#  Oscillating photonic Bell state from a semiconductor quantum dot for
quantum key distribution

Matteo Pennacchietti  Institute for Quantum Computing and Department of
Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario
N2L 3G1, Canada  Brady Cunard  Institute for Quantum Computing and Department
of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada  Shlok Nahar  Institute for Quantum Computing and Department of Physics
and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Mohd
Zeeshan  National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada
Sayan Gangopadhyay  Institute for Quantum Computing and Department of Physics
and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Philip J. Poole  National Research Council of Canada, Ottawa, Ontario K1A 0R6,
Canada  Dan Dalacu  National Research Council of Canada, Ottawa, Ontario K1A
0R6, Canada  University of Ottawa, Ottawa, Ontario, Canada K1N 6N5  Andreas
Fognini  Single Quantum B.V., Delft 2628 CJ, The Netherlands  Klaus D. Jöns
Institute for Photonic Quantum Systems (PhoQS), Center for Optoelectronics and
Photonics Paderborn (CeOPP) and Department of Physics, Paderborn University,
33098 Paderborn, Germany  Val Zwiller  Department of Applied Physics, Royal
Institute of Technology, Albanova University Centre, 106 91, Stockholm, Sweden
Thomas Jennewein  Institute for Quantum Computing and Department of Physics
and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Norbert Lütkenhaus  Institute for Quantum Computing and Department of Physics
and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Michael E. Reimer  Institute for Quantum Computing and Department of
Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario
N2L 3G1, Canada  Institute for Quantum Computing and Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

###### Abstract

An on-demand source of bright entangled photon pairs is desirable for quantum
key distribution (QKD) and quantum repeaters. The leading candidate to
generate entangled photon pairs is based on spontaneous parametric down-
conversion (SPDC) in a non-linear crystal. However, there exists a fundamental
trade-off between entanglement fidelity and efficiency in SPDC sources due to
multiphoton emission at high brightness, which limits the pair extraction
efficiency to 0.1% when operating at near-unity fidelity. Quantum dots in
photonic nanostructures can in principle overcome this trade-off; however, the
quantum dots that have achieved entanglement fidelities on par with SPDC
sources (99%) have poor pair extraction efficiencies of 0.01%. Here, we
demonstrate a 65-fold increase in the pair extraction efficiency compared to
quantum dots with equivalent peak fidelity from an InAsP quantum dot in a
photonic nanowire waveguide. We measure a raw peak concurrence and fidelity of
95.3% $\pm$ 0.5% and 97.5% $\pm$ 0.8%, respectively. Finally, we show that an
oscillating two-photon Bell state generated by a semiconductor quantum dot can
be utilized to establish a secure key for QKD, alleviating the need to remove
the quantum dot energy splitting of the intermediate exciton states in the
biexciton-exciton cascade.

Developing a bright, deterministic source of entangled photon pairs for
applications in photonic quantum information processing [1], quantum
communication [2] and networks [55], and enhanced imaging techniques [4, 5, 6]
has been a long-standing scientific and technological challenge. One promising
candidate is based on III-V semiconductor quantum dots (QDs) in photonic
nanostructures [7, 8, 9, 10]. Such QDs have shown deterministic polarization-
entangled photon pair emission via the biexciton (XX)-exciton (X) cascade and
high pair-extraction efficiencies over 30% [9, 8, 10]. However, since
semiconductor QDs are hosted in a solid-state environment, dephasing of the
exciton spin was a common concern due to interactions with the surrounding
nuclear spins of the atomic lattice and spins of free/trapped charges [11,
12]. Therefore, it was presumed that because of indium’s large nuclear spin of
9/2, indium-based QDs were unlikely to match the near-unity degree of
entanglement (i.e., concurrence $\geq$ 95%) of GaAs QDs, whose nuclei both
have smaller nuclear spins of 3/2 [13, 14].

Figure 1: Resonant excitation of the nanowire quantum dot. (a) Illustration of
the InAsP QD in an InP tapered nanowire waveguide, which emits the biexciton
(XX) and exciton (X) into the same Gaussian mode. (b) Illustration of TPRE
generating a biexciton in the QD. The intermediate exciton energy level has a
finite fine structure splitting (S) between the H-polarized ($X_{H}$) and the
V-polarized ($X_{V}$) states. (c) The emission spectrum from the nanowire QD
under $\pi$-pulse TPRE. Exciton and biexciton emission lines are located at
893.25 nm and 894.67 nm, respectively. The small central peaks are from
remnants of the filtered laser pulse midway between X and XX. We observe a
small contribution from the negatively charged exciton (X-). (d) Biexciton and
exciton state population as a function of pulse area where a pulse area of
$\pi$ corresponds to the maximum population. The dots represent experimental
data and the solid curves are the fitted curves (see Supplementary Note S1.1).
(e) Hanbury-Brown-Twiss autocorrelation measurements of both the X and XX
emission, demonstrating low multi-photon emission probability.

In contrast, it was suggested by Ref. [51] that indium-based QDs could reach
near-unity degrees of entanglement by reducing the multi-photon emission
probability, the single-photon detector dark count rate, and the detection
system timing jitter. The latter metric is critical when the QD exciton fine-
structure splitting (FSS) is non-zero, which causes the two-photon entangled
state to oscillate between two Bell states. Since previous studies did not
combine an optimized detection system with two-photon resonant excitation
(TPRE) to reduce the multiphoton emission probability [51], a concurrence of
95% or greater has not yet been achieved with an indium-based QD.

In this article, we show near-unity degrees of entanglement from an InAsP QD
in a site-selected tapered InP nanowire waveguide (NW-QD), see Figure 1a. We
measure a raw peak concurrence and fidelity of 95.3 $\pm$ 0.5% and 97.5 $\pm$
0.8%, respectively. This near-unity concurrence was achieved by performing a
‘time-resolved’ quantum state tomography experiment (QST) using TPRE (Figure
1b) and superconducting nanowire single-photon detectors (SNSPDs) with ultra-
low timing jitter ($<$ 20 ps) and dark count rate ($\sim$ 1 Hz). The time-
resolved nature of the experiment enabled us to measure high values of
concurrence over the entire exciton lifetime, resulting in a lifetime-weighted
concurrence and fidelity of $90.2\pm 0.2$ % and 94.0 $\pm$ 0.1 %,
respectively. These results correspond to a 27 % increase in the peak
concurrence and a 48% increase in the lifetime-averaged concurrence from
previous measurements on the same NW-QD entangled photon source without using
TPRE [51]. The stark increase validates the prediction that optimized
experimental conditions will increase the measured concurrence on the same QD,
and substantiates the initial claim that spin dephasing was not the factor
limiting the measured entanglement fidelity in previous studies of NW-QD
entangled photon sources [51, 16, 17, 7]. Furthermore, we propose a novel
“time-resolved” quantum key distribution (QKD) protocol that is agnostic to
the oscillations of the polarization entanglement caused by the energy
splitting of the intermediate exciton states in the biexciton-exciton cascade.
We further rigorously prove the security of our proposed time-resolved QKD
protocol with more detailed optical models as compared to previous QKD
experiments with QDs [18, 19, 20].

Remarkably, our measurements show that these NW-QDs can outperform other
indium-based QDs, and even those with no indium content [21, 22, 23]. Indeed,
our raw peak measured concurrence is the highest in the literature to date.
Additionally, the pair-extraction efficiency of the source is over an order of
magnitude higher than existing III-V QD sources with comparable entanglement
fidelity [13, 24]. This is an important finding because these NW-QD sources
offer a number of attractive features, including deterministic site-selected
growth [25], intrinsically low exciton FSS [26], potential pair-extraction
efficiencies of over 90% [27, 28], near-unity fabrication yields of bright QDs
[29], Gaussian emission profile with $93\%$ coupling efficiency of single
photons to a single-mode fibre [30], and wavelength emission variance from QDs
of less than 5 nm [31]. These facts together with the high entanglement
fidelity presented in this work illuminate a clear path for the deployment of
large arrays of bright, entangled photon pair sources for quantum photonic
technologies.

## RESULTS

Two Photon Resonant Excitation. Experimental results of TPRE on the NW-QD
using laser pulses of $\sim$13 ps (see Methods) are shown in Figure 1. In
Figure 1c, we observe that the exciton (X) and biexciton (XX) dominate the
emission spectrum with the small peaks between X and XX corresponding to
remnants of the laser excitation pulse. We observe three full Rabi cycles in
the count rates vs. the pulse area in Figure 1d, confirming that we are
coherently exciting the biexciton state. The oscillations dampen out and
settle to the 0.5 population point due to exciton-phonon interactions [32].

We fit the count rate vs. pulse area data to determine an estimate for the
probability of occupying the XX state after excitation (see Methods). At the
$\pi$-pulse, we find a XX and X population of 0.82 $\pm$ 0.01 and 0.77 $\pm$
0.01, respectively, resulting in a pair generation efficiency of 0.63 $\pm$
0.01%. Using the measured count rates of 150 kHz for XX and 145 kHz for X
under pulsed excitation (76.2 MHz), and the optical setup efficiency of 2.4%,
we estimate a pair extraction efficiency at the first lens of 0.65% $\pm$
0.02% (see Methods). The pair extraction efficiency is a 65-fold increase from
a GaAs QD with comparable peak concurrence [33].

To quantify the multi-photon emission probability under TPRE we perform a
Hanbury-Brown-Twiss experiment. The resulting histogram is displayed in Figure
1e. Using only the nearest neighbour peaks we calculate
$g^{(2)}_{X}(0)=0.0055\pm 0.0003$ and $g^{(2)}_{XX}(0)=0.0028\pm 0.0003$. We
observe that $g^{(2)}_{XX}(0)$ is reduced by over an order of magnitude by
TPRE as compared to previous work on the same NW-QD using quasi-resonant
excitation ($g^{(2)}_{XX}(0)=0.10\pm 0.01$) [51].

Influence of Detection System. In addition to the excitation technique, there
are two main performance metrics of the detector that degrades the measured
degree of entanglement. The first is detector dark counts, which is well
understood and adds uncorrelated coincidences similar to multi-photon
emission. The second is the finite timing resolution of the detection system
(i.e., timing jitter), which is largely unexplored and can reduce the measured
concurrence when the exciton FSS is non-zero. The FSS is a splitting of the
two intermediate exciton states [24] that causes the exciton spin state to
oscillate between $\ket{\uparrow\Downarrow}$ (-1 angular momentum) and
$\ket{\downarrow\Uparrow}$ (+1 angular momentum) with a period of $T_{S}=h/S$,
where $S$ is the energy of the FSS. The resulting polarization state of the
photon pair emitted via the biexciton-exciton cascade will evolve temporally
as:

$\begin{split}\ket{\psi(\tau)}&=\frac{1}{\sqrt{2}}\left[\cos\big(S\tau/2\hbar\big{missing})\Big{(}\ket{RL}+\ket{LR}\Big{)}\right.\\\
&\left.+i\sin\big(S\tau/2\hbar\big{missing})\Big{(}\ket{RR}+\ket{LL}\Big{)}\right]\\\
&=\frac{1}{\sqrt{2}}\left(\ket{HH}+e^{iS\tau/\hbar}\ket{VV}\right),\end{split}$
(1)

where $\tau=t_{X}-t_{XX}$ is the time delay between the emission of the
exciton photon and the biexciton photon. The first and second letter denotes
the polarization of the biexciton and exciton photon, respectively. The effect
of the FSS is a purely local unitary transformation of the exciton spin,
implying the state in Eq. (1) is maximally entangled over the entire exciton
lifetime [51]. However, if the finite timing resolution of the detection
system is larger than the FSS oscillation period, then the measured
concurrence will be reduced since the phase relationship between $\ket{HH}$
and $\ket{VV}$ in Eq. 1 will be averaged out. This effect can be circumvented
by using single-photon detectors with a timing resolution much less than the
FSS oscillation period.

Figure 2: Time-resolved quantum state tomography with SNSPDs. (a) Coincidence
counts versus time delay ($\tau$) for the HH and HV polarization projection
measurements. (b) Coincidence counts versus time delay ($\tau$) for the RL and
RR polarization projection measurements. (c) Combinations of coincidence
measurements for the states in Eq. 1 versus time delay. The combinations of
circular basis measurements $RL+LR-(RR+LL)$ display quantum oscillations
between two Bell states. The red curve is the best fit of the oscillations in
the circular basis to $\cos{\left(S\tau/\hbar\right)}$ convolved with a
Gaussian distribution of fixed FWHM of 30 ps to account for the small amount
of blurring from the SNSPD detection system.

Quantum State Tomography. We carried out an over-complete quantum state
tomography (QST) [76] experiment using TPRE and fast SNSPDs with individual
detector timing jitters of less than 20 ps and very low dark counts rates of
$N_{d}=1$ $\pm 1$ Hz. Taking into both SNSPDs and the correlation electronics,
we estimated a total system timing jitter (FWHM) of 30 ps (see Methods), which
is much faster than the exciton FSS period of the QD under study ($T_{S}=1.26$
ns). The QST experiment consisted of a total of 36 separate projective
measurements on the joint biexciton-exciton polarization state (see
Supplementary Figure S1 and S2). A subset of the 36 projection measurements
captured using the SNSPDs is plotted in Figure 2a (HH, HV) and Figure 2b (RL,
RR). The coincidence counts from HH follow an exponential decay since the
exciton spin state $\ket{\uparrow\Downarrow}$ \+ $\ket{\downarrow\Uparrow}$ is
a stationary state of the exchange Hamiltonian when the FSS is present [35].
We see high suppression of coincidences in the HV basis as compared to the HH
basis, as expected for a highly entangled state. In contrast, the RL and RR
states are not eigenstates of the exchange Hamiltonian when the FSS is
present. Thus, the coincidences histogram for the RR and RL basis, displayed
in Figure 2b oscillates $\pi/2$ out of phase with each other.

Eq. 1 predicts the oscillation between two Bell states,
$(\ket{RL}+\ket{LR})/\sqrt{2}$ and $(\ket{RR}+\ket{LL})/\sqrt{2}$, with
frequency $S/h$. We found that plotting the individual measured coincidences
in the form of $RL\,+LR\,-\,(RR\,+\,LL)$ reveals “quantum oscillations”
between these two Bell states (Figure 2c). We extract the FSS by fitting the
quantum oscillations (red curve in Figure 2c), yielding $3.226\pm 0.004$
$\mu$eV ($780.0\pm 1.0$ MHz). In contrast, the combination of $HH+VV-(HV+VH)$
follows an exponential decay. By fitting this decay, we extract the radiative
exciton lifetime of $\tau_{X}=0.777\pm 0.003$ns (see Supplementary Figure S3).

Figure 3: Near-unity concurrence from a nanowire quantum dot. (a) Concurrence
from reconstructed density matrices as a function of time delay for
experimental data and simulated data with time windows of $50$ps. Data is
presented for NW-QDs excited under TPRE at $\pi$-pulse using both SPADs and
SNSPDs. The shaded regions represent one standard deviation. (b) Percentage of
total coincidence counts as a function of time delay.

Concurrence under Resonant Excitation. With the 36 time-resolved coincidence
measurements we reconstruct the “instantaneous” density matrix, within a time
window of 50 ps as a function $\tau$, using an algorithm based on the maximum
likelihood estimation (MLE) [76] implemented in Python [36]. From these
reconstructed density matrices we calculate the time-resolved concurrence
[37]. In Figure 3a we plot the concurrence versus time delay for measurements
taken using TPRE and SNSPDs with low dark counts and precise timing jitter
(green curve). As a comparison, we also plot the measured concurrence (blue
curve) on the same QD using TPRE and single-photon avalanche diodes (SPADs),
which have significantly higher dark count rates and timing jitter (see
Methods). The prominent effect of the detection system on the measured
concurrence on the same QD is immediately evident from the difference in the
two measured curves acquired using SNSPDs versus SPADs.

Due to the large timing jitter of the SPADs (see Supplementary Figure S4),
XX-X coincidences are recorded when $\tau<0$. As $\tau$ increases and more
XX-X coincidences are sampled, there is an initial slow rise of the SPAD
concurrence. The increase then slows and reaches a peak of $80\pm 3\%$ due to
phase averaging or “blurring” of the quantum oscillations, i.e., time-
dependent phase information between the two Bell states in Eq. 1. The
concurrence then steadily decreases as the detection system samples more of
the quantum oscillations, eventually reaching the “flat” region where the
entire detection response function has sampled the temporal evolution of the
two-photon entangled state. In stark contrast, there is no distinction between
the “top” and “flat” regions in the SNSPD concurrence curve. Instead, at
$\tau=0$ the concurrence (fidelity) peaks at $\mathcal{C}_{p}=95.3\pm 0.5\,\%$
($F_{p}=97.5\pm 0.8\,\%$), an increase of 24% in the concurrence from the SPAD
experiment. We found that taking into account our measured multi-photon
emission probability has little effect on the peak concurrence (95.6 $\pm$
0.7%) and fidelity (97.7 $\pm$ 0.4%). For the time-resolved entanglement
fidelity, see Supplementary Figure S5. The concurrence then remains relatively
flat for $\tau>0$. As a result, the lifetime weighted concurrence (fidelity)
with the SNSPDs was $\tilde{\mathcal{C}}=90.2\pm 0.2\,\%$ ($\tilde{F}=94.0\pm
0.1\,\%$), which is significantly higher than the lifetime weighted
concurrence of $50.8\pm 0.2\,\%$ acquired with the SPADs.

After $\tau\geq 2$ ns we enter the “roll-off” region where the concurrence
reduces because the signal-to-dark count ratio begins to decrease as dark
counts dominate the coincidences. Again, we notice a distinct difference
between the SPAD and SNSPD curves, where the former decreases much more than
the latter because the dark count rate of the SPADs (tens to hundreds per
second) is significantly higher than the dark count rate of the SNSPDs (one
per second).

Additionally, in Figure 3a we plot the time-resolved concurrence versus time
delay calculated from a “dephasing-free” model for the SPADs (red) and SNSPDs
(yellow). The dephasing-free model accounts for only experimentally measured
parameters: the multi-photon emission probability, the measured count rates of
X and XX, the detector dark count rate, and the detector system timing
response (see Supplementary Note S1.8). We see excellent agreement between the
model and the experimental concurrence for the SPADs, with the predicted peak
concurrence of $78\pm 3$ %. In contrast, the simulation for the SNSPDs differs
from the experimental data in three areas: (1) at $\tau=0$ ns; (2) $0<\tau<2$
ns; and (3) $\tau>2$ ns.

(1) At $\tau=0$ ns. The small discrepancy between the experimental peak
concurrence and the $99.1\pm 0.04\,\%$ concurrence predicted by the model
could be due to the combination of leakage excitation laser photons (see
Supplementary Figure S6), the interaction of the exciton spin with phonons
[38, 39] and/or the large energy splitting caused by resonant excitation via
the AC Stark Shift [40]. The first mechanism would result in unwanted
correlations from the detection of H-polarized laser photons. The latter two
mechanisms act on time scales less than the timing resolution of our detection
system, thereby resulting in a small reduction in the measured concurrence.

(2) $0<\tau<2$ ns. We note that there is a small oscillation in the measured
concurrence between $\tau=0$ and $\tau=1.3$ ns, which coincides with the FSS
period. This drop in concurrence is not from spin-dephasing since it fully
recovers around 1.3 ns. Instead, the concurrence dip might be attributed to
the QD emission being slightly polarized due to asymmetry in the nanowire
shape and/or the waveplates not being perfect quarter and half.

(3) $\tau>2$ ns. The divergence of the experimental data from the model in the
“roll-off” region can be explained by spin-scattering of the exciton spin
before recombination with a characteristic interaction time of $\sim 25$ ns
(Supplementary Figure S7). Such a dephasing time is over an order of magnitude
greater than the exciton lifetime, which is in agreement with previous
experimental evidence from the neutral exciton in InGaAs QDs [41, 11]. We note
that with the SNSPDs we were able to observe spin-dephasing from InAsP QDs in
nanowires, which was not possible with SPADs due to their higher timing
jitter. Additionally, as illustrated in Figure 3b, only 8% of the exciton
photons are emitted in this region where the concurrence drops.

Time-resolved Entanglement-based QKD. Given that we observe high entanglement
fidelity over the entire exciton lifetime, we ask: Is it possible to perform a
quantum information task using the oscillating quantum state from Eq. 1
without temporal post-selection? Of course, if the quantum information
protocol of interest requires one specific Bell state, the oscillation caused
by the FSS is an obstacle. However, for quantum key distribution (QKD) a
specific Bell state is not required.

Indeed, we can devise a “time-resolved” QKD protocol where access to a
maximally entangled state of the form
$\left(\ket{HH}+\exp(iS\tau/\hbar)\ket{VV}\right)/\sqrt{2}$ is sufficient to
generate a secret key. In this case, we would map $H$ polarized light to the
bit $0$, and $V$ to the bit $1$. This key map from quantum to classical data
is expectedly agnostic to the phase of the entangled state. As detailed in
Supplementary Note S1.9, in addition to the key map, the procedure to estimate
the state shared between Alice and Bob in the QKD protocol is similar to
quantum state tomography. Therefore, since a detection system with low-timing
jitter compared to the FSS oscillation period allows us to observe a high
concurrence through tomographic reconstruction, we prove in Supplementary Note
S1.9 the state at every $\tau$ is able to generate a “time-resolved key rate”.
We further show that our time-resolved analysis is valid for any time-
dependent states.

As an example, we consider a time-resolved optical implementation of the
6-state protocol [69]. We compute the time-resolved key rate for this
protocol, i.e., the number of secret bits produced per two-fold coincidence as
a function of time delay $\tau$. This is visualized in Figure 4, where we
compare the key rates for the raw data from the NW-QD (blue curve) with the
ideal state given in Eq. 1 (black curve) and the SNSPD simulation from Figure
3a (red curve). From Figure 4, we see that the time-resolved key rate is
positive up to time delays between the biexciton and exciton emission of
$5\tau_{X}$ ns, implying that all photon pairs from the NW-QD are useful for
key generation.

Figure 4:  Time-resolved key rate. Theoretical key rate for: experimental data
(blue) with optimal choice of measurement bases states, dephasing-free model
(red) and “perfectly oscillating” state given by Eq. 1 (black).

The deviation of the simulated and experimental time-resolved key rate from
the perfect state ($r(\tau)=0.99$) shows that the time-resolved key rate is a
more sensitive metric than concurrence since $r(\tau)$ is dependent on the
closeness to maximally entangled states in a subspace spanned by Alice and
Bob’s key map choices. For example, the black curve in Figure 4 for the
perfect state gives the maximum possible key rate as the state oscillates
perfectly in the plane spanned by $HH$ and $VV$. However, the oscillations in
the blue curve imply there is no such choice of key map (see Supplementary
Notes S1.5 and S1.9) for the experimental data. Similarly, the non-zero multi-
photon emission probability and finite dark count rate in the simulation,
causes the red curve to sit below the perfect state.

We can calculate the number of bits of the secret key produced per excitation
pulse $R$ by calculating a lifetime-weighted time-resolved key rate. When
averaging from $\tau=0$ ns to $\tau=5\tau_{X}$ ns we find $R=0.88$ and
$R=0.64$ for the dephasing-free model and the experimental data, respectively.
Note that we have assumed a perfect channel for our proof-of-principle
analysis, so the key rate exactly reflects the utility of the source for QKD.
In practice, the channel would add some errors and loss which would further
degrade this key rate.

## DISCUSSION

It is well understood that the exciton FSS will reduce the measured
entanglement fidelity defined with respect to one specific Bell state unless
temporal post-selection is used [12, 9, 14, 43]. Thus, if the FSS is not
directly removed, then one must settle with either a less efficient entangled
photon source or a lower measured entanglement fidelity. In both cases, the
overall performance of the entangled photon source is reduced.

Our QKD analysis shows that this trade-off between efficiency and fidelity can
be overcome by removing the assumed constraint of requiring only one specific
Bell state. Indeed, we constructed a time-resolved QKD protocol where all
photon pairs emitted by a QD with non-zero FSS can be used in secret key
generation. This protocol works only when the detection system’s temporal
resolution is much smaller than the FSS period. By implementing our protocol,
the key rates in previous QKD experiments with QD entangled photon pair
sources [18, 19, 20] could be improved since these experiments did not account
for the FSS. Additionally, unlike previous security analyses that assume
perfect qubit states [18, 19, 20], we rigorously bound the effect of any
multi-photon components of the optical state on the key rate, which is more
applicable to practical implementations.

The core experimental challenge with this time-resolved QKD scheme is ensuring
accurate timing synchronization on the order of 10-20 ps between the two
parties. Such accurate timing synchronization is within reach of current
technology since it was recently shown that two distant quantum-networked
nodes were synchronized to $<$ 10 ps over a few kilometres [44]. Thus, it may
be less challenging to satisfy the synchronization requirements instead of
directly removing the FSS which requires complex fabrication methods [43, 45]
or external optics [46]. Therefore, our time-resolved protocol could
significantly relax the technical requirements for using bright QD entangled
photon sources for QKD.

We found that our QD entangled photon source suffers from significant
luminescence blinking (see Supplementary Figure S9) due to background n-doping
of the nanowire (see Supplementary Information of Ref. [47]). It has been
shown in previous work that blinking can be completely removed via the
application of an electric field to sweep away excess carriers from the
vicinity of the QD [73]. If blinking is successfully removed, our pair
extraction efficiency would increase almost six-fold (see Supplementary Note
S1.11). Additionally, techniques to improve the pair generation rate from 80 %
to unity can be achieved by optimizing the TPRE pulse [49]. Finally, the
addition of a gold mirror at the nanowire base should lead to a further two-
fold boost of the pair extraction efficiency [28].

In summary, we have shown that NW-QD emitters are a bright source of entangled
photon pairs with concurrence values that are on par with other leading
semiconductor QD devices [22, 23, 21, 24, 8, 10]. Our results demonstrated
that the low concurrence values found in earlier studies of NW-QD emitters
were not due to spin dephasing from the large 9/2 nuclear spin of indium.
Instead, imperfect experimental conditions, namely multi-photon emission,
detector dark counts, and timing jitter, were the limiting factors.
Additionally, we found that any residual spin dephasing occurred over time
scales that were much longer than the exciton’s radiative lifetime. Our
findings lay the foundation for creating large arrays of deterministically-
positioned NW-QDs for QKD networks.

## METHODS

#### Quantum Dot Source

Please refer to Ref. [16] for a description of the nanowire quantum dot growth
procedure.

#### Tomography Apparatus

A standard cryogenic micro-photoluminescence (PL) apparatus was used to
maintain the nanowire sample at 4.5K and enable the optical interfacing via a
top window and mirror. Excitation pulses were directed to a 70:30
beamsplitter, with the reflection passing to the micro-PL cryostat. The
excitation pulse power was monitored with a photodiode on the transmission
port of the 70:30 beamsplitter. After excitation, the photons from exciton and
biexciton transition, which are separated by 1 nm in wavelength, coupled to
the same waveguide mode of the nanowire and then propagate out into free
space. The emission from the nanowire is collimated with the objective lens
and directed up out of the cryostat. The emission travels through the 70:30
beamsplitter and then three notch filters which filter out the light from the
resonant pulse. Then, a transmission grating is used to separate the exciton
and biexciton photons into different spatial modes. With the photons
separated, we implemented a quantum state tomography setup similar to [17].
Independent polarization transformations were performed on each photon using a
$\lambda$/4 followed by a $\lambda$/2 waveplate. Projective polarization
measurements were then performed using a polarizer. All four waveplates were
mounted inside high-precision, stepper-motorized rotating stages which allowed
for automated data-taking and the precise setting of the waveplate principle
axes to implement each measurement. The beam was then coupled to a single
model fibre and passed through a fibre-based fibre-based bandpass filter (FWHM
of 0.07 nm) to the single-photon detectors. The electrical signals produced by
the detection events are then sent to the 2-channel time-tagging electronics.
The histogram bin width was set to 16 ps and the integration time for
capturing each histogram was 300 seconds. Please see Supplementary Figure
S1.12 for an illustration of the experimental apparatus. Please see
Supplementary Note S1.13 for information on how we calibrated the waveplates
and determined the measurement settings.

#### TPRE Pulse

For TPRE, a 3 ps pulse from a Mira 900P Ti:Sapphire laser was extended to an
approximately 13 ps pulse using a standard 4-f pulse shaper with two 1800
grooves/mm gratings. We note that the temporal duration was not measured
directly, instead, we are assuming that the pulse is Fourier-limited, a
reasonable assumption to make in the absence of any mechanism introducing
chirp into the setup. We have measured a spectral linewidth (FWHM) of 0.08 nm
at 894 nm, thus by assuming a Gaussian pulse envelope, the time-bandwidth
product gives $\Delta t\approx 0.441/\Delta\nu$.

#### Detection System

For the detection system, the two SNSPDs (Single Quantum Eos) had a timing
resolution of 19 ps and 18 ps, respectively, with the correlation electronics
(PicoQuant PicoHarp 300) having a timing resolution of 10 ps. Each exhibits a
Gaussian distribution resulting in an overall timing resolution of
$\sqrt{19^{2}+18^{2}+2*10^{2}}\approx 30$ ps. The overall detection system
timing resolution when using the APDs was 488 $\pm$ 1 ps, see the
Supplementary Note S1.4 for the calculation of this value. The two APDs had
dark count rates of 34 $\pm$ 18 and 306 $\pm$ 51, respectively.

#### Source Efficiency

To calculate the efficiency under TPRE, we sent the emission of the nanowire
sample directly to a high-resolution diffraction spectrometer. A single SPAD
detector was placed at the output of a controllable output split inside the
spectrometer to control the selected wavelength of the detector. The
efficiency of the collection objective and top mirror of the cryostat is 80%
and 95%, respectively. The transmission of the beamsplitter used for mixing in
the excitation pulse is 65%. The total optical efficiency after the 70:30
beamsplitter (see Supplementary Figure S10) to the SPAD detector after the
spectrometer was estimated to be 2.41 $\pm$ 0.07%. The unpolarized count rates
for both the X and XX emission line were sent to the spectrometer, and the
slit was used to select either to record the total count rates in each line.
We found 150k counts/sec for the biexciton and 145k counts/sec with the
exciton. This corresponds to a pair extraction efficiency at the first lens of
0.65 $\pm$ 0.02%.

#### Lifetime-Averaged Metrics

To calculate the lifetime average we use the equation,

$\tilde{\mathcal{C}}=\frac{\sum_{\tau_{n}}N_{\tau_{n}}\mathcal{C}(\rho(\tau_{n}))}{\sum_{\tau_{n}}N_{\tau_{n}}}$
(2)

where $\tau_{n}$ is the time bin of the coincidence histogram,
$\mathcal{C}(\rho(\tau_{n}))$ is the concurrence of the density matrix $\rho$
reconstructed at time bin $\tau_{n}$ and
$N_{\tau_{n}}=\sum_{ij}N_{\tau_{n}}^{ij}$ is the sum of all coincidences in
the 36 polarization measurements $i,j\in\\{H,V,D,A,R,L\\}$. Equivalently, we
use a similar formula to calculate the key rate

$R=\frac{\sum_{\tau_{n}}N_{\tau_{n}}r(\tau_{n})}{\sum_{\tau_{n}}N_{\tau_{n}}}$
(3)

where $r(\tau_{n})$ is the time-resolved key rate calculated for time delay
$\tau_{n}$. The theoretical basis for this is expanded on in Supplementary
S1.9.

### Acknowledgements

This research was undertaken thanks in part to funding from the Canada First
Research Excellence Fund, NSERC (programs Canadian Graduate Scholarship -
Masters and Discovery), CMC Microsystems and the Mitacs Accelerate program. We
acknowledge the generosity of Single Quantum B.V. for providing us with the
superconducting nanowire single-photon detection system for this work.

### Author contributions

M.P. performed the experiments with significant contributions from B.C. and
M.Z. Data analysis and modelling were performed by M.P. with code developed by
A.F. The development and fabrication the quantum dot emitter was done by D.D.
and P.J.P. The development of the QKD protocol was led by S.N. under the
supervision of N.L. with assistance from S.G. Two-photon resonant excitation
of the quantum dot was implemented by K.D.J. M.E.R. supervised the project and
wrote the paper with M.P. and S.N. with input from all authors.

### Competing Interests

A.F. is employed by Single Quantum and may profit financially. V.Z. is the co-
founder and chief scientific officer of Single Quantum and may profit
financially. The other authors declare no competing interests.

## References

  * [1] P Walther et al.  “Experimental one-way quantum computing”  In _Nature_ 434.7030, 2005, pp. 169–176  DOI: 10.1038/nature03347
  * [2] A K Ekert  “Quantum cryptography based on Bell’s theorem”  In _Phys. Rev. Lett._ 67.6, 1991, pp. 661–663  DOI: 10.1103/PhysRevLett.67.661
  * [3] Michele B Rota, Francesco Basso Basset, Davide Tedeschi and Rinaldo Trotta  “Entanglement Teleportation With Photons From Quantum Dots: Toward a Solid-State Based Quantum Network”  In _IEEE J. Sel. Top. Quantum Electron._ 26.3 Institute of ElectricalElectronics Engineers Inc., 2020, pp. 1–16  DOI: 10.1109/JSTQE.2020.2985285
  * [4] Tomohisa Nagata et al.  “Beating the Standard Quantum Limit with Four-Entangled Photons”  In _Science_ 316.5825, 2007, pp. 726–729  DOI: 10.1126/science.1138007
  * [5] M. Müller et al.  “Quantum-Dot Single-Photon Sources for Entanglement Enhanced Interferometry”  In _Phys. Rev. Lett._ 118 American Physical Society, 2017, pp. 257402  DOI: 10.1103/PhysRevLett.118.257402
  * [6] Gabriela Barreto Lemos et al.  “Quantum imaging with undetected photons”  In _Nature_ 512.7515, 2014, pp. 409–412  DOI: 10.1038/nature13586
  * [7] Klaus D Jöns et al.  “Bright nanoscale source of deterministic entangled photon pairs violating Bell’s inequality”  In _Sci. Rep._ 7.1 Nature Publishing Group, 2017, pp. 1700  DOI: 10.1038/s41598-017-01509-6
  * [8] Jin Liu et al.  “A solid-state source of strongly entangled photon pairs with high brightness and indistinguishability”  In _Nat. Nanotechnol._ 14.6, 2019, pp. 586–593  DOI: 10.1038/s41565-019-0435-9
  * [9] Yan Chen et al.  “Highly-efficient extraction of entangled photons from quantum dots using a broadband optical antenna”  In _Nat. Commun._ 9.1, 2018, pp. 2994  DOI: 10.1038/s41467-018-05456-2
  * [10] Hui Wang et al.  “On-Demand Semiconductor Source of Entangled Photons Which Simultaneously Has High Fidelity, Efficiency, and Indistinguishability”  In _Phys. Rev. Lett._ 122.11, 2019, pp. 113602  DOI: 10.1103/PhysRevLett.122.113602
  * [11] Andreas V Kuhlmann et al.  “Charge noise and spin noise in a semiconductor quantum device”  In _Nat. Phys._ 9.9 Nature Publishing Group, 2013, pp. 570–575  DOI: 10.1038/nphys2688
  * [12] Takashi Kuroda et al.  “Symmetric quantum dots as efficient sources of highly entangled photons: Violation of Bell’s inequality without spectral and temporal filtering”  In _Phys. Rev. B Condens. Matter_ 88.4 American Physical Society, 2013, pp. 041306  DOI: 10.1103/PhysRevB.88.041306
  * [13] Daniel Huber et al.  “Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots”  In _Nat. Commun._ 8.1 Nature Publishing Group, 2017, pp. 15506  DOI: 10.1038/ncomms15506
  * [14] Robert Keil et al.  “Solid-state ensemble of highly entangled photon sources at rubidium atomic transitions”  In _Nat. Commun._ 8, 2017, pp. 15501  DOI: 10.1038/ncomms15501
  * [15] A Fognini et al.  “Dephasing Free Photon Entanglement with a Quantum Dot”  In _ACS Photonics_ 6.7 American Chemical Society, 2019, pp. 1656–1663  DOI: 10.1021/acsphotonics.8b01496
  * [16] Marijn A M Versteegh et al.  “Observation of strongly entangled photon pairs from a nanowire quantum dot”  In _Nat. Commun._ 5.1 Springer ScienceBusiness Media LLC, 2014, pp. 5298  DOI: 10.1038/ncomms6298
  * [17] Tobias Huber et al.  “Polarization entangled photons from quantum dots embedded in nanowires”  In _Nano Lett._ 14.12 American Chemical Society, 2014, pp. 7107–7114  DOI: 10.1021/nl503581d
  * [18] Christian Schimpf et al.  “Quantum cryptography with highly entangled photons from semiconductor quantum dots”  In _Sci Adv_ 7.16, 2021  DOI: 10.1126/sciadv.abe8905
  * [19] Francesco Basso Basset et al.  “Quantum key distribution with entangled photons generated on demand by a quantum dot”  In _Sci Adv_ 7.12, 2021  DOI: 10.1126/sciadv.abe6379
  * [20] B Dzurnak et al.  “Quantum key distribution with an entangled light emitting diode”  In _Applied Physics Letters_ 107.26 AIP Publishing LLC, 2015, pp. 261101
  * [21] J Huwer et al.  “Quantum-Dot-Based Telecommunication-Wavelength Quantum Relay”  In _Phys. Rev. Applied_ 8.2 American Physical Society, 2017, pp. 024007  DOI: 10.1103/PhysRevApplied.8.024007
  * [22] Katharina D Zeuner et al.  “On-Demand Generation of Entangled Photon Pairs in the Telecom C-Band with InAs Quantum Dots”  In _ACS Photonics_ 8.8, 2021, pp. 2337–2344  DOI: 10.1021/acsphotonics.1c00504
  * [23] Caspar Hopfmann et al.  “Maximally entangled and gigahertz-clocked on-demand photon pair source”  In _Phys. Rev. B Condens. Matter_ 103.7 American Physical Society, 2021, pp. 075413  DOI: 10.1103/PhysRevB.103.075413
  * [24] Daniel Huber et al.  “Strain-Tunable GaAs Quantum Dot: A Nearly Dephasing-Free Source of Entangled Photon Pairs on Demand”  In _Phys. Rev. Lett._ 121.3, 2018, pp. 033902  DOI: 10.1103/PhysRevLett.121.033902
  * [25] Dan Dalacu et al.  “Ultraclean emission from InAsP quantum dots in defect-free wurtzite InP nanowires”  In _Nano Lett._ 12.11, 2012, pp. 5919–5923  DOI: 10.1021/nl303327h
  * [26] Ranber Singh and Gabriel Bester  “Nanowire quantum dots as an ideal source of entangled photon pairs”  In _Phys. Rev. Lett._ 103.6, 2009, pp. 063601  DOI: 10.1103/PhysRevLett.103.063601
  * [27] Julien Claudon, Niels Gregersen, Philippe Lalanne and Jean-Michel Gérard  “Harnessing light with photonic nanowires: fundamentals and applications to quantum optics”  In _Chemphyschem_ 14.11 John Wiley & Sons, Ltd, 2013, pp. 2393–2402  DOI: 10.1002/cphc.201300033
  * [28] Michael E Reimer et al.  “Bright single-photon sources in bottom-up tailored nanowires”  In _Nat. Commun._ 3.1 Nature Publishing Group, 2012, pp. 737  DOI: 10.1038/ncomms1746
  * [29] Patrick Laferrière et al.  “Unity yield of deterministically positioned quantum dot single photon sources”  In _Sci. Rep._ 12.1, 2022, pp. 6376  DOI: 10.1038/s41598-022-10451-1
  * [30] Gabriele Bulgarini et al.  “Nanowire waveguides launching single photons in a Gaussian mode for ideal fiber coupling”  In _Nano Lett._ 14.7 American Chemical Society, 2014, pp. 4102–4106  DOI: 10.1021/nl501648f
  * [31] Yan Chen et al.  “Controlling the exciton energy of a nanowire quantum dot by strain fields”  In _Appl. Phys. Lett._ 108.18 American Institute of Physics, 2016, pp. 182103  DOI: 10.1063/1.4948762
  * [32] Sebastian Lüker and Doris E Reiter  “A review on optical excitation of semiconductor quantum dots under the influence of phonons”  In _Semicond. Sci. Technol._ 34.6 IOP Publishing, 2019, pp. 063002  DOI: 10.1088/1361-6641/ab1c14
  * [33] Note that the QD in Ref. [24] had zero FSS and they did not use a time-resolved measurement technique. Instead, they calculated the concurrence using a single wide temporal window that included all emitted photons. As a result, we are comparing their overall, single concurrence value to our peak concurrence from a time-resolved measurement.
  * [34] Daniel F V James, Paul G Kwiat, William J Munro and Andrew G White  “Measurement of qubits”  In _Phys. Rev. A_ 64.5 American Physical Society, 2001, pp. 052312  DOI: 10.1103/PhysRevA.64.052312
  * [35] M Bayer et al.  “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots”  In _Phys. Rev. B Condens. Matter_ 65.19 American Physical Society, 2002, pp. 195315  DOI: 10.1103/PhysRevB.65.195315
  * [36] Tjeerd Fokkens, Andreas Fognini and Val Zwiller  “Optical Quantum Tomography Code. Available at https://github.com/afognini/Tomography/”  URL: https://github.com/afognini/Tomography/
  * [37] William K Wootters  “Entanglement of formation and concurrence”  In _Quantum Inf. Comput._ 1.1 Rinton Press, 2001, pp. 27–44  DOI: 10.26421/qic1.1-3
  * [38] Ulrich Hohenester, Gernot Pfanner and Marek Seliger  “Phonon-assisted decoherence in the production of polarization-entangled photons in a single semiconductor quantum dot”  In _Phys. Rev. Lett._ 99.4, 2007, pp. 047402  DOI: 10.1103/PhysRevLett.99.047402
  * [39] M Cygorek et al.  “Comparison of different concurrences characterizing photon pairs generated in the biexciton cascade in quantum dots coupled to microcavities”  In _Phys. Rev. B Condens. Matter_ 98.4 American Physical Society, 2018, pp. 045303  DOI: 10.1103/PhysRevB.98.045303
  * [40] T Seidelmann et al.  “Two-Photon Excitation Sets Limit to Entangled Photon Pair Generation from Quantum Emitters”  In _Phys. Rev. Lett._ 129.19 American Physical Society, 2022, pp. 193604  DOI: 10.1103/PhysRevLett.129.193604
  * [41] M Sénés et al.  “Spin Dynamics of Neutral and Charged Excitons in InAs/GaAs Quantum Dots”  In _Optical Properties of 2D Systems with Interacting Electrons_ Springer Netherlands, 2003, pp. 79–88  DOI: 10.1007/978-94-010-0078-9“˙6
  * [42] Dagmar Bruß  “Optimal eavesdropping in quantum cryptography with six states”  In _Physical Review Letters_ 81.14 APS, 1998, pp. 3018
  * [43] Daniel Huber et al.  “Semiconductor quantum dots as an ideal source of polarization-entangled photon pairs on-demand: a review”  In _J. Opt._ 20.7 IOP Publishing, 2018, pp. 073002  DOI: 10.1088/2040-8986/aac4c4
  * [44] T. Gerrits et al.  “White Rabbit-assisted Quantum Network Node Synchronization with Quantum Channel Coexistence”  In _Conference on Lasers and Electro-Optics_ San Jose, California: Optica Publishing Group, 2022, pp. FM1C.2  DOI: 10.1364/CLEO˙QELS.2022.FM1C.2
  * [45] M Zeeshan et al.  “Proposed Scheme to Generate Bright Entangled Photon Pairs by Application of a Quadrupole Field to a Single Quantum Dot”  In _Phys. Rev. Lett._ 122.22 American Physical Society, 2019, pp. 227401  DOI: 10.1103/PhysRevLett.122.227401
  * [46] A Fognini et al.  “Universal fine-structure eraser for quantum dots”  In _Opt. Express_ 26.19, 2018, pp. 24487–24496  DOI: 10.1364/OE.26.024487
  * [47] M E Reimer et al.  “Overcoming power broadening of the quantum dot emission in a pure wurtzite nanowire”  In _Phys. Rev. B Condens. Matter_ 93.19 American Physical Society, 2016, pp. 195316  DOI: 10.1103/PhysRevB.93.195316
  * [48] Liang Zhai et al.  “Low-noise GaAs quantum dots for quantum photonics”  In _Nat. Commun._ 11.1, 2020, pp. 4745  DOI: 10.1038/s41467-020-18625-z
  * [49] Timo Kaldewey et al.  “Coherent and robust high-fidelity generation of a biexciton in a quantum dot by rapid adiabatic passage”  In _Phys. Rev. B Condens. Matter_ 95.16 American Physical Society, 2017, pp. 161302  DOI: 10.1103/PhysRevB.95.161302

## References

  * [50] Mark Fox  “Quantum optics: an introduction”  Oxford: Oxford Univ. Press, 2006
  * [51] A Fognini et al.  “Dephasing Free Photon Entanglement with a Quantum Dot”  In _ACS Photonics_ 6.7 American Chemical Society, 2019, pp. 1656–1663  DOI: 10.1021/acsphotonics.8b01496
  * [52] Frank Verstraete and Henri Verschelde  “Fidelity of mixed states of two qubits”  In _Phys. Rev. A_ 66 American Physical Society, 2002, pp. 022307  DOI: 10.1103/PhysRevA.66.022307
  * [53] Pericles S. Theocaris and Emmanuel E. Gdoutos  “Matrix Theory of Photoelasticity”  Springer Berlin Heidelberg, 1979  DOI: 10.1007/978-3-540-35789-6
  * [54] Julia Neuwirth et al.  “Multipair-free source of entangled photons in the solid state”  In _Phys. Rev. B Condens. Matter_ 106.24 American Physical Society, 2022, pp. L241402  DOI: 10.1103/PhysRevB.106.L241402
  * [55] Michele B Rota, Francesco Basso Basset, Davide Tedeschi and Rinaldo Trotta  “Entanglement Teleportation With Photons From Quantum Dots: Toward a Solid-State Based Quantum Network”  In _IEEE J. Sel. Top. Quantum Electron._ 26.3 Institute of ElectricalElectronics Engineers Inc., 2020, pp. 1–16  DOI: 10.1109/JSTQE.2020.2985285
  * [56] A J Hudson et al.  “Coherence of an entangled exciton-photon state”  In _Phys. Rev. Lett._ 99.26 American Physical Society, 2007, pp. 266802  DOI: 10.1103/PhysRevLett.99.266802
  * [57] Renato Renner  “Symmetry of large physical systems implies independence of subsystems”  In _Nature Physics_ 3.9 Nature Publishing Group, 2007, pp. 645–649
  * [58] Matthias Christandl, Robert König and Renato Renner  “Postselection technique for quantum channels with applications to quantum cryptography”  In _Physical review letters_ 102.2 APS, 2009, pp. 020504
  * [59] Renato Renner  “Security of quantum key distribution”  In _International Journal of Quantum Information_ 6.01 World Scientific, 2008, pp. 1–127
  * [60] Igor Devetak and Andreas Winter  “Distillation of secret key and entanglement from quantum states”  In _Proceedings of the Royal Society A: Mathematical, Physical and engineering sciences_ 461.2053 The Royal Society, 2005, pp. 207–235
  * [61] Patrick J Coles, Eric M Metodiev and Norbert Lütkenhaus  “Numerical approach for unstructured quantum key distribution”  In _Nature communications_ 7.1 Nature Publishing Group, 2016, pp. 1–9
  * [62] Adam Winick, Norbert Lütkenhaus and Patrick J Coles  “Reliable numerical key rates for quantum key distribution”  In _Quantum_ 2 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2018, pp. 77
  * [63] Chi-Hang Fred Fung, HF Chau and Hoi-Kwong Lo  “Universal squash model for optical communications using linear optics and threshold detectors”  In _Physical Review A_ 84.2 APS, 2011, pp. 020303
  * [64] Oleg Gittsovich et al.  “Squashing model for detectors and applications to quantum-key-distribution protocols”  In _Physical Review A_ 89.1 APS, 2014, pp. 012325
  * [65] Nicky Kai Hong Li and Norbert Lütkenhaus  “Improving key rates of the unbalanced phase-encoded BB84 protocol using the flag-state squashing model”  In _Physical Review Research_ 2.4 APS, 2020, pp. 043172
  * [66] Twesh Upadhyaya, Thomas Himbeeck, Jie Lin and Norbert Lütkenhaus  “Dimension reduction in quantum key distribution for continuous-and discrete-variable protocols”  In _PRX Quantum_ 2.2 APS, 2021, pp. 020325
  * [67] Nicky Kai Hong Li  “Application of the Flag-State Squashing Model to Numerical Quantum Key Distribution Security Analysis”, 2020
  * [68] Norbert Lütkenhaus  “Estimates for practical quantum cryptography”  In _Physical Review A_ 59.5 APS, 1999, pp. 3301
  * [69] Dagmar Bruß  “Optimal eavesdropping in quantum cryptography with six states”  In _Physical Review Letters_ 81.14 APS, 1998, pp. 3018
  * [70] Varun Narasimhachar  “Study of realistic devices for quantum key-distribution”, 2011
  * [71] Shlok Nahar, Twesh Upadhyaya and Norbert Lütkenhaus  “Imperfect Phase-Randomisation and Generalised Decoy-State Quantum Key Distribution”  In _arXiv preprint arXiv:2304.09401_ , 2023
  * [72] Charles Santori et al.  “Submicrosecond correlations in photoluminescence from InAs quantum dots”  In _Phys. Rev. B Condens. Matter_ 69.20 American Physical Society, 2004, pp. 205324  DOI: 10.1103/PhysRevB.69.205324
  * [73] Liang Zhai et al.  “Low-noise GaAs quantum dots for quantum photonics”  In _Nat. Commun._ 11.1, 2020, pp. 4745  DOI: 10.1038/s41467-020-18625-z
  * [74] Jingzhong Yang et al.  “Photoneutralization of charges in GaAs quantum dot based entangled photon emitters”  In _Phys. Rev. B Condens. Matter_ 105.11 American Physical Society, 2022, pp. 115301  DOI: 10.1103/PhysRevB.105.115301
  * [75] Jan-Philipp Jahn et al.  “An artificial Rb atom in a semiconductor with lifetime-limited linewidth”  In _Phys. Rev. B Condens. Matter_ 92.24 American Physical Society, 2015, pp. 245439  DOI: 10.1103/PhysRevB.92.245439
  * [76] Daniel F V James, Paul G Kwiat, William J Munro and Andrew G White  “Measurement of qubits”  In _Phys. Rev. A_ 64.5 American Physical Society, 2001, pp. 052312  DOI: 10.1103/PhysRevA.64.052312

## SUPPLEMENTARY INFORMATION

###  S1.1 Fitting Rabi Oscillations

To fit the Rabi oscillations to a population of the biexciton state we used
the formula

$|c_{e}|^{2}=\frac{1}{2(1+2\xi^{2})}\left(1-\left(\cos(\Omega^{\prime}t)+\frac{3\xi}{\sqrt{4-\xi^{2}}}\sin(\Omega^{\prime}t)\right)e^{-3\gamma
t/2}\right)$ (S4)

from Ref. [50], where $\Omega^{\prime}=\Omega_{R}\sqrt{1-\xi^{2}/4}$ and
$\xi=\gamma/\Omega_{R}$, with $\gamma$ representing a general damping rate
parameter and $\Omega_{R}$ being the standard Rabi frequency. The Rabi
oscillation data in Figure 1a) of the main text is recorded as a function of
the average excitation pulse power at a fixed temporal pulse width. Thus, the
Eq. S4 above is not useful as it depends on time, a parameter we do not have
direct access to. Recall, that in the regime of pulsed resonant excitation, we
define the pulse area

$\Theta=\absolutevalue{\frac{\mu_{21}}{\hbar}\int\mathcal{E}_{0}(t)dt}$ (S5)

where $\mu_{21}$ representing the dipole moment of a given resonant transition
and $\mathcal{E}_{0}(t)$ is the temporal envelope of the excitation pulse. For
a constant field $\mathcal{E}_{0}$, the pulse area is therefore exactly
$\Omega_{R}t$ and thus, in Eq. S4 we can take $\Omega_{R}t\rightarrow\Theta$.
Therefore we can substitute
$\Omega^{\prime}t\rightarrow\Theta\sqrt{1-\xi^{2}/4}$ and $\gamma
t=\Omega_{R}\xi t\rightarrow\Theta\xi$ into Eq. S4 and find

$|c_{e}|^{2}=\frac{1}{2(1+2\xi^{2})}\left(1-\left(\cos(\Theta)+\frac{3\xi}{\sqrt{4-\xi^{2}}}\sin(\Theta)\right)e^{-3\Theta\xi/2}\right).$
(S6)

This equation depends only on the parameters $\Theta$ and $\xi$, and can
therefore be used to fit our data directly.

###  S1.2 Raw Projections

Figure S1: Raw coincidence histograms of each of the 36 projective
measurements using the SNSPDs. Figure S2: Raw coincidence histograms of each
of the 36 projective measurements using the SPADs.

###  S1.3 Exciton Lifetime

To determine the lifetime of the exciton state, we must look at biexciton-
exciton cross-correlations vs. the detection time delay, i.e., the time
between the detection of a biexciton and the subsequent detection of an
exciton: $\tau=t_{X}-t_{XX}$. The causal nature of the cascade, i.e., in the
case of resonant an exciton can only form after the biexciton is emitted,
means that the decay of coincidences over this time difference will correspond
directly to the true radiative lifetime of the exciton state. In Figure S3 we
plot the sum of the coincidences corresponding to all four possible
configurations that the cascade can emit into, i.e., $RR+LL+RL+LR$. We fit
these correlations to a simple exponential decay function
$A\exp(-\tau/\tau_{X})$ where $t$ is the time delay discussed and $\tau_{X}$
is the lifetime of the exciton. As stated in the main text we find a value of
$0.777\pm 0.003$ ns. The exponential decay model shows very close agreement
with the data.

Figure S3: A plot of the raw biexciton-exciton cross-correlation coincides vs.
time delay $t_{X}-t_{XX}$ with the coincidences on a log scale. Raw data is
indicated by the blue dots and the exponential fit by the orange curve.

###  S1.4 SPAD Timing Response

Our avalanche photodiodes (SPADs) are not the same as those employed in Ref.
[51], but the procedure for extracting the timing resolution of the detection
system is the same. The results of the analysis done on our SPAD detection
system are displayed in Figure S4. The $HH$ and $VV$ biexciton-exciton cross-
correlation coincidence histograms should follow a simple exponential decay
function for $t>0$ with a Heaviside step function $\Theta(t)$ to represent the
causal nature of the cascade, i.e., the exciton is always created after the
recombination of the biexciton. The histogram for the sum $HH$ \+ $VV$
histogram is then the convolution of the above function with the detection
timing response $g(t)$

$f(t)=\Theta(t)e^{-t/\tau_{X}}*g(t).$ (S7)

The detection response function will thus be imprinted onto the coincidence
histogram data at time delays $t\leq 0$, i.e., the rising edge of the curve,
where $\Theta(t)$ dictates the shape. We can therefore limit ourselves to
histogram data points where $t\leq 0$,

$f(t\leq
0)\approx\Theta(t)*g(t)=f(t_{-})\implies\frac{df(t_{-})}{dt}=d\left[\Theta(t)*g(t)\right]=d\left[\Theta(t)\right]*g(t)=\delta(t)*g(t)=g(t).\,^{,}$
(S8)

Thus, as described above mathematically, the derivative of the histogram at
time $t<0$ should directly give the detection system timing response function
$g(t)$. In Figure S4a) the raw $HH+VV$ coincidence histogram is plotted
against the time delay $t$ and the red curve is the raw data filtered by a
Savitzki-Golay filter to reduce the impact noise. This filtered coincidence
curve of the raw curve is represented by $f(t)$ in the above model. The
derivative of $f(t)$ is displayed in Figure S4b). We call the points where
$t\leq 0$ is displayed in red, $\tilde{g}(t)$, which fully describes the
detection system timing response. Given that the impulse response of each SPAD
follows the same function (see Ref. [51]), the timing response for the cross-
correlation histogram should be a symmetric function. Thus, the raw data
$\tilde{g}(t)$ is mirrored across the $t=0$ axis to give $g(t)$ in terms of
raw data points. Then, these points are fitted with the function

$g_{f}(t,t_{0},A,\sigma)=\frac{A}{\left(\cos{\left(\frac{t-t_{0}}{\sigma}\right)}\right)^{2}}$
(S9)

in order to extract the parameters for $g(t)$ to be used in the simulations.
We find the FWHM to be $\sigma=488\pm 1$ ps. The $R^{2}$-value for this fit is
0.9971. The fitted curve now represents $g(t)$ — the timing response for our
coincidence detection system when using SPADs. For the SNSPDs, the method
discussed above is difficult to use because the system now has very high
temporal precision. Thus, there are only a few points around the slope at
$t=0$ making the derivative of the coincidence histogram an unreliable method
for finding $g(t)$. Instead, we estimate the timing resolution just using the
FWHM of the jitter histograms for the SNSPDs and the time tagger stated by the
manufacturer, and then assume all timing response functions are Gaussian. See
Methods in the main body of the article.

Figure S4: a) Raw $HH+VV$ projection coincidence counts (blue dots) with a
Savitzki-Golay filter (red curve) with a polynomial degree of 6 and a window
length of 65, applied in order to smooth the raw data. b) Numerical derivative
of the filtered curve (blue dots) with only points where $t\leq 0$ plotted in
red and denoted as $\tilde{g}$. c) Mirrored version of $\tilde{g}$ across the
x-axis to generate the raw data that represents $g(t)$ (blue points), which is
then fitted to Eq. S9 (red curve). All plots are against the coincidence time
delay, denoted as t.

###  S1.5 Time-resolved Entanglement Fidelity

Aside from the concurrence, we also calculated the time-resolved fidelity to
the set of maximally entangled states ($F_{max}$) [52] and the fidelity with
respect to a particular Bell state, displayed in Figure S5a. The effect of the
FSS becomes clear when we plot the fidelity with respect to
$\ket{\Phi^{+}}=\left(\ket{HH}+\ket{VV}\right)/\sqrt{2}$ in Figure S5a. As per
Eq. 1 in the main body, the pair emitted by the QD should start in the
$\ket{\Phi^{+}}$ state and return to the $\ket{\Phi^{+}}$ state after
$\tau=T_{S}$. This is precisely the behaviour that we observed, as shown in
Figure S5a. We see very clear, high-visibility oscillations with the fidelity
to $\ket{\Phi^{+}}$ remaining above 80$\%$ up to 4 ns, at which point more
than 99% of photon pairs have been emitted.

Just as with the concurrence we see small oscillations in $F_{max}$ that
result in a reduction of the maximum fidelity from its initial peak at
$\tau=0$ of $96.6\pm 0.2\%$ to a minimum at $92.6\pm 0.6\%$ at $\tau=0.625$
ns. However, just like the concurrence, the fidelity comes back to $95.0\pm
0.8\%$ at $\tau=1.25$ ns which is nearly within $1\%$ of the initial peak
value. Interestingly, we see that the fidelity to the
$\ket{\Phi^{-}}=\left(\ket{HH}-\ket{VV}\right)/\sqrt{2}$ state reaches a
maximum of $77\%$ at halfway point of the oscillation period ($\tau=0.625$
ns). If instead, we plot fidelity to the closest maximally entangled pure
state at $\tau=0.625$ ns, denoted as $|{\tilde{\Phi}}\rangle$, we see that it
reaches the black dots exactly. This implies the state is not oscillating
exactly as $\left(\ket{HH}+\exp(i\theta)\ket{VV}\right)/\sqrt{2}$ but rather
in a rotated version of this state. Indeed, if we apply a different unitary
transformation to each qubit individually $U_{1}(\theta_{1},\phi_{1})\otimes
U_{2}(\theta_{2},\phi_{2})$, we can transform the state $|\tilde{\Phi}\rangle$
to the state $\ket{\Phi^{-}}$. The unitary transformation we apply is in the
form of a general waveplate, with parameters $\theta_{1}=0.722,\phi_{1}=1.956$
radians and $\theta_{2}=1.111,\phi_{2}=2.142$ radians [53].

In Figure S5b we plot the reconstructed density matrix at $\tau=0$. The
density matrix exhibits exactly the characteristics of the $\ket{\Phi^{+}}$
Bell state, namely negligible imaginary components and four positive real
components. Similarly, in Figure S5c we plot the density matrix of the rotated
$|\tilde{\Phi}\rangle$ state, namely $\left[U(\theta,\phi)\otimes
U(\theta,\phi)\right]|\tilde{\Phi}\rangle$. We see what is expected from the
$\ket{\Phi^{-}}$ state: negligible imaginary components; and, two positive
real components and two negative real components. The form of the density
matrix in Figure S5c confirms that we are indeed able to rotate the initially
reconstructed state $|\tilde{\Phi}\rangle$ into the expected $\ket{\Phi^{-}}$
state.

Figure S5: Two-photon state as a function of the exciton lifetime. a) Fidelity
of reconstructed density matrix to the set of maximally entangled two-qubit
state (black), the $\ket{\Phi^{+}}$ Bell state (red), the $\ket{\Phi^{-}}$
Bell state (blue) and the closest maximally entangled pure state at
$\tau=0.625$ (orange) as a function of time delay $\tau$. b) Reconstructed
density matrix at $\tau=0$ ns. c) Reconstructed density matrix at $\tau=0.625$
ns after rotating qubit one by $U_{1}(\theta_{1},\phi_{1})$ and qubit two by
$U_{2}(\theta_{2},\phi_{2})$ such that it closely resembles the state
$\ket{\Phi^{-}}$.

###  S1.6 Dephasing-Free Model

The model used here is based on the work of Ref. [51] which estimates the two-
photon density matrix reconstructed from performing quantum state tomography
on a QD emitter. In this particular model, we assume that the QD emitter is in
the “dephasing-free” regime, i.e., the two-photon entangled state emitted by
the QD is in the maximally entangled state
$\ket{\psi(\tau)}=\left(\ket{HH}+\exp(iS\tau/\hbar)\ket{VV}\right)/\sqrt{2}$
where $S$ is the finite exciton fine structure splitting (FSS). As a result,
the probability of detecting a coincidence in a particular two-photon
polarization state $\ket{ij}\,i,j\in\\{H,V,D,A,R,L\\}$ at a given time delay
is

$p_{ij}(\tau)=|\bra{ij}\ket{\psi(\tau)}|^{2}\,\frac{1}{\tau_{X}}e^{-\tau/\tau_{X}}\,\Theta(\tau)$
(S10)

where $\tau_{X}$ is the radiative lifetime of the exciton and $\Theta(\tau)$
is the Heaviside step function. To estimate the total number counts in each
histogram bin after an experiment we use

$N_{0}=N_{X}\,N_{XX}\,\frac{T_{\textnormal{bin}}\,\Delta\tau}{f_{\textnormal{rep}}}$
(S11)

where $N_{X}$, $N_{XX}$ are the measured count rates of the exciton and
biexciton, respectively, $T_{\textnormal{exp}}$ is the experiment runtime, and
$f_{\textnormal{rep}}$ is the excitation laser repetition rate. However, even
though the model assumes the QD emitter experiences no dephasing, we still
include the non-zero multiphoton emission probability of the QD emitter, and
the finite dark count and timing resolution of the single-photon detectors
used in the experiment.

To incorporate the effect of detector dark counts which are unpolarized and
therefore add unwanted coincidences to all polarization basis measurements. We
estimate the expected number of coincidences from dark counts with the
equation

$N_{d}^{cc}\simeq\Big{[}N_{XX}\,N_{d_{X}}+N_{X}\,N_{d_{XX}}\Big{]}\,\frac{T_{\textnormal{bin}}\,\Delta\tau}{f_{\textnormal{rep}}}$
(S12)

where $N_{d_{X}}$ and $N_{d_{XX}}$ are the dark count rates for the single
photon detectors measuring exciton and biexciton photons, respectively. The
finite timing resolution of the detection system is included by convolving the
time-dependent count probability distribution with the detection system
response function $g(\tau)$. The coincidence count histogram versus time delay
for a given polarization basis measurement $ij$ is thus

$N_{ij}(\tau)=\left(N_{0}\,p_{ij}(\tau)+N_{d}\right)*g(\tau)\,\Delta\tau$
(S13)

where $*$ is the convolution operation. With the simulated $N_{ij}(\tau)$
coincidence histogram, the density matrix can be reconstructed in 50 ps width
time bins to give $\rho(\tau)_{QD}$. Finally, the effect of multi-photon
emission on the reconstructed density matrix is included by adding the 4x4
identity matrix $I/4$, i.e., uncorrelated light. The final simulated density
matrix is then

$\rho_{\textnormal{sim}}(\tau)=(1-p_{m})\,\rho(\tau)_{QD}+p_{m}\,\frac{I}{4}$
(S14)

where $p_{m}\simeq(\,\tilde{g}^{(2)}_{XX}(0)+\tilde{g}^{(2)}_{X}(0)\,)/2$
[54]. Time slices of $\rho_{\textnormal{sim}}(\tau)$ with a width of 50 ps,
were then used to calculate the time-resolved concurrence curves for the SPAD
and SNSPDs displayed in Figure 3.

###  S1.7 Polarized HBT Measurement

Figure S6: Hanburry Brown Twiss autocorrelation histograms taken just after
performing quantum state tomography, when the emission from the quantum dot is
polarized to either H or V. We see a significantly higher number of
coincidences at $t=0$ in the H polarized histogram, which is likely due to
stray photons that were not successfully filtered.

###  S1.8 Dephasing Model

To explore the effects that a dephasing mechanism would have on the
experimental time-resolved concurrence curve, we combined the model described
in Ref. [55] with the dephasing free-model outline in the Supplementary Note
S1.6. The model incorporates two types of spin-dephasing: 1) spin-flips (i.e.,
spin-scattering) of the exciton spin; and 2) cross-dephasing between the
H-polarized and V-polarized exciton spin [56]. Mathematically, spin-flips are
modelled as the addition of uncorrelated light (i.e., the identity) over an
interaction period of $\tau_{SS}$, and cross-dephasing as the addition of
$\rho_{HV}=\left(\ket{HH}\bra{HH}+\ket{VV}\bra{VV}\right)/2$ (i.e., the loss
of coherence between $\ket{HH}$ and $\ket{VV}$) with an interaction period of
$\tau_{HV}$. The final simulated density matrix when dephasing is included is
written as,

$\begin{split}\rho_{\textnormal{sim}}(\tau)=k\,e^{-\tau/\tau_{SS}}e^{-\tau/\tau_{HV}}\rho(\tau)_{QD}+\left(1-k\,e^{-\tau/\tau_{SS}}\right)\frac{I}{4}\,+k\,e^{-\tau/\tau_{SS}}(1-e^{-\tau/\tau_{HV}})\,\rho_{HV}\end{split}$
(S15)

where $k=1-p_{m}$. With this model for spin-dephasing, we can now quantify the
effect of the detection system (included in $\rho(\tau)_{QD}$) and multi-
photon emission (incorporated through $k$) plus the effect of spin-scattering
and cross-dephasing. We explore the effect of both spin-flip and cross-
dephasing on the time-resolved concurrence by looking at each mechanism
individually and comparing it to the concurrence curve from the experimental
data (see Figure S7 in Sec S1.8). Specifically, two distinct limits are
examined: varying the spin-flip coherence time $\tau_{SS}$ with no cross-
dephasing ($\tau_{HV}\rightarrow\infty$); and varying the cross-dephasing time
$\tau_{HV}$ with no spin-flip dephasing ($\tau_{SS}\rightarrow\infty$). As
expected, the curves displayed in Figure S7 do not account for the dip in the
concurrence between $\tau=0$ ns and $\tau=1.3$ ns, implying that it is not
spin-scattering and cross-dephasing that are responsible. However, the model
indicates that dephasing times about an order of magnitude larger than the
lifetime could explain the slow reductions in the experimental concurrence
when $\tau>2$ ns. Still, these dephasing values are much greater than the
radiative lifetime, implying that their impact will not degrade the lifetime-
average concurrence, and therefore the source could still be considered
“dephasing-free”.

Figure S7: Time-resolved concurrence curve from the dephasing model for
different characteristic interaction times. a) In the limit of no cross-
dephasing, the time-resolved concurrence is calculated for different spin-
scattering interaction times. b) In the limit of no spin-scattering, the time-
resolved concurrence is calculated for different cross-dephasing interaction
times.

###  S1.9 Using quantum dots with FSS for QKD

In this supplement, we describe a general method to calculate secret key rates
for entanglement-based (EB) QKD experiments that use fast detectors. The idea
is to break each round of the protocol into ${n_{\textrm{bin}}}$ time-bins
where the detectors record the click patterns in each time-bin. We then use
this to calculate key rates for the QD source with non-zero FSS.

In our security proof, we assume that the threshold detectors used have no
memory effects (detector dead times, afterpulsing, etc.), and all detectors
have the same detection efficiency. Any security proof for independent and
identically distributed (iid) states can be lifted to a security proof for
general states via the quantum de Finetti theorem [57] or the postselection
technique [58], which are asymptotically tight. As we only consider asymptotic
key rates, we can thus without loss of generality assume that Alice and Bob’s
shared state is iid i.e.
$\rho_{A^{{n_{\textrm{round}}}}B^{{n_{\textrm{round}}}}}=\rho_{AB}^{\otimes{n_{\textrm{round}}}}$,
where ${n_{\textrm{round}}}$ is a large number of protocol rounds. In the rest
of this supplement we focus on proving security for these iid states.

####  S1.9.1 QKD Protocol Steps

Here we outline the steps in a generic time-resolved entanglement-based (EB)
protocol with ${n_{\textrm{round}}}$ rounds.

  1. 1.

Signal Transmission: Eve prepares optical states $\rho_{AB}$ to send to Alice
and Bob.

  2. 2.

Measurement: Alice and Bob each have sets of POVM elements
$\\{\Gamma^{A}_{{\vec{i}}}\\}$ and $\\{\Gamma^{B}_{{\vec{j}}}\\}$. Here, the
elements $i_{k}$ and $j_{k}$ of the vectors ${\vec{i}}$ and ${\vec{j}}$
represent the click patterns observed in time-bin $k$.

After repeating the above steps ${n_{\textrm{round}}}$ times, we proceed to
the next part of the protocol.

  1. 3.

Acceptance Testing: Alice and Bob randomly choose a subset of the rounds for
testing. For the rounds chosen for testing, they both publicly announce their
measurement results ${\vec{i}}$ and ${\vec{j}}$ to form a distribution of the
frequency of events. They then check if this frequency distribution belongs to
the acceptance set agreed upon before running the protocol. If it does, they
proceed with the protocol after discarding the test results. Otherwise, they
abort.

  2. 4.

Announcements and Sifting: Alice and Bob make announcements over the
authenticated classical channel. They sift the non-tested data based on the
announcements made i.e. they choose a subset of signal and measurement data to
keep and discard the rest based on the announcements.

  3. 5.

Key Map: Alice uses her measurement results ${\vec{i}}$ as well as the
announcements to map her data into a key string $x$. This is called the raw
key. We assume here that the key is a bit string for simplicity, but all the
steps can be applied more generally.

  4. 6.

Error Correction: Alice and Bob then perform error correction over the
authenticated classical channel to make Bob’s measurement outcomes ${\vec{j}}$
match with Alice’s bit string $x$. We denote the data communicated per key bit
to Eve in this process as $\delta_{\text{leak}}$.

  5. 7.

Privacy Amplification: Alice and Bob produce their final secret key by
applying an appropriate hash function on the raw key (Theorem 5.5.1 of [59]).

####  S1.9.2 Key rate calculation

  * •

The secret key rate in the asymptotic limit in the iid case can be found using
the Devetak-Winter formula [60]: $R=H(Z|E)-\delta_{\text{leak}}$ where $Z$ is
the key register, $E$ is Eve’s register, and $\delta_{\text{leak}}$ is the
number of bits per round leaked to Eve during Step 6 of the protocol.

    * –

A lower bound for the key rate can be found by minimising $H(Z|E)$ over Eve’s
marginal states consistent with the observed statistics.

    * –

Following [61, 62], this minimization can be reformulated as an SDP

$\displaystyle R=\underset{\rho_{AB}}{\textrm{min }}$ $\displaystyle
D\left(\mathcal{G}\left(\rho_{AB}\right)||\mathcal{Z}\left(\mathcal{G}\left(\rho_{AB}\right)\right)\right)-\delta_{\textrm{leak}}$
(S16) s.t.
$\displaystyle\textrm{Tr}\left[\left(\Gamma_{{\vec{i}}}\otimes\Gamma_{{\vec{j}}}\right)\rho_{AB}\right]=p_{{\vec{i}}{\vec{j}}}\quad\quad\forall{\vec{i}},{\vec{j}}$
$\displaystyle\rho_{AB}\geq 0,$

where $A$ and $B$ are Alice and Bob’s registers. The observed statistics
$p_{{\vec{i}}{\vec{j}}}$ is the probability of Alice and Bob observing
outcomes ${\vec{i}}$ and ${\vec{j}}$ respectively. Note that in phrasing the
optimization problem in this way, we have implicitly chosen a point acceptance
set (Step 3 of protocol steps) to be exactly the statistics from the honest
implementation. This choice is meaningful as we work in the asymptotic limit.

    * –

Here, the relative entropy
$D\left(\mathcal{G}\left(\rho_{AB}\right)||\mathcal{Z}\left(\mathcal{G}\left(\rho_{AB}\right)\right)\right)-\delta_{\text{leak}}$
is the objective function where $\mathcal{G}$ is a map that represents the
protocol (including announcements), and $\mathcal{Z}$ is a map that can be
constructed from the key map. Since we do not use most of the specific details
of these maps, we abstract the objective function as $f(\rho_{AB})$. For more
details, see [62, 61].

  * •

The aim is to reduce this to a problem that is numerically tractable. There
are two obstacles to doing this.

    * –

The first is that each time-bin has an infinite-dimensional Fock space
associated with it. This is a well-studied problem in QKD [63, 64, 65, 66].
Any of these methods can be used to reduce the problem to solving finite-
dimensional SDPs in each time-bin.

    * –

The dimension of the space after this process is still large due to the
multiple time-bins. Thus, we will simplify Eq. (S16) to a minimization on each
time-bin individually to obtain a “time-resolved key rate”.

####  S1.9.3 Time-resolved key rate

  * •

Since we assume that the threshold detectors have no memory effects, each of
Alice and Bob’s POVM elements can be modeled as
$\Gamma_{\vec{i}}=\bigotimes_{k=1}^{n_{\textrm{bin}}}\Gamma_{i_{k}}$.

    * –

For threshold detectors such as the SNSPDs, $\Gamma_{i_{k}}$ is block diagonal
in the total number of photons in time-bin $k$ i.e.

$\Gamma_{i_{k}}=\bigoplus_{n_{k}=0}^{\infty}\Gamma_{i_{k}}^{n_{k}}.$ (S17)

    * –

Thus, the POVM elements $\Gamma_{\vec{i}}$ are block-diagonal as

$\Gamma_{\vec{i}}=\bigoplus_{\vec{n}}\Gamma^{n_{1}}_{i_{1}}\otimes\Gamma^{n_{2}}_{i_{2}}\otimes\ldots\otimes\Gamma^{n_{n_{\textrm{bin}}}}_{i_{n_{\textrm{bin}}}}.$
(S18)

    * –

As a result, we can without loss of generality (see for instance, Observation
8 from [64]) assume that Alice and Bob’s joint state has the same block-
diagonal structure

$\rho_{AB}=\bigoplus_{{\vec{n}},{\vec{m}}}\rho_{AB}^{{\vec{n}}{\vec{m}}},$
(S19)

where $\rho_{AB}^{{\vec{n}}{\vec{m}}}$ are sub-normalized states.

  * •

Following Eq. (D.9) from [67], we can exploit the block-diagonal structure in
the state to write
$f(\rho_{AB})=\sum_{{\vec{n}},{\vec{m}}}f(\rho_{AB}^{{\vec{n}}{\vec{m}}})$,
where $f(\rho_{AB})$ is the objective function of the SDP in Eq. (S16). Thus,
$R\geq\sum_{{\vec{n}},{\vec{m}}}R^{{\vec{n}}{\vec{m}}}$ where

$\displaystyle R^{{\vec{n}}{\vec{m}}}=\underset{\rho_{AB}}{\textrm{min }}$
$\displaystyle f\left(\rho_{AB}^{{\vec{n}}{\vec{m}}}\right)$ (S20) s.t.
$\displaystyle\textrm{Tr}\left[\left(\Gamma_{{\vec{i}}}\otimes\Gamma_{{\vec{j}}}\right)\rho_{AB}\right]=p_{{\vec{i}}{\vec{j}}}\quad\quad\forall{\vec{i}},{\vec{j}}$
$\displaystyle\rho_{AB}\geq 0$
$\displaystyle\rho_{AB}=\bigoplus_{{\vec{n}},{\vec{m}}}\rho_{AB}^{{\vec{n}}{\vec{m}}}.$

Since each $R^{{\vec{n}}{\vec{m}}}$ is positive, taking finitely many of these
terms is sufficient to lower bound the key rate. In particular, we consider
those terms where all but one time-bin has no photons on each of Alice and
Bob’s sides.

    * –

Note that
$\rho_{AB}^{{\vec{n}}_{k_{c}}{\vec{m}}_{l_{c}}}=\rho_{AB}^{n_{k_{c}}m_{l_{c}}}\bigotimes_{k\neq
k_{c},l\neq l_{c}}\outerproduct{0}{0}_{k}\otimes\outerproduct{0}{0}_{l}$ where
$k_{c}$ and $l_{c}$ are the time-bins with photons on Alice and Bob’s side
respectively.

  * •

If the detection efficiency of all detectors is identical, it is sufficient to
prove security for lossless detectors instead [68] by considering loss to be a
part of the channel. In this case, for all Alice’s events ${\vec{i}_{k_{c}}}$
that have detection events in just time-bin $k_{c}$,
$\Gamma_{{\vec{i}_{k_{c}}}}=\Gamma_{i_{k_{c}}}\bigotimes_{k\neq
k_{c}}\outerproduct{0}{0}_{k}$. Similarly,
$\Gamma_{{\vec{j}_{l_{c}}}}=\Gamma_{j_{l_{c}}}\bigotimes_{l\neq
l_{c}}\outerproduct{0}{0}_{l}$ for all Bob’s events ${\vec{j}_{l_{c}}}$ that
have detection events in just time-bin $l_{c}$. Thus,

$\displaystyle\textrm{Tr}\left[\left(\Gamma_{{\vec{i}_{k_{c}}}}\otimes\Gamma_{{\vec{j}_{l_{c}}}}\right)\rho_{AB}\right]$
$\displaystyle=\textrm{Tr}\left[\left(\Gamma_{{\vec{i}_{k_{c}}}}\otimes\Gamma_{{\vec{j}_{l_{c}}}}\right)\sum_{n_{k_{c}},m_{l_{c}}=0}^{\infty}\rho_{AB}^{{\vec{n}}_{k_{c}}{\vec{m}}_{l_{c}}}\right]$
$\displaystyle=\textrm{Tr}\left[\left(\Gamma_{i_{k_{c}}}\otimes\Gamma_{j_{l_{c}}}\right)\rho_{AB}^{{k_{c}}{l_{c}}}\right],$
(S21)

where ${\vec{n}}_{k_{c}}$ (${\vec{m}}_{l_{c}}$) are vectors where all but the
$k_{c}$ ($l_{c}$) entry is 0, and
$\rho_{AB}^{{k_{c}}{l_{c}}}=\sum_{n_{k_{c}},m_{l_{c}}=0}^{\infty}\rho_{AB}^{n_{k_{c}}m_{l_{c}}}$.
Note that the right hand side of Eq. (S21) is described in terms of operators
that act on the space for a single time-bin $k_{c}$, $l_{c}$ for Alice and Bob
respectively.

    * –

For all events ${\vec{i}}$, ${\vec{j}}$ with clicks in more than one time-bin,
$\textrm{Tr}\left[\left(\Gamma_{{\vec{i}}}\otimes\Gamma_{{\vec{j}}}\right)\rho_{AB}^{{\vec{n}}_{k_{c}}{\vec{m}}_{l_{c}}}\right]=0$.

  * •

Additionally, we define the function
$g\left(\rho_{AB}^{{k_{c}}{l_{c}}}\right)\coloneqq\sum_{n_{k_{c}},m_{l_{c}}=0}^{\infty}f\left(\rho_{AB}^{{\vec{n}}_{k_{c}}{\vec{m}}_{l_{c}}}\right)$
acting on the space for a single time bin for Alice and Bob. Thus, the SDP in
Eq. (S20) can be simplified as

$\displaystyle
R^{k_{c}l_{c}}=\underset{\rho_{AB}^{{k_{c}}{l_{c}}}}{\textrm{min }}$
$\displaystyle g\left(\rho_{AB}^{{k_{c}}{l_{c}}}\right)$ (S22) s.t.
$\displaystyle\textrm{Tr}\left[\left(\Gamma_{i_{k_{c}}}\otimes\Gamma_{j_{l_{c}}}\right)\rho_{AB}^{{k_{c}}{l_{c}}}\right]=p_{{\vec{i}_{k_{c}}}{\vec{j}_{l_{c}}}}\quad\quad\forall{\vec{i}_{k_{c}}},{\vec{j}_{l_{c}}}$
$\displaystyle\rho_{AB}^{{k_{c}}{l_{c}}}\geq 0.$

  * •

We define
$r(\tau)=\frac{\sum_{\absolutevalue{k_{c}-l_{c}}=\tau}R^{k_{c}l_{c}}}{\textrm{Tr}\left[\rho_{AB}^{\tau}\right]}$
to be the time-resolved key rate where
$\rho_{AB}^{\tau}=\sum_{\absolutevalue{k_{c}-l_{c}}=\tau}\rho_{AB}^{{k_{c}}{l_{c}}}$.
Thus, we have simplified the key rate SDP in Eq. (S16) to the time-resolved
key rate SDPs described in Eq. (S22) which act on the spaces corresponding to
a single time-bin $k_{c}$, $l_{c}$ for Alice and Bob respectively.

  * •

The time-resolved key rate SDPs described in Eq. (S22) are still infinite-
dimensional. As discussed above, any of the techniques from [63, 64, 65, 66]
to reduce this to a finite-dimensional optimization can be used to obtain
lower bounds on the time-resolved key rate. For the purposes of the example we
consider in Section S1.9.4, we use the dimension reduction method from [66].

####  S1.9.4 Proof-of-principle experiment

So far, we have described a generic framework to compute key rates with fast
detectors i.e. detectors with low timing jitter and dead times to resolve
subsequent time-bins with independent detection events. We now describe how we
used this framework to obtain the plots in Figure 4, which clearly demonstrate
the utility of a QD with non-zero FSS. Note that in the proof-of-principle
experiment, we consider the case where only single-click events, i.e. a click
only in a single detector time-bin in each protocol round for Alice and Bob
each. As we expect our source to primarily send out single-photon states to
Alice and Bob each, this would be close to what we expect to observe in a real
experiment.

  * •

We consider a polarization-based 6-state protocol [69] where both Alice and
Bob measure their state in 3 conjugate polarization bases which we term the
‘X’, ‘Y’ and ‘Z’ bases. A schematic of the setup is shown in Figure S8. We set
the probability of measuring in the Z-basis to be $\sqrt{0.99}$ each. After
testing, they announce their basis data and discard all signals not in the
Z-basis. Alice then maps single-clicks in the Z-basis into bits. Finally, they
do error correction and privacy amplification.

Figure S8: Schematics of proof-of-principle experimental setup. PBS:
polarisation beamsplitter; RNG: random number generator; SNSPD:
superconducting nanowire single photon detector; TT: time tagger; CLK: clock.
The basis choice polarisation rotator together with the RNG randomly choose
between three mutually unbiased bases to measure in. The choice of which three
mutually unbiased bases are used for the protocol is made by the fixed
polarisation rotator. The clock rate must be faster than the FSS oscillations
of the quantum dot.

  * •

For the key rate calculation, we simulate the observations we expect to see
from the characterized nanowire QD entangled photon source.

    * –

We use the dimensional reduction method [66] to construct finite-dimensional
SDPs from Eq. (S22) which allows us to consider projections of measurments and
states in a subspace only. For this, we need the weight
$W=\textrm{Tr}\left[\Pi_{1}\rho^{k_{c}l_{c}}\Pi_{1}\right]$ outside the
projected subspace. Thus, the time-resolved key rate can be lower bounded by
the solution to the following finite-dimensional SDP:

$\displaystyle R^{k_{c}l_{c}}\geq\underset{\rho_{AB}^{11}}{\textrm{min }}$
$\displaystyle g\left(\rho_{AB}^{11}\right)$ (S23) s.t. $\displaystyle
p_{{\vec{i}_{k_{c}}}{\vec{j}_{l_{c}}}}-W\leq\textrm{Tr}\left[\left(\Gamma_{i_{k_{c}}}^{1}\otimes\Gamma_{j_{l_{c}}}^{1}\right)\rho_{AB}^{11}\right]\leq
p_{{\vec{i}_{k_{c}}}{\vec{j}_{l_{c}}}}\quad\quad\forall{\vec{i}_{k_{c}}},{\vec{j}_{l_{c}}}$
$\displaystyle\rho_{AB}^{11}\geq 0,$

where $\Gamma^{1}=\Pi_{1}\Gamma\Pi_{1}$ is the projection of the POVM elements
into the single-photon subspace $\Pi_{1}$. Note that since the POVM elements
commute with our chosen projection, we have used the tighter Eq. (49) from
[66].

Here, the weight outside the subspace $W$ can be estimated from the multi-
click events (see for e.g.- [70, 65, 71] for some protocol-dependent
techniques to do this). In our case, since we have assumed to observe only
single-click events, this can be calculated exactly to be $W=0$. Thus, the
time-resolved key rate SDP simplifies to numerically optimising an SDP in the
qubit space.

  * •

Additionally since we are performing the 6-state experiment, we have a
tomographically complete set of measurements on the qubit space. Thus, the
state $\rho_{AB}^{11}$ can be determined exactly from the observations for
each time-bin. As a result, we can compute the time-resolved key rates
$r(\tau)$ by simply computing the objective functions for the qubit density
matrix for the time-bins. So no SDP optimizations are required for our simple
example.

  * •

Finally, as described in Supplementary Note S1.5, the actual entangled photon
pair does not rotate in the $HH-VV$ plane as described by Eq. (1). Thus, our
initial key map consisting of $H$ and $V$ polarizations might not be optimal.
We instead run an optimization over all choices of conjugate polarization
bases to measure in. Specifically, just as in Supplementary Note S1.5, we
apply two independent unitary transformations
$U_{1}(\theta_{1},\phi_{1})\otimes U_{2}(\theta_{2},\phi_{2})$ to the
reconstructed two-photon state, and find the set of parameters that maximizes
the key rate $R$. We use the result of this optimization to create the blue
curve in Figure 4. Practically, this can be implemented in the protocol by
adding appropriate waveplates before Alice and Bob’s entire measurement setup
as depicted in Figure S8.

###  S1.10 Blinking

For a QD emitter to be considered truly “on-demand”, a photon should be
emitted for every incident excitation pulse. Consequently, if excitation
pulses are incident at a regular, periodic interval $T_{\textnormal{rep}}$ on
the QD, then photons should be emitted in a matching temporal pulse train
pattern. Thus, the coincidence histogram from the HBT autocorrelation
experiment will consist of coincidence peaks every $T_{\textnormal{rep}}$
except near the $\tau\rightarrow 0$ as a true single photon source should
exhibit strong anti-bunching. The height of the histogram side peaks at
$\tau\neq 0$ represents the probability of detecting a photon at some time
delay $\tau$ after an initial photon has been detected [72]. Thus for a true
on-demand source, the heights of these $\tau\neq 0$ peaks should all be equal,
as there is an equal probability of a coincidence occurring between any
photons emitted in the train. This situation where all peaks are of equal
height is known as the “Poisson” limit, with the corresponding peak heights
known as the “Poisson” level [73]. Typically, the phenomenon of luminescence
blinking leads to a deviation from these flat peaks for all $\tau\neq 0$.
Instead, the height of the peaks in the histogram will increase as the time
delay $\tau$ decreases. This means that, given the detection of an initial
photon emitted from the QD, there is a higher probability of detecting a
photon from the excitation pulse that directly followed, compared to the
Poisson limit where photons are equally likely to arrive at any $\tau$ [72].
Such behaviour can be shown to result from the QD transitioning between a
state that allows for the generation of the biexciton via two-photon resonant
excitation, to a state that forbids it and thus forbids the generation of an
entangled photon pair. In other words, the QD will transition from a biexciton
“ON” state to a biexciton “OFF” state. For example, if a charge carrier
tunnels into an empty QD (ground state $g$) between excitation pulses, then
the QD will then be in some charge state $c$. This extra charge will block the
generation of a biexciton via a resonant two-photon absorption, and hence
there will be no subsequent emission of an exciton or a biexciton photon [74].
In order for optical excitation to occur, the charge must then tunnel out of
the QD or recombine with an opposing carrier. This inherently takes some
finite amount of time during which the QD will not be producing any biexciton
or exciton photons. Hence, a simple random telegraph (ON-OFF) model can be
applied to quantify the autocorrelation function that is observed in the
presence of luminescence blinking [75],

$g^{(2)}(\tau)=1+\frac{1-\beta}{\beta}e^{-\tau/\tau_{b}}.$ (S24)

Here, $\beta$ is the probability of the QD being in the ON state at any given
time and $1/\tau_{b}=1/\tau_{g}+1/\tau_{c}$ is the transition rate between the
two states discussed above, i.e., the rate at which the QD transitions from
the ground state to the charge state and then vice versa, i.e.,
ON$\rightarrow$OFF and OFF$\rightarrow$ON, respectively [74].

Figure S9: a) Raw data from the Hanburry-Brown-Twiss autocorrelation histogram
for the biexciton. The orange x’s represent the location of the maximum of
each peak. b) Fit of Eq. S24 to the peak heights extracted from the plot to
the left as a function of time delay $\tau$.

In Figure S9, we can see the results of the fit of Eq. S24 to the HBT
autocorrelation histogram for biexciton emission, from which we extract
$\beta_{XX}=0.167\pm 0.006$ and $1/\tau_{b,XX}=2.86+/-0.07$ MHz. When fitting,
we scale the entire Eq. S24 function by a factor $A_{P}$, which represents the
peak heights of the raw histogram at the Poisson level, i.e., the peak heights
at long $\tau$ where $g^{(2)}(\tau)=1$. From the fit, we find the $A_{P}=84\pm
3$ for the biexciton. When performing the same fit procedure for the exciton
HBT histogram we find $\beta_{X}=0.186\pm 0.007$ and
$1/\tau_{b,X}=2.88+/-0.09$ MHz. Using the extracted $\beta$ values, we can
calculate the true $\tilde{g}^{(2)}(0)$ values, which must be normalized to
the Poisson level when blinking is present, with the expression
$g^{(2)}(0)=\tilde{g}^{(2)}(0)/\beta$ where $\tilde{g}^{(2)}(0)$ is the value
normalized to the nearest neighbour peaks we quoted in the main body. Doing
this, we find $g^{(2)}_{X}(0)=0.029\pm 0.002$ and $g^{(2)}_{XX}=0.032\pm
0.002$. We note the reason for stating the smaller $\tilde{g}^{(2)}(0)$ values
in the main body is because it is used to calculate the multi-photon emission
probability, which is the relevant metric for quantum state tomography [54].

###  S1.11 Estimating Pair Extraction Efficiency

An estimate for the pair extraction efficiency of the source can be found by
examining $\eta_{\textnormal{int}}$ and $\eta_{\textnormal{ext}}$ the
“internal” and “external” efficiencies of the system, respectively. Internal
efficiency is the probability that a given excitation pulse will generate an
exciton and a biexciton. It is determined by the exciton and biexciton
preparation probabilities from the fits to the Rabi oscillations and the
blinking ratio:
$\eta_{\textnormal{int}}=\eta_{\textnormal{prep},X}\,\eta_{\textnormal{prep},XX}\,\eta_{\textnormal{blink}}$.
The preparation fidelity and blinking ratios are
$\eta_{\textnormal{prep},XX}=0.82\pm 0.01$,
$\eta_{\textnormal{prep},X}=0.774\pm 0.009$ and
$\eta_{\textnormal{blink}}=0.167\pm 0.006$, respectively, and their
calculation were presented earlier in Sec. S1.1 and in Sec. S1.10,
respectively. Next, we need to estimate the external efficiency of the
nanowire, which is the probability that a biexciton-exciton photon pair
emitted by the QD will be subsequently coupled out of the nanowire into the
propagating free-space optical mode collected by the objective lens. We
approximate this efficiency by using the detected count rates for the exciton
and biexciton at saturation under pulsed quasi-resonant excitation (from Ref.
[51]),

$\eta_{\textnormal{NW}}=\frac{N_{X}N_{XX}}{(\eta_{\textnormal{opt}}\Gamma)^{2}}=\frac{(942.89\textnormal{kHz})(401.29\textnormal{kHz})}{(0.063)^{2}(76.2\textnormal{MHz})^{2}}=0.016\,.$
(S25)

The overall predicted efficiency is then
$\eta_{est}=\eta_{\textnormal{int}}\,\eta_{\textnormal{ext}}=0.17\pm 0.01\%$,
which is within the same order of magnitude as the experimental pair
extraction efficiency of $\eta_{\textnormal{exp}}=0.65\pm 0.02\%$. We note
that the estimate of 0.17 is likely an underestimate because we are not
including any of the other minor exciton complexes, which appear during quasi-
resonant exciton, in the calculation of the nanowire extraction efficiency. If
this modest contribution was included in the extraction efficiency of the
nanowire then the two numbers will likely be in closer agreement.

###  S1.12 Experimental Apparatus

The experimental setup consisted of three main parts: generating the
excitation pulse, quantum state tomography of the photon pairs from the
nanowire QD sample, and single-photon detection. An illustration of the entire
experimental apparatus can be seen in Figure S10. See the Methods section for
a written description.

Figure S10: Acronyms: TG — transmission grating, BPF — fiber coupled bandpass
filter, PM — right-angle prism mirror, NFs — notch filters, SNSPDs —
superconducting nanowire single-photon detectors, Pol — linear polarizer, HWP
— half-waveplate, QWP — quarter-wave plate.

###  S1.13 Apparatus Calibration

To calibrate the experimental setup pictured above, we first found the
principles axes of the $\lambda/2$ and $\lambda/4$ waveplates with respect to
$H$-polarized light in the lab frame. This was done by sending in a weak $H$
polarized laser signal through the QD excitation path, from which it reflected
off the device substrate and then followed the path of the QD emission through
both waveplates and the linear polarizer, which has its polarizing axis fixed
at 0 degrees to the horizontal, as illustrated in Figure S10. With the
$H$-polarized laser running, the photo-intensity coupled into the single-mode
fiber was monitored while one waveplate was at a fixed angle and the other was
rotated. For example, the $\lambda/2$ waveplate rotated, while the $\lambda/4$
waveplate remained fixed at a specified angle. After a full 90-degree rotation
the $\lambda/2$ waveplate, the angle corresponding to the intensity maximum
$\theta$, was recorded. Next, the action of the two waveplates was swapped,
i.e., the $\lambda/2$ waveplate remained fixed at the angle ($\theta$) and the
$\lambda/4$ waveplate was rotated 90 degrees. Again, the angle corresponding
to the intensity maximum ($\phi$) was recorded. Finally, the action of the
waveplates was again swapped, with the $\lambda/4$ remaining fixed at $\phi$
and the $\lambda/2$ waveplate rotating. This iterative process was then
repeated until the angles at maximum intensity ($\theta,\phi$) no longer
change after a few repetitions of the procedure. This was done for both the
exciton and biexciton emission wavelengths. The end result of this calibration
procedure was four waveplate angles, i.e., one for each waveplate on both the
exciton and biexciton path, which corresponded to the waveplate settings for
projecting onto the $H$ polarization state. The waveplate setting (angles) for
projective measurements on all other polarization states ($V,A,D,R,L$) are
simply the $H$ settings shifted by some calculable amount [76]. Assuming
perfect waveplates, with the equation

$\ket{\psi_{proj}}=\hat{U}_{\lambda/4}(\phi)\hat{U}_{\lambda/2}(\theta)\ket{H}=\frac{1}{2}\begin{pmatrix}\cos(\phi)^{2}+i\sin(\phi)^{2}&\frac{(1-i)}{2}\sin(2\phi)\\\
\frac{(1-i)}{2}\sin(2\phi)&\sin(\phi)^{2}+i\cos(\phi)^{2}\end{pmatrix}\begin{pmatrix}\cos(2\theta)&\sin(2\theta)\\\
\sin(2\theta)&-\cos(2\theta)\end{pmatrix}\begin{pmatrix}1\\\ 0\end{pmatrix}$
(S26)

we determine the angles $\theta$, $\phi$ required to implement a projective
measurement along the state $\ket{\psi_{proj}}$, given that the waveplates are
calibrated to $\ket{H}=\begin{pmatrix}1&0\end{pmatrix}^{\intercal}$. The
angles we used are listed below in Table S1. We verified the waveplate
settings by performing a full 36-measurement quantum state tomography trial on
the $H$-polarized laser. With these calibrated settings, the system
reconstructed the $H$-polarized laser input to a highly-pure state with a
purity Tr$\left(\rho^{2}\right)=99.8\pm 0.2\%$. The density matrix of the
reconstructed state is plotted in Figure S11.

$\ket{\psi_{proj}}$ | H | V | A | D | R | L
---|---|---|---|---|---|---
$\phi$ (QWP) | 0 | 0 | 45 | 45 | 90 | 0
$\theta$ (HWP) | 0 | 45 | -22.5 | 22.5 | 22.5 | 22.5
Table S1:  Waveplate angle settings used in the quantum state tomography
experiments.  Figure S11: Reconstructed density matrix from performing quantum
state tomography on an H-polarized laser signal. The single column in the real
part of $\rho$ at $\ket{HH}\bra{HH}$ demonstrates that the system was well
calibrated with respect to H polarized photons.