./figures/

# LEO Clock Synchronization with Entangled Light
††thanks: Approved for Public Release: Distribution is Unlimited;
$\\#$23-0869.

Ronakraj Gosalia1, Robert Malaney1, Ryan Aguinaldo2, Jonathan Green2 and Peter
Brereton3  1University of New South Wales, Sydney, NSW 2052, Australia.  2
Northrop Grumman Corporation, San Diego, CA 92128, USA.  3 NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA.

###### Abstract

Precision navigation and timing, very-long-baseline interferometry, next-
generation communication, sensing, and tests of fundamental physics all
require a highly synchronized network of clocks. With the advance of highly-
accurate optical atomic clocks, the precision requirements for synchronization
are reaching the limits of classical physics (i.e. the standard quantum limit,
SQL). Efficiently overcoming the SQL to reach the fundamental Heisenberg limit
can be achieved via the use of squeezed or entangled light. Although
approaches to the Heisenberg limit are well understood in theory, a practical
implementation, such as in space-based platforms, requires that the advantage
outweighs the added costs and complexity. Here, we focus on the question: can
entanglement yield a quantum advantage in clock synchronization over lossy
satellite-to-satellite channels? We answer in the affirmative, showing that
the redundancy afforded by the two-mode nature of entanglement allows
recoverability even over asymmetrically lossy channels. We further show this
recoverability is an improvement over single-mode squeezing sensing, thereby
illustrating a new complexity-performance trade-off for space-based sensing
applications.

##  I Introduction

In the pursuit of achieving a highly-synchronized network of clocks on low-
Earth-orbit (LEO) satellites, a solution that is optimized for size, weight,
and power (SWaP) is critical. Current state-of-the-art synchronization
protocols, such as optical two-way time and frequency transfer [1], suffer
from a precision limit known as the standard quantum limit (SQL).
Specifically, the scaling of temporal resolution with input resources is given
by $\Delta t\propto 1/\sqrt{N}$ [2, 3], where $N$ is the total number of
photons exchanged during the measurement time. Fortunately, it is known that a
better scaling is possible via the use of squeezed light or entangled light,
both of which can theoretically reach a scaling of $\Delta t\propto 1/N$ [2,
3, 4, 5, 6]. This scaling is formally known as the Heisenberg limit. In this
work, we refer to any timing measurement that, for a given $N$, is smaller
than the SQL as a _quantum advantage_. On the technological roadmap, however,
it is an open question as to whether any quantum advantage will be feasible in
space under SWaP constraints.

In our recent work [7], we showed how a squeezed light setup on two LEO
satellites could deliver a quantum advantage in clock synchronization despite
typical sources of loss and noise such as beam diffraction, satellite-pointing
jitter, and detector inefficiency. We conduct a similar assessment here, but
this time with entangled light. Although, entanglement for clock
synchronization has been extensively studied in the discrete variable regime
before, in the form of single-photon entanglement (e.g. [8, 9, 10]),
entanglement within the continuous variable regime has received relatively
little attention. We investigate here the latter regime and study a system
model to achieve quantum-enhanced clock synchronization between LEO satellites
via the exchange of two-mode squeezed vacuum (TMSV) states. Through a detailed
study that incorporates typical loss sources in LEO, we assess whether a
quantum advantage exists in practice for such a system. We note the TMSV state
is a workhorse for continuous-variable quantum communications [11, 12, 13],
interferometry [14], and quantum computing [15].

The remainder of this paper is as follows. In Section II we detail our
formalism, while Section III describes the system model. In Section IV, TMSV
simulations are discussed and in Section V a comparison with single-mode
squeezed vacuum (SMSV) is provided. Section VI concludes our work.

##  II Temporal modes

The system model consists of a transmitter, Alice, who repeatedly emits
picosecond-duration pulses in the TMSV state at a repetition frequency of
order $100$ MHz synchronized to her clock. The receiver, Bob, mixes Alice’s
pulses with his coherent local oscillator (LO) pulses (having the same
duration and repetition frequency as Alice) synchronized to his clock. Our
objective is to synchronize Alice and Bob’s clocks by measuring offsets
between their pulses to a precision beyond the SQL. To achieve this objective,
the temporal modes (TMs) of the TMSV state can be exploited — a state
originally centered around a single TM will excite higher order TMs when a
non-zero offset exists [16]. In the following, we formalize the effect of
offsets on a TM.

###  II-A Formalism

In this section, we show that Alice’s TMSV state can be fully described by a
single TM. In our model, the TMSV state propagates in free space along the
longitudinal $z$-axis with the following positive-frequency111Denoted in this
work with a superscript (+). electric field operator form [17]:

$\displaystyle\hat{E}^{(+)}(u)=i\int_{-\infty}^{\infty}\sqrt{\frac{\hbar\omega}{2\epsilon_{0}}}\hat{a}(\omega)e^{-i\omega
u}d\omega,$ (1)

where $\hat{E}^{(-)}(u)=(\hat{E}^{(+)}(u))^{\dagger}$ and
$\hat{E}(u)=\hat{E}^{(+)}(u)+\hat{E}^{(-)}(u)$. Here, $u=t-z/c$ is the
spatial-temporal coordinate (also referred to as the mean light-cone variable
[16]) along the $z$-axis and $t$-domain, $\omega$ is the angular frequency,
$\hbar$ is the reduced Planck constant (we set $\hbar=2$ from Section III
onwards), $\epsilon_{0}$ is the permittivity of free space, $c$ is the speed
of light in vacuum, and $\hat{a}(\omega)$ is the monochromatic-mode
annihilation operator. The monochromatic operators satisfy the commutation
relation:
$[\hat{a}(\omega),\hat{a}(\omega^{\prime})]=\delta(\omega-\omega^{\prime})$.
We can then define the $j$-th discrete annihilation superoperator
$\hat{A}_{j}$ as
$\hat{A}_{j}=\int_{-\infty}^{\infty}f_{j}^{*}(\omega)\hat{a}(\omega)d\omega$
where $\quad\hat{a}(\omega)=\sum_{j=0}^{\infty}f_{j}(\omega)\hat{A}_{j}$.
Here, $f_{j}(\omega)$ is a weight function describing the spectral shape.
Next, we apply a Fourier transform to obtain the time domain version:
$\tilde{\hat{A}}_{j}=\int_{-\infty}^{\infty}\tilde{f}_{j}(u)\tilde{\hat{a}}(u)dt$,
where
$\tilde{f}_{j}(u)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f_{j}(\omega)e^{-i\omega
u}d\omega$ is the time-domain form of $f_{j}(\omega)$ and
$\tilde{\hat{a}}(u)=(2\pi)^{-1}\int_{-\infty}^{\infty}\hat{a}(\omega)e^{-i\omega
u}d\omega$. Note, the weight functions are subject to the orthogonality
relation [18]:
$(2\pi)^{-1}\int_{-\infty}^{\infty}f_{j}^{*}(\omega)f_{k}(\omega)d\omega=\int_{-\infty}^{\infty}\tilde{f}_{j}^{*}(u)\tilde{f}_{k}(u)dt=\delta_{jk}$
and this imposes bosonic commutation on the superoperators such that
$[\hat{A}_{j},\hat{A}_{k}^{\dagger}]=\delta_{jk}$. In this superoperator form,
the electric field is a superposition over the entire TM set $\\{v_{j}\\}$:

$\displaystyle\hat{E}^{(+)}(u)=\sum_{j=0}^{\infty}\hat{A}_{j}v_{j}(u),v_{j}(u)=i\int_{-\infty}^{\infty}\sqrt{\frac{\hbar\omega}{2\epsilon_{0}}}f_{j}(\omega)e^{-i\omega
u}d\omega.$ (2)

Notably, the current TM basis does not satisfy the orthogonality condition,
that is $\int_{-\infty}^{\infty}v_{j}^{*}(u)v_{k}(u)du\neq 0,\forall j,k$ due
to the factor of $\sqrt{\hbar\omega}$ in Eq. 2 [19]. However, making the
assumption that the spectral profile is approximately monochromatic (i.e.
$\omega\approx\omega_{0}$, where $\omega_{0}$ is the laser central frequency)
provides a scenario where the orthogonality condition holds. Hence,
$v_{j}(u)\approx\mathcal{E}\int_{-\infty}^{\infty}f_{j}(\omega)e^{-i\omega
u}d\omega$, where $\mathcal{E}=i\sqrt{\hbar\omega_{0}/(2\epsilon_{0})}$. Now,
we define a new TM basis set, $\\{y_{\ell}\\}$, by applying a unitary
transform: $y_{\ell}(u)=\sum_{j=0}^{\infty}\hat{U}_{\ell}^{j}v_{j}(u)$, with
the fundamental mode [17, 20]
$y_{0}(u)=\sqrt{1/(\sum_{j=0}^{\infty}||v_{j}(u)||)}\sum_{j=0}^{\infty}v_{j}(u)$,
where $||v_{j}(u)||=\int_{-\infty}^{\infty}|v_{j}(u)|^{2}du$. We denote the
annihilation and creation operators for this new TM basis as $\hat{b}_{\ell}$
and $\hat{b}^{\dagger}_{\ell}$, respectively, and modify Eq. 2 to:

$\displaystyle\hat{E}^{(+)}(u)=\mathcal{E}\hat{b}_{0}y_{0}(u),$ (3)

where the TMSV state is completely defined by the fundamental TM $y_{0}$.

###  II-B Spatial-temporal fluctuation

Now, we want to find the effect an offset $\Delta u$ has on the TM formalism.
Specifically, an offset will change Eq. 3 to: $\hat{E}^{(+)}(u-\Delta
u)=\mathcal{E}\hat{b}_{0}y_{0}(u-\Delta u)$. Here, a Taylor series expansion
about $u=0$ for small $\Delta u$ ($\Delta u\ll 1$ s) gives [16]:

$\displaystyle\hat{E}^{(+)}(u-\Delta
u)\approx\mathcal{E}\hat{b}_{0}\left(y_{0}(u)+\frac{\Delta
u}{u_{0}}z_{1}(u)\right),$ (4)

where $u_{0}=\sqrt{1/(\omega_{0}^{2}+\Delta\omega^{2})}$ is a normalization
factor, $\Delta\omega$ is the frequency spread [16], and

$\displaystyle z_{1}(u)=\frac{1}{\sqrt{\Omega^{2}+1}}\left(y_{1}(u)+i\Omega
y_{0}(u)\right),$ (5)

denotes the timing mode, and $\Omega=\omega_{0}/\Delta\omega$. The expectation
of Eq. 4 is [16]:

$\displaystyle\langle\hat{E}^{(+)}(u-\Delta
u)\rangle=\mathcal{E}\sqrt{N_{in}}e^{i\theta}\left(y_{0}(u)+\frac{\Delta
u}{u_{0}}z_{1}(u)\right),$ (6)

where $N_{in}$ is the total number of photons in Alice’s state and $\theta$ is
the global phase. Using the expectation we can equivalently express Eq. 4 as

$\displaystyle\hat{E}^{(+)}(u-\Delta
u)\equiv\mathcal{E}\left(\hat{b}_{0}y_{0}(u)+\hat{b}_{1}y_{1}(u)\right),$ (7)

with $\langle\hat{b}_{0}\rangle=(1+i\omega_{0}\Delta
u)\sqrt{N_{in}}e^{i\theta}$ and $\langle\hat{b}_{1}\rangle=(\Delta\omega\Delta
u)\sqrt{N_{in}}e^{i\theta}$. Our analysis thus far shows that Alice’s state
can be fully described with TM $y_{0}$ when there is a zero offset ($\Delta
u=0$), and as a superposition of $y_{0}$ and the timing mode $z_{1}$ for small
$\Delta u$. Bob can use the excitation of $z_{1}$ to measure $\Delta u$ by
shaping his LO to $z_{1}$ [16]. We note here that an offset $\Delta u$ between
Alice and Bob generalizes over the spatial domain $\Delta z$ and the temporal
domain $\Delta t$, and for this reason we denote $\Delta u$ as the spatial-
temporal offset.

##  III System model

Figure 1: Alice, the transmitter in this one-way link, generates a TMSV state
via a nonlinear down-conversion process consisting of an SHG and an OPO. The
two modes of her TMSV state are emitted over spatially separate paths with
effective transmissivities $\eta_{1}$ and $\eta_{2}$ to Bob. Bob mixes the
TMSV state with his coherent state LO shaped in the TM $z_{1}$ and performs
BHD. The BHD photocurrents are then post-processed to take advantage of the
quadrature-correlations present in Alice’s TMSV state and maximize the signal-
to-noise ratio of the $\Delta u$ offset estimate.

The system model is shown in Fig. 1 where two one-way inter-satellite links
between Alice and Bob are detailed. Alice emits the two modes of a TMSV state
over separate spatial paths 1 and 2. At the receiver, Bob performs two sets of
balanced homodyne detections (BHD) on the two modes of the TMSV state.
Finally, on board post-processing is used to retrieve the variance of the
correlated $\hat{X}$ quadrature thereby taking advantage of the entanglement
within a TMSV state. In the analysis to follow, both BHD setups are assumed to
be on the same satellite (Bob) with the measurement currents post-processed
on-board. However, more generally, the two BHDs can be deployed on two
separate satellites, each processing one of the two modes of Alice’s TMSV
state. The only additional assumptions we need for this case is that both
receiver satellites are time synchronized, and can communicate their
measurements over a classical link for post-processing. The analysis in this
latter case would require only slight modifications to that given here.

###  III-A Alice’s entangled state

At Alice, a second harmonic generator (SHG), an optical parametric oscillator
(OPO), and a beam splitter (BS) are used to generate and coherently mix two
single-mode squeezed vacuum (SMSV) states (squeezed in alternating quadratures
$\hat{X}$ and $\hat{P}$), and generate a TMSV state. The SHG-OPO configuration
is detailed further in [21, 22, 16, 23]. The TMSV state has correlated
$\hat{X}$ quadratures and anticorrelated $\hat{P}$ quadratures that can be
taken advantage of during measurement [24]. Although the setup used here is
one of several ways to generate a TMSV state, the chosen approach allows for
direct comparisons with an SMSV setup which is discussed later.

We simplify Alice’s TMSV state representation by expressing it over only the
first two TMs $\\{y_{0},y_{1}\\}$ as follows:

$\displaystyle\ket{\Psi}_{a}=\bigotimes_{\ell=0}^{1}\hat{D}_{\ell}(\sqrt{{N}_{1}}e^{i\theta_{1}})\hat{D}_{\ell}(\sqrt{{N}_{2}}e^{i\theta_{2}})\hat{S}_{\ell}(\xi)\ket{00}.$
(8)

Here, $\hat{D}_{\ell}(\cdot)$ is the displacement operator in TM $\ell$,
defined as
$\hat{D}_{\ell}(\sqrt{{N}_{1}}e^{i\theta_{1}})=\exp[\sqrt{{N}_{1}}(e^{i\theta_{1}}\hat{b}_{1,\ell}^{\dagger}-e^{-i\theta_{1}}\hat{b}_{1,\ell})]$,
where $\hat{b}_{1,\ell}$ is the annihilation operator in mode $1$ of the TMSV
state and TM $\ell$, ${N}_{1}$ is the number of photons emitted from Alice to
Bob along path 1, and $\theta_{1}$ is the corresponding phase. Similar
expressions for path $2$ are found by substituting in index $2$. Further,
$\hat{S}_{\ell}(\cdot)$ is the TMSV operator in TM $\ell$, defined as [24]
$\hat{S}_{\ell}(\xi)=\exp\left(\xi\hat{b}_{1,\ell}^{\dagger}\hat{b}_{2,\ell}^{\dagger}-\xi^{*}\hat{b}_{1,\ell}\hat{b}_{2,\ell}\right)$,
where $\xi=re^{i\phi}$ is the squeezing parameter and we set $\phi=0$ here to
simplify $\xi=r$ as a real quantity. Finally, the operators are applied to the
two-mode vacuum state $\ket{00}$.

###  III-B Bob and Charlie’s balanced homodyne detection

BHD is a technique used to optimally measure a specific quadrature of a signal
by mixing it with a shaped LO. Here, in our system model, we mix each of the
two modes of Alice’s TMSV state separately with Bob’s LO. We assume that the
time of arrival of Alice’s TMSV state is known a priori at Bob to an accuracy
of order $10^{-12}$ s — the technique we discuss here improves upon this
accuracy.

Bob’s LO are shaped in the TM $z_{1}$ and split into $\hat{E}_{1}$ and
$\hat{E}_{2}$, where each have the positive-frequency electric field
description:

$\displaystyle\hat{E}^{(+)}_{1}(u)=\left(\hat{l}_{0}y_{0}(u)+\hat{l}_{1}y_{1}(u)\right)=\hat{E}^{(+)}_{2}(u).$
(9)

Here, we use the expanded form of $z_{1}$ from Eq. 5 and $\hat{l}_{\ell}$
denotes the LO annihilation operator in a TM $\ell$. For optimal detection,
Bob’s LO needs to be in a coherent state:
$\ket{\Psi}_{b,1}=\bigotimes_{\ell=0}^{1}\hat{D}_{\ell}(\sqrt{{N}_{lo}}e^{i\theta_{lo}})\ket{0}=\ket{\Psi}_{b,2}$
(shown in Fig. 1). Here, we assume that BHD 1 and 2 both receive the same
number of photons from the LO and have the same phase, and thus use the
general variables ${N}_{lo}$ and $\theta_{lo}$. Similar to [16], the
expectation of Eq. 9 is:

$\displaystyle\langle\hat{E}^{(+)}_{1}(u)\rangle=\frac{\mathcal{E}\sqrt{{N}_{lo}}e^{i\theta_{lo}}}{\sqrt{\Omega^{2}+1}}\left(i\Omega
y_{0}(u)+y_{1}(u)\right)=\langle\hat{E}_{2}^{(+)}(u)\rangle,$ (10)

where we have set
$\langle\hat{l}_{0}\rangle=i\Omega\sqrt{{N}_{lo}/(\Omega^{2}+1)}e^{i\theta_{lo}}$
and $\langle\hat{l}_{1}\rangle=\sqrt{{N}_{lo}/(\Omega^{2}+1)}e^{i\theta_{lo}}$
[16]. As output, the BHD 1 measurement will generate a photocurrent given by
$\hat{I}_{1}=|\mathcal{E}|^{2}\sum_{\ell=0}^{1}\left(\hat{b}_{1,\ell}^{\dagger}\hat{l}_{\ell}+\hat{b}_{1,\ell}\hat{l}^{\dagger}_{\ell}\right)$
[16]. The expectation of the photocurrent is then

$\displaystyle\langle\hat{I}_{1}\rangle=2|\mathcal{E}|^{2}\sqrt{{N}_{1}{N}_{lo}}\bigg{[}\frac{\Delta
u}{u_{0}}\cos(\theta_{1}-\theta_{lo})+$
$\displaystyle\frac{\Omega}{\sqrt{\Omega^{2}+1}}\sin(\theta_{1}-\theta_{lo})\bigg{]},$
(11)

and a similar process would find that we can substitute in $N_{2}$ for
$\langle\hat{I}_{2}\rangle$. We can also derive the variances (equal for both
photocurrents), which differ from [16] and can be written:

$\displaystyle\langle\delta\hat{I}^{2}_{1}\rangle=|\mathcal{E}|^{4}{N}_{lo}\cosh{2r}=\langle\delta\hat{I}^{2}_{2}\rangle.$
(12)

Although not required (as the individual timing measurements could be averaged
directly), it will be useful to include a post-processing step whereby the
individual photocurrents are added to give a final photocurrent:
$\hat{I}_{\text{TMSV}}=\hat{I}_{1}+\hat{I}_{2}$. We will henceforth denote
$\hat{I}_{\text{TMSV}}$ as the post-processed photocurrent. The ideal222Here
ideal refers to the no loss scenario, and contrasts with real where loss is
considered. post-processed photocurrent has expectation (assuming
$\theta=\theta_{lo}$):

$\displaystyle\langle\hat{I}_{\text{TMSV},ideal}\rangle=\frac{2|\mathcal{E}|^{2}\Delta
u\sqrt{{N}_{lo}}}{u_{0}}\left(\sqrt{{N}_{1}}+\sqrt{{N}_{2}}\right).$ (13)

Similarly, with some algebra we can find the variance,

$\displaystyle\langle\delta\hat{I}_{\text{TMSV},ideal}^{2}\rangle=\langle\delta\hat{I}_{1}^{2}\rangle+\langle\delta\hat{I}_{2}^{2}\rangle+2(\langle\hat{I}_{1}\hat{I}_{2}\rangle-\langle\hat{I}_{1}\rangle\langle\hat{I}_{2}\rangle)$
$\displaystyle=2|\mathcal{E}|^{4}{N}_{lo}\left(\cosh{2r}-\frac{\Omega^{2}-1}{\Omega^{2}+1}\cos(2\theta_{lo})\sinh{2r}\right).$
(14)

Here, again assuming approximately monochromatic laser sources at Alice and
Bob, $\Omega\gg 1$ (i.e. $\omega_{0}\gg\Delta\omega$), and by setting the
measurement phase to $\theta_{lo}=n\pi$ for $n\in\mathbb{Z}$ — corresponding
to an $\hat{X}$ quadrature measurement — the final variance will be minimized
to

$\displaystyle\langle\delta\hat{I}_{\text{TMSV},ideal}^{2}\rangle=2|\mathcal{E}|^{4}{N}_{lo}e^{-2r}.$
(15)

The variance is now phase-sensitive and reduces with increasing $r$ as
desired. This reduced variance will minimize noise in our measurement of
$\Delta u$ allowing us to reach a precision beyond the SQL.

###  III-C Spatial-temporal offset estimate — ideal

Let us now derive $\Delta u$ in the ideal case by considering a signal-to-
noise ratio (SNR) of 1 [16], 333In this work (similar to previous studies e.g.
[16]) we set the signal-to-noise ratio to the worst-case of 1 — the point
where the signal is equal to the noise level, to find the minimum measurable
spatial-temporal offset. where
$\text{SNR}=\langle\hat{I}\rangle/\sqrt{\langle\delta\hat{I}^{2}\rangle}$
[16]. Using Eq. 13 and Section III-B, we find

$\displaystyle\Delta
u_{\text{TMSV},ideal}=\frac{e^{-r}}{\sqrt{2}(\sqrt{{N}_{1}}+\sqrt{{N}_{2}})\sqrt{\omega_{0}^{2}+\Delta\omega^{2}}}.$
(16)

Now, we assume photon conservation during the generation of the TMSV state and
set $N_{in}=N_{1}+N_{2}$. Additionally, we assume that Alice’s photons are
evenly divided across the two modes of her TMSV state so $N_{1}=N_{2}$, giving

$\displaystyle\Delta
u_{\text{TMSV},ideal}=\frac{e^{-r}}{2\sqrt{{N}_{in}}\sqrt{\omega_{0}^{2}+\Delta\omega^{2}}}.$
(17)

This $\Delta u_{\text{TMSV},ideal}$ represents the minimum offset that is
measurable by a setup using TMSV state of $r$ squeezing level, $N_{in}$
photons, $\omega_{0}$ central frequency and $\Delta\omega$ frequency spread.
However, so far we have not considered the effect of any losses and
inefficiencies in the system.

###  III-D Spatial-temporal offset estimate — realistic

We now extend our analysis to a realistic inter-satellite link by
incorporating various sources of loss. We make use of the following BS unitary
operator to model loss [24]

$\displaystyle\hat{\text{BS}}=\begin{bmatrix}\sqrt{\eta}&-\sqrt{1-\eta}\\\
\sqrt{1-\eta}&\sqrt{\eta}\end{bmatrix},$ (18)

where $\eta$ is the effective transmissivity, $0\leq\eta\leq 1$, and loss is
$1-\eta$. Our model encapsulates the effect of photonic loss due to beam
diffraction, satellite pointing misalignment and detector inefficiencies into
the general term $\eta$ (further details are in our previous work [7]). The
photocurrent in BHD 1 with BS operator applied is then

$\displaystyle\hat{I}_{1,real}=|\mathcal{E}|^{2}\sum_{\ell=0}^{1}\left(\sqrt{\eta_{1}}\hat{b}_{1,\ell}^{\dagger}-\sqrt{1-\eta_{1}}\hat{v}_{1,\ell}^{\dagger}\right)\hat{l}_{\ell}~{}+$
$\displaystyle\left(\sqrt{\eta_{1}}\hat{b}_{1,\ell}-\sqrt{1-\eta_{1}}\hat{v}_{1,\ell}\right)\hat{l}_{\ell}^{\dagger}.$
(19)

Here, $\eta_{1}$ denotes the effective transmissivity along path 1, $\hat{v}$
[$\hat{v}^{\dagger}$] denotes the annihilation [creation] operator of the
vacuum state. $\hat{I}_{2,real}$ can be found by substituting in index $2$.
The expectation of the real post-processed photocurrent is then

$\displaystyle\langle\hat{I}_{\text{TMSV},real}\rangle=\langle\hat{I}_{\text{TMSV},ideal}\rangle\left(\sqrt{\eta_{1}}+\sqrt{\eta_{2}}\right)/2,$
(20)

with variance given by

$\displaystyle\langle\delta\hat{I}_{\text{TMSV},real}^{2}\rangle=2|\mathcal{E}|^{4}{N}_{lo}\bigg{(}(\eta_{1}+\eta_{2})\sinh^{2}{r}+1+$
$\displaystyle\sqrt{(1-\eta_{1})(1-\eta_{2})}-\sqrt{\eta_{1}\eta_{2}}\sinh{2r}\bigg{)}.$
(21)

Our analysis can be verified by setting $\eta_{1}=\eta_{2}=1$ in Eq. 20 and
Section III-D to retrieve Eq. 13 and Section III-B, respectively. Finally, we
derived $\Delta u$ with propagation loss at $\text{SNR}=1$:

$\displaystyle\Delta u_{\text{TMSV},real}=~{}$
$\displaystyle\frac{\sqrt{Q}}{\left(\sqrt{\eta_{1}}+\sqrt{\eta_{2}}\right)\sqrt{{N}_{in}\left(\omega_{0}^{2}+\Delta\omega^{2}\right)}},$
(22) $\displaystyle Q=~{}$
$\displaystyle\left(\eta_{1}+\eta_{2}\right)\sinh^{2}{r}+1+$
$\displaystyle\sqrt{\left(1-\eta_{1}\right)\left(1-\eta_{2}\right)}-\sqrt{\eta_{1}\eta_{2}}\sinh{2r}.$

Again, setting $\eta_{1}=\eta_{2}=1$ retrieves the ideal case in Eq. 17. Eq.
(22) is one of the main results of our study as it represents the minimum
offset measurable by a setup using a TMSV state under lossy conditions.

In order to quantify the quantum advantage of our TMSV state, we note the SQL
of a general single transmit/two receiver setup, which is found by setting
$r=0$ in Eq. 22:

$\displaystyle\Delta
u_{\text{SQL},real}=\frac{\sqrt{1+\sqrt{(1-\eta_{1})(1-\eta_{2})}}}{\left(\sqrt{\eta_{1}}+\sqrt{\eta_{2}}\right)\sqrt{{N}_{in}\left(\omega_{0}^{2}+\Delta\omega^{2}\right)}}.$
(23)

Configurations that allow $\Delta u_{\text{TMSV},real}<\Delta
u_{\text{SQL},real}$ have a quantum advantage and optimize the available
$N_{in}$.

##  IV Results

Figure 2: Minimum measurable spatial-temporal offset, $\Delta u$, vs.
effective transmissivity, $\eta$. Here, $\eta_{1}=\eta_{2}\equiv\eta$,
$\lambda_{0}=815$ nm, $\Delta\omega=2\pi\times 10^{6}$ rad/s and
$N_{in}=10^{3}$ photons. The gray highlights classical region with lower
boundary being the SQL. Evidently, TMSV states with $3~{}$dB squeezing and
more can achieve a quantum advantage since $\Delta u$ measurement below the
SQL is possible when $\eta\gtrsim 0.4$. Notably, increasing the squeezing
level is beneficial only when $\eta\gtrsim 0.7$ to achieve even lower $\Delta
u$.

We now conduct a numerical simulation using system parameters representative
of a LEO satellite link, namely: $\lambda_{0}=815$ nm ($\omega_{0}=2\pi
c/\lambda_{0}$), $\Delta\omega=2\pi\times 10^{6}$ rad/s, and $N_{in}=10^{3}$
photons. The chosen laser wavelength allows for use of SWaP-efficient silicon-
based detectors [25]; while $N_{in}$ is 5 orders of magnitude smaller than the
expected number of pump photons for a $10$ W $10$ ps laser system to represent
generation inefficiency (discussed further in Section V).

In the first simulation, Alice generates four different TMSV states with
squeezing levels ranging from $r=3$ to $15$ dB444The squeezing parameter $r$
in dB is defined as $r_{\text{dB}}=-10\log_{10}\left(e^{-2r}\right)$.. The
measurable offset, $\Delta u$, is plotted for each $r$ and compared against
the SQL in Fig. 2. Note, both path transmissivities are equal
$\eta_{1}=\eta_{2}=\eta$. The results show a quantum advantage when
$\eta\gtrsim 0.4$ since $\Delta u_{\text{TMSV},real}<\Delta
u_{\text{SQL},real}$, even for small $r$. Additionally, increasing the
squeezing level is advantageous only at high transmissivity ($\eta\gtrsim
0.8$). Finally, in extremely lossy conditions, $\eta<0.3$, a quantum advantage
does not exist as $\Delta u_{\text{TMSV},real}\approx\Delta
u_{\text{SQL},real}$.

Figure 3: The quantum advantage check, $\Delta u_{\text{SQL},real}-\Delta
u_{\text{TMSV},real}$, for various $\eta_{1}$ and $\eta_{2}$. Here, a constant
squeezing level of $r=5$ dB is used and all other parameters are as per Fig.
2. Regions shown in color exhibit quantum advantage with contour lines
indicating equal levels of quantum advantage. A minimum effective
transmissivity of $\eta_{1},\eta_{2}\gtrsim 0.1$ is required.

In the second simulation, we generalize the transmissivity terms considering
$\eta_{1}$ and $\eta_{2}$ not equal (e.g two receiver satellites), and use a
constant $r=5$ dB squeezing level. The results in Fig. 3 show a quantum
advantage exists as long as $\eta_{1},\eta_{2}\gtrsim 0.1$. The contour lines
show regions with the same level of quantum advantage. For example, following
the $6\times 10^{-9}$ s contour line, when $\eta_{1}=\eta_{2}=0.695$ the
quantum advantage is equal to when $\eta_{1}=0.585$ and $\eta_{2}=0.825$.
Hence, a TMSV state can take advantage of asymmetrically lossy channels and
compensate for one channel using another less lossy channel.

Another example of the performance advantage over asymmetric channels is when
$\eta_{2}=0.5$, $\Delta u_{\text{SQL},real}>\Delta u_{\text{TMSV},real}$ as
long as $\eta_{1}>0.05$ — this scenario shows that a quantum advantage is
possible even in extremely lossy conditions. Although not visualized here, for
$r>5$ dB, a larger quantum advantage is achievable, but at the cost of
stricter requirements for $\eta_{1}$ and $\eta_{2}$. Overall, the unique
advantage of the TMSV setup over asymmetrically lossy channels is not present
in the SMSV setup in [16] where only a single channel exists, nor in a
classical setup where the quadratures are uncorrelated. During LEO, channel
transmissivity will be a dynamically varying parameter and thus this effect
would be advantageous.

##  V Relation to other work

Figure 4: Minimum measurable spatial-temporal offset, $\Delta u$, vs.
effective transmissivity, $\eta$, for setups with a TMSV and SMSV states. For
equivalent (5 dB) squeezing, the SMSV state yields a lower $\Delta u$ than the
TMSV state and thus a greater level of quantum advantage when
$0.05<\eta<0.95$. Note, when $\eta=1$ both setups are equivalent and below the
SQL.

We can now compare our analysis with previous studies in [16, 23], where an
SMSV state over a one-way link was used instead. Our derivation in Section III
reveals that $\Delta u_{\text{TMSV},ideal}\equiv\Delta u_{\text{SMSV},ideal}$,
where the latter is from eq. (11) in [16]. This important finding shows that a
TMSV state, in principle, yields the same level of precision (and in turn
quantum advantage) as an SMSV state over an ideal channel. However, this
equivalence in $\Delta u$ holds true only when all loss and noise effects are
ignored. In contrast, we can find $\Delta u_{\text{SMSV},real}$ following the
same process as before:

$\displaystyle\Delta
u_{\text{SMSV},real}=\frac{1}{2}\sqrt{\frac{\eta_{1}e^{-2r}+(1-\eta_{1})}{\eta_{1}{N}_{in}(\omega_{0}^{2}+\Delta\omega^{2})}}.$
(24)

A third simulation is then conducted with results summarized in Fig. 4. For
$r=5$ dB squeezing, we find $\Delta u_{\text{SMSV},real}<\Delta
u_{\text{TMSV},real}$ when $0.05\lesssim\eta\lesssim 0.95$. Hence, SMSV in
fact outperforms TMSV over symmetrically lossy channels.

Although a TMSV state exhibits advantageous performance over asymmetrically
lossy channels, however, the trade-offs in SWaP between both states further
complicate the matter. In our analysis, equal power was maintained by using
equal $N_{in}$ across the approaches considered, since laser energy is
proportional to the number of photons emitted. Additionally, we assume that
the efficiency of converting $N_{in}$ to $N_{1}$ and $N_{2}$ is equal for the
SMSV and TMSV states since the same SHG-OPO optical system is used. However,
the TMSV state requires one additional SHG-OPO path for the second mode which
adds to the required SWaP. Additionally, a co-located second receiver aperture
and post-processing stage are needed for the TMSV state. Hence, the additional
SWaP requirement for a TMSV state needs to be traded-off with any expected
gains over asymmetric channels in practice.

Furthermore, a classical setup may have higher pulse transmit efficiency than
both the SMSV and TMSV states due to a simpler optical system. For example,
current 3U CubeSats have a $10$ W optical power budget [25], which over a $10$
ps duration pulse with $\lambda_{0}=815$ nm is approximately $4.1\times
10^{8}$ pump photons/pulse. In a TMSV state, these pump photons are
spontaneously down-converted via the SHG-OPO setup into signal and idler
photons in a probabilistic manner. In our study, we integrated the effect of
this probabilistic conversion process by setting $N_{in}=10^{3}$ photons.
However, in a classical state, where there is no parametric down-conversion,
the pump photons convert directly into signal/idler photons with much higher
efficiency, and thus have a higher effective transmit power. Having said this,
our analysis does ensure a fair comparison by using the same $N_{in}$ for the
SQL, TMSV and SMSV approaches.

##  VI Conclusion

In this study, we presented an analysis of a space-based quantum-enhanced
synchronization system using entangled light. We derived the minimum
measurable spatial-temporal offset, $\Delta u$, for a TMSV state over lossless
and lossy channels. Also, by comparing with the SQL, we found practical
scenarios where the quantum approach is indeed advantageous. The simulation
results presented show system and channel configurations that enable a quantum
advantage at squeezing levels as low as $3$ dB for constant $N_{in}$. Our
results also found a unique advantage of a setup using a TMSV state whereby
redundancy in the two-modes allows for recoverability over asymmetrically
lossy channels. This redundancy supersedes both the classical and SMSV
approaches and would be useful on inter-satellite links. Finally, a comparison
of sensitivity between the SMSV and TMSV states reveals equivalence in
performance under ideal channel conditions, as well as the superior
performance of SMSV in certain inter-satellite channels. This last point
highlights a new trade-off with regard to performance versus implementation
complexity.

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