixed $\theta$ these fidelities would be
impossible to reach, which can be seen from the sudden change of behavior in
Figures 9b and c.

##  XI Conclusions and discussion

In conclusion, we have performed a detailed theoretical and experimental
investigation of the single-photon entanglement protocol. We have developed a
general model for states heralded using the single-photon protocol, tailored
the model to our experimental setting, NV centers in bulk diamond, and
experimentally verified the effect of several experimental parameters.

We have studied the effect of the bright state population $\alpha$ on the
generated entangled state and the success probability to herald a state. We
have demonstrated the entangled state phase dependence on MW pulse phases
$\vartheta$, the optical phase stabilization setpoint $\delta\varphi$ and the
detection time of the heralding photon detection in combination with an
emission frequency difference between the qubits. We have shown the
observation of transform-limited spectral linewidths, using a Charge-Resonance
(CR) check to remove any spectral shift. Nonetheless, our data on remote
entanglement is consistent with a residual fluctuating frequency shift between
the emitters, this will be a subject for future work. We have observed a
decrease of the double excitation probability for lowered optical power of the
excitation laser pulses. Additionally, we have shown that reducing the optical
laser power can also lead to different heralded state fidelities dependent on
which detector heralded the state. Lastly, we have shown that the optical
laser power, together with the initial bright state population, can be used to
optimize the state fidelity and success probability.

Finally, while the experiments carried out in this work involved the nitrogen-
vacancy center as a qubit platform, the conclusions presented here are readily
applicable to other qubit platforms, such as other solid state defects and
quantum dots. The insights gained in this work will be crucial in improving
the entangled state fidelities using the single-photon entanglement protocol.

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## Acknowledgments

We acknowledge financial support from the EU Flagship on Quantum Technologies
through the project Quantum Internet Alliance (EU Horizon 2020, grant
agreement no. 820445); from the European Research Council (ERC) through an ERC
Consolidator Grant (grant agreement no. 772627 to R.H.); from the Netherlands
Organisation for Scientific Research (NWO) through a VICI grant (project no.
680-47-624) and the Zwaartekracht program Quantum Software Consortium (project
no. 024.003.037/3368). S.B. acknowledges support from an Erwin-Schrödinger
fellowship (QuantNet, no. J 4229-N27) of the Austrian National Science
Foundation (FWF).

##  Appendix A Average heralded density matrix

In this work we derive a general theoretical model for two-qubit states
heralded by the single-photon entanglement protocol. The different steps and
the corresponding unitaries are given in the main text. In this Appendix we
write the expressions for the resulting density matrices for the different
detection patterns.

The average heralded density matrix for a photon detection in port C of the
beam splitter is given by

$\begin{split}\rho_{C}=&\frac{1}{p_{\text{click,C}}}(\rho_{1}+\rho_{2}+\rho_{\text{incoherent}}+\rho_{\text{noise}})\\\
\end{split}$ (A.2)

with the success probability $p_{\text{click,C}}$

$\begin{split}p_{\text{click,C}}=&\Tr(\rho_{1})+\Tr(\rho_{2})+\Tr(\rho_{\text{incoherent}})+\Tr(\rho_{\text{noise}}).\\\
\end{split}$ (A.3)

Single photon In the case of a single detected photon and no loss, the density
matrix is given by

$\begin{split}\rho_{1}=&\ket{\Psi_{4,1}}\bra{\Psi_{4,1}}\\\
=&\frac{1}{2}\begin{pmatrix}a_{00}&a_{01}&a_{02}&0\\\
a_{01}^{\ast}&a_{11}&a_{12}&0\\\ a_{02}^{\ast}&a_{12}^{\ast}&a_{22}&0\\\
0&0&0&0\\\ \end{pmatrix}\end{split}$ (A.4)

with elements

$\displaystyle\begin{split}a_{00}=&\alpha_{A}\alpha_{B}(c_{0,A}^{2}\eta_{B}|\zeta_{B}|^{2}+c_{0,B}^{2}\eta_{A}|\zeta_{A}|^{2}+c_{0,A}c_{0,B}\sqrt{\eta_{A}}\sqrt{\eta_{B}}(\zeta_{B}\zeta_{A}^{\ast}+\zeta_{A}\zeta_{B}^{\ast}))\end{split}$
(A.5)
$\displaystyle\begin{split}a_{01}=&\alpha_{A}\sqrt{\alpha_{B}}\sqrt{1-\alpha_{B}}e^{i\vartheta_{B}}(c_{0,A}\sqrt{\eta_{B}}\zeta_{A}^{\ast}\zeta_{B}+c_{0,B}\sqrt{\eta_{A}}|\zeta_{A}|^{2})\end{split}$
(A.6)
$\displaystyle\begin{split}a_{02}=&\sqrt{\alpha_{A}}\sqrt{1-\alpha_{A}}\alpha_{B}e^{i\vartheta_{A}}(c_{0,A}\sqrt{\eta_{B}}|\zeta_{B}|^{2}+c_{0,B}\sqrt{\eta_{A}}\zeta_{A}\zeta_{B}^{\ast})\end{split}$
(A.7)
$\displaystyle\begin{split}a_{11}=&\alpha_{A}(1-\alpha_{B})\eta_{A}|\zeta_{A}|^{2}\end{split}$
(A.8)
$\displaystyle\begin{split}a_{12}=&\sqrt{\alpha_{A}}\sqrt{1-\alpha_{A}}\sqrt{\alpha_{B}}\sqrt{1-\alpha_{B}}\sqrt{\eta_{A}}\sqrt{\eta_{B}}e^{-i(\vartheta_{B}-\vartheta_{A})}\zeta_{A}\zeta_{B}^{\ast}\end{split}$
(A.9)
$\displaystyle\begin{split}a_{22}=&(1-\alpha_{A})\alpha_{B}\eta_{B}|\zeta_{B}|^{2}\end{split}$
(A.10)

Here we use a short-hand notation for the photonic modes as defined in
Equation (5), where$\zeta_{i}$ represents the detected mode at time
$t^{\prime}$.

Two photons When two photons arrive at the beam splitter without any lost
photon, we accept an heralding event when both photons are being detected in
port C of the beam splitter because we assume non-number resolving detectors.
The first photon is detected at time $t^{\prime}$ and the second photon at
time $t^{\prime\prime}$. For these states we obtain the following density
matrix

$\begin{split}\rho_{2}=&\ket{\Psi_{4,2}}\bra{\Psi_{4,2}}\\\
=&\frac{1}{4}\begin{pmatrix}a_{00}&a_{01}&a_{02}&0\\\
a_{01}^{\ast}&a_{11}&a_{12}&0\\\ a_{02}^{\ast}&a_{12}^{\ast}&a_{22}&0\\\
0&0&0&0\\\ \end{pmatrix}\end{split}$ (A.11)

with elements

$\displaystyle\begin{split}a_{00}=&\alpha_{A}\alpha_{B}(c_{0,B}^{2}\eta_{A}^{2}|\zeta_{AA}|^{2}+c_{0,A}^{2}\eta_{B}^{2}|\zeta_{BB}|^{2}+c_{0,A}c_{0,B}\eta_{A}\eta_{B}(\zeta_{AA}\zeta_{BB}^{\ast}+\zeta_{BB}\zeta_{AA}^{\ast})\\\
&+\eta_{A}\eta_{B}(|\zeta_{A}(t^{\prime})\zeta_{B}(t^{\prime\prime})+\zeta_{A}(t^{\prime\prime})\zeta_{B}(t^{\prime})|^{2})\\\
&+c_{0,A}\sqrt{\eta_{A}}\eta_{B}\sqrt{\eta_{B}}(\zeta_{BB}\zeta_{A}^{\ast}(t^{\prime})\zeta_{B}^{\ast}(t^{\prime\prime})+\zeta_{BB}\zeta_{A}^{\ast}(t^{\prime\prime})\zeta_{B}^{\ast}(t^{\prime})+\zeta_{A}(t^{\prime})\zeta_{B}(t^{\prime\prime})\zeta_{BB}^{\ast}+\zeta_{A}(t^{\prime\prime})\zeta_{B}(t^{\prime})\zeta_{BB}^{\ast})\\\
&+c_{0,B}\eta_{A}\sqrt{\eta_{A}}\sqrt{\eta_{B}}(\zeta_{AA}\zeta_{A}^{\ast}(t^{\prime})\zeta_{B}^{\ast}(t^{\prime\prime})+\zeta_{AA}\zeta_{A}^{\ast}(t^{\prime\prime})\zeta_{B}^{\ast}(t^{\prime})+\zeta_{A}(t^{\prime})\zeta_{B}(t^{\prime\prime})\zeta_{AA}^{\ast}+\zeta_{A}(t^{\prime\prime})\zeta_{B}(t^{\prime})\zeta_{AA}^{\ast}))\end{split}$
(A.12)
$\displaystyle\begin{split}a_{01}=&\alpha_{A}\sqrt{\alpha_{B}}\sqrt{1-\alpha_{B}}e^{i\vartheta_{B}}\\\
&(c_{0,A}\eta_{A}\eta_{B}\zeta_{AA}^{\ast}\zeta_{BB}+\eta_{A}\sqrt{\eta_{A}}\sqrt{\eta_{B}}\zeta_{AA}^{\ast}(\zeta_{A}(t^{\prime})\zeta_{B}(t^{\prime\prime})+\zeta_{A}(t^{\prime\prime})\zeta_{B}(t^{\prime}))+\eta_{A}^{2}|\zeta_{AA}|^{2})\end{split}$
(A.13)
$\displaystyle\begin{split}a_{02}=&\sqrt{\alpha_{A}}\sqrt{1-\alpha_{A}}\alpha_{B}e^{i\vartheta_{A}}\\\
&(c_{0,B}\eta_{A}\eta_{B}\zeta_{BB}^{\ast}\zeta_{AA}+\sqrt{\eta_{A}}\eta_{B}\sqrt{\eta_{B}}\zeta_{BB}^{\ast}(\zeta_{A}(t^{\prime})\zeta_{B}(t^{\prime\prime})+\zeta_{A}(t^{\prime\prime})\zeta_{B}(t^{\prime}))+\eta_{B}^{2}|\zeta_{BB}|^{2})\end{split}$
(A.14)
$\displaystyle\begin{split}a_{11}=&\alpha_{A}(1-\alpha_{B})\eta_{A}^{2}|\zeta_{AA}|^{2}\end{split}$
(A.15)
$\displaystyle\begin{split}a_{12}=&\sqrt{\alpha_{A}}\sqrt{1-\alpha_{A}}\sqrt{\alpha_{B}}\sqrt{1-\alpha_{B}}\eta_{A}\eta_{B}e^{-i(\vartheta_{B}-\vartheta_{A})}\zeta_{AA}\zeta_{BB}^{\ast}\end{split}$
(A.16)
$\displaystyle\begin{split}a_{22}=&(1-\alpha_{A})\alpha_{B}\eta_{B}^{2}|\zeta_{BB}|^{2}\end{split}$
(A.17)

Again, here we have used a short hand notation for the photonic modes. The
joint modes $\zeta_{ii}$ denote the detected modes at time $t^{\prime}$ and
$t^{\prime\prime}$.

Lost photons We consider all the detection patterns for which at least one
photon is being detected in port C of the beam splitter and at least one
photon is lost. We sum over all the individual detection patterns and arrive
at

$\begin{split}\rho_{\text{incoherent}}=&\sum_{i=1,2}\ket{\Psi_{4,i,r}}\bra{\Psi_{4,i,r}}\\\
=&\frac{1}{2}\begin{pmatrix}a_{00}&0&0&0\\\ 0&a_{11}&0&0\\\ 0&0&a_{22}&0\\\
0&0&0&0\\\ \end{pmatrix}\end{split}$ (A.18)

with elements

$\displaystyle\begin{split}a_{00}=&\alpha_{A}\alpha_{B}(\\\
&\eta_{A}|\zeta_{A}(t^{\prime})|^{2}((1-\eta_{B})|\zeta_{B}(t_{r})|^{2})+c_{ee,B}^{2}-\eta_{B}^{2}|\zeta_{BB}(t^{\prime\prime},t^{\prime\prime\prime})|^{2})\\\
&+\eta_{B}|\zeta_{B}(t^{\prime})|^{2}((1-\eta_{A})|\zeta_{A}(t_{r})|^{2})+c_{ee,A}^{2}-\eta_{A}^{2}|\zeta_{AA}(t^{\prime\prime},t^{\prime\prime\prime})|^{2})\\\
&+\eta_{A}(1-\eta_{A})|\zeta_{AA}(t^{\prime},t_{r})|^{2}(c_{0,B}^{2}+c_{e,B}^{2}+c_{ee,B}^{2})\\\
&+\eta_{A}(1-\eta_{A})|\zeta_{AA}(t_{r},t^{\prime})|^{2}(c_{0,B}^{2}+c_{e,B}^{2}+c_{ee,B}^{2})\\\
&+\eta_{B}(1-\eta_{B})|\zeta_{BB}(t^{\prime},t_{r})|^{2}(c_{0,A}^{2}+c_{e,A}^{2}+c_{ee,A}^{2})\\\
&+\eta_{B}(1-\eta_{B})|\zeta_{BB}(t_{r},t^{\prime})|^{2}(c_{0,A}^{2}+c_{e,A}^{2}+c_{ee,A}^{2})\\\
&+\eta_{A}^{2}|\zeta_{AA}(t^{\prime},t^{\prime\prime})|^{2}(|\zeta_{B}(t_{r})|^{2}+c_{ee,B}^{2}-\eta_{B}^{2}|\zeta_{BB}(t^{\prime\prime\prime},t^{\prime\prime\prime\prime})|^{2})\\\
&+\eta_{B}^{2}|\zeta_{BB}(t^{\prime},t^{\prime\prime})|^{2}(|\zeta_{A}(t_{r})|^{2}+c_{ee,A}^{2}-\eta_{A}^{2}|\zeta_{AA}(t^{\prime\prime\prime},t^{\prime\prime\prime\prime})|^{2}))\\\
\end{split}$ (A.19)
$\displaystyle\begin{split}a_{11}=&\alpha_{A}(1-\alpha_{B})\eta_{A}(1-\eta_{A})(|\zeta_{AA}(t^{\prime},t_{r})|^{2}+|\zeta_{AA}(t_{r},t^{\prime})|^{2})\\\
\end{split}$ (A.20)
$\displaystyle\begin{split}a_{22}=&(1-\alpha_{A})\alpha_{B}\eta_{B}(1-\eta_{B})(|\zeta_{BB}(t^{\prime},t_{r})|^{2}+|\zeta_{BB}(t_{r},t^{\prime})|^{2})\\\
\end{split}$ (A.21)

Here we have used the relations
$\eta_{i}|\zeta_{i}(t^{\prime})|^{2}+(1-\eta_{i})|\zeta_{i}(t_{r})|^{2}=|c_{e,i}|^{2}$
and
$\eta_{i}^{2}|\zeta_{ii}(t^{\prime},t^{\prime\prime})|^{2}+(1-\eta_{i})\eta_{i}|\zeta_{ii}(t^{\prime},t_{r})|^{2}+(1-\eta_{i})\eta_{i}|\zeta_{ii}(t_{r},t^{\prime})|^{2}+(1-\eta_{i})^{2}|\zeta_{ii}(t_{r},t_{r})|^{2}=|c_{ee,i}|^{2}$
to simplify the matrix elements.

Noise counts In case of a false heralding event by a noise count, the average
density matrix consists of two parts $\rho_{0}$ and $\rho_{\text{lost}}$

$\rho_{\text{noise}}=p_{d}(\rho_{0}+\rho_{\text{lost}})$ (A.22)

$\rho_{0}$ corresponds to the situation where no photon is emitted and
$\rho_{\text{lost}}$ where all emitted photons are lost. $\rho_{0}$ is given
by

$\begin{split}\rho_{0}=&\ket{\Psi_{4,0}}_{A}\bra{\Psi_{4,0}}_{A}\otimes\ket{\Psi_{4,0}}_{B}\bra{\Psi_{4,0}}_{B}\\\
=&\begin{pmatrix}a_{00}&a_{01}\\\ a_{01}^{\ast}&a_{11}\\\
\end{pmatrix}\otimes\begin{pmatrix}b_{00}&b_{01}\\\ b_{01}^{\ast}&b_{11}\\\
\end{pmatrix}\end{split}$ (A.23)

with elements

$\displaystyle\begin{split}a_{00}=\alpha_{A}c^{2}_{0,A}\end{split}$ (A.24)
$\displaystyle\begin{split}a_{01}=\sqrt{\alpha_{A}}\sqrt{1-\alpha_{A}}c_{0,A}e^{i\vartheta_{A}}\end{split}$
(A.25) $\displaystyle\begin{split}a_{11}=(1-\alpha_{A})\end{split}$ (A.26)

and similar elements for $b_{ij}$. $\rho_{r}$ represents the density matrix in
case all photons are lost

$\begin{split}\rho_{\text{lost}}=&\sum\ket{\Psi_{4,0,r}}\bra{\Psi_{4,0,r}}\\\
=&\begin{pmatrix}a_{00}&0&0&0\\\ 0&a_{11}&0&0\\\ 0&0&a_{22}&0\\\ 0&0&0&0\\\
\end{pmatrix},\end{split}$ (A.27)

with elements

$\displaystyle\begin{split}a_{00}=&\alpha_{A}\alpha_{B}(c^{2}_{0,A}((1-\eta_{B})|\zeta_{B}(t_{r})|^{2}+(1-\eta_{B})^{2}|\zeta_{BB}(t_{r},t_{r})|^{2})\\\
&+c^{2}_{0,B}((1-\eta_{A})|\zeta_{A}(t_{r})|^{2}+(1-\eta_{A})^{2}|\zeta_{AA}(t_{r},t_{r})|^{2})\\\
&+(1-\eta_{A})(1-\eta_{B})|\zeta_{A}(t_{r})|^{2}|\zeta_{B}(t_{r})|^{2})\\\
\end{split}$ (A.28)
$\displaystyle\begin{split}a_{11}=\alpha_{A}(1-\alpha_{B})((1-\eta_{A})|\zeta_{A}(t_{r})|^{2}+(1-\eta_{A})^{2}|\zeta_{AA}(t_{r},t_{r})|^{2})\\\
\end{split}$ (A.29)
$\displaystyle\begin{split}a_{22}=(1-\alpha_{A})\alpha_{B}((1-\eta_{B})|\zeta_{B}(t_{r})|^{2}+(1-\eta_{B})^{2}|\zeta_{BB}(t_{r},t_{r})|^{2}).\\\
\end{split}$ (A.30)

##  Appendix B Experimental parameters for simulations

In Figures 4 and 6 we provide simulations for the average fidelity with
respect to the maximally entangled states using the model developed in Section
IV. The parameters we use for these simulations are listed in Table B2, both
for the AB and BC links.

Table B2: Table with experimental parameters. The start of the detection window is with respect to the maximum intensity of the optical pulse and the detection probability for each setup is integrated over the detection window. Furthermore, these experiments used the same optical pulse as used in References [29, 30] and is different from the optical pulse shape indicated in Figure 7a. References [29, 30] use a different arbitrary waveform generator (AWG), the Tektronix 5014, which causes the double excitation probabilities of those specific optical excitation pulses to be larger.  | AB | BC
---|---|---
Excited state lifetime | 12.4 ns | 12.4 ns
Detection window start | 4 ns | 5 ns
Detection window duration | 15 ns | 15 ns
Bright state population $\alpha_{A}$ | 0.07 | -
Bright state population $\alpha_{B}$ | 0.05 | 0.05
Bright state population $\alpha_{C}$ | - | 0.1
Detection probability $\eta_{A}\int\mathcal{E}^{2}dt$ | 3.8e-4 | -
Detection probability $\eta_{B}\int\mathcal{E}^{2}dt$ | 5.2e-4 | 4.6e-4
Detection probability $\eta_{C}\int\mathcal{E}^{2}dt$ | - | 2.8e-4
Double excitation probability | 0.06 | 0.08
Noise count rate | 10 Hz | 30 Hz
Phase stability $\sigma_{\delta\varphi}$ | $30^{o}$ | $21^{o}$
FWHM frequency difference $\Delta f_{\text{FWHM}}$ | 13 MHz | 13 MHz
Polarization mismatch | $8^{o}$ | $8^{o}$