statistical certainty on the violation of Bell’s inequality, represented on
the colormap values, drives the instrument performance requirements. The
colormap corresponds to a statistical confidence between $1\sigma$ and
$8\sigma$. No violation at all can be achieved unless $p>\tfrac{1}{\sqrt{2}}$.

As a starting point, we consider the design example from Section II.1.2, which
for the COW tests predicted optimum performance around 1,500-km altitude
orbits, using $0.3\,$-m and $1.0\,$-m diffraction-limited telescopes at
$1550\,$nm. This flight system, upgraded with a suitable entangled photon pair
source operating at 1% pair production probability with a $1\,$GHz clock rate,
and a second $0.3\,$-m telescope pointed to a second $1.0\,$-m aperture, would
perform the Bell tests indicated in Figure 17.

Figure 17: System performance to achieve an $n\sigma$ violation (color coded)
of Bell’s inequality across orbital distances represented on the y-axis. An
Earth-Spacecraft link is assumed up to Geostationary orbit. The three dotted
lines on the top of the chart represent 1.0, 0.5, and 0.33 times the Earth-
Moon mean orbital radius. For simplicity, the source-to-Alice and source-to-
Bob channels are assumed identical. The color scale represents the achievable
statistical significance of the experiment, in $n\sigma$ . Note: for y-axis
values greater than Geostationary, the integration time was clamped to 1 hour.

It is evident that this system as described would be insufficient to support
the human-decision Bell tests as described. An upgraded system, with improved
pointing and larger receiver apertures, as well as a higher rate photon
source, is required. For example, Figure 18 shows the predicted performance
assuming $2.0\,$-m aperture transmitters, with a $2.0\,$-m Lunar-vicinity
receiver and a $10\,$-m aperture on Earth.

Figure 18: System performance to achieve $3\sigma$ violation of Bell’s
inequality across orbital distances represented on the y-axis. Up to
Geostationary orbit (the first horizontal white-dashed line), an Earth-
Spacecraft link is assumed. The three dashed lines on the top of chart
represent $1.0,0.5$, and $0.33$ times the Earth-Moon mean orbital radius. Note
there is an optimal contour for high $p$ values near the y-value of $6\cdot
10^{3}\,$km. The shapes of contour boundaries 6, 7, and 8 are due to the
integration time associated with the orbital flyby. This upgraded link
scenario will meet requirements for the midway-between-Earth-and-Moon Bell
test.

In both of the above examples, the x-axis of Figures 17 and 18 captures both
the fidelity of source and receiver parameters. Now, we can combine Equation
56 with the contours of Figures 17 and 18 to derive system performance
parameters.

The Bell test requires transmission of entangled photons to two receivers,
characterized in Eq. 43. For a source-to-receiver separation corresponding to
the diameter of the Earth, applying Equation 38 with $\lambda=810$ nm,
$D_{Tx}=0.5$ m, $D_{Rx}=3.5$ m, $M^{2}=1.05$, and $\eta_{x}=0.1$ leads to a
one-channel link efficiency of $0.003$. The two-channel efficiency,
characterizing the likelihood that both photons from an entangled pair are
recorded at their respective (equidistant) receivers, is $0.003^{2}\simeq
10^{-5}$ for the assumptions stated. Assuming a source clock rate of 1 GHz and
corresponding pair production probability of 1%, the rate of success is about
100 transmitted photon pairs per second. If the source telescope diameter is
reduced to 22 cm, and the receiver telescope diameters are reduced to 1.0 m,
the success rate drops to 0.03 transmitted photon pairs per second; the time
to measure 500 events would then increase from about 5 seconds to 4.5 hours,
the latter of which might require integration over multiple orbital passes.
Per Figure 16, with 500 successful measurement events a statistical confidence
of $3\sigma$ is achieved for parameter $p\geq$ 0.85. Assuming a source
fidelity of 0.90, Eq. 64 then constrains the noise probability, $\Delta
t_{R}\cdot N_{noise}$, to be less than 0.06 over the measurement interval. The
critical point in evaluating this trade study is that $\Delta t_{R}$ is much
less than the integration period. Using the upper limit of $\Delta
t_{R}\approx 1/(0.003\cdot 1$GHz$)=333\,$ns, the requirement for $p>0.85$ is
satisfied for $N_{noise}<170\,$kcps.

It is worth noting that, e.g., through the 5 s interval of the experiment,
roughly 500 signal counts are resolved against 800 K noise events through the
application of temporal filtering – count events occurring outside the
expected time-of-arrival bins of signal events are excluded from further
analysis. This technique requires time synchronization between source and
receiver, conveniently represented in units of picoseconds of drift per second
of integration. In the case requiring 5 s of integration, the timing
requirement is maintaining temporal synchronization between source and
receiver to reduce the net drift to less than $100\,$ps$/5\,$s$=20\,$ps/s. In
the link scenario requiring 4.5 hours of total integration, the residual drift
is required to be less than $100\,$ps$/4.5\,$hr$=6\,$fs/s; although this
sounds very low indeed, one would incorporate active locking to stay
synchronized, e.g., by distributing a classical clock signal. These figures
can be further partitioned into phase noise, long-term stability, and time-
transfer requirements for local clock systems on each network node of the Bell
test.

We consider 1/2 Earth-Moon baseline for the human-decision Bell tests. In this
scenario, $\lambda=780\,$nm, $D_{Tx}=1.0\,$m, $D_{Rx}=1.0\,$m, $M^{2}=1.05$,
and $\eta_{x}=0.1$. The low efficiency (10-100 photons per hour, depending on
other link assumptions) drives a further requirement of multiplexing the
source to improve the transmitted signal photon count rate to 2-5 photon
counts per second (about the maximum rate that an astronaut involved in a
human Bell test can reasonably accommodate). While multiplexed entangled
photon pair sources are an active area of research Meyer-Scott  _et al._
(2020); Dhara  _et al._ (2021), the high-order multiplexing required to
perform the human-decision Bell tests should be considered a new area of
research.

The overall efficiency of the human-decision Bell test could be increased by
using an even larger aperture receiver on the Earth-side. If an 8.3-m
effective aperture is used (such as NASA’s RF-Optical Hybrid telescope Mohageg
_et al._ (2021, 2020)), the net photon-pair rate improves to 0.2 counts per
second. Without requiring source multiplexing, the detector dark noise
requirement to achieve a $3\sigma$ result is 0.095 noise counts per second.
Such detector performance was realized in Mueller (2021), with a corresponding
detection efficiency of 0.75.

####  II.2.6 Bell Tests: Summary

The proposed Bell test measurement scenarios will test the assumption that
there are no local hidden variable processes at work between inertial frames,
or across long baselines. Executing highly statistically significant tests
across planetary baselines is possible but challenging using existing
technologies. The most ambitious of the human-decision Bell tests require
development of multiplexed source technology, and a commitment to deploying
large, diffraction-limited telescopes with exceptionally low-noise detection
systems in high-earth and Lunar orbits; they would further benefit from
ground-based, large-diameter telescope infrastructure. Performing Bell tests
with high statistical confidence in the low-to-mid orbital regimes would be
possible using a $10^{8}$ pair-per-second source of high fidelity entangled
photon pairs.

###  II.3 Long-Baseline Quantum Teleportation

The third pillar of DSQL science is to perform quantum teleportation over long
distances in space, which acts as a pathfinder for future quantum
communication networks as well as a testbed for studying the interplay between
quantum entanglement and gravity. Planned and operational space-based quantum
optics experiments, most notably Micius Ren  _et al._ (2017), use long
baseline quantum teleportation to test a basic assumption of quantum
mechanics: that quantum correlations of entangled photon pairs, shielded from
the environment, are maintained across any baseline. The overarching goal of
the proposed DSQL quantum teleportation experiments is to test this assumption
across ever-longer baselines, and between inertial frames.

Micius demonstrated the distribution of entangled photon pairs across a
1200-km baseline, and successful uplink of a teleported photon from Earth to
space up to a 1400-km baseline Ren  _et al._ (2017). How much longer should a
new quantum teleportation baseline be to advance the art? We consider four
phenomenological benchmark distances. First, in the context of a future
network of quantum sensors coupled together using a teleportation swapping
system (as described in Ref. Gottesman  _et al._ (2012); Kómár  _et al._
(2014); Kimble (2008)), a global-scale network will require the teleportation
protocol to function across a global baseline. The first baseline benchmark is
thus the diameter of the Earth. The second benchmark range corresponds to the
distance between a geostationary spacecraft and the surface of the Earth. This
range is of practical importance— if future technology development improves
the rate of usable quantum entanglement distribution, geostationary spacecraft
could serve a valuable role in tomorrow’s quantum networks Bedington  _et al._
(2017).

High clock-rate entangled photon pair generation, coupled with high timing
resolution photon time-of-arrival detection, are necessary tools to uncover
relativistic effects in quantum measurement. As these timing parameters
improve, the departure of the gravitational model of the long baseline link
from Newtonian to Schwarzschild, and from Schwarzschild to multi-body, becomes
measurable. Hence, the third benchmark baseline for quantum teleportation
occurs where the experimental timing resolution provides sensitivity to multi-
body gravitational effects. Roughly speaking, this threshold is reached at the
first Lagrange point of the Earth-Moon system. The final benchmark for
teleportation corresponds to the maximum range available given state-of-the-
art technology, using the simple link expression described in Appendix
Appendix A: DSQL System Design. Currently, this range is on the order of the
Earth-Moon mean orbital distance.

Testing teleportation between inertial frames is the other, equally important
motivation for this proposed set of DSQL experiments. Generally, the available
timing performance of the system must be high enough to allow sensitivity to
these inertial effects. In analogy to benchmarking quantum teleportation
baselines based on phenomenological thresholds, the sensitivity to time
dilation between frames is benchmarked against timing resolution. The first
benchmark is when the total predicted time dilation of a given experiment
exceeds the available timing resolution of the measurement apparatus. The
second benchmark is when the contribution to time dilation from relative
velocity (special relativistic effects) and from gravitational-potential
difference (general relativistic effects) are both greater than the system
timing precision. This presents a logical path towards driving future
experiments with ever-improving system timing performance, enabling
sensitivity to higher-order general relativistic effects, such as frame
dragging, Shapiro time delay, and gravitational deflection.

In the Section that follows, the basic concept of quantum teleportation and
its variants like entanglement swapping and atomic state teleportation are
presented. Next, we highlight the subtleties of performing such experiments in
the Earth-Moon system and review the influence of gravitational effects. We
conclude this Section by briefly evaluating the requirements on a flight
quantum memory – which would substantially enhance the viability of the
proposed experiments – and how it could be realized with current technology.

####  II.3.1 The Quantum Teleportation Protocol

Teleportation Bennett  _et al._ (1993); Bouwmeester  _et al._ (1997) is a
quantum protocol where an unknown quantum state is transferred from one system
to another, possibly far away, by using two channels: a) a maximally entangled
state and b) a classical signal. Because of the use of non-classical resources
this protocol does not have an equivalent classical counterpart. In the
photonics domain, teleportation can be described as follows Bouwmeester  _et
al._ (1997): first, an entangled pair is generated, one photon is sent to
Alice, and the other to Bob (see Figure 19). Alice then performs a Bell State
Measurement (BSM) Bell (1964) on her part of the entangled state and the
unknown quantum state, thereby projecting the - innately uncorrelated - two
photons into an entangled state. The BSM will project Bob’s part of the state
onto one of four different possible states, depending on the BSM result
111Note that with linear optical elements (and no ancilla photons), the
projection is not perfect: only 2 of the 4 Bell states give unambiguous
experimental signatures, the other 2 giving the same signature as each other
Calsamiglia and Lütkenhaus (2001). As a consequence, all photonic
teleportation experiments to date have been limited to an efficiency of
$50\%$. Adding extra single-photon input states to the quantum circuit can
boost this efficiency Olivo and Grosshans (2018); Grice (2011); Ewert and van
Loock (2014), and complete (i.e., $100\%$ efficient) Bell-state analysis can
be achieved using various matter-based qubits.. Alice communicates her BSM
result to Bob and based on this information he will suitably rotate the state
he has allowing him to recover the original unknown input state. It is worth
noting that neither Alice nor Bob obtain any knowledge on the input state at
any time, and that the final unitary transformation does not depend on the
input state but only on the result of the BSM. The ‘original’ particle loses
all quantum information during this process. This consequently implies that
quantum teleportation does not permit creating a copy or clone of the original
state - in accordance with the no-cloning theorem in quantum physics Park
(1970); Wootters and Zurek (1982).

Figure 19: Scheme of quantum teleportation Bennett  _et al._ (1993). An
unknown input state is transferred with perfect fidelity using a combination
of distributed entanglement created in an entangled photon source (EPS) and a
classical signal conveying the outcome of the Bell-state measurement (BSM).
The final step is a unitary operation (U), which is contingent upon the BSM
result, and is applied to the entangled twin particle.

Quantum teleportation also applies to the transfer of an entangled particle,
arguably the ultimate unknown state. This leads to a protocol called
entanglement swapping Żukowski  _et al._ (1993); Pan  _et al._ (1998), which
generalizes to multi-partite entanglement Bose  _et al._ (1998). Amazingly,
quantum information and quantum entanglement can be seamlessly transferred or
relayed through a network of quantum particles using quantum entanglement and
classical information as the resources.

The teleportation protocol is not only highly interesting from a fundamental
perspective, but also a crucial concept for multi-user quantum networks which
could be used for secure communications, interlinking quantum computers, or
distributed quantum sensors. The ultimate application of quantum teleportation
or entanglement swapping will make use of quantum memories with ultra-long
storage times (hours, days, or even years). For instance, a space craft could
carry a register of stored, entangled quantum bits (qubit), and gradually use
them up for quantum communication tasks such as quantum networking, secure
communications or super-dense-coding 222In superdense coding one is able to
transmit up to twice the normal amount of information on a successfully
transmitted photon. However, because the protocol must be implemented at the
single-photon level (i.e., the maximum attempt rate is limited by the inverse
bandwidth of the photons), and long space links imply very high loss, it is
nearly always preferable to send bright optical pulses instead, if the goal is
simply to transmit classical bits. proposed by Bennett and Wiesner Bennett and
Wiesner (1992). However, while in principle a quantum memory could store
information indefinitely, current quantum memory systems typically operate
with storage times of order milliseconds or less; see Section II.3.4.

Standard quantum theory places no bound on the distance over which
entanglement or teleportation may be accomplished. However, the required
classical and entanglement channels impose limits on a teleportation protocol.
First, the transfer speed for quantum teleportation is limited to luminal
signaling due to the classical channel: even if the entangled “receiving”
particle is already at its destination, the correct input state can only be
retrieved once the (classical) information about Alice’s Bell State
Measurement has arrived. Second, the protocol would suffer from any
decorrelation or decoherence in the entangled channel used as a communication
resource, and thus any impact of curved space time on entanglement could
impact the teleportation fidelity, see Section II.1.6. The DSQL would allow
studies of these unwanted effects in a realistic environment.

####  II.3.2 State-of-the-Art Long-Range Teleportation

Long-range teleportation was achieved outside a single laboratory starting in
2002, when a signal was teleported between two stations separated by $55\,$m
using optical fiberMarcikic  _et al._ (2003). In 2003, the first long-range
teleportation with an active unitary operation at the receiver was
demonstrated over 500 m Ursin  _et al._ (2004), sending the entangled photon
through optical fiber at $2/3$ the speed of light, while the BSM result was
radioed above ground and “overtook” the entangled photon to arrive in time for
an electro-optic modulator to rapidly apply the correct unitary operation. In
subsequent years, quantum teleportation was demonstrated over increasingly
further distances including demonstrations over 100 km Yin  _et al._ (2012),
over 144 km Ma  _et al._ (2012a), and in 2017 from ground to space Ren  _et
al._ (2017) demonstrating the teleportation of independent single-photon
qubits from ground to a low-Earth-orbit satellite, through an uplink channel,
over distances of up to 1,400 kilometres. These demonstrations represent major
advances, yet all photons involved in the experiments were generated by the
same laser pulse on the same optical bench, and only after their creation was
the receiver photon transmitted over a large distance, i.e., the actual
entangling BSM operation occurred while all the photons were technically still
in or very close to the original lab, which would largely defeat the purpose
in a practical quantum networking application.

A more general implementation of quantum teleportation – entanglement
swapping, which is also used for quantum repeaters Briegel  _et al._ (1998) –
would be implemented with photons generated from independent sources. This is
experimentally much more challenging compared to the previous realizations,
because the different photons must be spectrally and temporally
indistinguishable in order to achieve a high-quality BSM, which is based on
two-photon interference. Typically, entangled photons have a temporal
coherence of $\approx 200-500$ fs, which is right at the limit of
synchronizing lasers. A first demonstration of two-photon interference using
two synchronized femtosecond lasers was reported by Kaltenbaek et al. in 2006
Kaltenbaek  _et al._ (2006), and entanglement swapping shown in 2009
Kaltenbaek  _et al._ (2009).

Another method to realize truly independent optical sources uses two entangled
photon sources operated with continuous wave lasers, and very narrow-band
filtered photons Halder  _et al._ (2007), with associated coherence times of
several hundred picoseconds, longer than the timing resolution of detectors. A
particularly promising approach is to generate entangled photon inside high-
finesse optical resonators; such sources are intrinsically narrow-band, and do
not require frequency filtering, which otherwise severely reduces the
achievable rates. For example, photon pairs were generated in lithium niobate
whispering gallery resonators with coherence lengths tunable roughly between
10 – 20 ns Förtsch  _et al._ (2013). Furthermore, these sources can be
engineered to match the wavelengths and bandwidths of atomic transitions
Schunk  _et al._ (2015), an important factor for implementation of a quantum
repeater 333One important drawback of using such narrow-band sources is that
the relative motion of the platforms will lead to spectral Doppler shifts that
will need to be compensated for, e.g., a relative velocity of 10,000 km/s
leads to a frequency shift of $3.3\times 10^{-5}$, or $\sim$6.4 GHz; if that
is comparable to the bandwidth of the light, it will need to be corrected
before coupling the photon to a quantum memory, or interfering it with a non-
shifted photon..

Another interesting aspect of long-range quantum teleportation is the fidelity
reduction due to the emission statistics of typical realistic photon sources,
including entangled photon sources based on spontaneous parametric down
conversion (SPDC) Kwiat  _et al._ (1995); Zhang  _et al._ (2021) and four-wave
mixing Chen  _et al._ (2005); Camacho (2012). Here the thermal statistics of
the source constrain the probability of creating exactly one photon pair in a
pulse to be $\leq$ 1/4; the empty pulses lead to inefficiency while pulses
with two or more pairs lead to noise; one method to ameliorate this problem
uses multiplexing Meyer-Scott  _et al._ (2020). Alternatively, the Jennewein
group proposed in 2013 Bourgoin  _et al._ (2012) that quantum teleportation
implemented with single emitters (e.g., quantum dots) could greatly improve
teleportation fidelity for ground and space links. The technical challenges
around such emitters make this approach challenging to implement; however,
recently high-efficiency coupling of photons from quantum dot sources into
optical fibers has been realized Zhai  _et al._ (2021), as has generation
(though not yet efficient extraction) of high quality polarization-entangled
photon pairs Huber  _et al._ (2018); Schimpf  _et al._ (2021).

####  II.3.3 Teleportation in the Earth-Moon System

Expanding quantum teleportation and entanglement swapping over large distance
scales would demonstrate truly quantum communication protocols at
unprecedented scales and provide crucial insights into the validity of quantum
mechanics, leading the path towards deep-space quantum networking and quantum
computing.

As stated above, long range “passive” teleportation Pirandola  _et al._ (2015)
from ground to space was accomplished with the 2017 Micius mission Ren  _et
al._ (2017), transferring one of the entangled photons from an SPDC source
from a ground station at very high elevation (around 5000 m) to a receiver on
board Micius, at around 600 km altitude. With the DSQL we want to extend this
range and perform quantum teleportation experiments on the Earth-Moon distance
by, for instance, connecting the International Space Station (ISS) and the
Lunar Gateway (LG) NASA (2020, 2021). While atmospheric photon scattering can
be avoided in outer space, the losses in photons due to the diffraction of
optical beams traversing such a distance will be challenging (see sections
Appendix Appendix A: DSQL System Design). Furthermore, the travel time of a
light signal from Earth’s surface to the Moon is about 1.3 seconds. This
implies that, to complete the protocol of quantum teleportation from Alice on
ISS to Bob on LG, the quantum state carried by Bob’s photon entangled with
Alice’s must be kept longer than a time scale of order 1 second in Bob’s
quantum memory waiting for the final operation (assuming that Bob already
possessed his half of the entangled state before Alice made her Bell state
measurement, i.e., assuming that the entanglement was pre-shared as shown in
the original picture of quantum teleportation (Figures 19), which has not been
the case for most teleportation experiments to date). Fortunately, this may be
achievable by emerging technology Rančić  _et al._ (2018).

In the conventional ground-based experiments requiring transmission of
multiple entangled photons (e.g., for a Bell test or quantum teleportation
verified by full quantum state tomography on a large ensemble of systems),
because of the limited spatial separation of the transmitter and receiver, the
late events in each agent’s worldline would be inside the future lightcones of
the early events of the other agent’s worldline (e.g., Giustina  _et al._
(2015); Shalm  _et al._ (2015); Rauch  _et al._ (2018)), and thus could
causally depend on the outcomes and settings of the early events (e.g., the
events $M^{\prime\prime}_{B}$ and $M_{A}$ in Figure 21 (left)), potentially
opening up a “memory loophole” Barrett  _et al._ (2002). The O(1)-second
travel time of light signals along our long baseline offers the possibility to
perform sufficiently many resolvable runs within this travel time, so that a
whole set of outcomes by one agent for ensemble averaging can each be
spacelike-separated from the measurement events in the same period by the
other agent (Figure 21 (middle) and (right)). To achieve this, however, the
photon emission rate of the source of the entangled photon pairs has to be
large enough to compensate for the high transmission loss of photons over this
large length scale. In the Bell tests this will eliminate the two-sided memory
of the early measurements by the other agent, thereby closing the memory
loophole Barrett  _et al._ (2002) without the need to suppress it by
performing sufficiently many runs of an experiment with memory Gill (2003);
Larsson (2014).

The long-term quantum memories that Alice and Bob would ideally use to store
their quantum state may experience additional effects which can be treated
independently as interactions with their respective environments at finite
temperatures. For example, if Alice’s quantum memory on the (accelerating) ISS
is coupled to the vacuum state of quantum fields with respect to the Earth, it
will experience the Unruh effect Unruh (1976), seeing an effective temperature
due to the acceleration; however, for the ISS acceleration this temperature is
only $\sim 4\times 10^{-20}\,$K, which is much lower than the temperature of
the ambient environment and thus negligible. With a smaller acceleration,
Bob’s quantum memory will see an even lower Unruh temperature in the vacuum
state of the fields.

Since the coupling between photons and gravitational waves is extremely weak,
the gravitational effect on photons in the quantum optical experiments at the
ISS-LG scale are mainly those for electromagnetic fields in a fixed spacetime
background: 1) the gravitational Doppler shift ($\Delta\lambda/\lambda_{0}\sim
10^{-9}$) and 2) the Wigner rotation of polarization, where gravitational
field provides a classical background Lin  _et al._ (2015); Schleich and
Scully (1982); Brodutch and Terno (2011); Brodutch  _et al._ (2015). These can
be negligible compared to similar effects due to the relative motion (the
radial Doppler shift $\Delta\lambda/\lambda_{0}\sim 10^{-5}$ and the
transverse Doppler shift $\Delta\lambda/\lambda_{0}\sim 10^{-10}$), which can
either be suppressed by executing the experiments during periods when the
relative radial motion is minimal, or by dynamically correcting according to
the reference laser beams from the photon sources Ren  _et al._ (2017).

In the passive teleportation Pirandola  _et al._ (2015) achieved by the Micius
mission, the operation supposed to be done by Bob according to the classical
signal from Alice is performed not physically, but virtually via data analysis
Ren  _et al._ (2017) to obtain the fidelity of quantum teleportation. In this
approach Bob does not need to maintain the quantum coherence of his photon or
quantum memory until he receives Alice’s classical signals. To obtain the
fidelity more efficiently, Bob can perform the measurement immediately
[$M_{B}$ in gray in Figure 20 (middle)], randomly choosing which measurement
to perform on different photons in the ensemble. Bob can even perform the
measurement on $B$ before Alice’s joint measurement on $A$ and $C$ in the
bookkeeper coordinates. In this case, Alice can also make a “delayed choice”
on performing the joint measurement on $A$ and $C$ or not Ma  _et al._
(2012b).

Entanglement swapping is more difficult to achieve than a Bell test or
teleportation because more participating photons and observers are involved.
In particular, both Alice and Bob may store part of their states [carried by
$A$ ($B$) of the entangled $Aa$-pair ($Bb$-pair) in Figure 20 (right)] in a
local memory until they perform their local measurements. In each run a joint
measurement on two photons [$a$ and $b$ in Figure 20 (right)], each belonging
to an entangled photon pair produced by Alice or Bob, may be performed by
Diana. Just like the incomplete passive quantum teleportation, the time order
of the measurements by Alice, Bob, and Diana in different bookkeeper’s
coordinates can be different once they are spacelike separated. In particular,
the decision of performing joint measurement or not can be made by Diana later
than both local measurements by Alice and Bob in bookkeeper’s coordinates, so
that quantum entanglement of the photons of Alice and Bob appears to be
determined after the fact by Diana’s delayed choice, as seen in that reference
frame Peres (2000).

With potentially all parties (Alice, Bob, Charlie/Diana) in high relative
motion with respect to each other, it is important to consider the frame
dependence, which is innate with canonical quantization, where one first
chooses a coordinate system and specifies the time coordinate. One then writes
down the Hamiltonian and the Schrödinger equation, and assigns the quantum
state evolved accordingly. Note that simultaneity is relative, and quantum
states in different reference frames are in general incommensurate when their
associated time slices are different. Suppose Charlie in our quantum
teleportation experiment and Diana in our entanglement swapping experiment are
placed on a transfer vehicle between the Earth and the Moon. When the transfer
vehicle is going from the Earth to the Moon, two separate events on the ISS
and the LG that are considered as simultaneous by Alice on the ISS (or Bob on
the LG) will not occur at the same time when perceived on the transfer
vehicle: for Charlie and Diana the LG event would occur before the ISS event.
If the transfer vehicle is instead returning from the Moon to the Earth, the
same events will have an opposite time order, the event on the ISS occurring
first according to Charlie and Diana. The time coordinate of each event in the
transfer vehicle’s frame is determined by the radar signals, namely, after the
transfer vehicle receives the echo of the radar signal emitted earlier by
itself D‘Inverno (1992). In Figure 20 the green dashed lines represent the
time-slices in the reference frame of a transfer vehicle moving from the Earth
(Alice) to the Moon (Bob), while the blue lines represent the time-slices in
the reference frame of a bookkeeper who is roughly at rest both for Alice and
Bob.

Figure 20: Spacetime diagrams of the Bell test ($a$), quantum teleportation
($b$) and delayed-choice entanglement swapping, where Diana may choose to
perform a joint measurement ($c$) or not ($d$). Here, “$\times$” and “$\circ$”
represent local measurement and operation events, respectively. The black
dotted lines and thick solid lines represent the worldlines of the
participating quantum objects.

Figure 21: In a conventional experiment of the Bell test (left), a series of
identical processes are done while a later measurement event can be in the
future lightcone of earlier events (e.g., $M^{\prime\prime}_{B}$ and $M_{A}$).
Between ISS and LG we may be able to achieve the Bell test (middle) and
incomplete quantum teleportation (right) with all Alice’s measurements $M_{A}$
spacelike separated from all Bob’s $M_{B}$ in the period of sampling.

####  II.3.4 Flight Quantum Memory

Quantum memories are essential ingredients for implementing large-scale
quantum networks or long-distance quantum communication channels, where they
are critical for quantum repeaters enhancing the transmission range. In
addition, quantum memories based on atomic or solid-state systems enable
fundamental physics research, like testing atomic teleportation (see Sections
II.3.1 and II.3.3) and performing loophole-free Bell tests (see Section II.2)
over long distances. In all these applications the role of the quantum memory
is to store single or multiple entangled photons in a long-lived state and to
retrieve them reliably on demand.

Throughout the last two decades there has been an enormous effort in
developing and improving quantum memories for photonic qubits, relying on a
variety of physical systems and concepts Simon  _et al._ (2010); Pirandola
_et al._ (2015); Heshami  _et al._ (2016). Most systems studied today can be
assigned to one of the following categories: rare-earth-ion doped solids,
color centers in diamonds, crystalline solids, hot and cold atomic vapors,
molecules, and switchable optical delay lines. The performance of different
quantum memories can be compared by characteristic properties like the
efficiency to retrieve a photon when requested, the fidelity that the
retrieved photon is in the same state as the previously stored one, and the
storage time. Other key parameters include the repetition rate and the ability
to store multiple photons simultaneously. Furthermore, when coupling to an
optical fiber or transmitting device is required, the wavelength and mode
structure of the stored and retrieved photons plays an important role. In
general, most systems show good performance in one or more of these aspects
but have limitations in others, so that the choice of the optimal quantum
memory strongly depends on the planned application.

In order to support complete atomic state teleportation and Bell tests
covering the distance between the Earth and the Moon, the DSQL requires a
quantum memory with storage times longer than one second. Since free-space
transmission is used, the wavelength and mode structure have to be compatible
with the available transceivers (or suitable wavelength/bandwidth converters
must be used). The requirements on the efficiency, fidelity and repetition
rate depend on the details of the respective protocols and are specified in
the corresponding sections. Moreover, size, weight, and power (SWaP) are
limiting resources aboard any spacecraft and have to be accounted for when
choosing a quantum memory platform for the DSQL.

Due to the required long storage times, quantum memories based on molecules
and crystalline solids (including semiconductor quantum dots) are currently
not suited to being used for experiments spanning the Earth-Moon distance.
However, systems employing rare-earth-ion doped solids, diamond color centers,
and atomic vapors are promising candidates and will be briefly discussed in
the following. For a more detailed comparison we refer to the excellent review
articles on this topic Simon  _et al._ (2010); Heshami  _et al._ (2016).

Rare earth ion-doped solids combine long coherence times and good optical
access to collective electronic and nuclear spins. The energy difference
between the ground and excited state of the memory is typically in the low MHz
range. Experiments have demonstrated coherence times of the order of one
second Holzäpfel  _et al._ (2020) up to one minute Heinze  _et al._ (2013),
and even of several hours Zhong  _et al._ (2015). With
${}^{167}\mathrm{Er}^{3+}\mathrm{:Y}_{2}\mathrm{SiO}_{5}$ there is also a
material available that operates close to the telecom bandwidth Rančić  _et
al._ (2018). One downside of these systems is the need for cryogenic cooling
in the regime of 1-4 K, which could limit its implementation in space
missions. However, first steps towards space-compatible cryostats have been
made You  _et al._ (2018).

Vacancy centers in diamond enable the storage of qubits in single electron and
nuclear spins, with the latter providing storage times of up to one second at
room temperature Maurer  _et al._ (2012), or even one minute within cryostats
Bradley  _et al._ (2019). In these systems neighboring spins can interact with
each other, allowing for multi-qubit storage and the two-qubit operations
Waldherr  _et al._ (2014); Bradley  _et al._ (2019) necessary for advanced
quantum repeater applications. In most experiments either neutral or
negatively charged nitrogen or silicon are used to create the defect centers,
leading to optical wavelengths between 700 and $750\,$nm and frequency
differences ranging from 100s of kHz to a few MHz. In order to address single
spins the photon-coupling typically needs to be enhanced with resonators and
cavities Sipahigil  _et al._ (2016); Bogdanović  _et al._ (2017); Häußler  _et
al._ (2019).

Atomic vapors made of alkali metals provide large optical depths even at room
temperature and are therefore well-suited for photonic quantum memories Yu
_et al._ (2020). The qubit is stored in the collective excitation of the atoms
with energy differences of several GHz between the ground and the excited
state of the memory. Since the lifetime is mainly limited by atomic motion –
the photonic state is mapped onto the distributed states of the atoms at the
particular locations when the photon was absorbed – cooling the atomic
ensemble and employing dipole traps or optical lattices can improve the
storage properties, enabling lifetimes of one second Katz and Firstenberg
(2018) and beyond Dudin  _et al._ (2013). In addition, mode matching is a
crucial step for atomic vapor systems and can be enhanced by employing
cavities Bao  _et al._ (2012) or by placing the atomic cloud inside of
nanofibers Sprague  _et al._ (2014). Fundamental atomic physics experiments
generating Bose-Einstein condensates have been realized on a sounding rocket
Becker  _et al._ (2018) and on the ISS Aveline  _et al._ (2020) demonstrating
the general feasibility of such an apparatus in space.

####  II.3.5 Teleportation Mission Design

The procedure outlined in Appendix Appendix A: DSQL System Design is applied
to the quantum teleportation process, characterized by the resultant fidelity
of the teleported state compared to the initial qubit state, as determined by
state tomography Altepeter  _et al._ (2005), for an initial entangled resource
of the form

$\hat{\rho}=p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)\frac{1}{4}\hat{I}.$ (37)

Using maximum likelihood estimation techniques444In this specific example,
Bayesian estimation did not provide a meaningful benefit to calculation time.,
we calculate the resultant fidelity of the teleportation state in the presence
of loss and noise events; see Figure 22. The results are an average over 10
tomographies per data point, with the counts sampled from a Poissonian
distribution to take into account normal counting noise.

Figure 22: The fidelity of simulated tomographies with an input state being
the Werner state in Eq. 37 , and the target state being a maximally entangled
state. The x-axis is parameter $p$ ($p=1$ implies all signal, while $p=0$
implies all noise) and the y-axis is the total number of successful count
events (i.e., for all measurements ) integrated over the measurement duration.
The colorscale ranges from a tomographic fidelity of 0 to 1. Tomographic
fidelities greater than 0.66 are only possible through quantum correlations.

The Bell tests described above require up to two simultaneous optical
channels. The single-channel link efficiency expression in Equation (38)
characterizes the one-way losses of a teleportation experiment. Figure 22
represents the fidelity of quantum tomography as a function of the noise
parameter (x-axis) and the number of successful measurement events (y-axis).
In analogy to the derivation of the instrument requirements for the Bell
tests, we start by deciding what tomographic fidelity is required to meet
experimental goals, then derive instrument requirements from the corresponding
count rate and noise parameter for a successful demonstration of long-baseline
quantum teleportation. For example, achieving a quantum tomography fidelity of
0.90 requires, e.g., a noise parameter of 0.95 and signal counts in excess of
1700. Consider an $F_{clock}$ = 1-GHz clock rate source generating entangled
photon pairs at $810\,$nm with P(1) = 1% pair production per pulse efficiency,
used to close the link between a $0.5\,$-m transmitter aperture and a
$1.0\,$-m receiver aperture across the baseline of the Earth’s diameter, with
$10\,$dB of additional losses assumed contributing to the net link efficiency
(see Appendix Appendix A: DSQL System Design). In this configuration, roughly
250 events per second are expected, requiring $6.8\,s$ of integration to
obtain the desired counting statistics. Achieving a purity in excess of $0.95$
means that for every 20 signal events, there is at most 1 noise event.
Accounting for 2-fold and 4-fold noise counts in a detection system with
$500\,ps$ resolution, the required noise count rate is about $N_{noise}$ =
11.25 noise events per second, commensurate with the capabilities of state-of-
the-art detector systems Mueller (2021), with a photon-time-of-arrival
resolution $\Delta t_{R}$ (and residual timing synchronization error) less
than or equal to the optical pulse width of signal photons. Increasing the
aperture sizes to e.g., $0.5\,$m and $3.5\,$m, results in a higher flux of
signal photons and relaxes the noise requirement commensurately.

####  II.3.6 Quantum Teleportation: Summary

The experiments propose to perform a completely quantum mechanical process—the
teleportation of the quantum state of one photon to another—in a regime where
relativistic effects impact the results. The DSQL will empirically test
whether teleportation across long-range links between inertial frames is
successful, as predicted by standard theory. Successful demonstration of the
quantum teleportation experiments described in this Section will thus provide
critical empirical justification for what are currently untested assumptions
of QFTCST in the weak-field regime. These experimental regimes are not
otherwise achievable in laboratory analog experiments, and truly require
spacecraft links.

Using one or more quantum memories may enhance the teleportation system
performance. The key figures of merit of a quantum memory are its bandwidth
and wavelength, which should be compatible with the signal photons; its read
and write efficiency, which need to be high to avoid introducing more loss to
an already lossy channel; its coherence time and storage time, which are
linked to the efficiencies and need to be of comparable magnitude to the time
of flight between nodes in the network (or the round-trip light time between
two nodes); and the number of storable modes, which needs to be high given the
high clock rates and low link efficiencies of the long baseline channels.
Furthermore, the quantum memory modes should be individually addressable and
exhibit continuous read-out capability.

Ultimately, the performance parameters of a space-qualified quantum memory
that would enhance a proposed DSQL experiment are beyond the current state of
the art. Ground station quantum memory systems are marginally more mature.
While no specific implementation plan is proposed at this time, the general
philosophy of continuing engineering design of the DSQL mission is to ensure
system compatibility with future, ground-based quantum memory systems.

###  II.4 Potential Applications of Squeezed Light

Squeezed states of light are the quantum states that offer a reduced quantum
uncertainty in one quadrature of the electromagnetic mode phase space $(x,p)$
while having an increased uncertainty in the conjugate quadrature. If the area
in the phase space representing the squeezed state retains the minimal values,
the same as for the vacuum or coherent state (such minimum-uncertainty states
are sometimes called the intelligent states), then the squeezed state is said
to have a unity purity and it remains a pure quantum state. If, however, the
phase space area is increased, i.e., the anti-squeezing exceeds the squeezing,
as is often the case in experiment, the squeezed state is (partially) mixed,
and is characterized by a purity value below unity. The purity is therefore an
important parameter in the context of squeezed states applications in quantum
information processing.

One particularly important example of squeezed states is the so-called
squeezed vacuum. This is the state centered on the phase-space diagram so that
$\langle x\rangle=\langle p\rangle=0$, which means that its Fock
representation contains only even photon-number states $|n\rangle$. In spite
of having a zero mean field, the squeezed vacuum carries finite optical
energy, which is uniquely determined by the degree of squeezing.

It should be noted that squeezed states are fragile quantum states that
decohere quickly under loss or other external coupling. Usually squeezed
states are measured with continuous variable measurement techniques, i.e., by
measuring the continuous spectrum of the electromagnetic field variables. The
field variables are typically measured by optical homodyne or heterodyne
detection, while the amplitude squeezing can be measured by directly observing
the reduced optical power fluctuations. The homodyne detection is mode-
selective, i.e., it measures the optical mode defined by a reference beam (the
local oscillator). Hence, any changes in the classical mode structure of a
beam travelling long distances (e.g., red shifts or change of bandwidth or
polarization) will be noticed as decoherence leading to a reduced
interferometric visibility. By systematic modification of the reference beam,
one can deduce a change in the classical mode structure. This technique will
enable one to distinguish between various effects that may act on the optical
fields along their path in space, and effects that are acting on the field
excitation, e.g., the photon statistics.

Squeezed states of light have been thoroughly investigated in the context of
sensitive interferometric measurements surpassing the shot-noise limit. The
first such application of squeezed vacuum was done back in 1987 Xiao  _et al._
(1987). A very prominent modern demonstration of this technology geared for
gravitational wave detection was performed in the context of the LIGO project
The LIGO Scientific Collaboration (2011). In this work a 10-dB squeezed vacuum
source was coupled to the dark port of the LISA interferometer, leading to a
3.5-dB noise suppression below the shot noise level. This limited noise
suppression is due to the loss in the optical system – a technical problem
that needs to be mitigated in all squeezed light applications. More recently
both the LIGO and VIRGO projects have demonstrated sensitivity enhancements
using squeezed light Yap  _et al._ (2020); Acernese and et al. (2019).

Squeezed states and closely related continuous-variable (CV) entangled states
of light have been found useful in the area of quantum information processing.
For example, these states can serve as a building block for linear quantum
computation Knill  _et al._ (2001); Menicucci  _et al._ (2006). The quantum
network applications include CV quantum key distribution protocols Yuen and
Shapiro (1978); Shapiro  _et al._ (1979); Elser  _et al._ (2009), CV quantum
teleportation Yonezawa  _et al._ (2004, 2007a); Ma  _et al._ (2012a) and
entanglement swapping Takei  _et al._ (2005); Yonezawa  _et al._ (2007b). The
CV quantum teleportation typically has higher efficiency (albeit lower
fidelity) then the discrete-value quantum teleportation, and may become the
protocol of choice in situations where the efficiency is a stretched resource
(e.g., over large distances)111Note, however, that discrete variable (DV)
quantum communication (in which quantum states are encoded on single photons)
can often be operated in a post-selected regime, i.e., only keeping cases in
which the photon actually successfully made it through a lossy channnel; in
this case the DV protocols can tolerate much higher loss Boaron  _et al._
(2018). . Another interesting aspect of quantum teleportation, also highly
relevant to space applications, is that it can be achieved not only in
bipartite but also in tripartite systems, in which case Bob receives classical
communications both from Alice and a third party Yonezawa  _et al._ (2007b).
There may be an even larger number of communicating parties, which presents an
opportunity for building a quantum network and implementing various
multipartite quantum communication protocols.

Besides interferometric sensors, quantum communications, and computations
applications, squeezed light can be used in spectroscopy Polzik  _et al._
(1992); Ribeiro  _et al._ (1997), biological research Taylor  _et al._ (2013),
photochemistry, high-resolution imaging Treps  _et al._ (2002, 2003), and
calibration of light detector and sources Brida  _et al._ (2006); Vahlbruch
_et al._ (2016).

Spectroscopic and photochemical applications rely on the fact that squeezed
light has unusual two-photon absorption properties. It is predicted that one
can achieve a linear (rather than quadratic) dependence of the absorption rate
on the optical intensity for weak fields, significantly different absorption
rates for phase- and amplitude-squeezed beams of the same power, and the
possibility of a decreasing absorption rate with increasing intensity Gea-
Banacloche (1989); Javanainen and Gould (1990). Conversely, the enhanced
intensity fluctuations of an anti-squeezed state can enhance the two-photon
absorption compared to coherent or thermal light Gea-Banacloche (1989); Landes
_et al._ (2021).

Calibration of photo-detectors is enabled by strong correlation of the
intensity fluctuations in two-mode squeezed optical beams. Originally proposed
by D.N. Klyshko, and often associated with his name, this method uses two
photon states (e.g., from SPDC) to calibrate a pair of photon-counting
detectors Klyshko (1980): Treating the detection of one of the photons as a
herald guarantees the presence of the other photon directed to the detector
being calibrated: after subtracting noise counts, the detection efficiency is
simply the coincidence rate divided by the singles rate at the heralding
detector (whose efficiency then cancels out) Rarity  _et al._ (1987); Kwiat
_et al._ (1994); Polyakov and Migdall (2007). The method now has been
generalized to a pair of analog detectors Brida  _et al._ (2006); Vahlbruch
_et al._ (2016), and even to a CCD array Brida  _et al._ (2010), in which
cases the two-photon light source is replaced by a two-mode squeezed light
source, and the photocounts are replaced by the photocurrents’ fluctuations.

Finally, squeezed states can potentially act as “probe states”. Here the idea
is that because of the reduced intrinsic noise, any signature imprinted on
such states can be recovered with better fidelity. In the DSQL settings, this
may help determine if the fragile quantum states travel long distances and
along changing gravitational fields without decoherence or dephasing, and
whether it is possible to distinguish different mechanisms of decoherence.
These questions could be investigated by deploying a squeezed light source on
a lunar orbit, and a homodyne detection setup on a second satellite around
lunar orbit (or on Earth). However, the feasibility of such a measurement
depends on whether one can still detect squeezing given the low collection
efficiency typical for such distances, which can be viewed as an additional
high loss of the quantum link. Perhaps a positive answer can be obtained by
changing the approach to the squeezed states measurement from a quantitative
statistical measurement of the squeezed states properties (e.g., variance
measurements) to instead merely distinguish the quantum states, i.e., asking
whether a measured state is more likely to be a squeezed state (with coherence
preserved) or a classical state (e.g., coherent state).

##  III The DSQL Mission

###  III.1 A Sequence of Missions

The experiments described in this manuscript can be achieved by a phased
deployment of spacecraft and ground infrastructure.

  1. 1.

Phase 1: Elliptical orbit with multiple ground stations

  2. 2.

Phase 2: Spacecraft array with multiple ground stations

  3. 3.

Phase 3: Lunar node with extremely large aperture ground station

As indicated throughout the text, spacecraft occupying elliptical orbits are
well suited for explorations of relativistic effects. Phase 1 of DSQL could
involve a single spacecraft in such an orbit. The spacecraft would be
outfitted with an optical payload consisting of: a pair of independently
gimballed telescopes; a high-rate entangled photon pair source 222Note: Many
of the COW-type tests described in Section II.1 are agnostic as to whether the
source is in space with detectors on the ground or vice versa, but there may
be technical reasons to prefer one over the other, i.e., not wanting to fly a
high repetition-rate pump laser, or detectors that require cryogenic cooling.;
a high performance single-photon detection system capable of performing
photonic state tomography; a stabilized fiber optical delay line; and a
reconfigurable optical switch array. The flight terminal requires exceptional
pointing accuracy to leverage larger apertures for high efficiency links.
Recent flight missions have demonstrated performance commensurate with the
requirements Fesq  _et al._ (2019); Israel (2016); Yin  _et al._ (2017a). A
summary of key technology items is provided in Section III.2 below.

The Phase 1 system would enable COW tests, tests of quantum teleportation, and
a subset of the Bell tests between inertial frames. An array of ground
stations, potentially located around the world, could establish quantum
communication links with the spacecraft in support of the experiments
described here, as well as supporting new experiments and technology
demonstrations by a user community.

Phase 2 adds additional spacecraft to the network, in complementary elliptical
orbits with lower orbital period (greater orbital semimajor axis) than the
Phase 1 spacecraft. This array of spacecraft will perform COW tests at larger
baselines, and allow the full range of inertial frames required to achieve the
Bell tests and quantum teleportation tests. One or more of the spacecraft
would be located at a point suitable to support a future human-decision Bell
test, either in a 9-day period orbit (roughly corresponding to a “midway
between Earth and moon” configuration), or in orbit about the fourth/fifth
Lagrange point of the Earth-Moon system.

Phase 3 of DSQL provides the capability to perform quantum optical tests well
into the regime of 2-body gravitational physics, with baselines long enough to
finally perform human-decision Bell tests. Astronauts and large-aperture
telescopes on or near the Moon are assumed, e.g., the link analysis given in
Appendix Appendix A: DSQL System Design considers a 1-m aperture, nearly
diffraction-limited telescope on/near the moon. A ground system on Earth will
need to be established with an extremely large aperture telescope; there are a
number of development efforts underway to produce Earth-based 10-m class
telescopes, e.g., JPL is engaged in the design and deployment of an 8.3-m
class telescope suitable for supporting deep-space classical optical
communications Charles  _et al._ (2011). The ground system further requires a
large collection area coupled efficiently to low-jitter single-photon
detectors. Superconducting nanowire single-photon detectors are a technology
that has demonstrated system detection efficiencies of 98% Reddy  _et al._
(2020), photon number resolution Cahall  _et al._ (2017); Schmidt  _et al._
(2019), dark count rates below $10^{-4}$ cps Shibata  _et al._ (2017), and
timing jitter below 3 ps Korzh  _et al._ (2020) (though not yet all in a
single device).

###  III.2 DSQL Technology Challenges

The DSQL experiments considered and discussed here are at the bleeding edge of
what current technology can accomplish, as is evident from the photon rates
estimated in Appendix Appendix A: DSQL System Design. Therefore, in addition
to the scientific research, we would like to summarize some of the quantum
technology developments that could immensely improve the feasibility of the
proposals discussed in this article.

The challenges involved in technology advances can be divided into three
groups:

  1. 1.

Tasks involved with Research and Development

     * •

Deterministic, high-rate sources of single and entangled photons. For
instance, the currently considered probabilistic sources (e.g. SPDC-based) can
only operate at a fraction of the clock rate due to inherent multi-pair
emissions (see Appendix Appendix D: Obtaining the Parametric Model for Bell
Tests). Solid-state systems such as quantum dots have made huge advances
recently Zhai  _et al._ (2021); Huber  _et al._ (2018); Schimpf  _et al._
(2021), however their integration into flight-suitable, cryogenic packaging
poses challenges.

The long-range teleportation experiments would also benefit from the improved
photon statistics from deterministic single-photons and entangled photons; see
Section II.3.2.

     * •

High-rate sources for entangled photons suitable for multiplexing many optical
channels using spread-spectrum techniques. SPDC entangled photon sources have
already been flown in space, and can be tailored for wide-band operations.
Combined with multi-channel filtering, a large pair-rate enhancement could be
obtained; see Appendix Appendix D: Obtaining the Parametric Model for Bell
Tests.

     * •

Various of the experiments proposed would highly benefit from advanced quantum
memories. While there are many types and configurations of quantum memories
that are explored for ground applications, the DSQL requirements may be
different than the usual ground-based applications, as a benefit is possible
as long as the memory efficiency is higher than a direct link. Although the
COW experiments (see Section II.1.2) may be implementable with shorter delay-
line type memories (i.e., in free space or optical fiber delays), most DSQL
proposals could benefit from long-term flying memories. For example, long-
range teleportation and long-range entanglement Bell-tests would benefit from
a fixed, long-term delay memory (about 1 second), see, e.g., Figure 20, while
the relativistic observer experiments would require a quantum memory with
shorter maximum storage time (few milliseconds) but with adjustable readout
times; see Figure 10.

  2. 2.

Tasks dedicated to Technology Development

     * •

Ultra-high-speed and high-efficiency detectors for single photons (e.g., based
on superconducting nanowires (SNSPD)) have achieved tremendous performance
Reddy  _et al._ (2020); Cahall  _et al._ (2017); Schmidt  _et al._ (2019);
Shibata  _et al._ (2017); Korzh  _et al._ (2020). However, integrating these
devices into a space-amenable system requires further advancement in small-
scale cryogenics, and advancing multi-mode optical fiber interfaces.

     * •

Large-scale space telescopes for transmitting or receiving the DSQL quantum
signals could benefit from larger apertures, and novel approaches including
segmented mirrors or “deployable” systems could be beneficial. For ground
applications, segmented arrays of optical receivers are another option that
could be considered, as the large area of a “photon bucket” may be more
important than precise optical imaging.

     * •

The COW tests require sets of fiber-optic delay lines on different
communications nodes. These delay lines are used in a measurement of photon
phase shift induced by gravity. This effect is on the order of several waves,
which requires a commensurate length stability of the local delay lines.
Stabilization may be achieved through active feedback to an atomic reference.
We note here that this concept for stabilizing the fiber delay line is very
close to an optical clock, which may also be used as a complimentary tool to
explore relativistic effects.

     * •

Precision measurement of satellite range and velocity are required. The COW
tests, Bell tests between inertial frames, and tests of quantum teleportation
all demand precise accounting for the range and the relative velocities
between nodes. As described at length in Section II.1.2, these kinematic
contributions to phase and frequency are orders of magnitude larger than the
relativistic effects DSQL aims to investigate; if left uncorrected, they will
dominate the measurements; advanced spacecraft ranging and velocity
measurements are thus required to implement these experiments.

     * •

Time synchronization between nodes is required to perform quantum entanglement
swapping and teleportation, since the two photons incident on the beam
splitter are required to overlap temporally as well as spatially. The
allowable time-of-arrival error is less than the optical pulsewidth. In the
extreme case, an optical pulsewidth of 1 ps propagating over a 1.3-s light
path (between the Moon and Earth) at 1-10 GHz repetition rate sets the time
synchronization requirement.

  3. 3.

Ground system infrastructure

     * •

Ground-based systems involving multi-meter aperture telescopes need to be
adapted and interfaces for quantum subsystems developed and demonstrated. In
particular, given the limited access to such facilities, a very efficient and
fast DSQL system should be devised.

###  III.3 Other Applications of DSQL

The instrumentation required to achieve the scientific goals described above
is useful for other scientific and technical applications.

For example, a local stabilized laser system is required at all nodes for the
Einstein equivalence principle test. This subsystem could form the basis of an
optical clock, opening the doors to classical clock-comparison experiments.
Used in conjunction with the infrastructure required for the quantum
teleportation experiments, the fundamental elements of a quantum network of
clocks (Re. Kómár  _et al._ (2014)) will be hosted by the DSQL.

As noted above, the extremely long baseline quantum channels require extremely
low noise detection systems to achieve meaningful statistical significance.
The low-noise channel could be exploited in a demonstration of purely
classical optical communication to achieve performance close to the asymptotic
channel capacity limit Shannon (1948); Holevo (1973).

The DSQL telescopes could be directed towards astronomical light sources,
where the high-rate, single photon-sensitive receiver could be used for
narrow-band, high-speed astronomy Shahbaz (2018). Along similar lines, the
quantum state tomography system could be used to assess other astronomical
sources by testing for correlations in polarized light emission. The pair of
telescopes at each node could also be used for a demonstration of a quantum
telescope array Gottesman  _et al._ (2012), where one telescope from each node
is directed towards an astronomical target, and all telescopes are fed by
coherent non-local single-photon states. Coincidence counts between different
telescopes are then used to determine the coherence of the astronomical light
coming to the telescopes (as a function of their baseline separations), and
thereby information about the spatial distribution of the source itself.

##  IV Conclusion

The evolution of quantum states can be predicted using a variety of means
(e.g., the Wigner function formalism), none of which are fully compliant with
Lorentz invariance, as demanded by General relativity for all measurements.
This fundamental discrepancy is at the heart of modern physics and motivates
the body of experiments proposed in this manuscript. As stated in the
introduction, QFTCST is a successful theoretical framework supported strongly
by astrophysical measurement. The proposed DSQL experiments present a means of
testing QFTCST in a complimentary, weak-field setting local to the Earth. The
results of the DSQL tests will have a significant bearing on theories outside
of QFTCST that express coupling between gravitation and quantum states Ralph
_et al._ (2009); Blencowe (2013); Bruschi  _et al._ (2014).

We have proposed a set of experiments that conduct quantum optical
measurements in a regime where relativistic effects are strong and measurable.
The EEP tests propose to assess a hitherto untested prediction of general
relativity – that quantum states of light accumulate the expected phase when
propagating along geodesic paths defined by the local spacetime. The Bell
tests propose to measure violation of Bell’s inequality across extremely long
baselines, and between relatively moving inertial frames Lin and Hu (2021).
The latency associated with the long baseline is sufficiently high to close
the “free-will” loophole through the involvement of astronauts; such human-
decision Bell tests have important philosophical and psychological
implications as well. Finally, the quantum teleportation tests will validate
the prediction that quantum entanglement is maintained over the long baselines
associated with proposals to establish global quantum networks Simon (2017);
Gündoğan  _et al._ (2021).

One to four spacecraft, and one or more optical ground stations, would be
required to execute some or all of the listed experiments. The technology
required to execute the experiments is mostly present. Key quantum technology
development areas are: high-rate, high-purity, multiplexed entangled photon
pair sources; simultaneously high efficiency, low dark-noise, high count-rate
single-photon detector systems; and addressable, high efficiency, high repeat-
fidelity, and long storage time quantum memories. Key classical technology
development areas are large diameter flight telescopes, radiation-hard high-
speed read-out electronics, and modified existing optical ground stations,
upgraded with the infrastructure required to close quantum optical links. The
technology development, instrument development, and mission execution will
benefit immensely from international cooperation and long-term strategic
planning.

##  V Abbreviations

National Aeronautics and Space Administration (NASA), Deep Space Quantum Link
(DSQL), Lunar Gateway (LG), International Space Station (ISS), General
Relativity (GR), Quantum Field Theory in Curved Spacetime (QFTCST), Colella-
Overhauser-Werner (COW), Hong-Ou-Mandel (HOM), Einstein’s Equivalence
Principle (EEP), Universality of Free Fall (UFF), Local Lorentz Invariance
(LLI), Universality of Gravitational Redshift (UGR), Low Earh Orbit (LEO),
Medium Earth Orbit (MEO), Geostationary Earth Orbit (GEO), Global Positioning
System (GPS), Mach-Zehnder interferometers (MZI), Ghirardi-Rimini-Weber-Pearle
(GRW-P), Diosi-Penrose (DP), nastopoulos-Blencowe-Hu (ABH), Alternative
Quantum Theories (AQT), Newton-Schrödinger equation (NSE), Bell State
Measurement (BSM), Spontaneous Parametric Down Conversion (SPDC), Field Of
View (FOW), Clauser, Horne, Shimony, and Holt (CHSH).

##  VI Acknowledgments

This research was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (80NM0018D0004). ©2021. All rights reserved. This work
was supported by the Biological and Physical Science Division of NASA
headquarters through a contract with the Jet Propulsion Laboratory, California
Institute of Technology. B. L. Hu is supported by NASA/JPL grant 301699-00001.
A. Roura is supported by the Q-GRAV Project within the Space Research and
Technology Program of the German Aerospace Center (DLR). T. Jennewein
acknowledges support by the Canadian Space Agency (FAST). G. Vallone and P.
Villoresi are supported by the Agenzia Spaziale Italiana (2018-14-HH.0, CUP:
E16J16001490001, Q-SecGroundSpace, 2020-19- HH.0 CUP: F92F20000000005 Italian
Quantum CyberSecurity I-QKD); INFN MoonLIGHT-2; European Union’s Horizon 2020
Framework research and development program (857156, OpenQKD), Ministero
dell’Istruzione, dell’Università e della Ricerca (Departments of Excellence,
Law 232/2016).

## Appendix A: DSQL System Design

The basic link expressions characterizing optical transmission paths are
expressed in this Section. These are developed following the procedure of
Klein and Degnan (1974); Degnan and Klein (1974) and partially carried out in
analogy to a previously published evaluation of low earth orbiting satellite-
enabled quantum key distribution Bourgoin  _et al._ (2012); Polnik  _et al._
(2020); Mazzarella  _et al._ (2020b).

The basic one-way link efficiency for coupling a Gaussian mode through
perfectly aligned circular apertures is

$\eta_{L}(\vec{R})=\eta_{x}(\vec{R})\left(1-\exp\left(\frac{-2\pi^{2}D^{2}_{Tx}D^{2}_{Rx}}{\pi^{2}D^{4}_{Tx}+\left(M^{2}\right)^{2}|\vec{R}|^{2}\lambda^{2}}\right)\right).$
(38)

In Equation 38 Johnson (2021), light of wavelength $\lambda$ is transmitted
through an aperture of size $D_{Tx}$ with diffraction limit factor $M^{2}\geq
1$. The light is directed towards a receiver aperture of diameter $D_{Rx}$
across a displacement vector of $\vec{R}$. Most of the DSQL experiments are
reasonably characterized using the assumption $|\vec{R}|\gg\pi
D^{2}_{Tx}/\lambda$, i.e. the diffracted spot at the receiver is much bigger
than the receiver aperture. In this limit Equation 38 then reduces to the
ratio of the receiver aperture to the spot size formed by the transmitter
telescope at the receiver plane:

$\eta_{L}(\vec{R})\approx\eta_{x}(\vec{R})\left(\frac{D^{2}_{Rx}}{\left(M^{2}\left(\frac{\lambda}{D_{Tx}}\right)|\vec{R}|\right)^{2}}\right),$
(39)

where the prefactor $\eta_{x}$ characterizes other loss factors:

$\eta_{x}=\eta_{Rx}\eta_{D}\eta_{Tx}\eta_{atm}(\vec{R})\eta_{margin}.$ (40)

Here, $\eta_{Rx}$ characterizes the receiver efficiency (except for the
detection efficiency) Degnan and Klein (1974), $\eta_{D}$ is the total photon
detection efficiency of the receiver, $\eta_{Tx}$ characterizes the
transmitter efficiency, clipping efficiency and pointing errors Klein and
Degnan (1974), $\eta_{atm}(\vec{R})$ characterizes absorption through the
atmosphere (which is a strong function of horizon angle), and $\eta_{margin}$
accounts for any additional inefficiencies of the link.

Equation 38 parametrically describes the link efficiency of the quantum
channel in terms of the indicated instrumentation performance parameters. For
single photons created at rate $F_{clock}$, the received photon flux $N_{s}$
in units of counts per second is

$N_{s}=F_{clock}\cdot\eta_{L}.$ (41)

Similarly, if only one photon is transmitted from an entangled photon pair
source operating at clock rate $F_{clock}$ and a per-pulse photon pair
production probability $p(1)$, the number of received photons is

$N_{e}=F_{clock}\cdot p(1)\cdot\eta_{L}.$ (42)

Finally, closing a simultaneous link to a pair of receivers, located at
$\vec{Z_{1}}$ and $\vec{Z_{2}}$ relative to the source, has an expected total
rate of successful events

$N_{2e}=F_{clock}\cdot p(1)\cdot\eta_{L}(\vec{Z_{1}})\eta_{L}(\vec{Z_{2}}).$
(43)

Note that in the limit of perfect spatial and temporal acquisition, the total
number of successfully recovered photons is the integral of either Equation 42
or 43 over the time that the spacecraft maintains clear line-of-sight with the
other ends of the network 333Due to increased effects of atmospheric
turbulence, it is generally assumed that a stable optical link cannot be
established for elevation angles less than $20\deg$ above the horizon
($\theta_{m}$ in Figure 23).. This can be approximated using the product of
the relevant rate equation and the total integration time per orbital passage.
The integration time can be extracted through geometry and numerical analysis
of the various orbital configurations described in the sections above Johnson
(2021). Link expressions such as Equation 43 require line of sight between the
source and two other nodes, which must be satisfied simultaneously.

The integration time is determined by the orbital dynamics. Our example
scenario is an Earth-orbiting spacecraft closing link with a single Earth
ground station, though this does not capture the diversity of spacecraft-to-
spacecraft links, links between Earth and Moon orbiters, or links to multiple
ground stations. Nevertheless, since many of the experiments described in this
report do rely on links between ground stations on Earth and Earth-orbiting
satellites, it is an instructive example. This situation is depicted in Figure
23. The total integration time $T$ is obtained through geometry and orbital
dynamics. In the regime of perturbation-free, circular orbits about a circular
Earth, the integration time can be computed using Equation 44.

$\centering T=2\frac{\sqrt{R_{e}^{2}+a^{2}-R_{e}a\cdot
sin(\theta_{m})}}{\sqrt{\frac{GM_{e}}{a^{3}}}-\Omega_{e}}\frac{cos(\theta_{m})}{a}.\@add@centering$
(44)

Figure 23: A greatly simplified orbital model to calculate integration time
for quantum optical experiments. A ground station is located on Earth, which
rotates at rate $\Omega_{e}$. The spacecraft is in an orbit characterized by
orbital altitude $h$, or semimajor axis $a=h+R_{e}$, where $R_{e}$ is the
Earth mean radius. The orbital frequency of the spacecraft is $\omega_{s}$.
The ground station is limited to view $\theta_{m}$ above the local horizon
angle. The shaded area in the diagram represents the part of the orbit where
line of sight is maintained, in a reference frame rotating with the Earth.

A separate calculation estimates the total noise count rate incurred during
the measurement process. Three sources of noise events are considered: the
intrinsic dark count rate of the receiver $D_{r}$; the rate of extra photon
events from the source $S_{n}$, and background photons counted by the
detection system $B_{sky}$. The total noise count rate, $N_{noise}$, in units
of counts per second, is then:

$N_{noise}=\eta_{Rx}\cdot\left(W\cdot A\cdot\frac{(FOV)^{2}}{4}\cdot
BW+S_{n}\right)+D_{r},$ (45)

where $FOV$ is the telescope linear field of view, A is the primary mirror
collection area, and $BW$ the filtering bandwidth; $W$ is the background
radiance in units of photon flux per area-solid angle per unit bandwidth.

## Appendix B: The Human-Decision Bell Tests in the Context of Free Will

A human-decision Bell test requires humans on both sides of the Bell test- a
human “Bob” and a human “Alice”. In the context of a future NASA space
mission, this may involve astronauts on the International Space Station
(“Bob”) and astronauts on or near the surface of the Moon (“Alice,” or,
perhaps “Artemis”.) A local explanation only needs to predict or influence the
detector settings on one side to violate Bell’s Inequality and match the
predictions of quantum mechanics, i.e., a local scheme can thus still violate
Bell’s inequality even if only one side’s random number generator is perfectly
free will, independent, and unpredictable. Any amount of unpredictability will
suffice to show a Bell violation, but the significance of the violation
depends on the predictability.

While in this experiment, the human choices are not strictly required to pass
tests for randomness, these free-will choices must be unpredictable to the
measurement on the other side in a way that is different from all prior
schemes of generating randomness for Bell tests. At minimum, this requires one
of two assumptions, as we now describe.

First, one can take the scientifically accepted materialist view that human
choices are results of physical processes in the brain, i.e., some
probabilistic combination of deterministic computation and randomness, from
the external environment or internal thermal, quantum, or chaotic processes.
For human (or, perhaps animal) choices to make a difference, one must assume
that the complexity of the process that links physical inputs in the brain’s
past light cone to a decision must exceed some threshold such that the other
side cannot compute, predict, or influence the decision.

Alternatively, one can drop the assumption that human choices are purely
results of physical processes in the brain, and instead adopt a stance like
Cartesian dualism444https://plato.stanford.edu/entries/dualism/, where one
invokes some external non-physical mind that is somehow able to inject events
into our 3+1 dimensional spacetime while not being part of it. This is similar
to the assumption that is required to close the freedom-of-choice loophole
when quantum random number generators are used in the “loophole-free”
experiments cited above—a truly novel bit of information enters the world such
that a 0 and 1 are both perfectly compatible with identical past light cones.
The only difference here would be that somehow “will” is involved, not just
“freedom”.

The experiment must be implemented in a way that ensures the participants feel
their choices come from their own free will, while simultaneously being
unpredictable; otherwise, the physical state of their brains might already be
zeroing in on a choice. For example, the participants could be asked a series
of questions that they did not know or even consider in advance. For an
astronaut on the moon, it could be questions about their next series of
meals—something that matters to them enough to feel they are exercising free
will, but something where their answers are not predictable, e.g., “instant
coffee or instant tea,” to use an astronaut version of Sam Harris’ opening
question in his book about free will Harris (2012). Their answers would need
to be registered and turned into polarizer settings in a way that is space-
like separated from the source and the measurement on the other side. The
composition and phrasing of the questions asked of the astronauts would need
to be carefully considered by experts in behavioral psychology and philosophy
of the mind.

Other than choosing basis settings, there is a second way in which the human
Bell test addresses foundational principles of quantum theory, going back to
Wigner’s suggestion Wigner (1961) that quantum collapse could somehow be
caused by conscious minds. There is barely enough time in an Earth-Moon
experiment for measurement results to be shown to participants such that each
becomes conscious of the results in a way that is space-like separated from
the other. If collapse only happens in conscious minds, no experiment to date
has actually closed the locality loophole. One may even consider moving
macroscopic masses based on the measurement results, to address Penrose’s
suggestion Penrose (1989) that collapse takes place between macroscopically
distinct gravitational fields.

David Hume states that the question of free will is “the most contentious
question of metaphysics” Hume (1748). An age-old discussion, beginning in the
ancient Western philosophical texts of Plato and Aristotle, and continuing to
the present day, the question of whether human decisions are based in genuine
free will or are deterministic, is still open. Determinism is the idea that
everything in the universe is determined by causal laws. This means that
anything in the universe that happens at any given moment is the result of
some antecedent cause. Thus, determinism maintains that there is no such thing
as an uncaused event. The idea that every event is caused, is one of the
fundamental presuppositions of science Pojman and Fieser (2008). According to
determinism, since human actions are events, no human action is uncaused, and
therefore are not free, and instead are simply the result of some causal
process. Additionally, determinism precludes randomness – since everything is
the effect of some previous cause, nothing is truly random.

If human beings can exercise free will, then humans are able to perform
uncaused acts. For instance, at time $t_{1}$, an agent $S$ can perform either
act $A_{1}$ or act $A_{2}$. So, $S$’s action at $t_{1}$ will determine what
the world looks like after $t_{1}$, regardless of the pre-existing conditions
at $t_{1}$ Pojman and Fieser (2008). An uncaused act should be unpredictable,
non-hysteretical, and otherwise stochastic. There is a subtle difference
between making a “knee-jerk” or instinctual reaction to some stimuli versus
taking time to internally deliberate decision before action Lamont (1969).
This distinction underscores the timing latency requirement in human-decision
Bell tests. It also suggests alternative testing schemes with either shorter
decision time intervals (forcing an instinctual decision) or longer time
intervals (allowing some level of thought before decision making), or even
using non-human animals as the decision-making agents.

Free will may be non-random and predictable by the local observer themself.
Free will is independent of the observed system, assuming that our universe
consists of 1) free will of observers and 2) the observed world; both
assumptions follow from causality.

The question of free will versus determinism is based in the question of
causality: whether every event must have a cause, or if there are events that
are causally undetermined, a question that impacts the value of scientific
inquiry. In this broader context, a Bell test involving human decision-making
creates an empirical framework with which to assess the idea of free will, and
to explore the relationship between human decision making and the causal
trajectory in nature leading to the moment of decision.

## Appendix C: Obtaining the Parametric Model of the COW Tests

The state of the interferometer output (see Figure 3) can be written as:

$|\psi\rangle=\frac{1}{2}\left((e^{i(\phi_{GR}+\theta)}-1)|0,1\rangle+i(e^{i(\phi_{GR}+\theta)}+1)|1,0\rangle\right),$
(46)

where the expression for $\phi_{GR}$ is given in Equation 8, and $\theta$ is
an additional controllable phase that can be tuned to improve the measurement
precision, by biasing to the linear part of the fringe. We can model the
experimental imperfection by assuming the preparation contains our target
state $|\psi\rangle$ with probability $p$, and a noise photon with probability
$(1-p)$. The flux of noise photons $N_{noise}$ has been defined in Equation
$(\ref{eq:NoiseEq})$. The parameter $p$ can be interpreted as the experiment
quality factor and can be linked to the system parameters such as timing,
receiver aperture, and spectral filtering bandwidth as follows:

$p=(1-N_{noise}\Delta t_{R})F.$ (47)

Here $(N_{noise}\Delta t_{R})$ is the probability of recording a count due to
a noise photon within the expected coincidence time window $\Delta t_{R}$, and
$F$ is the fidelity of the source, assumed here to be 0.95. We further assume
all the system parameters in $N_{noise}$ to be fixed apart from the spectral
filtering bandwidth that needs to change according to the signal photon
bandwidth, so that $p=p(\sigma)$. Keeping into account the finite bandwidth
$\sigma$ of the photon having frequency $\omega_{0}$, the probability of
having a detection at detector A is given by:

$P_{A}=\frac{p(\sigma)}{2}(1-e^{-2\sigma^{2}\tau^{2}}\cos(\phi))+\frac{(1-p(\sigma))}{2},$
(48)

where $\phi=\phi_{GR}+\theta$ and $p$ should be regarded as an overall quality
factor for the experiment. A more complete analysis should also take into
account error sources such as path length mismatch and attitude determination
error; these as well other source of imperfection will be explored in detail
elsewhere.

Let $M$ be the operator denoting a count at detector A; the error on the
gravitationally induced phase shift between the interferometric arms is then
given by:

$\displaystyle\Delta\phi(\phi,\sigma)\equiv\frac{\Delta
M(\phi)}{\left|\frac{\partial\left<\hat{M}\right>}{\partial{\phi}}\right|}=\frac{\sqrt{\left<\hat{M^{2}}\right>-\left<\hat{M}\right>^{2}}}{\left|\frac{\partial\left<\hat{M}\right>}{\partial{\phi}}\right|}$
$\displaystyle=\frac{\sqrt{1-p^{2}e^{-4\frac{\sigma^{2}\phi^{2}}{\omega_{0}^{2}}}\cos^{2}(\phi)}}{pe^{-2\frac{\sigma^{2}\phi^{2}}{\omega_{0}^{2}}}\left|4\frac{\phi^{2}}{\omega_{0}^{2}}\sigma^{2}\cos(\phi)+\omega_{0}\sin(\phi)\right|}.$
(49)

Where we have used $\tau=\phi/\omega_{0}$ .The quantity $\Delta
M(\phi)\equiv\sqrt{\left<\hat{M^{2}}\right>-\left<\hat{M}\right>^{2}}$ is
often referred to as an “estimator” for the unknown quantity $\phi$. Given an
experiment with a certain quality factor $p(\sigma)$, we can choose the
overall relative phase $\phi$ in order to minimize the previous expression.
More formally, we can solve the optimization problem:

$\Delta\phi_{opt}(\sigma)=\min_{\phi}\Delta\phi(\phi,\sigma).$ (50)

The optimized phase error can then be propagated in order to obtain the error
on the parameter $\alpha$, characterizing violations of UGR. The result of the
optimization is shown in Figure 24, showing that $\Delta\phi_{opt}$ is
essentially constant for the range of the photon bandwidth considered as
expected – as long as the interferometers are well matched, so that the path
imbalance is less than the coherence length, the fringe visibility will be
very high.

Figure 24: Plot of the phase error, $\Delta\phi_{opt}$, versus the photon
bandwidth.

## Appendix D: Obtaining the Parametric Model for Bell Tests

A simple but somewhat old-fashioned way to estimate the statistical
significance of a Bell test is to assume uncorrelated trials along with
unbiased and uncorrelated random basis selection, though this approach does
not address the detector efficiency or memory loopholes. Under these
assumptions, there are two types of effects: systematic imperfections which
lower the intrinsic and/or measured entanglement fidelity of the produced
entangled Bell state, and purely statistical fluctuations on the measurements
due to Poisson photon counting statistics. A lower measured entanglement
fidelity means a CHSH parameter $S$ that is less than the quantum mechanical
maximum of $2\sqrt{2}$, but still hopefully greater than 2, the maximum value
that local realism allows. Statistical fluctuations would lead to a measured
value of $S$ drawn from a distribution centered around an expectation value
(proportional to the entanglement fidelity of the measured quantum state) with
a standard deviation of $\sigma$. A 5-sigma result would mean that the
measured value of $S$ was 5$\sigma$ above the local-realist limit of 2.

Many effects can degrade the measured entanglement fidelity: intrinsic
entanglement fidelity of the source, detector dark counts, sky background, and
noise from multi-photon events, i.e., having photons from different entangled
pairs each arrive within the coincidence window. In all of these cases, the
measurement results on each side of the experiment are completely random and
uncorrelated with each other. Experimentally, they produce results that are
indistinguishable from those produced by a completely incoherent “mixed”
state. As in the previous section, to model the mixed state as measured by the
pair of detectors, we take a fraction $p$ of a particular Bell sate, say
$|\Phi^{+}\rangle$, and a fraction $(1-p)$ of the completely incoherent state
of dimension 4. The mixed state that is actually measured, where the $(1-p)$
contribution includes both the source’s intrinsic incoherence and the external
noise, is

$\hat{\rho}=p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)\frac{1}{4}\hat{I}.$ (51)

Given this mixed state, we will first calculate the expected value of the CHSH
parameter $S$ and its statistical uncertainty $\sigma$, assuming $N$ total
coincidence measurements. To this end, we first define an experimentally
measured correlation coefficient $E(a,b)$ as a function of measurement basis
settings $a$ and $b$; its value ranges from $-1$ to $+1$:

$E(a,b)\equiv\frac{N(a,b)+N(a_{\perp},b_{\perp})-N(a,b_{\perp})-N(a_{\perp},b)}{N(a,b)+N(a_{\perp},b_{\perp})+N(a,b_{\perp})+N(a_{\perp},b)},$
(52)

where $N(a,b)$ is the number of coincidences where Alice’s photon passes
through an analyzer with setting $a$ (e.g., for polarization entanglement, a
polarizing beam splitter oriented at angle $a$), and Bob’s photon passes
through an analyzer with setting $b$. Similarly, $N(a,b_{\perp})$ is the
number of coincidences where Bob’s photon is instead detected in the
$b_{\perp}$ output of his measurement apparatus (e.g., his polarizing beam
splitter). There will be 16 such coincidence measurements; the sum of all 16
counts is $N:=\sum_{i}N_{i}$.

The CHSH parameter is then defined as the sum of four correlation
coefficients, with the sign flipped on the coefficient with the widest
separation between the measurement basis angles:

$S=E(0,\tfrac{\pi}{8})+E(0,-\tfrac{\pi}{8})+E(\tfrac{\pi}{4},\tfrac{\pi}{8})-E(\tfrac{\pi}{4},-\tfrac{\pi}{8}).$
(53)

For the mixed state in Eq. (51), the expected CHSH parameter $\langle
S\rangle=2\sqrt{2}p$ 555One could also consider state imperfections of the
form $p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)(|HH\rangle\langle
HH|+|VV\rangle\langle VV|)/2$, i.e., a partially entangled state, with perfect
correlations only in one basis. In this case the CHSH parameter has an
expected value of $\langle S\rangle=2\sqrt{2}p+2(1-p)$.. This reaches the
quantum mechanical maximum of $2\sqrt{2}$ for the pure Bell state and 0 for
the completely incoherent state, where there are no correlations.

Assuming each of the 16 counts $N_{i}$ is drawn from a Poisson distribution
whose standard deviation $\sigma_{i}$ is $\sqrt{N_{i}}$, the variance of $S$
can be calculated through propagation of error as

$\sigma^{2}=\sum_{i=1}^{16}\sigma_{i}^{2}\left(\frac{\partial S}{\partial
N_{i}}\right)^{2}=\sum_{i=1}^{16}N_{i}\left(\frac{\partial S}{\partial
N_{i}}\right)^{2},$ (54)

whose expectation value for the state in (51) is

$\left<\sigma^{2}\right>=\frac{8}{N}(2-p^{2}).$ (55)

To claim an $n\cdot\sigma$ violation of Bell’s inequality,
$S_{\textrm{measured}}-2>n\sigma$. The expectation value of $n$, the number of
$\sigma$’s of Bell violation, is then

$\textrm{\\# of $\sigma$
violation}=\left<n\right>=\sqrt{N}\frac{p-\tfrac{1}{\sqrt{2}}}{\sqrt{2-p^{2}}}.$
(56)

A contour plot of this expected number of $\sigma$ violation as a function of
Bell-state fraction $p$ and total coincidence counts $N$ is shown in Figure
16. If $p\leq\tfrac{1}{\sqrt{2}}$, no Bell violation would occur, and the
result would be compatible with local realism no matter how large $N$ is. Near
this threshold, quality (high $p$) wins over quantity (high $N$). Above this
threshold, the significance of the result scales as $\sqrt{N}$ as might be
expected. In practice, once the entanglement fidelity threshold is crossed,
accumulating enough data does not take very long. This is qualitatively
different than many other physics experiments, where one can “average down”
the noise.

Now we proceed to estimate $p$ for various space scenarios, determine how many
coincidences $N$ are required to achieve a given $\sigma$-level of Bell
violation, and estimate the time required to achieve this. An important
parameter is $t$, the time window inside of which two photons will be counted
as a coincidence. This should be as short as possible to avoid accidental
coincidences from dark counts, sky background, or incorrectly paired entangled
photons. However, it does not help for $t$ to be shorter than the combined
effect of the time resolution of the detectors, the jitter in the amount of
atmospheric delay, and the timing jitter of the optical and electronic
systems; otherwise, the signal of true coincidences is also reduced. The
intrinsic jitter of the superconducting nanowire single-photon detectors
(SNSPD) is 0.1 ps, determined in part by material parameters of the nanowire.
The best reported performance is 3 ps, but more typically 30 ps for optimized
nanowire and electronics. The atmospheric jitter is typically 10 ps Kral  _et
al._ (2005). Thus, accounting for 10-100 ps of excess atmospheric jitter
captures the expected dynamics. Next we will see that this window $t$ sets the
scale for maximum source brightness and maximum allowable background.

If photons (entangled or not) arrive at a rate $r$ and are Poisson
distributed,

$P(k\textrm{ photons in time window }t)=\frac{(rt)^{k}e^{-rt}}{k!}.$ (57)

For small rates such that $rt\ll 1$,

$P(\textrm{1 or more photons in time window }t)=1-e^{-rt}\approx rt.$ (58)

If Alice is receiving photons at a rate $r_{a}$ and Bob at a rate $r_{b}$, the
probability of recording (though not necessarily detecting) an accidental
coincidence within the small coincidence window $t$ is
$P_{\mathrm{accidental}}=(r_{a}t)(r_{b}t)$. The rate that accidental
coincidences occur is $r_{\mathrm{accidental}}=r_{a}r_{b}t$.

Next we turn to an imperfect source of entangled photons. Without knowing more
about the source itself and any potential dependence on measurement basis
choices, we model the source itself as producing pure entangled photons at a
rate $r_{e}$ along with completely incoherent photon pairs at a rate $r_{i}$.
For good sources, $r_{i}$ will be <5% of $r_{e}$. If noiseless detectors were
to measure the source directly with no other background, $p$ would be
$r_{e}/(r_{e}+r_{i})$. Both of these rates are reduced by $\eta_{a}$ for
Alice’s link and by $\eta_{b}$ for Bob’s link.

Finally, we turn to backgrounds counts and dark counts, whose rates can be
added together. For nanowire detectors at the wavelengths of interest, dark
counts can be less than 1 per second, assuming (great) care is taken to
exclude thermally excited blackbody radiation. Depending on the wavelength,
observatory location, phase of the moon, and size of the telescope, the total
rates of noise counts $n_{a}$ and $n_{b}$ are somewhere between 0.1 and 1000
counts per second.

The full accidental-pairing rate should take into account mispairings between
any combination of photons from different entangled pairs, incoherent photons
emitted in pairs by the source, background photons, and dark counts. Each
observer’s photon detection rate is the sum of background noise counts $n_{a}$
or $n_{b}$ and photon pairs, which are created at the source at a rate
$(r_{e}+r_{i})$, and which reach the observers with efficiencies $\eta_{a}$
and $\eta_{b}$. This makes

$\displaystyle r_{a}$ $\displaystyle=$
$\displaystyle(r_{e}+r_{i})\eta_{a}+n_{a},\quad\textrm{and}$ $\displaystyle
r_{b}$ $\displaystyle=$ $\displaystyle(r_{e}+r_{i})\eta_{b}+n_{b}.$

The accidental coincidence rate is then

$r_{\mathrm{accidental}}=(r_{e}\eta_{a}+r_{i}\eta_{a}+n_{a})(r_{e}\eta_{b}+r_{i}\eta_{b}+n_{b})t,$
(59)

where different terms on each side might dominate for asymmetric links.

This is to be compared to the rate of proper entangled pairs arriving.
Assuming the coincidence window is set so that the vast majority of entangled
pairs that do make it arrive within the window, that rate is just

$r_{\mathrm{entangled}}=r_{e}\eta_{a}\eta_{b}.$ (60)

With these definitions, the purity of the state $p$ as measured at the noisy
detectors exposed to background light is

$p=\frac{r_{\mathrm{entangled}}}{r_{\mathrm{entangled}}+r_{\mathrm{accidental}}}=\frac{1}{1+r_{\mathrm{accidental}}/r_{\mathrm{entangled}}}.$
(61)

Unless $p>\tfrac{1}{\sqrt{2}}$, no amount of data taking will violate Bell’s
Inequalities. This corresponds to
$r_{\mathrm{accidental}}/r_{\mathrm{entangled}}<0.41$. If this holds and $p$
is high enough, the number of photon pairs required to achieve an $n$-$\sigma$
result can be computed using Equation (56) or looked up in Figure 16. For
example, per Figure 16, conducting a Bell test to demonstrate a CHSH
inequality violation of $3\sigma$ requires 500 counts with entanglement
fidelity of $p=0.85$. To conduct the same test to a violation of $6\sigma$
requires 1000 counts with $p=0.90$. Achieving this level of performance
requires increasingly stringent instrumentation performance requirements as
the baseline is increased, since the signal falls off while the detector noise
does not.

The two parameters represented in Figure 16, $N$ and $p$, are partitioned into
high-level subsystem performance requirements sufficient to achieve the
desired level of statistical confidence. The y-axis value, $N$, is computed by
multiplying Eq. 42 or Eq. 43 by the duration of the experiment.

Next we consider, the effects of noise counts. Equation 51 characterizes
measurements in terms of a noise parameter $p$. This parameter is composed of
a contribution of external noise events (Eq. 45) and the quantum fidelity $F$
of the source. First, the a source with fidelity $F$ is:

$\hat{\rho}_{in}=F|\Phi^{+}\rangle\langle\Phi^{+}|+(1-F)\frac{1}{4}\hat{I}.$
(62)

The receiver sums photon counts over a time interval $\Delta t_{R}$,
corresponding to a frame-period, word-length, or user-defined integration
window. The smallest value $\Delta t_{R}$ could take would be the total timing
jitter of the receiver. The largest value would be the effective frame-rate of
the receiver system, which would be no smaller than the inverse of the product
of transmitter clock rate and single-side channel loss. The probability of
counting a noise event within this period is $P_{N}=\Delta t_{R}\cdot
N_{noise}$. A limiting case describing the worse-case effect of noise events
on the measurement process is expressed in Equation 63:

$\hat{\rho}_{out}=P_{N}\hat{\rho}_{in}+(1-P_{N})\frac{1}{4}\hat{I}.$ (63)

Combining Eq. 62 and Eq. 63 with Eq. 51, the noise parameter $p$ can be
reduced to source fidelity, noise count rate, and coincidence window time
using

$p=(1-N_{noise}\Delta t_{R})F.$ (64)

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