# Violation of Bell’s inequality with quantum-dot single-photon sources

Eva M. González-Ruiz<EMAIL_ADDRESS>Sumanta K. Das  Peter Lodahl  Anders
S. Sørensen  Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute
University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

(September 3, 2024)

###### Abstract

We investigate the possibility of realizing a loophole-free violation of
Bell’s inequality using deterministic single-photon sources. We provide a
detailed analysis of a scheme to achieve such violations over long distances
with immediate extensions to device-independent quantum key distribution. We
investigate the effect of key experimental imperfections that are unavoidable
in real-world single-photon sources including the finite degree of photon
indistinguishability, single-photon purity, and the overall source efficiency.
We benchmark the performance requirements to state-of-the-art deterministic
single-photon sources based on quantum dots in photonic nanostructures and
find that experimental realizations appear to be within reach. We also
evaluate the requirements for a post-selected version of the protocol, which
relaxes the demanding requirements with respect to the source efficiency.

††preprint: APS/123-QED

##  I Introduction

Violation of Bell’s inequality is of fundamental significance for our
understanding of nature and has been the subject of a broad experimental
endeavour for the last few decades. A complete violation is highly challenging
due to the presence of the locality and detection loopholes [1], which must be
closed simultaneously to realize loophole-free experiments [2, 3, 4]. While
the locality loophole only requires measurement stations to be sufficiently
separated, overcoming the detection loophole demands high transmission
efficiencies [5], which is particularly challenging at the large distances
required to close the locality loophole. In addition to its fundamental
significance, closing the loopholes enables important technological
applications. In particular, a detector-loophole-free violation of Bell’s
inequality is required for device-independent quantum key distribution (DIQKD)
[5], which allows ultimately secure communication protected even against
hacking attempts on the applied hardware. Although these applications do not
necessarily need to close the locality loophole, they impose similar demands
since cryptography is based on the communicating parties to be far apart.

Recently three different experiments have successfully closed both loopholes
simultaneously. On one side, Bell violation has been achieved based on a
heralding scheme with NV centers [4], but the operation of these systems is
rather complicated and typically slow leading, e.g., to limited key rates for
DIQKD. Faster operations have been achieved in purely photonic systems by
exploiting efficient superconducting detectors, but at the cost of only
working over short distances ($<200$ m) [2, 3].

A solution to overcome these problems has been proposed in Ref. [6] based on a
photonic approach using deterministic single-photon sources. This method
exploits heralding measurements at a central station and is thus applicable to
long distances with limited transmission probability. If the photons are
transmitted through vacuum this could allow the implementation of loophole-
free violation of Bell’s inequality. For photons transmitted through optical
fibers on the hand, the slower propagation speed in the fibers potentially
open the locality loophole (discussed in detail below). Nevertheless the
proposal still enables closing the detector loophole over long distances and
thus the application of the scheme for DIQKD. The scheme could also be
implemented with spontaneous parametric down conversion (SPDC) sources, but
this was found to have a less favorable scaling with the transmission
efficiency [6]. For instance it achieves lowers a key rate in DIQKD and this
makes an implementation with on-demand single-photon sources more attractive.
Real single photons, however, have a number of imperfections which could
potentially prevent the violation of Bell’s inequality. The influence of these
imperfections thus needs to be carefully assessed to determine the
applicability of existing single-photon sources.

In this article we theoretically investigate the performance of real single-
photon sources for the (detector-)loophole-free Bell test of Ref. [6] taking
into account the quality of the single-photon sources and the efficiency of
the local stations. We focus on InGaAs quantum dots embedded in a photonic-
crystal waveguide as single-photon sources. These have recently shown near
ideal performance, generating on-demand single photons with up to a 99.4%
purity and 96% indistinguishability [7, 8]. The results derived here 111The
codes used in this study are available at the University of Copenhagen public
repository ERDA. DOI:
https://doi.org/10.17894/ucph.6b93ed79-4a9d-4d34-b5ea-07479288d11d are also
applicable to other types of single-photons sources. We investigate the
generic imperfections and analyse the relation between the achievable
violation of Bell’s inequality and the indistinguishability of the generated
photons through the Hong-Ou-Mandel (HOM) visibility, their second-order
correlation function ($g^{(2)}$) and the single-photon source efficiency. We
hope that the analysis will motivate experimental demonstrations of long-
distance photonics based (detector-)loophole-free Bell inequality violations
in the near future.

##  II Ideal implementation

Figure 1:  a) Set-up proposed in Ref. [6], consisting of two local stations
(Alice and Bob) and a central heralding station (CHS). The different optical
components involved are indicated: beam splitters with transmittances $T$ (at
Alice and Bob’s stations) and 50% (at the CHS); half and quarter-waveplates
(HWP and QWP), switchable mirrors (SM), polarizing beam splitters (PBS) and
single-photon sources (SPS). The photodetectors are labeled as $D_{1}$,
$D_{2}$, $D_{3}$, and $D_{4}$ at the CHS and $A_{H}$ ($B_{H}$) and $A_{V}$
($B_{V}$) for Alice’s (Bob’s) detectors, with the sub index indicating the
polarization of the corresponding incoming photons. The operators that model
the creation of photons at each section of the set-up are indicated as well.
b) Example of quantum-dot single-photon source and its energy level schemes.
The light-matter interaction between a quantum-dot (yellow dot) and a
waveguide mode is enhanced by the nanostructure realizing a deterministic
single-photon source. The quantum dot is driven by a resonant excitation pulse
$(\Omega_{i})$ and subsequently emits a single photon into the waveguide by
spontaneous emission. Insert: The optical transitions are subject to two
different types of decoherence: 1) rapidly fluctuating phonon interactions
induce pure dephasing with a rate $\gamma_{d}$ and 2) slow drifts of the
levels induce a slowly varying detuning $\Delta_{i}$.

We investigate the scheme originally put forward in Ref. [6]. It is based on a
heralding scheme in which two parties, Alice and Bob, each generate a pair of
single photons with orthogonal polarization (see Fig. 1). Here we will
consider an implementation based on having only a single on-demand single-
photon source at each station. As we will show below an implementation based
on subsequent emissions from the same source is highly advantageous since
several imperfections associated with slow drifts in the sources naturally
cancel out.

The single-photon sources each emit a pair of photons with vertical
polarization but with one of them delayed in time with respect to the other.
The photons are sent to different optical paths by means of a switchable
mirror. The early photon is sent through the longest arm, while the late
photon is sent to the short arm and is rotated to horizontal polarisation with
a half-wave plate. The early and late photons arrive simultaneously at a
polarizing beam-splitter (PBS), which merges the two inputs generating a state
described by

$\displaystyle\begin{split}\hat{a}_{s,H}^{\dagger}\hat{a}_{s,V}^{\dagger}\ket{\emptyset}_{A}&=\ket{1_{H}}\ket{1_{V}}_{A},\\\
\hat{b}_{s,H}^{\dagger}\hat{b}_{s,V}^{\dagger}\ket{\emptyset}_{B}&=\ket{1_{H}}\ket{1_{V}}_{B}\,,\end{split}$
(1)

where $\hat{a}_{s}^{\dagger}$ ($\hat{b}_{s}^{\dagger}$) denotes the photon
creation operator acting on the vacuum state $\ket{\emptyset}$ either at
Alice’s (A) or Bob’s (B) station, with the second index denoting horizontal
($H$) or vertical ($V$) polarization.

After the PBS, the photon pair is sent towards a beam-splitter of
transmittance T. The reflected field is then sent to a detection set-up,
whereas the transmitted field is sent towards a central heralding station
(CHS) placed between Alice and Bob. At the CHS the photons pass through
additional optical components until they are finally detected by four single
photon detectors $D_{l}$ with $l=1,..4$. As we will show later, entanglement
will be generated if the photons detected at the CHS have orthogonal
polarization. This restricts the possible accepted detection patterns to
clicks on pairs of detectors $D_{1}D_{2}$, $D_{3}D_{4}$, $D_{1}D_{4}$ and
$D_{2}D_{3}$ (see Fig. 1).

Ideally, two photons (one from each side) are transmitted to the CHS while the
other photons remain in the station where they were generated. At the CHS,
Bell state measurements on the received photons are performed [10] thereby
creating a polarization entangled state between the photons that remained in
the local stations. Importantly, the protocol is still applicable even for a
very low transmission probability to the CHS, since photon loss only reduces
the heralding rate but not the performance of the protocol. Finally, Alice and
Bob measure their respective qubits by means of a pair of half-wave and
quarter-wave plates (HWP and QWP) and a pair of photodetectors.

The basic operational principle of the set up is that of the Hong-Ou-Mandel
(HOM) effect [11]. In each station, beam splitters of transmittance $T$
transmit single photons from Alice and Bob to the CHS. Conditioning on two
clicks at the CHS means that at least two photons were transmitted, and by
choosing a low transmittance $T\ll 1$ we ensure that there is a negligible
probability of transmitting more than two. In this case there will thus be two
photons at Alice and Bob’s stations for the final Bell’s test with a very high
probability if $T\ll 1$. The two photons at the CHS photons could, however, be
from the same station, such that either Alice or Bob will have two photons
while the other would have none. To exclude this possibility a HWP is inserted
in each of the arms leading to the CHS. This HWP rotates each of the
polarizations by 45 degrees. Seen in the horizontal-vertical basis the HWP
essentially acts as a beam-splitter transformation between the two
polarization states described by

$\hat{a}_{H}^{\dagger}\rightarrow\frac{1}{\sqrt{2}}\left(\hat{a}_{H}^{\dagger}+\hat{a}_{V}^{\dagger}\right),\quad\hat{a}_{V}^{\dagger}\rightarrow\frac{1}{\sqrt{2}}\left(\hat{a}_{H}^{\dagger}-\hat{a}_{V}^{\dagger}\right)\,.$
(2)

Thereby it acts as a beam splitter in a HOM-like setup such that two photons
in the same arm will bunch and never end up with opposite polarization when
measured in the horizontal-vertical basis. Conditioning on photons with
different polarization will thus ensure that the detected photons came from
different stations and that there is always a photon at both Alice and Bob’s
stations.

If the two photons transmitted to the heralding station have the same
polarization they will bunch at the 50:50 central beam splitter due to their
indistinguishability, leading to the possible detector combinations
$D_{1}D_{2}$ and $D_{3}D_{4}$. These patterns do not, however, provide any
information about the initial polarization of the photons before the HWP
except that they were identical. Thus, the state of the pair that Alice and
Bob will share in this case is the Bell state
$\ket{\phi^{-}}=\frac{1}{\sqrt{2}}\left(\hat{a}_{H}^{\dagger}\hat{b}_{H}^{\dagger}-\hat{a}_{V}^{\dagger}\hat{b}_{V}^{\dagger}\right)\ket{\emptyset}$
(with the minus sign coming from a more detailed analysis of the protocol).

On the other hand, if the pair of photons arriving at the CHS has opposite
polarizations, they are distinguishable and therefore will not show the HOM
bunching effect at the beam splitter. If they are transmitted to the same port
of the beam splitter they will click on the same detector since the PBS acts
as a HOM setup for the photons encoded in diagonal polarizations by the HWPs
and will be discarded. On the contrary, if they are transmitted to different
PBS, the combinations $D_{1}D_{4}$ and $D_{2}D_{3}$ can occur and since their
polarizations were initially different, the state that Alice and Bob share
must be proportional to
$\ket{\psi^{-}}=\frac{1}{\sqrt{2}}\left(\hat{a}_{H}^{\dagger}\hat{b}_{V}^{\dagger}-\hat{a}_{V}^{\dagger}\hat{b}_{H}^{\dagger}\right)\ket{\emptyset}$
(again the minus sign is found from a more detailed analysis).

From the above analysis we thus find that recording clicks in detectors with
opposite polarizations implies that there will be a photon at each of Alice
and Bob’s stations prepared in an entangled state conditioned on the outcome
at the CHS. Subsequent measurements in different bases implemented by rotating
the polarization with the HWPs and QWPs at the local stations can then be used
to demonstrate the violation of Bell’s inequality. The combinations that
generate entanglement ($D_{1}D_{2}$, $D_{3}D_{4}$, $D_{1}D_{4}$ and
$D_{2}D_{3}$) represent 50$\%$ of the total number of detection events. This
is in accordance with the fact that any linear-optical circuit that performs
Bell-state measurement can only generate distinguishable entangled states with
at most 50$\%$ probability [12]. In particular, in this work we analyse the
state heralded by the detection patterns $D_{1}D_{4}$ and $D_{2}D_{3}$, which
is $\ket{\psi^{-}}$.

From the above idealized description it is clear that the protocol relies
heavily on high-quality single-photon sources. In practice, real sources will
always have small multi-photon components and non-perfect
indistinguishability. In addition, optical systems are prone to losses. To
assess the performance of the protocol for real sources we thus need to
evaluate the effect of these imperfections.

##  III Model for real sources

The ultimate purpose of this work is to analyse the influence that the quality
of a realistic single-photon source can have on the success of the protocol.
In this section we present the main parameters that describe the source. We
will account both for the loss of indistinguishability of photons due to
decoherence processes and the purity of the source, as well as the efficiency
of the set-up.

###  III.1 Dephasing

Figure 2: Degree of indistinguishability of photons from different sources,
i.e. the HOM visibility $V_{\beta}^{(0)}$. The indistinguishability depends on
the width $\sigma$ of the probability distribution for the frequency
fluctuations $\Delta$ as well as the rate of pure dephasing $\gamma_{d}$
relative to the spontaneous emission rate $\gamma$.

The single-photon source $i$ emits a photon in a state
$\hat{a}_{i}^{\dagger}\ket{\emptyset}$ . Here the single mode operators
$\hat{a}_{i}$ fulfill the single mode commutation relation
$[\hat{a}_{i},\hat{a}_{j}^{\dagger}]=\delta_{ij}$ and are defined by

$\hat{a}_{i}=\int_{-\infty}^{\infty}f_{i}(t)\hat{a}_{i}(t)dt\,,$ (3)

where $\hat{a}_{i}(t)$ is a photon annihilation operator for a photon at time
$t$ fulfilling the continuous time commutation relations
$\left[\hat{a}(t),\hat{a}^{\dagger}(t^{\prime})\right]=\delta(t-t^{\prime}),\quad\left[\hat{a}(t),\hat{a}(t^{\prime})\right]=\left[\hat{a}^{\dagger}(t),\hat{a}^{\dagger}(t^{\prime})\right]=0$.
We have here introduced the function $f_{i}(t)$, with $i\in\\{1,2,3,4\\}$,
(different for every source in which the photons are generated) which
describes the shape of each photons wavepacket. $f_{i}(t)$ convolutes each
photon’s creation operator differently in time, according to the properties of
the emitter. Ideally $f_{i}(t)$ will be the same for all sources, resulting in
fully indistinguishable photons. Variations between emitters and temporal
fluctuations, however, imply that they may vary between different emitters and
between different experimental runs. These functions thus encode the relevant
photon coherences.

The functions $f_{i}(t)$ satisfy the normalization
$\int_{-\infty}^{\infty}|f_{i}(t)|^{2}dt=1$ and their overlap defines the
degree of indistinguishability of the photons, that we define as $\alpha_{ij}$
or $\beta_{ij}$ depending on whether the pair of photons was produced at the
same or different stations:

$\displaystyle\begin{split}\expectationvalue{\hat{a}_{i}(t)\hat{a}_{j}^{\dagger}(t^{\prime})}&=\int_{-\infty}^{\infty}f_{i}^{*}(t)f_{j}(t)dt\\\
&\equiv\begin{cases}\alpha_{ij},\quad\text{if}\quad ij=12,34\\\
\beta_{ij},\quad\text{if}\quad ij=13,14,23,24\end{cases}\,.\end{split}$ (4)

This distinction between photons generated at the same or opposite stations is
important, since their degree of indistinguishability is expected to be
different ($\alpha_{ij}>\beta_{ij}$) because sources at the same station are
likely easier to be identical. We can relate $\alpha_{ij}$ or $\beta_{ij}$ to
the raw HOM visibility $V_{\alpha,\beta}^{(0)}$ by calculating the probability
of detecting coincidence counts after a 50:50 beam splitter, obtaining
$P_{cc}=\frac{1}{2}\left(1-|\alpha_{ij}|^{2}\right)$ (or $\beta_{ij}$, if
photons were generated at different stations). Given that the maximum
probability of coincidence counts that can be achieved is $P_{max}=1/2$, we
obtain the relation between visibility and mode overlap [13]:

$V_{\alpha}^{(0)}=1-\frac{P_{cc}}{P_{max}}=\overline{|\alpha_{ij}|^{2}}\,,$
(5)

where we have defined $\overline{X}$ to be the average of the quantity $X$
over all experimental runs due to the variability of $f_{i}(t)$.

The convoluting functions $f_{i}(t)$ contain the decoherence processes of the
quantum dot that decrease the indistinguishability of the emitted photons.
Generally decoherence arises due to a number of different mechanisms acting on
different time scales. We divide these processes into two categories: slow and
fast processes relative to the decay rate of the emitter. For quantum dot
sources the fast process originates from phonon dephasing, whereas intrinsic
charge or nuclear spin noise as well as drift in the experimental setup will
contribute to the slow detuning processes [14]. We assume that the two photons
generated at each station are obtained by multiplexing photons emitted from
the same quantum dot [15], i.e. photons from the same stations are insensitive
to slow processes, which will have the same influence on two consecutive
photons, but will be affected by fast processes which change the mode function
$f_{i}(t)$ between the two photons. On the contrary, photons emitted from
different stations will also be affected by slow drift depending on the degree
to which these can be stabilized between the distant emitters.

The fast processes are modeled as white noise with a pure dephasing rate
$\gamma_{d}$ corresponding to the typical model used to describe phonon
dephasing [16, 17]. In Appendix A we show that this leads to a visibility

$V_{\alpha}^{(0)}=\overline{|\alpha_{ij}|^{2}}=\frac{\gamma}{\gamma+2\gamma_{d}}\,,$
(6)

where $\gamma$ is the spontaneous decay rate of the quantum dot.

The crossed visibility, $V_{\beta}^{(0)}$, is also affected by the slow
dephasing processes that creates an energy difference $\Delta_{ij}$ in the
frequency splitting between quantum dots. This difference is assumed to follow
a Gaussian distribution with a width $\sigma$ when averaging over the
experimental runs. Evaluating the visibility for detuned emitters and
performing the average we obtain the averaged visibility

$V_{\beta}^{(0)}=\overline{|{\beta_{ij}}|^{2}}=\sqrt{\frac{\pi}{2}}\frac{\gamma}{\sigma}e^{\frac{(\gamma+2\gamma_{d})^{2}}{2\sigma^{2}}}\text{erfc}\left(\frac{\gamma+2\gamma_{d}}{\sqrt{2}\sigma}\right)\,.$
(7)

A similar result was obtained in Ref. [18].

In Fig. 2 we show the relation between visibility and width of the
distribution for various dephasing rates. The relations derived in Eqs. (6)
and (7) between the intrinsic parameters of the quantum dot and the
indistinguishability of the emitted photons can be used to calibrate the width
$\sigma$ to the measured HOM visibility through Eq. (5). For the remaining of
this article all plots will thus be shown as a function of the experimentally
accessible visibilities $V_{\alpha}^{(0)}$ and $V_{\beta}^{(0)}$ rather than
$\gamma_{d}$ and $\sigma$. We note however, that expressing it in terms of
visibilities is dependent on the specific model we have assumed for the noise.
Below we shall also need higher order moments e.g.
$\expectationvalue{\alpha_{ij}\beta_{ik}\beta_{jk}}$. Such higher order
correlations cannot be directly related to the visibility, which only depends
on second order moments. Any relation between higher order terms and the HOM
visibility will thus always be model dependent.

###  III.2 Optical loss

Although the heralding scheme ensures that the outcome of the protocol is not
affected by the transmission efficiency between stations $\eta_{t}$, we must
account for losses occurring locally since such losses potentially open the
detection loophole. We define $\eta_{1,i}$ as the probability to reach the
first beam splitter, and $\eta_{2,i}$ as the probability for the reflected
photons to be detected (including detection efficiency) as sketched in Fig. 3.
The additional index $i=1,2,3,4$ specifies the source that generated the
photon. This distinction between sources is for instance relevant in
demultiplexing schemes, which can have different efficiencies in the different
channels of transmission [15]. Including the losses at the beam splitter, the
local efficiency $\eta_{l,i}$ is therefore given by
$\eta_{l,i}\equiv\eta_{1,i}\eta_{2,i}(1-T)$.

Experimentally, the coupling of light from quantum dots to photonic
nanostructures has been proven to reach up to 98% [19]. As it will be
quantified below, the main challenge will be to couple the photons from the
nanostructure and to a detector with near-unity efficiency. With a
demultiplexed single-photon source this would require highly-efficient
coupling of the photon source to a fiber combined with efficient switching and
detection. A significant step towards this demanding goal was the recent
demonstration of a fiber-coupled device with $57\%$ overall efficiency [8].
Another implementation would require operating two quantum dots per station
that are mutually interfered. In such a configuration the two sources and
detectors could potentially be integrated on a single chip, which likely would
be the most efficient approach. A bottleneck for this implementation would be
the need to make two quantum-dot sources indistinguishable, but encouragingly
$93\%$ HOM visibility was recently reported between two remote quantum dots
[20].

Figure 3:  Sketch of the efficiencies $\eta_{i}$ and definitions of the
creation operators of the lost photons $\hat{c}^{\dagger}$.

###  III.3 Purity

Finally, we consider that the single-photon sources might in fact emit two or
more photons at once with a certain probability. This generates the state
$\hat{\rho}=P_{0}\hat{\rho}_{0}+P_{1}\hat{\rho}_{1}+P_{2}\hat{\rho}_{2}+\mathcal{O}(P_{3})$,
where $\hat{\rho}_{k}$ is the density matrix representing the state of the
k-th photon component. The multi-photon component would originate in an
experiment either from imperfect suppression of the optical pulses used to
pump the quantum dot or from multi-photon emission due to the finite duration
of the excitation pulse.

We furthermore assume that the probability for the quantum dot not to emit a
photon is negligible $P_{0}\simeq 0$, so that $P_{1}+P_{2}=1$. This limit is
valid as long as the lack of creation of photons by the quantum dot is
included in the efficiency parameters $\eta_{1,i}$.

The effect of multi-photon generation can be measured via the second order
correlation function $g^{(2)}$

$g^{(2)}=\frac{\expectationvalue{\iint
d\tau_{1}d\tau_{2}\hat{a}^{\dagger}(\tau_{1})\hat{a}^{\dagger}(\tau_{2})\hat{a}(\tau_{2})\hat{a}(\tau_{1})}}{\expectationvalue{\int
d\tau_{1}\hat{a}^{\dagger}(\tau_{1})\hat{a}(\tau_{1})}\expectationvalue{\int
d\tau_{2}\hat{a}^{\dagger}(\tau_{2})\hat{a}(\tau_{2})}}\,,$ (8)

which, for the multi-photon state $\hat{\rho}$ yields

$g^{(2)}=\frac{2P_{2}}{\left(P_{1}+2P_{2}\right)^{2}}\simeq\frac{2P_{2}}{P_{1}^{2}}\,,$
(9)

to first order in $P_{2}$. Throughout this paper we consider the limit
$g^{(2)}\ll 1$ such that the probability that more than one photon is emitted
by more than one of the sources is negligible. This means that all results
derived here should be considered lowest order expansion in $g^{(2)}$ and are
only applicable in the limit $g^{(2)}\lesssim 0.1$. Moreover, the second-
emitted photon is modelled to be completely distinguishable from the other
four photons. The purity of the source will affect the measured HOM visibility
$V_{\alpha,\beta}$ with a small contribution proportional to $g^{(2)}$:

$V_{\alpha,\beta}=V_{\alpha,\beta}^{(0)}-\kappa g^{(2)}\,,$ (10)

where in general $1<\kappa<3$, depending on the distinguishability of the 1
and 2-photon components. For the model considered here, $\kappa=2$ where the
additional photon is assumed to be fully distinguishable. [21, 22]. In the
following, when plotting quantities as a function of $V_{\alpha,\beta}^{(0)}$
we refer to the value which would be obtained in the absence of any two photon
contribution. Hence $V_{\alpha,\beta}^{(0)}$ will be higher than the values
measured experimentally.

##  IV Analysis

Having defined the model used to describe the imperfections, we now turn to
the analysis of the performance of the protocol. We do this by analyzing how
each of the field operators transform in the Heisenberg picture under the
linear optics setup shown in Fig. 1 when the losses in Fig. 3 are taken into
account. After this transformation we then evaluate all expressions using the
model for the real single-photon sources.

After being generated, the two pairs of photons with orthogonal polarization
will encounter a beam splitter with transmittance $T$, that transforms the
creation operators according to

$\displaystyle\begin{split}\hat{a}_{s,{\varsigma}}^{\dagger}&=i\sqrt{1-T}\hat{a}_{\varsigma}^{\dagger}+\sqrt{T}\hat{a}_{t,{\varsigma}}^{\dagger}\,,\end{split}$
(11)

and similarly for Bob’s photons, where $\hat{a}_{\varsigma}^{\dagger}$
represents a reflected photon at the first beam splitter with polarization
$\varsigma\in\\{H,V\\}$ and $\hat{a}_{t,{\varsigma}}^{\dagger}$ a transmitted
one. Next, by fixing the angle $\phi=-\pi/8$ for both HWP, we see that the
transmitted photons are transformed as:

$\displaystyle\begin{split}\hat{a}_{s,{\varsigma}}^{\dagger}&=i\sqrt{1-T}\hat{a}_{\varsigma}^{\dagger}+\sqrt{\frac{\eta_{t}T}{2}}(\hat{a}_{t,H}^{{}^{\prime}\dagger}\pm\hat{a}_{t,V}^{{}^{\prime}\dagger})\,.\end{split}$
(12)

where we have inserted the probability of successful transmission between
Alice and Bob and the CHS, $\eta_{t}$. Since we include losses, this
transformation should, in principle, also include creation operators for the
lost photons. However, since we consider photon counting (and multi-photon
errors are considered separately below), such lost photons do not contribute
to the final results and we omit the lost photon terms for brevity. Once the
photons reach the heralded station they encounter another beam splitter, this
time with 50% transmittance, followed by polarizing beam splitters that direct
horizontally (vertically) polarized photons to detectors $D_{1}$ and $D_{3}$
($D_{2}$ and $D_{4}$).

We can now incorporate all the steps into a single transformation of the
initial operators $\hat{a}_{S}^{\dagger}$ ($\hat{b}_{S}^{\dagger}$), obtaining

$\displaystyle\begin{split}&\hat{a}_{{s,\varsigma},{i}}^{\dagger}\rightarrow\\\
&\sqrt{\frac{\eta_{1,i}\eta_{t}T}{2}}\hat{O}_{i}^{\dagger}+i\sqrt{\eta_{1,i}\eta_{2,i}(1-T)}\hat{a}_{{{\varsigma},{i}}}^{\dagger}+\hat{L}_{i}^{\dagger}\,,\end{split}$
(13)

and similarly for Bob’s operators. We have here defined creation operators for
photons at the CHS as

$\displaystyle\begin{split}\hat{O}_{1,2}^{\dagger}&\equiv\frac{1}{\sqrt{2}}\left(i\left(p_{H,{1,2}}^{\dagger}\pm
p_{V,{1,2}}^{\dagger}\right)+\left(q_{H,{1,2}}^{\dagger}\pm
q_{V,{1,2}}^{\dagger}\right)\right),\\\
\hat{O}_{3,4}^{\dagger}&\equiv\frac{1}{\sqrt{2}}\left(\left(p_{H,{3,4}}^{\dagger}\pm
p_{V,{3,4}}^{\dagger}\right)+i\left(q_{H,{3,4}}^{\dagger}\pm
q_{V,{3,4}}^{\dagger}\right)\right)\,.\end{split}$ (14)

Eq. (13) expresses that a photon can be either successfully transmitted to the
CHS ($\hat{O}_{i}^{\dagger}$), reflected at the first beam splitter and
detected locally ($\hat{a}_{{\varsigma},i}^{\dagger}$) or lost
($\hat{L}_{i}^{\dagger}$). The definition of these operators simplify the
transformation applied to the initial state and is described in detail in
Appendix B.

Eq. (13) allows us to calculate the density matrix $\hat{\rho}$ shared by
Alice and Bob after the heralding by just tracing the CHS and loss operators
out from the initial state conditioned on the correct detection patterns:

$\small\begin{split}\hat{\rho}=\Tr_{\text{loss},CHS}\left\\{\hat{a}_{s,H}^{\dagger}\hat{a}_{s,V}^{\dagger}\hat{b}_{s,H}^{\dagger}\hat{b}_{s,V}^{\dagger}\ket{\emptyset}\bra{\emptyset}\hat{a}_{s,H}\hat{a}_{s,V}\hat{b}_{s,H}\hat{b}_{s,V}\right\\}\,,\end{split}$
(15)

where the partial trace of the CHS is already conditioned to the accepted
measurement outcomes $D_{1}D_{4}$ and $D_{2}D_{3}$.

Figure 4: a) Violation of the CHSH inequality $S\geq 2$, as a function of the
HOM visibility of the local sources $V_{\alpha}^{(0)}$. The curves represent
both the case in which photons from Alice and Bob’s stations and equally
indistinguishable ($V_{\alpha}^{(0)}=V_{\beta}^{(0)}$) and when there is
inhomogeneous broadening with a width $\sigma\neq 0$ so that
$V_{\alpha}^{(0)}>V_{\beta}^{(0)}$. The threshold for CHSH violation $S=2$
corresponds to the visibilities
$(V_{\alpha}^{(0)},V_{\beta}^{(0)})=\left\\{(0.79,0.79),(0.86,0.67),(0.93,0.51),(0.96,0.40)\right\\}$.
b) Contour plot of the CHSH value $S$ as a function of the visibilities
$V_{\alpha}$ and $V_{\beta}$. The red isoline $S=2$ delimits the regions with
and without violation of Bell’s inequality.

Alice and Bob measure this state by means of two consecutive QWP and HWP, with
the total transformation
$\hat{U}_{tot}(\theta,\phi)=\hat{U}_{HWP}(\phi)\hat{U}_{QWP}(\theta)$
(Appendix B). It transforms the creation operators of the photons into the
operators $\hat{A}_{H}^{\dagger}$ and $\hat{A}_{V}^{\dagger}$
($\hat{B}_{H}^{\dagger}$ and $\hat{B}_{V}^{\dagger}$), which represent the
creation operators at the corresponding detectors (see Fig. 1):

$\displaystyle\begin{split}\mathbf{\hat{A}}\equiv\begin{pmatrix}\hat{A}_{H}\\\
\hat{A}_{V}\end{pmatrix}&=\hat{U}_{tot}(\theta_{A},\phi_{A})\begin{pmatrix}\hat{a}_{H,{1}}\\\
\hat{a}_{V,{2}}\end{pmatrix}\equiv\hat{U}_{tot}(\theta_{A},\phi_{A})\mathbf{\hat{a}}\\\
\end{split}\,,$ (16)

and similarly with Bob’s operators. We further define the measurement
operators
$\hat{M}_{{+},{A}}\equiv\mathbb{I}-\ket{\emptyset}\bra{\emptyset}_{A,{H}}$ and
$\hat{M}_{{-},{A}}\equiv\mathbb{I}-\ket{\emptyset}\bra{\emptyset}_{A,{V}}$,
and similarly for Bob. These operators project into the subspace in which any
number of clicks occurs at the detectors. We thus assume that the detectors
cannot distinguish how many photons arrived but only measure the presence or
absence of photons. Events in which there is a wrong number of photons
involved are thus also included. We identify a click in the detectors $A_{H}$
and $B_{H}$ with an outcome $+$, and $A_{V}$ and $B_{V}$ with $-$. We can then
calculate the joint probability $P_{x,y}$ that the outcomes
$x,y\in\left\\{+,-\right\\}$ are simultaneously measured by Alice and Bob as

$\displaystyle\begin{split}P_{x,y}=\Tr{\hat{M}_{x,y}\hat{\rho}^{\prime}}\,,\end{split}$
(17)

where $\hat{M}_{x,y}=\hat{M}_{{x},{A}}\otimes\hat{M}_{{y},{B}}$ and we have
defined:

$\displaystyle\begin{split}\hat{\rho}^{\prime}=\hat{U}_{tot}(\theta_{A},\theta_{B},\phi_{A},\phi_{B})\hat{\rho}\hat{U}_{tot}^{\dagger}(\theta_{A},\theta_{B},\phi_{A},\phi_{B})\,,\end{split}$
(18)

and
$\hat{U}_{tot}(\theta_{A},\theta_{B},\phi_{A},\phi_{B})=\hat{U}_{tot}(\theta_{A},\phi_{A})\otimes\hat{U}_{tot}(\theta_{B},\phi_{B})$.
Note that for convenience the probabilities defined here, include the
probability of detecting the photons at the CHS. We therefore need to
normalize the probabilities with the success probability when we want to
evaluate results conditioned on detections at the CHS. This allows us to
calculate the CHSH [23] correlations
$C(\textbf{a},\textbf{b})=\textbf{a}\textbf{b}$, where a and b are two unitary
observables that can take the values $\\{-1,1\\}$, as a function of the
projected probabilities:

$\displaystyle\begin{split}&C(\textbf{a},\textbf{b})=\frac{P_{+,+}-P_{+,-}-P_{-,+}+P_{-,-}}{P_{success}}\,,\end{split}$
(19)

where the choice of a and b is determined by the HWP and QWP angles
$\theta_{A}$, $\theta_{B}$, $\phi_{A}$, and $\phi_{B}$. Note that the
denominator of Eq. (19), $P_{success}$, corresponds to the total success
probability for the photons to arrive at the heralding station conditioned on
the correct measurement outcomes $D_{1}D_{4}$ and $D_{2}D_{3}$. We can now
analyse how the violation of the CHSH inequality

$S=|C(\mathbf{a},\mathbf{b})+C(\mathbf{a^{\prime}},\mathbf{b})+C(\mathbf{a},\mathbf{b^{\prime}})-C(\mathbf{a^{\prime}},\mathbf{b^{\prime}})|\leq
2\,,$ (20)

is affected by all the processes related to real sources described in Section
III.

###  IV.1 Effect of distinguishability of photons

We start by analysing solely the influence of decoherence of the emitter,
ignoring multi-photon generation ($g^{(2)}=0$) and local losses
($\eta_{l,i}=1$). Furthermore, we assume for now that apart from the
decoherence the protocol behaves ideally: Alice and Bob keep one photon each
and send one to the CHS, which is detected in the desired patterns.
Physically, this corresponds to post selecting only events where there are
always two detected photons at the CHS and one at each station. Applying these
conditions to the transformation in Eq. (13) and operating on the initial
state, we obtain:

$\displaystyle\begin{split}\ket{\psi}=-&\frac{{\eta_{t}}{T}{(1-T)}}{2}(\hat{O}_{2}^{\dagger}\hat{O}_{4}^{\dagger}\hat{a}_{H,1}^{\dagger}\hat{b}_{H,3}^{\dagger}+\hat{O}_{2}^{\dagger}\hat{O}_{3}^{\dagger}\hat{a}_{H,1}^{\dagger}\hat{b}_{V,4}^{\dagger}+\\\
&\hat{O}_{1}^{\dagger}\hat{O}_{4}^{\dagger}\hat{a}_{V,2}^{\dagger}\hat{b}_{H,3}^{\dagger}+\hat{O}_{1}^{\dagger}\hat{O}_{3}^{\dagger}\hat{a}_{V,2}^{\dagger}\hat{b}_{V,4}^{\dagger})\ket{\emptyset}\,.\end{split}$
(21)

We then calculate the partial trace over the the CHS described by the
operators $\hat{O}_{i}^{\dagger}$. Contrary to the ideal case, the state
shared by Alice and Bob after conditioning on the desired detection pattern is
no longer a pure state due to the imperfect indistinguishability of the
photons. This can be seen from the (unnormalized) density matrix obtained by
conditioning on the correct detection patterns
$\expectationvalue{\hat{O}_{i}\hat{O}_{j}\hat{O}_{k}^{\dagger}\hat{O}_{l}^{\dagger}}_{D1D4,D2D3}$.
The full results are given in Appendix C. For simplicity we here only
reproduce the density matrix in a simpler form assuming
$\alpha_{ij}\equiv\alpha$ and $\beta_{ij}\equiv\beta$ for all $i,j$:

$\small{\hat{\rho}=\frac{{\eta_{t}}^{2}{T}^{2}{(1-T)}^{2}}{4}\begin{pmatrix}1-{|\beta|^{2}}&0&0&{|\beta|^{2}}-{|\alpha|^{2}}\\\
0&1+{|\beta|^{2}}&-\left({|\alpha|^{2}}+{|\beta|^{2}}\right)&0\\\
0&-\left({|\alpha|^{2}}+{|\beta|^{2}}\right)&1+{|\beta|^{2}}&0\\\
{|\beta|^{2}}-{|\alpha|^{2}}&0&0&1-{|\beta|^{2}}\end{pmatrix}}\,,$ (22)

where we have applied the definitions for indistinguishability introduced in
Eq. (4). The density matrix shown in Eq. (22) is written in a basis that is
dependent on polarization, as well as the mode functions which may vary from
shot to shot of the experiment. This dependence will be traced out once the
measurement at the local stations is performed. Note that the state described
by Eq. (22) is a mixed state in contrast to the indistinguishable limit
($\alpha=\beta=1$), in which we in fact recover the pure state
$\ket{\psi^{-}}$.

We can now evaluate the results of the measurement with the density matrix
$\hat{\rho}$ and calculate the different joint probability contributions by
means of Eqs. (17) and (18). We note, however, that the density matrix in Eq.
(22) is expressed in terms of photon operators which also contain mode
functions that vary in time due to noise. All expectation values will involve
a total of 8 mode functions $f_{i}$, which we need to average over. Therefore,
this results in higher order moments in $\alpha_{ij}$ and $\beta_{ij}$, which
have to be averaged e.g. giving terms of the form
$\overline{\alpha^{2}\beta^{2}}$.

We calculate the value of the CHSH Bell parameter $S$ as a function of the HOM
visibility $V_{\alpha}^{(0)}$, that is, the HOM visibility of the sources from
the same station (see Fig. 4(a)). We study two different limits: first, we
consider the limit in which all photons are equally indistinguishable
($\alpha=\beta$). An optimization over the measurement angles is carried out
for each value of the visibility in other to maximise $S$. In this limit a HOM
visibility of at least $\approx 79\%$ is needed to violate Bell’s inequality.
We further study the limit in which the cross visibility $V_{\beta}^{(0)}$
from both stations is lower than the visibility from the same station
$V_{\alpha}^{(0)}$ by varying the inhomogeneous broadening $\sigma$. As shown
in Fig. 4(a), with increasing slow spectral diffusion a higher visibility of
the photons from the same stations is required to achieve a Bell violation. In
Fig. 4(b) we show the size of the violation for all sets of visibilities,
keeping in mind that $V_{\alpha}^{(0)}\geq V_{\beta}^{(0)}$. Note that if a
good local visibility $V_{\alpha}^{(0)}$ is achieved, the requirement for the
crossed visibility $V_{\beta}^{(0)}$ between stations is rather limited.
Remarkably and as an exemplary case, for the experimentally realized value of
$V_{\alpha}^{(0)}=96\%$ [7] it suffices to reach $V_{\beta}^{(0)}>40\%$ to
violate Bell’s inequality. This is an encouraging requirement that seems well
within reach with quantum-dot sources.

###  IV.2 Losses and multi-photon errors

Figure 5:  Evaluation of CHSH threshold for a postselected protocol and with finite multi-photon contributions. In both figures the CHSH parameter $S$ is shown as a function of the second order correlation function for different values of the HOM visibility, in the limit in which the transmission and local efficiencies are low ($\eta_{t}=\eta_{2,i}=0.1$) and for post-selected events. As post-selection is performed, we set $T=0.5$, which will increase the heralding rate without affecting the performance of the protocol. a) The crossed and local HOM visibilities $V_{\alpha}$ and $V_{\beta}$ are identical. b) The local HOM visibility is fixed to $V_{\alpha}=0.95$ and $V_{\beta}$ varies between 0.6 and $V_{\alpha}$. | $P(A=x,B=y)$ | $P(A=x,B=\emptyset)$ | $P(A=\emptyset,B=y)$ | $P(A=\emptyset,B=\emptyset)$
---|---|---|---|---
4-photon contributions | $P_{2110}$ | $P_{{2200},{SD}}$ |  | $P_{{2020},{SD}}$ |  | $P_{2002}$ | $P_{{2020},{DD}}$
$(i+j+k+l=4)$ |  | $P_{2101}$ |  | $P_{2011}$ |  | $P_{3001}$ |
|  | $P_{3100}$ |  | $P_{3010}$ |  | $P_{{2200},{DD}}$ |
5-photon contributions | $P_{2111}$ | $P_{2102}$ | $P_{{3200},{SD}}$ | $P_{2012}$ | $P_{{3020},{DD}}$ | $P_{2003}$ | $P_{{2030},{DD}}$
($i+j+k+l=5$) | $P_{2210,{SD}}$ | $P_{{2201},{SD}}$ | $P_{2120,{DD}}$ | $P_{{2021},{SD}}$ | $P_{2210,{DD}}$ | $P_{3002}$ | $P_{{3200},{DD}}$
| $P_{2120,{SD}}$ | $P_{{2300},{SD}}$ |  | $P_{{2030},{SD}}$ |  | $P_{{2201},{DD}}$ | $P_{{3020},{SD}}$
| $P_{3110}$ | $P_{3101}$ |  | $P_{3011}$ |  | $P_{{2021},{DD}}$ | $P_{{2300},{DD}}$
Table 1:  Different contributions to the probabilities $P(A=x,B=y)$,
$P(A=x,B=\emptyset)$, $P(A=\emptyset,B=y)$ and $P(A=\emptyset,B=\emptyset)$,
where $x$ and $y$ stand for the outcome of Alice and Bob’s measurement,
respectively. Note that, in the case in which more than one photon is detected
at the local stations, we must distinguish between them clicking at the same
(denoted with subscript $SD$) or different ($DD$) detectors. The first case is
seen as an acceptable outcome $x$ or $y$, since we assume that detectors are
not number resolving, whereas $DD$ cases are recognized as wrong events by
Alice and Bob and are thus assigned to an erroneous outcome $\emptyset$.

When losses and multi-photon errors are included the detection of the correct
patterns at the CHS does not guarantee that Alice and Bob share the state
predicted by Eq. (22). For instance, Alice and Bob might get a positive
message from the CHS station, but when attempting to measure the state, Bob
does not detect anything at his measurement station. Several events can lead
to this situation, depending on whether one of the sources generated more than
one photon. Therefore we must identify all of these cases. To simplify the
analysis, we introduce the notation $P_{ijkl}$ to denote the probability
associated with each of these events. The indices stand for $i$ photons
arriving at the CHS, $j$ and $k$ photons being detected by Alice and Bob,
respectively, and $l$ photons lost at any point in the set-up. To first order
in $g^{(2)}$ and with vacuum emission included in the local efficiency, we can
restrict the analysis to total photons numbers of 4 and 5 so that $4\leq
i+j+k+l\leq 5$. Moreover, $i\geq 2$, since the accepted detection patterns at
the CHS require two photons being detected. Table 2 in Appendix D shows all
the different contributions that contribute to each probability term.

We now divide the contributions into four categories: $P(A=x,B=y)$ with
$x,y=\pm 1$ includes all the terms in which both Alice and Bob measure a click
in one and only one of their detectors; $P(A=x,B=\emptyset)$ and
$P(A=\emptyset,B=y)$ describe events where one of them does not detect any
photon and $P(A=\emptyset,B=\emptyset)$ is the probability that none of them
detected anything, as specified in Table 1. It is important to note that when
more than one photon is reflected at the first beam splitter, they can either
be detected at the same (denoted with subscript $SD$) or different ($DD$)
detectors. In the first situation, this probability will contribute to
$P(A=x,B=\emptyset)$ and $P(A=\emptyset,B=y)$, since we assume that detectors
are not number resolving and the two photons at the CHS must have originated
from the same source. On the other hand, for the $DD$ cases Alice or Bob will
know that the result is incorrect. Here we will just assign it to the same
category as a lost photon. Furthermore, the two photons from the other station
must again have gone to the CHS and these events thus contribute to
$P(A=\emptyset,B=\emptyset)$.

We start by considering the effect of multi-photon emission, in the situation
typically encountered in current experiments, where the local efficiency is
limited and a violation of Bell’s inequality can only be obtained by post
selection (thereby not closing the detection loophole). In this limit we
discard all events in which Alice or Bob did not detect any photon (i.e., the
probability terms $P(A=x,B=\emptyset)$, $P(A=\emptyset,B=y)$ and
$P(A=\emptyset,B=\emptyset)$). Since this situation is insensitive to local
losses it is desirable to set the transmittance to $T=0.5$, which increases
the heralding rate. Therefore, compared to the situation in Sec. IV.1, the
only new contributions to the CHSH correlations are the 5-photon terms
$P_{2111}$, $P_{2210}$, $P_{2120}$ and $P_{3110}$ (see Table 2).

As mentioned above we will only go to lowest order in the two photon emission
probability $P_{2}$ (or equivalently $g^{(2)}$ according to Eq. (9)). We can
thus restrict the analysis to events where there are 4 and 5 photons in total
and the final probability distributions are therefore given by a new
probability distribution for the accepted events

$\displaystyle\begin{split}P_{T}&(A=x,B=y)=\\\
&\frac{P_{1}^{4}P_{4}(A=x,B=y)+P_{1}^{3}P_{2}P_{5}(A=x,B=y)}{P_{1}^{4}+4P_{2}P_{1}^{3}}\,,\end{split}$
(23)

where $P_{4}(A=x,B=y)$ and $P_{5}(A=x,B=y)$ correspond to the probability of 4
and 5 photon event contributions, respectively. Here $P_{5}(A=x,B=y)$ accounts
for the possibility of any of the four single-photon sources emitting two
photons. Applying Eq. (9) we obtain that the total probability that accounts
for all the events equals

$\displaystyle\begin{split}P_{T}&(A=x,B=y)=\\\
&\frac{P_{4}(A=x,B=y)+\frac{1}{2}g^{(2)}P_{5}(A=x,B=y)}{1+2g^{(2)}}\,.\end{split}$
(24)

This allows us to calculate the CHSH S-parameter by means of Eq. (19) as a
function of the second order correlation function $g^{(2)}$, the efficiencies
of every channel, and the indistinguishability of the photons through the
moments of $\alpha_{ij}$ and $\beta_{ij}$.

The value of the CHSH parameter $S$ is plotted as a function of the second
order correlation function $g^{(2)}$ in Fig. 5. We observe a Bell violation
for low values of $g^{(2)}$ that ceases once the two-photon probability
exceeds a certain value of not more than 10 %. The robustness towards multi-
photon contributions is dependent on the HOM visibility of the photons. The
predicted values are compatible with state-of-the-art single-photon sources,
where $g^{(2)}\lesssim 2\%$ is currently achieved by quantum-dot single-photon
sources embedded in photonic nanostructures [24, 8, 7]. In the figure we focus
on the regime corresponding to a typical post-selected experiment where the
local and transmission efficiencies are rather small
$\eta_{t}=\eta_{2,1}=0.1$. These presented results thus roughly represent the
limit $\eta\rightarrow 0$. For the situations considered below with higher
higher local efficiency $\eta_{2,i}\approx 1$ we find that the scheme is more
robust to $g^{(2)}$.

Figure 6:  CHSH parameter $S$ as a function of the local efficiency
$\eta_{l}=\eta_{1,i}\eta_{2,i}(1-T)$ of Alice and Bob’s stations for different
HOM visibilities. To focus on the effect of the local efficiency we consider a
situation with a very limited transmission to the CHS ($T=10^{-3}$ and
$\eta_{t}=0.1$) and vary the efficiency of the final arm $\eta_{2}$ with
$\eta_{1,i}=1$.

Finally we consider the detection loophole by accepting all events (including
those in which no photons are detected, i.e. $P(A=\emptyset,B=y)$,
$P(A=x,B=\emptyset)$, $P(A=\emptyset,B=\emptyset)$). In the derivation of the
CHSH inequality it is assumed that the measurement can only take on two values
-1 and 1. Experimentally we will also have the possibility of an inconclusive
outcome $\emptyset$, so we must decide on a strategy for dealing with those
outcomes. Here we chose that every time Alice and Bob do not detect a photon
they assign a determined outcome to it [5] such that there are only two
possible outcomes of the experiment that we can assign to $\pm 1$, as required
by the CHSH inequality. In particular, Alice and Bob assign a positive
detection $x,y=\\{+\\}$ whenever they do not detect any photon. Thus, we
arrive at a probability $\tilde{P}$ distribution

$\displaystyle\begin{split}&\tilde{P}(A=x,B=y)=\\\
&P(A=x,B=y)+\delta_{y,+}P(A=x,B=\emptyset)\\\
&+\delta_{x,+}P(A=\emptyset,B=y)+\delta_{x,+}\delta_{y,+}P(A=\emptyset,B=\emptyset)\,.\end{split}$
(25)

This allows us to study the effect of the local efficiency on the performance
of the protocol (see Fig. 6) independently of the other effects, by setting
$g^{(2)}=0$, choosing different values of HOM visibility, and calculating the
effective probabilities by applying Eq. (25) to Eqs. (23) and (24).

To mimic a situation corresponding to long distance communication where there
is limited transmittance and a negligible probability to detect more than one
photon at the CHS, we consider a very low transmittance of the beam splitter,
$T=10^{-3}$, and a transmittance to the CHS $\eta_{t}=0.1$. We vary the local
efficiency by changing the loss rate in the detection arm $\eta_{2}$ with
$\eta_{1}=1$. These choices, however, have very little influence on the
results as long as we can neglect multiphoton events at the CHS. For perfect
HOM visibility, the threshold that delimits $S=2$ is $\eta_{l}=82.8\%$, which
corresponds to the lowest efficiency that any CHSH Bell test can tolerate for
loophole-free violation [25]. The simple strategy for dealing with null
detections, as introduced above, thus have similar performance as the best
achievable strategy. As we introduce additional errors, see Fig. 7, the
required local efficiency increases and goes beyond 90% for realistic values
of $g^{(2)}$ and HOM visibility. Such high local efficiency is not yet
achievable in state-of-the-art implementations, although a lot of progress in
this direction has been achieved in recent years [8, 7]. Note that compared to
the post-selected limit (Fig. 5), the violation of the inequality is less
sensitive to $g^{(2)}$. This is due to the difference in the local efficiency
in the two plots. In the post selected limit events in which two photons are
emitted have a higher chance of being accepted relative to the single-photon
events, because either of the two photons can make it to the detectors. Had we
considered post selection in the limit of high local efficiency, it would be
less sensitive to $g^{(2)}$ since some undesired results, e.g. $DD$ events,
can be discarded.

Figure 7:  Threshold for the violation of the CHSH Bell’s inequality $(S\leq
2)$. We vary $\eta_{2,i}$ for different values of the second order correlation
function and determine the threshold visibility $V_{\beta}^{(0)}$ required to
violate the CHSH inequality. The local HOM visibility has been set to
$V_{\alpha}^{(0)}=1$ and we assume a low transmission efficiency ($T=10^{-3}$,
$\eta_{t}=0.1$).

###  IV.3 Optimizing the probability of transmission $T$

In previous sections we have calculated and optimised the CHSH value $S$
within the limit of a very small transmission to the CHS. This ensures a
higher local efficiency and thus allows us to investigate how well the
protocol might ideally work as well as identify thresholds for the success of
the protocol. A low probability of transmission, however, also implies that
the number of successful heralding events will be low. In this subsection we
find the optimal transmittance $T$ of the beam splitters at the local stations
(see Fig. 1(a)) to violate Bell’s inequality with the highest number of
standard deviations $\sigma_{S}$.

Figure 8:  Effect of finite transmittance $T$. a) CHSH parameter $S$ as a
function of the transmittance $T$ of the beam-splitters at the end stations
for several sets of single-source error parameters: visibility
($V_{\alpha}^{(0)}=V_{\beta}^{(0)}$), purity ($g^{(2)}$), and the local
efficiency ($\eta_{2}$ with $\eta_{1,i}=1$). A large transmission probability
decreases the performance of the protocol since there will be more events
where no photons are detected at end stations. b) Number of standard
deviations $Z^{\prime}$ with which a violation of Bell’s inequality is
achieved (normalised to the number of experimental runs) as a function of the
transmittance $T$. In an experiment with $n$ experimental runs the CHSH
inequality is violated by $Z^{\prime}\sqrt{n}$ standard deviations. An optimal
$T$ value that maximises the Bell violation can be found for each parameter
set. Note that the legend is divided into two parts inserted in part a) and b)
but applies to both subfigures a) and b), i.e. half the lines are defined by
the legend in a), the others in b).

The CHSH parameter (Eq. (20)) is the sum of four independently measured
correlations $C(\mathbf{a},\mathbf{b})$. This allows us to write the standard
deviation $\sigma_{S}$ of the CHSH parameter $S$ as

$\displaystyle\begin{split}\sigma_{S}^{2}&=\expectationvalue{S^{2}}-\expectationvalue{S}^{2}\\\
&=\sigma_{C(\mathbf{a},\mathbf{b})}^{2}+\sigma_{C(\mathbf{a}^{\prime},\mathbf{b})}^{2}+\sigma_{C(\mathbf{a},\mathbf{b}^{\prime})}^{2}+\sigma_{C(\mathbf{a}^{\prime},\mathbf{b}^{\prime})}^{2}\,.\end{split}$
(26)

Furthermore, given that
$C^{2}(\mathbf{a},\mathbf{b})=\mathbf{a}^{2}\mathbf{b}^{2}=1$ we obtain the
standard deviation of $S$ for a single run of the experiment

$\displaystyle\begin{split}\sigma_{S}^{2}=4-C(\mathbf{a},\mathbf{b})^{2}-C(\mathbf{a}^{\prime},\mathbf{b})^{2}-C(\mathbf{a},\mathbf{b}^{\prime})^{2}-C(\mathbf{a}^{\prime},\mathbf{b}^{\prime})^{2}\,.\end{split}$
(27)

After a number of independent experimental runs $n$, there have been
$N=nP_{CHS}$ successful events, where $P_{CHS}$ is the total probability of
acceptance, equal to the sum of all the probability contributions that return
a positive message from the CHS. The standard deviation of the average of $S$
yields

$\bar{\sigma}_{S}=\frac{\sigma_{S}}{\sqrt{N/4}}=\frac{2\sigma_{S}}{\sqrt{nP_{CHS}}}\,,$
(28)

where the factor of 4 arises from the four different measurement
configurations of the CHSH parameter $S$. We define $Z$ as the number of
standard deviations with which the Bell inequality can be violated

$Z=\frac{S-2}{\bar{\sigma}_{S}}=\frac{S-2}{2\sigma_{S}}\sqrt{nP_{CHS}}\,,$
(29)

where we have inserted Eq. (28) and take into account that the violation
occurs for $S>2$.

We have optimized the set of angles for the final measurement at each station
for various sets of visibility, losses and multi-photon parameters and a
variable transmittance T. Using Eq. (29) this allows us to calculate a
dimensionless number $Z^{\prime}\equiv Z/\sqrt{n}$ expressing the number of
standard deviations normalised by the number of experimental runs. The number
of standard deviations obtained after $n$ experimental runs is then
$Z^{\prime}\sqrt{n}$. As we can observe in Fig. 8(b), the compromise between a
strong violation and the probability of success manifests itself in different
optimal transmission coefficients that maximises $Z^{\prime}$ for different
parameter sets. Generally the optimal transmission decreases as other errors
become more significant, since in that case there is less room for errors
introduced by the transmission. Furthermore, in the limit of low transmission
efficiency ($\eta_{t}=0.1$), the main contributions to the probability of
acceptance at the CHS arise from events in which two photons arrive (and not
three). This leads to $P_{CHS}\propto T^{2}$ resulting in a linear increase of
$Z^{\prime}\propto\sqrt{P_{CHS}}\propto T$ for small $T$, as observed in Fig.
8(b).

Having calculated the optimal $T$, it is straightforward to evaluate the
number of experimental runs $n$ necessary to violate Bell’s inequality with a
certain number of standard deviations $Z$

$n=\left(\frac{Z}{Z^{\prime}}\right)^{2}\,.$ (30)

Current experiments can easily reach $V_{\alpha}^{(0)}\geq 0.9$ and
$g^{(2)}=0.02$ [8, 7] and for simplicity we assume that a similar visibility
is reached between different stations $V_{\beta}^{(0)}=V_{\alpha}^{(0)}$. A
much more challenging requirement is the local efficiency. If we assume
optimistic values of $\eta_{1}=1$ and $\eta_{2}=0.95$, we get
$Z^{\prime}=6.2\cdot 10^{-6}$ with $\eta_{t}=0.1$ at the optimal transmission.
This means that Bell’s inequality can be violated with three standard
deviations after $2.3\cdot 10^{11}$ runs. Note that for the sake of
simplicity, we use the number of standard deviations with which a Bell
violation is achieved as the figure of merit. For a more precise
characterization of the violation in an actual experiment it would be
desirable to consider the the p-value [26]. Considering the number of standard
deviations, however, allows for a simpler evaluation of the requirement to
achieve a violation by a different amount $Z$ by means of Eq. (30).

For a typical repetition rate of the laser that excites the quantum-dot
single-photon sources ($\simeq$ 75 MHz [7, 8, 27]) this approximately
corresponds to only 52 minutes. If we further assume an attenuation length of
20 km, given that the data was obtained for $\eta_{t}=0.1$, this allows for a
separation distance between Alice and Bob’s stations of more than 90 km. This
is a promising result for future experimental implementations.

###  IV.4 Memory considerations

For experiments based on photons emitted from a central source the locality
loophole can typically be closed by rapidly switching the measurement basis at
each station. The requirement for closing this loophole is then achieved by
having the measurement stations sufficiently separated such that no signalling
is possible, i.e. by placing the detectors outside their respective light
cones.

Note, however, that for protocols based on heralding this requirement concerns
the heralding station as well as Alice and Bob’s stations. For photons
transmitted through vacuum this can be achieved by adding only a minor delay
at each station corresponding to the time it takes to detect the photons at
the CHS. On the other hand if Alice and Bob are far apart it is desirable to
use optical fibers for the transmission of photons between the measurement
stations and the CHS. Since the speed of light is reduced when travelling in
the fiber, the heralding is likely to happen inside the light cone of the
stations, thus opening the locality loophole.

This problem can be solved by using local memories at the measurement stations
of Alice and Bob. They would then need to store their corresponding photons
until the heralding is outside the light cone. In this way, any signalling
produced by the CHS would not affect the measurement of the state of the
photon, thus closing the locality loophole. State-of-the-art quantum memories
have reached an efficiency above 85% [28] which is in principle sufficient to
allow an efficient violation using this setup, but this would put very
stringent requirements on the other local efficiencies. On the other hand for
the application of the considered setup for DIQKD, the locality loophole is
less important and the main challenge is to close the detection loophole,
which can still be achieved without memories.

##  V Conclusion and outlook

We have studied the feasibility of using single-photon sources for loophole-
free Bell test following the proposal of Ref. [6]. The method is general and
applies to any single-photon sources, but for concreteness we focus on sources
based on quantum dots. Since the considered protocol is completely photonic
and has a built-in robustness to losses in the transmission to the CHS, it is
highly promising for violating Bell inequalities over long distances, although
the reduced transmission speed of optical fibers will likely open the locality
loophole for fiber based implementations. Beyond their fundamental interest
such detection-loophole-free violations are of immense technological interest
since they allow DIQKD, providing ultimate security in communication.

The success of the protocol is strongly dependent on the visibility, purity,
and efficiency of the single-photon sources. We find that a HOM visibility of
at least 79% and second-order correlation function $g^{(2)}<10\%$ suffice for
violating the CHSH inequality when post-selecting events (not loophole-free).
Such metrics are already demonstrated with state-of-the-art quantum dot
sources [8, 7]. Performing a fully loophole-free violation of Bell’s
inequality, however, remains a challenge since this puts very stringent
requirements on the efficiency of the single-photon sources. For realistic
parameters we find that efficiencies on the order of $\eta=90\%$ are required.
This is beyond what has been achieved so far, but continuous improvements in
efficiencies have been achieved in recent years. We thus believe that sources
capable of achieving such efficiencies will be available in the future. Once
the necessary efficiencies are achieved it is highly encouraging that we find
the requirements for the crossed visibility between stations to be lower than
those for photons generated at the same station. For instance comparing Figs.
2 and 7 we see that if the local indistinguishability is high
($V_{\alpha}^{(0)}\approx 1$) we may obtain a violation even if the spectral
fluctuations between the two quantum dots are comparable to the decay rate
$\sigma\sim\gamma$. This highly relaxed performance reduces considerably the
experimental complexity.

The stringent requirements on the efficiencies may be remedied by designing
more advanced protocols. In particular it has been shown that by using non-
maximally entangled states and Eberhard’s inequality (or, equivalently, CHSH
with the assignation strategy used in the present work) the efficiency
threshold for Bell tests can be lowered to $\eta=66.7\%$ [29]. For the current
setup such non-maximally entangled states can by obtained by replacing the
beam-splitter at the CHS by a beam-splitter with a different transmission [6].
It is therefore very likely that a similar advantage can be gained for this
system. It has, however, recently been proven that the security of some
generalised DIQKD protocols involving non-maximally entangled states can be
bounded to a similar threshold as the CHSH inequality [30, 31]. It is thus
unclear if a similar advantage can also be obtained for DIQKD. In the future
it will be highly interesting to explore the precise relation between the
results we have obtained here and the conditions for performing DIQKD as well
as the possible advantages of changing to non-maximally entangled states.

###### Acknowledgements.

We would like to thank Matthias Christandl for fruitful discussions. We
acknowledge the support of Danmarks Grundforskningsfond (DNRF 139, Hy-Q Center
for Hybrid Quantum Networks).

##  Appendix A Dephasing in single-photon emitters

To evaluate the performance of the protocol we need to evaluate the
indistinguishability parameters $\alpha_{ij}$ and $\beta_{ij}$ (and higher
order moments). In this section we calculate these for a particular model of
the dephasing, which was also considered in [18]. We consider a Hamiltonian
that describes the dynamics of the system, containing the field, the emitter
and their interaction [32]:

$\hat{H}=\left(\omega_{eg}^{i}+\Delta_{i}+F_{i}(t)\right)\ket{e_{i}}\bra{e_{i}}+\int{dk\omega_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k}}-\int{dkg_{k}\ket{e_{i}}\bra{g_{i}}\hat{a}_{k}e^{ikz_{i}}+h.c.}\,,$
(31)

where $\omega_{eg}^{i}$ is the natural frequency splitting of each emitter,
perturbed by a slow frequency drift $\Delta_{i}$ and rapidly varying (e.g.
phonon-induced) random fluctuations represented by the uncorrelated force
$F_{i}(t)$. Solving the time evolution of the system through Schrödinger’s
equation with a suitable wavefunction ansatz [32] allows us to find the mode
functions $f_{i}(t)$ appearing in Eq. (3):

$\displaystyle
f_{i}(t,t_{0})=\sqrt{\gamma}e^{-\frac{\gamma}{2}(t-t_{0})}e^{-i\left(\Delta_{i}(t-t_{0})+\int_{t_{0}}^{t}F_{i}(t^{\prime})dt^{\prime}\right)}\equiv\sqrt{\gamma}e^{-\frac{\gamma}{2}(t-t_{0})}e^{-i\left(\Delta_{i}(t-t_{0})+\phi_{i}(t,t_{0})\right)}\,,$
(32)

where we account for the spontaneous decay (with spontaneous emission rate
$\gamma$) and dephasing of the emitter through the random phase $\phi_{i}$. We
assume that the uncorrelated force $F_{i}(t)$ satisfies
$\expectationvalue{F_{i}(t)}=0$ and [33]

$\expectationvalue{F_{i}(t)F_{j}(t^{\prime})}=2\gamma_{d}\delta_{ij}\delta(t-t^{\prime})\,,$
(33)

where $\gamma_{d}$ is the pure dephasing rate. This model for dephasing is
equivalent to the standard density matrix description where a decay rate
$\gamma_{d}$ is added to the off-diagonal elements of the density matrix and
has e.g. been shown to provide a good description for phonon induced dephasing
in quantum dots [16, 17].

We can now use the above model above to evaluate the desired correlation
functions. From Eq. (4) we have

$\displaystyle\begin{split}\expectationvalue{|{\alpha_{ij}}|^{2}}&=\int_{t_{0}}^{\infty}\int_{t_{0}}^{\infty}dtdt^{\prime}\expectationvalue{f_{i}^{*}(t,t_{0})f_{j}^{*}(t^{\prime},t_{0})f_{i}(t^{\prime},t_{0})f_{j}(t,t_{0})}\,.\end{split}$
(34)

Substituting Eq. (33) into 34, assuming that the energy level in the quantum
dot evolves slowly enough in time such that $\Delta_{i}=\Delta_{j}$, we find

$\displaystyle\begin{split}\expectationvalue{|{\alpha_{ij}}|^{2}}=\iint\limits_{t_{0}}^{\infty}&dtdt^{\prime}\gamma^{2}\exp{-\gamma(t+t^{\prime}-2t_{0})}\expectationvalue{\exp{-i\left(\phi_{i}(t^{\prime})+\phi_{j}(t)-\phi_{i}(t)-\phi_{j}(t^{\prime})\right)}}\,.\end{split}$
(35)

Since white noise is necessarily Gaussian, we have a Gaussian distribution for
$\phi_{i}(t)$, leading to

$\displaystyle\begin{split}&\expectationvalue{|{\alpha_{ij}}|^{2}}=\iint\limits_{t_{0}}^{\infty}dtdt^{\prime}\gamma^{2}\exp{-\gamma(t+t^{\prime}-2t_{0})}\exp{\expectationvalue{-\frac{\expectationvalue{\left(\phi_{i}(t^{\prime})+\phi_{j}(t)-\phi_{i}(t)-\phi_{j}(t^{\prime})\right)^{2}}}{2}}}\,.\end{split}$
(36)

Applying Eq. (33) the expectation value in the exponent yields

$\displaystyle\begin{split}&\expectationvalue{\left(\phi_{i}(t^{\prime})+\phi_{j}(t)-\phi_{i}(t)-\phi_{j}(t^{\prime})\right)^{2}}=4\gamma_{d}|t^{\prime}-t|\,,\end{split}$
(37)

finally after substituting in Eq. (36), this leads to

$\expectationvalue{|{\alpha_{ij}}|^{2}}=\frac{\gamma}{\gamma+2\gamma_{d}}\,.$
(38)

In a similar fashion, one can obtain higher order terms, that are relevant for
the three and four photon contributions:

$\displaystyle\begin{split}\expectationvalue{\alpha_{ij}\alpha_{ik}\alpha_{jk}}&=\frac{\gamma^{2}}{\left(\gamma+\gamma_{d}\right)\left(\gamma+2\gamma_{d}\right)},\quad\expectationvalue{\alpha_{ij}\alpha_{kl}\alpha_{ik}\alpha_{jl}}=\frac{\gamma^{3}\left(3\gamma+5\gamma_{d}\right)}{\left(\gamma+\gamma_{d}\right)\left(\gamma+2\gamma_{d}\right)^{2}\left(3\gamma+2\gamma_{d}\right)}\,,\end{split}$
(39)

where $i\neq j\neq k$. Note that
$\expectationvalue{\alpha_{ij}\alpha_{kl}}=\expectationvalue{\alpha_{ij}}\expectationvalue{\alpha_{kl}}$
for $i,j\neq k,l$, since the noise is uncorrelated.

On the other hand, when the photons are generated at different stations, we
can no longer assume that the energy splitting is zero during each run of the
experiment. The slow drift implies a splitting
$\Delta_{ij}\equiv\Delta_{j}-\Delta_{i}\neq 0$, yielding the following
expectation value of the indistinguishability parameter $\beta_{ij}$:

$\displaystyle\begin{split}\expectationvalue{|{\beta_{ij}}|^{2}}=\iint\limits_{t_{0}}^{\infty}&dtdt^{\prime}\gamma^{2}\exp{-\gamma(t+t^{\prime}-2t_{0})}\exp{-i\Delta_{ij}\left(t-t^{\prime}\right)-2\gamma_{d}|t^{\prime}-t|}\,,\end{split}$
(40)

where we have again calculated the expectation value as in Eq. (37). Solving
these integrals gives us

$\expectationvalue{|{\beta_{ij}}|^{2}}=\frac{\gamma\left(\gamma+2\gamma_{d}\right)}{\left(\gamma+2\gamma_{d}\right)^{2}+\Delta_{ij}^{2}}\,.$
(41)

Similarly we can obtain the higher order terms that are needed for calculating
the necessary correlations:

$\displaystyle\begin{split}&\expectationvalue{\alpha_{ij}\beta_{ik}\beta_{jk}}=\frac{\gamma^{2}\left(12\left(\gamma+\gamma_{d}\right)^{2}\left(\gamma+2\gamma_{d}\right)^{2}+\left(3\gamma^{2}+6\gamma\gamma_{d}+4\gamma_{d}^{2}\right)\Delta^{2}\right)}{\left(3\left(\gamma+\gamma_{d}\right)\left(\gamma+2\gamma_{d}\right)\left(4\left(\gamma+\gamma_{d}\right)^{2}+\Delta^{2}\right)\left(\left(\gamma+2\gamma_{d}\right)^{2}+\Delta^{2}\right)\right)}\,,\\\
\\\
&\expectationvalue{\alpha_{ij}\alpha_{kl}\beta_{ik}\beta_{jl}}=\frac{\gamma}{3\gamma+2\gamma_{d}}\left(-\frac{8\gamma_{d}\left(\gamma+\gamma_{d}\right)\left(2\gamma+\gamma_{d}\right)}{\left(\gamma+2\gamma_{d}\right)\left(4\left(\gamma+\gamma_{d}\right)^{2}+\Delta^{2}\right)}+\frac{3\gamma^{2}+6\gamma\gamma_{d}+2\gamma_{d}^{2}}{\left(\gamma+2\gamma_{d}\right)^{2}+\Delta^{2}}+\frac{2\gamma_{d}^{2}\left(3\gamma+2\gamma_{d}\right)^{2}}{\left(\gamma+2\gamma_{d}\right)^{2}\left(\left(3\gamma+2\gamma_{d}\right)^{2}+\Delta^{2}\right)}\right)\,.\end{split}$
(42)

Note that as we assume that all photons are generated by only two quantum
dots, there is in fact only one value of the detuning. Therefore we can reduce
it to a single value $\Delta$ regardless of the considered index.

To apply the above results we need to average over the slow variations of
$\Delta$. We assume a Gaussian distribution centered around $\Delta=0$ with
standard deviation $\sigma$ described by
$P(\Delta)=1/\left(\sqrt{2\pi}\sigma\right)\exp{\frac{-\Delta^{2}}{2\sigma^{2}}}$.
When we take the average we obtain:

$\displaystyle\begin{split}\expectationvalue{|{\beta_{ij}}|^{2}}&=\sqrt{\frac{\pi}{2}}\frac{\gamma}{\sigma}e^{\frac{(\gamma+2\gamma_{d})^{2}}{2\sigma^{2}}}\text{erfc}\left(\frac{\gamma+2\gamma_{d}}{\sqrt{2}\sigma}\right)\,,\\\
\expectationvalue{\alpha_{ij}\beta_{ik}\beta_{jk}}&=\sqrt{\frac{\pi}{2}}\frac{\gamma}{3\sigma(\gamma+\gamma_{d})(\gamma+2\gamma_{d})}\cdot\\\
&\left(e^{\frac{(\gamma+2\gamma_{d})^{2}}{2\sigma^{2}}}(\gamma+2\gamma_{d})(3\gamma+2\gamma_{d})\text{erfc}\left(\frac{\gamma+2\gamma_{d}}{\sqrt{2}\sigma}\right)-4e^{\frac{2(\gamma+\gamma_{d})^{2}}{\sigma^{2}}}\gamma_{d}(\gamma+\gamma_{d})\text{erfc}\left(\frac{\sqrt{2}(\gamma+\gamma_{d})}{\sigma}\right)\right)\,,\\\
\expectationvalue{\alpha_{ij}\alpha_{kl}\beta_{ik}\beta_{jl}}&=\sqrt{\frac{\pi}{2}}\frac{\gamma}{\sigma(3\gamma+2\gamma_{d})(\gamma+2\gamma_{d})^{2}}(2e^{\frac{(3\gamma+2\gamma_{d})^{2}}{2\sigma^{2}}}\gamma_{d}^{2}(3\gamma+2\gamma_{d})\text{erfc}\left(\frac{3\gamma+2\gamma_{d}}{\sqrt{2}\sigma}\right)+\\\
e^{\frac{(\gamma+2\gamma_{d})^{2}}{2\sigma^{2}}}&(\gamma+2\gamma_{d})(3\gamma^{2}+6\gamma\gamma_{d}+2\gamma_{d}^{2})\text{erfc}\left(\frac{\gamma+2\gamma_{d}}{\sqrt{2}\sigma}\right)-4e^{\frac{2(\gamma+\gamma_{d})^{2}}{\sigma^{2}}}\gamma_{d}(2\gamma+\gamma_{d})(\gamma+2\gamma_{d})\text{erfc}\left(\frac{\sqrt{2}(\gamma+\gamma_{d})}{\sigma}\right))\,.\end{split}$
(43)

Although exact, the above expressions give trouble numerically when
$\sigma\rightarrow 0$. Therefore, an asymptotic expansion of the complementary
error functions has been used for $\sigma<0.1$. We express it up to third non-
vanishing order

$\text{erfc}(x)\simeq\frac{1}{\sqrt{\pi}}e^{-x^{2}}\left(\frac{1}{x}-\frac{1}{2x^{3}}+\frac{3}{4x^{5}}\right)\,.$
(44)

For instance, $\expectationvalue{|\beta_{ij}|^{2}}$ becomes

$\expectationvalue{|\beta_{ij}|^{2}}\simeq\frac{\gamma}{\gamma+2\gamma_{d}}\left(1-\frac{\sigma^{2}}{(\gamma+2\gamma_{d})^{2}}+\frac{3\sigma^{4}}{(\gamma+2\gamma_{d})^{4}}\right)\,,$
(45)

which shows that
$\expectationvalue{|\beta_{ij}|^{2}}(\sigma=0)=\expectationvalue{|\alpha_{ij}|^{2}}$,
as expected. In a similar way, we calculate the expansion for higher order of
the indistinguishability parameters.

##  Appendix B Photon operator transformations

In this appendix we show explicitly how the optical set up transforms the
initial creation operators and give more details of the model we use to
describe the different errors mechanisms. From Eqs. (11) and 12 we obtain the
global transformation

$\displaystyle\begin{split}\hat{a}_{s,{\varsigma}}^{\dagger}&=\frac{\sqrt{\eta_{t}T}}{2}\left(i\left(\hat{p}_{H}^{\dagger}\pm\hat{p}_{V}^{\dagger}\right)+\left(\hat{q}_{H}^{\dagger}\pm\hat{q}_{V}^{\dagger}\right)\right)+i\sqrt{1-T}\hat{a}_{\varsigma}^{\dagger}\,,\end{split}$
(46)

and similarly for Bob’s photons, where $\varsigma\in\\{H,V\\}$. We now apply
Eq. (46) to all operators in the initial state of the system
$\hat{a}_{s,H}^{\dagger}\hat{a}_{s,V}^{\dagger}\hat{b}_{s,H}^{\dagger}\hat{b}_{s,V}^{\dagger}\ket{\emptyset}$
and trace out the CHS operators $\hat{p}_{H}^{\dagger}$,
$\hat{p}_{V}^{\dagger}$, $\hat{q}_{H}^{\dagger}$ and $\hat{q}_{V}^{\dagger}$
to obtain the state shared by Alice and Bob after the heralding. This state,
as explained in section II, will depend on the detection pattern at the CHS.
If we ignore all imperfections and restrict the state to those events where
photons lead to clicks corresponding to opposite polarizations

$\displaystyle\begin{split}\ket{\psi_{D1D4}}&=\frac{{\eta}_{t}{T}{(1-T)}}{\sqrt{2}}\ket{\psi^{-}}=-\ket{\psi_{D2D3}}\,,\end{split}$
(47)

whereas the state that is created after the patterns $D_{1}D_{2}$ and
$D_{3}D_{4}$ is

$\ket{\psi_{D1D2}}=\ket{\psi_{D3D4}}=\frac{i{\eta}_{t}{T}{(1-T)}}{\sqrt{2}}\ket{\phi^{-}}\,,$
(48)

agreeing with our HOM arguments from section II.

In this work we focus the analysis on the patterns that generate the state
$\ket{\psi^{-}}$ ($D_{1}D_{4}$ and $D_{2}D_{3}$) in order to simplify the
presentation of the results. Note that the transmission probability only
enters in the prefactor of the unnormalised state. After renormalization to
account for the state conditioned on the detection of the photons, the states
are independent of the transmission efficiency thanks to the heralding scheme.

The heralding station notifies Alice and Bob when the correct pattern of
clicks has been detected. In the ideal scenario these patterns guarantee that
the protocol succeeded in generating the desired state. To evaluate the
performance under non-ideal condition, however, we need to consider the error
mechanisms described in the previous section. To do this we start by
describing the transformation of the mode operators when a photon is lost:

$\hat{a}^{\dagger}_{i}\rightarrow\sqrt{\eta_{j,i}}\hat{a}^{\dagger}_{i}+\sqrt{1-\eta_{j,i}}\hat{c}_{j,i}^{\dagger}\,,$
(49)

where $\hat{c}_{j,i}^{\dagger}$ creates a photon that escapes the set up. The
index $j\in\\{1,2,t\\}$ indicates where the loss happened following the
notation from Fig. 3. Applying this transformation to each segment of the
setup leads us to the loss operator $\hat{L}_{i}^{\dagger}$:

$\displaystyle\begin{split}\hat{L}_{i}^{\dagger}\equiv\sqrt{1-\eta_{1,i}}\hat{c}_{1,i}^{\dagger}+i\sqrt{\eta_{1,i}(1-\eta_{2,i})(1-T)}\hat{c}_{2,i}^{\dagger}+\sqrt{\eta_{1,i}(1-\eta_{t})T}\hat{c}_{t,i}^{\dagger}\,.\end{split}$
(50)

The operator directly provides the total probability for the photon to escape
the set-up
$\expectationvalue{\hat{L}_{i}\hat{L}_{i}^{\dagger}}=1-\eta_{1,i}\eta_{2,i}(1-T)-\eta_{1,i}\eta_{t}T$.
Note that $\expectationvalue{\hat{L}_{i}\hat{L}_{j}^{\dagger}}_{i\neq j}=0$
and
$\expectationvalue{\hat{L}_{i}\hat{L}_{j}\hat{L}_{k}^{\dagger}\hat{L}_{l}^{\dagger}}_{i\neq
k,j\neq l}=0$ expressing that lost photons do not interfere. By means of Eqs.
(49) and 50 we can obtain the global transformation in the main text (13).

Finally, we detail the action of the consecutive QWP and HWP on the creation
operators $\hat{a}_{H,{1}}^{\dagger}$, $\hat{a}_{V,{2}}^{\dagger}$,
$\hat{b}_{H,{3}}^{\dagger}$ and $\hat{b}_{V,4}^{\dagger}$, represented by the
Jones matrices. They are described by the unitary operations [34]:

$\displaystyle\begin{split}\hat{U}_{QWP}(\theta)&=\frac{1}{\sqrt{2}}\begin{pmatrix}i-\cos(2\theta)&\sin(2\theta)\\\
\sin(2\theta)&i+\cos(2\theta)\end{pmatrix},\quad\hat{U}_{HWP}(\phi)=\begin{pmatrix}\cos(2\phi)&-\sin(2\phi)\\\
-\sin(2\phi)&-\cos(2\phi)\end{pmatrix}\,.\end{split}$ (51)

The total transformation is thus
$\hat{U}_{tot}(\theta,\phi)=\hat{U}_{HWP}(\phi)\hat{U}_{QWP}(\theta)$.

##  Appendix C Conditioning on the correct detection pattern

In this appendix we explicitly show the expectation values of the CHS
operators $\hat{O}_{i}$ up to three-photon detections as a function of the
indistinguishability parameters $\alpha_{ij}$ and $\beta_{ij}$. Selecting only
the combinations of operators that lead to a correct event, that is,
$D_{1}D_{4}$ and $D_{2}D_{3}$ and applying Eqs. (4) and (14), we obtain:

$\displaystyle\begin{split}&\expectationvalue{\hat{O}_{1}\hat{O}_{3}\hat{O}_{1}^{\dagger}\hat{O}_{3}^{\dagger}}_{D1D4,D2D3}=\\\
&=\frac{1}{2}\expectationvalue{\left(\left(\hat{p}_{H,{3}}\hat{q}_{H,{1}}+\hat{p}_{V,{3}}\hat{q}_{H,{1}}\right)-\left(\hat{p}_{H,{1}}\hat{q}_{H,{3}}+\hat{p}_{V,{1}}\hat{q}_{H,{3}}\right)\right)\cdot((\hat{p}_{H,{3}}^{\dagger}\hat{q}_{H,{1}}^{\dagger}+\hat{p}_{V,{3}}^{\dagger}\hat{q}_{H,{1}}^{\dagger})-(\hat{p}_{H,{1}}^{\dagger}\hat{q}_{H,{3}}^{\dagger}+\hat{p}_{V,{1}}^{\dagger}\hat{q}_{H,{3}}^{\dagger}))}=1-|\beta_{13}|^{2}\,.\end{split}$
(52)

Similarly, for the other combinations:

$\displaystyle\begin{split}\expectationvalue{\hat{O}_{1}\hat{O}_{4}\hat{O}_{1}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}&=2t(1-t)1+|\beta_{14}|^{2},\quad\expectationvalue{\hat{O}_{2}\hat{O}_{3}\hat{O}_{2}^{\dagger}\hat{O}_{3}^{\dagger}}_{D1D4,D2D3}=1+|\beta_{23}|^{2},\\\
\expectationvalue{\hat{O}_{2}\hat{O}_{4}\hat{O}_{2}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}&=1-|\beta_{24}|^{2},\quad\expectationvalue{\hat{O}_{1}\hat{O}_{2}\hat{O}_{1}^{\dagger}\hat{O}_{2}^{\dagger}}_{D1D4,D2D3}=1-|\alpha_{12}|^{2},\\\
\expectationvalue{\hat{O}_{3}\hat{O}_{4}\hat{O}_{3}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}&=1-|\alpha_{34}|^{2},\quad\expectationvalue{\hat{O}_{1}\hat{O}_{4}\hat{O}_{2}^{\dagger}\hat{O}_{3}^{\dagger}}_{D1D4,D2D3}=-\alpha_{12}\alpha_{34}-\beta_{13}\beta_{24},\\\
\expectationvalue{\hat{O}_{1}\hat{O}_{3}\hat{O}_{2}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}&=-\alpha_{12}\alpha_{34}+\beta_{14}\beta_{23}\,.\end{split}$
(53)

It can be easily checked that any other combination of CHS operators has
vanishing expectation value. If no conditioning on the detection pattern is
done, the sum of the expectation values are independent of the
indistinguishability of the photons and equal 4, reflecting that with the
current normalization this is proportional to the product of the number of
photons at each side. Note that in the limit of completely indistinguishable
photons ($\alpha_{12}=\alpha_{34}=1$),
$\expectationvalue{\hat{O}_{1}\hat{O}_{2}\hat{O}_{1}^{\dagger}\hat{O}_{2}^{\dagger}}_{D1D4,D2D3}$
and
$\expectationvalue{\hat{O}_{3}\hat{O}_{4}\hat{O}_{3}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}$
vanish, meaning that we never have a successful detection event with two
photons from the same station. As explained in Sec. II, this occurs because of
the HOM effect, since two photons from from the same station will bunch
together after the half-wave plate
($\hat{a}_{H,{1}}^{\dagger}\hat{a}_{V,{2}}^{\dagger}\rightarrow\hat{a}_{H}^{{}^{\prime}\dagger
2}-\hat{a}_{V}^{{}^{\prime}\dagger 2}$), and therefore no $D_{1}D_{4}$ or
$D_{2}D_{3}$ clicks can ever happen. Finally, note that the expectation values
in Eq. (53) correspond to the generalised matrix elements of the density
matrix in Eq. (22).

In a similar manner, we can calculate the expectation values for the
occurrence of three photons at the CHS. We only keep those terms that
correspond to combinations of clicks perceived as correct by the CHS,
$D_{1}D_{4}$ and $D_{2}D_{3}$, due to the detectors not being photon-number
resolving. For instance, a term such
$\hat{p}_{H,{1}}^{\dagger}\hat{q}_{V,{3}}^{\dagger}\hat{q}_{V,{4}}^{\dagger}$
is taken into consideration, while
$\hat{p}_{H,{1}}^{\dagger}\hat{p}_{V,{3}}^{\dagger}\hat{q}_{V,{4}}^{\dagger}$
is not. Applying once more Eqs. (4) and (14) we obtain

$\displaystyle\begin{split}\expectationvalue{\hat{O}_{1}\hat{O}_{3}\hat{O}_{4}\hat{O}_{1}^{\dagger}\hat{O}_{3}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}&=\frac{1}{2}\left(3-\absolutevalue{\alpha_{34}}^{2}-\absolutevalue{\beta_{13}}^{2}+3\absolutevalue{\beta_{14}}^{2}-2\alpha_{34}\beta_{13}\beta_{14}\right),\\\
\expectationvalue{\hat{O}_{2}\hat{O}_{3}\hat{O}_{4}\hat{O}_{2}^{\dagger}\hat{O}_{3}^{\dagger}\hat{O}_{4}^{\dagger}}_{D1D4,D2D3}&=\frac{1}{2}\left(3-\absolutevalue{\alpha_{34}}^{2}+3\absolutevalue{\beta_{23}}^{2}-\absolutevalue{\beta_{24}}^{2}-2\alpha_{34}\beta_{23}\beta_{24}\right),\\\
\expectationvalue{\hat{O}_{3}\hat{O}_{1}\hat{O}_{2}\hat{O}_{3}^{\dagger}\hat{O}_{1}^{\dagger}\hat{O}_{2}^{\dagger}}_{D1D4,D2D3}&=\frac{1}{2}\left(3-\absolutevalue{\alpha_{12}}^{2}-\absolutevalue{\beta_{13}}^{2}+3\absolutevalue{\beta_{23}}^{2}-2\alpha_{12}\beta_{13}\beta_{23}\right),\\\
\expectationvalue{\hat{O}_{4}\hat{O}_{1}\hat{O}_{2}\hat{O}_{4}^{\dagger}\hat{O}_{1}^{\dagger}\hat{O}_{2}^{\dagger}}_{D1D4,D2D3}&=\frac{1}{2}\left(3-\absolutevalue{\alpha_{12}}^{2}+3\absolutevalue{\beta_{14}}^{2}-\absolutevalue{\beta_{24}}^{2}-2\alpha_{12}\beta_{14}\beta_{24}\right)\,,\end{split}$
(54)

where again any other combination of three CHS operators has an expectation
value equal zero.

##  Appendix D Probability events $P_{ijkl}$

In this section we include Table 2, where we detail the contribution of all 4
and 5-photon events.

Event probability | CHS | Alice | Bob | Lost photons
---|---|---|---|---
$P_{2110}$ | AB | A | B | $\emptyset$
$P_{2200}$ | BB | AA | $\emptyset$ | $\emptyset$
$P_{2020}$ | AA | $\emptyset$ | BB | $\emptyset$
$P_{2101}$ | AB | A | $\emptyset$ | B
| BB | A | $\emptyset$ | A
$P_{2011}$ | AB | $\emptyset$ | B | A
| AA | $\emptyset$ | B | B
$P_{2002}$ | AB | $\emptyset$ | $\emptyset$ | AB
| AA | $\emptyset$ | $\emptyset$ | BB
| BB | $\emptyset$ | $\emptyset$ | AA
$P_{3100}$ | ABB | A | $\emptyset$ | $\emptyset$
$P_{3010}$ | AAB | $\emptyset$ | B | $\emptyset$
$P_{3001}$ | ABB | $\emptyset$ | $\emptyset$ | A
| AAB | $\emptyset$ | $\emptyset$ | B
$P_{2111}$ | AB | A | B | A
| AA | A | B | B
$P_{2210}$ | AB | AA | B | $\emptyset$
$P_{2120}$ | AA | A | BB | $\emptyset$
$P_{2201}$ | AB | AA | $\emptyset$ | B
| BB | AA | $\emptyset$ | A
$P_{2021}$ | AA | $\emptyset$ | BB | A
$P_{2102}$ | AB | A | $\emptyset$ | AB
| AA | A | $\emptyset$ | BB
| BB | A | $\emptyset$ | AA
$P_{2012}$ | AB | $\emptyset$ | B | AA
| AA | $\emptyset$ | B | AB
$P_{2300}$ | BB | AAA | $\emptyset$ | $\emptyset$
$P_{2030}$ | AA | BBB | $\emptyset$ | $\emptyset$
$P_{2003}$ | AB | $\emptyset$ | $\emptyset$ | AAB
| AA | $\emptyset$ | $\emptyset$ | ABB
| BB | $\emptyset$ | $\emptyset$ | AAA
$P_{3110}$ | AAB | A | B | $\emptyset$
$P_{3101}$ | AAB | A | $\emptyset$ | B
| ABB | A | $\emptyset$ | A
$P_{3011}$ | AAA | $\emptyset$ | B | B
| AAB | $\emptyset$ | B | A
$P_{3200}$ | ABB | AA | $\emptyset$ | $\emptyset$
$P_{3020}$ | AAA | $\emptyset$ | BB | $\emptyset$
$P_{3002}$ | AAA | $\emptyset$ | $\emptyset$ | BB
| AAB | $\emptyset$ | $\emptyset$ | AB
| ABB | $\emptyset$ | $\emptyset$ | AA
Table 2:  4-photon and 5-photon events that lead to a correct detection
pattern at the CHS. A and B indicate where the photons were generated, while
the header tells where they were detected.

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