# Protecting Classical-Quantum Signals in Free Space Optical Channels

E. Villaseñor,1 M. S. Winnel,2 T. C. Ralph,2 R. Aguinaldo,3 J. Green,3 and R.
Malaney.1
1 School of Electrical Engineering & Telecommunications,
The University of New South Wales, Sydney, NSW 2052, Australia.
2 Centre for Quantum Computation and Communication Technology,
School of Mathematics and Physics, University of Queensland, St Lucia,
Queensland 4072, Australia.
3 Northrop Grumman Mission Systems, San Diego, California, USA.

###### Abstract

Due to turbulence and tracking errors, free-space optical channels involving
mobile transceivers are characterized by a signal’s partial loss or complete
erasure. This work presents an error correction protocol capable of protecting
a signal passing through such channels by encoding it with an ancillary
entangled bipartite state. Beyond its ability to offer protection under
realistic channel conditions, novel to our protocol is its ability to
encompass both classical and quantum information on the encoded signal. We
show how, relative to non-encoded direct transmission, the protocol can
improve the fidelity of transmitted coherent states over a wide range of
losses and erasure probabilities. In addition, the use of ancillary non-
Gaussian entangled bipartite states in the signal encoding is considered,
showing how this can increase performance. Finally, we briefly discuss the
application of our protocol to the transmission of more complex input states,
such as multi-mode entangled states.

##  I Introduction

Building a global quantum network is a major technological challenge [1]. Any
quantum technology that relies on mature hardware may provide a practical path
forward. Continuous Variable (CV) quantum communications, which can be
implemented using off-the-shelf components, is one such technology [2, 3, 4].
In this technology, quantum information is extracted using homodyne or
heterodyne measurements - detection techniques previously developed to extract
classical information encoded via coherent communications [5, 6, 7].
Interestingly, CV quantum and coherent communications can coexist using the
same signal [8, 9, 10, 11, 12], and it is this interesting co-existence of
classical and quantum information we study here. Our main contribution will be
the introduction of a combined classical-quantum encoding protocol that
accommodates a wide range of partial loss and erasure conditions in the
channel. As we discuss later, such conditions are those anticipated for
practical free-space optical channels, where at least one of the transceivers
is mobile.

Optical free-space communications provide the basis for long-range
communications, both classical [13] and quantum [4], through horizontal
channels on the ground [2, 4] or more general channels to (or sometimes
between) flying objects such as drones [14], aircraft [15], or satellites [16,
17, 18]. However, beam deformation and wandering induced by the turbulent
atmosphere can have detrimental consequences [19, 20, 21, 22]. If at least one
of the transceivers is mobile and untethered, then further misalignment of the
optical beam introduced by transceiver motion (e.g., jitter) must also be
considered [23, 24]. Any error correction protocol that can alleviate such
combined effects would be useful for a pragmatic communication deployment.

In the context of quantum information alone, previous works have analyzed and
experimentally demonstrated how error correction of the pure erasure channel
(total loss in a channel) can be achieved via the introduction of ancillary
modes transmitted through independent channels [25, 26, 27, 28, 29]. Unlike
these previous works, however, here we present a new protocol capable of error
correction under more realistic conditions, as well as accommodating combined
classical-quantum information encoding. The realistic channel conditions we
consider encompass a range of loss conditions across the independent channels
- not just simple erasures. As we shall see, careful monitoring of the channel
losses via separate bright classical reference signals provides the feedback
mechanism we need to protect a three-mode-encoded classical-quantum signal
under our realistic channel conditions.

Our specific contributions in this work can be summarized as follows: (i) We
develop a three-channel error correction protocol capable of correcting any
amount of loss on one mode of a three-mode quantum state while allowing some
loss on the other modes, demonstrating how to optimize signal recovery. (ii)
We show that transmitted coherent states exhibit a substantial improvement in
fidelity over direct state transmission using one channel under our new
protocol. (iii) We demonstrate how transmission of classical information can
be embedded into the protocol by adding a displacement operation, quantifying
the displacement characteristics needed to achieve communications with a
specific bit-error rate. (iv) We show how non-Gaussian states used as
entangled ancillary states within the protocol can further increase the
fidelity of transmitted states. (v) We investigate whether the trends found
for coherent states also apply to the distribution of entanglement by the
protocol, discussing how care must be taken in interpreting the transmission
of different input states.

This work is organized as follows. In Section II, the protocol is introduced.
In Section III, simulation results of the state transmission via the protocol
are presented. The usefulness of non-Gaussian operations is presented in
Section IV. Potential modifications to the protocol are discussed in Section
V, as are discussions on the use of alternate input states to the protocol.
Our conclusions are drawn in Section VI.

##  II The protocol

Figure 1: The protocol to transmit quantum and classical information (a). The
protocol protects a single-mode quantum state against erasures by combining it
with an entangled bipartite state, $\ket{\psi_{\mathrm{AB}}}$. Classical
information is encoded using the displacement operation applied in mode 2. Bob
monitors the channels via the LOs. The syndrome measurement corresponds to a
dual homodyne measurement. A correction is applied to mode $1^{\prime}$ to
recover the original quantum state using the syndrome result. Dashed lines
represent the transmission of classical information required to optimize the
protocol. Turbulence simulations were obtained via phase screen simulations
(b). Here, $r_{t}$ and $r_{d}$ represent aperture radii, and $L$ is the
distance between transceivers.

In Fig. 1.a, our protocol is presented. The protocol consists of the
concurrent transmission of classical and quantum information embedded into an
error correction protocol. The error correction code is first described in the
following, followed by the transmission of classical information via the
protocol.

The following steps are followed to perform error correction in the protocol.
(i) Alice initially possesses the single-mode quantum state,
$\ket{\phi_{\mathrm{s}}}$ the “quantum signal” state to be error corrected in
mode $1^{\mathrm{o}}$, alongside an entangled bipartite state,
$\ket{\psi_{\mathrm{AB}}}$ in modes $2^{\mathrm{o}}$ and $3^{\mathrm{o}}$.
(ii) The state $\ket{\phi_{\mathrm{s}}}$ is combined with mode
$2^{\mathrm{o}}$ of the state $\ket{\psi_{\mathrm{AB}}}$ using a balanced Beam
Splitter (BS), shown as BS1, to create a three-mode encoded state. (iii) The
three modes of the encoded state are transmitted via the three fluctuating
channels (acting independently on the three modes). The loss applied to each
mode is characterized by the transmissivity coefficient $T_{j}$ with
$j=1,2,3$. A Local Oscillator (LO) is transmitted from Alice to Bob through
each channel (a bright classical beam of known amplitude multiplexed with each
mode). (iv) Bob receives the three modes, $1$, $2$, and $3$. (v) Bob obtains
$T_{j}$ information on the channels by monitoring the three LOs. (vi) Bob uses
a second BS (BS2), on modes $1$ and $2$ to decode the state. (vii) Applying a
third BS (BS3), dual homodyne measurements are then performed on modes
$2^{\prime}$ and $3^{\prime}$ (referred to as the syndrome measurements
$\tilde{x}$ and $\tilde{p}$). (viii) Bob uses his channel information,
combined with the syndrome measurements, to determine the optimal corrective
displacement, $\hat{D}_{\mathrm{corr}}$, that is applied to mode $1^{\prime}$
to obtain the output state $\ket{\phi_{\mathrm{out}}}$ of the protocol. If
error correction was successful then, $\ket{\phi_{\mathrm{out}}}$ will deviate
only marginally from $\ket{\phi_{\mathrm{s}}}$.

A critical component of the protocol is the classical computing required to
apply the correction $\hat{D}_{\mathrm{corr}}$. This classical computation can
be thought of as an algorithm that takes as inputs the measured $T_{j}$ in
each channel; the syndrome measurements, $\tilde{x}$ and $\tilde{p}$; and the
(pre-measured) excess noise in each channel; and determines the optimal value
of the gain, $g$, in $\hat{D}_{\mathrm{corr}}$ that will be applied to mode
$1^{\prime}$ so to optimize the fidelity between $\ket{\phi_{\mathrm{out}}}$
and $\ket{\phi_{\mathrm{s}}}$. The details of this optimization procedure are
non-trivial and are explained in detail below. However, suffice it to say this
procedure can be implemented in software _a priori_. In the following, we will
utilize one particular quantum information formalism to illustrate how the
needed calculations within the software can be implemented.

The simulation results that appear later will be based on two steps: (i) the
derivation of the analytical expressions for the transformations of quantum
states in each step of Fig 1.a; and (ii) the numerical simulations to
determine the probability distributions for the transmissivity of each channel
(based on models of the phase screens indicated in Fig. 1.b) required as input
to the analytical expressions.

###  II-A State transformation

A previous result [30] demonstrates that the outcome state of CV quantum
teleportation can be elegantly computed using the Characteristic Function (CF)
formalism. In this work, we present a similar result.

For any $n$-mode quantum state, $\hat{\rho}$, its CF is defined as

$\displaystyle\makebox{\large$\chi$}(\lambda_{1},\lambda_{2},...,\lambda_{n})=\text{Tr}\left\\{\hat{\rho}\hat{D}(\lambda_{1})\hat{D}(\lambda_{2})...\hat{D}(\lambda_{n})\right\\},$
(1)

where $\lambda_{i}\in\mathbb{C}$, and $\hat{D}$ is the displacement operator,

$\displaystyle\hat{D}(\lambda_{j})=e^{\lambda_{j}\hat{a}^{{\dagger}}_{j}-\lambda_{j}^{*}\hat{a}_{j}},$
(2)

where $\hat{a}_{j}$ and $\hat{a}^{{\dagger}}_{j}$ are respectively the
annihilation and creation operators of mode $j$, and ∗ represents the complex
conjugate. Using the CF formalism, linear optics operations can be expressed
by simply transforming the CF arguments while leaving the functions unchanged.
The effect of a loss-noise channel, with transmissivity $T_{j}$ and excess
noise $\epsilon$ on a single-mode quantum state, corresponds to the
transformation of its CF as,

$\displaystyle\makebox{\large$\chi$}\left(\lambda\right)\xrightarrow{\text{channel}}\makebox{\large$\chi$}\left(\sqrt{T_{j}}\lambda\right)\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{1-T_{j}+\epsilon}\lambda\right),$
(3)

where $\makebox{\large$\chi$}_{\ket{0}}$ is the CF of the vacuum state.

The full derivation of the CF of $\ket{\psi_{m}athrm{out}}$ is presented in
Appendix A. Given the CF of the quantum signal,
$\makebox{\large$\chi$}_{\mathrm{s}}(\lambda_{1})$, and the CF of the
entangled state,
$\makebox{\large$\chi$}_{\mathrm{AB}}(\lambda_{2},\lambda_{3})$, the CF of the
output state corresponds to,

$\displaystyle\makebox{\large$\chi$}_{\mathrm{out}}(\lambda)$
$\displaystyle=\makebox{\large$\chi$}_{\mathrm{s}}\left((T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
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} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\tilde{g}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}})\lambda\right)\times$
(4)
$\displaystyle\makebox{\large$\chi$}_{\mathrm{AB}}\left(\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\tilde{g}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)\lambda,\sqrt{T_{3}}\tilde{g}\lambda^{*}\right)$
$\displaystyle\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{(1+\epsilon)(2\tilde{g}^{2}+1)+2g^{2}(1-\eta^{2})-T^{\prime}}\lambda\right),$
(5)

with

$\displaystyle T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\frac{\sqrt{T_{1}}+\sqrt{T_{2}}}{2};~{}~{}~{}~{}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\frac{\sqrt{T_{1}}-\sqrt{T_{2}}}{2};$
$\displaystyle
T^{\prime}=\frac{T_{1}}{2}(1+\tilde{g})^{2}+\frac{T_{2}}{2}(1-\tilde{g})^{2}+T_{3}\tilde{g}^{2}.$
(6)

Here, $\tilde{g}=g\eta$, with $\eta^{2}$ the efficiency of the homodyne
measurements. The gain parameter when applying $\hat{D}_{\mathrm{corr}}$, $g$,
corresponds to a free parameter Bob can select at will (see Appendix A for
more detail). Error correction in the protocol works when Bob selects the
appropriate value of $g$ based on the knowledge he obtains by monitoring the
channels. By selecting the appropriate value of $g$, the output state can be
made independent of the loss affecting one of the three modes in the encoded
state. The ideal scenario occurs when two of the three modes are unaffected by
the loss. In this scenario, $g$ can always be selected to nullify the loss
effects in the remaining mode. For more insight, some idealized scenarios with
specific $g$ values are discussed in Appendix B.

_Transmission of classical information_ : The only additional requirements for
transmitting classical information are an agreed digital modulation scheme
between Alice and Bob and an extra displacement operation $\hat{D}(\Delta)$.
During the first step in the error correction, the operation $\hat{D}(\Delta)$
is used by Alice to encode a symbol on mode $2^{\mathrm{o}}$ of
$\ket{\psi}_{\mathrm{AB}}$ immediately after it has been prepared. The symbol
corresponds to one or multiple classical bits following a predetermined
digital modulation scheme. The symbol is then recovered by Bob automatically
from the values of the syndrome measurement results, $\tilde{x}$ and
$\tilde{p}$. Note, we adopt $\hbar=2$; and $\Delta$ , $\tilde{x}$ and
$\tilde{p}$ are dimensionless (the variance of the vacuum noise is one).

###  II-B Transmissivity calculations

The technique used here to model turbulence via phase screen simulations has
been detailed extensively in our previous works, with experimental validation
over a horizontal free-space channel of 1.5 km [31]. The simulation
methodology in the present work follows our previous works; details can be
seen in [31, 32].

Our phase screen simulations allow us to find the Probability Density Function
(PDF) of the transmissivity in the channel for any given communications setup.
Henceforth, we refer to this PDF as $\mathcal{P}$. The dominant parameters of
a communication setup are the propagation distance, $L$, the wavelength of the
light used, $\lambda$, the beam waist at the transmitter, $r_{\mathrm{t}}$,
and the size of the receiving aperture, $r_{\mathrm{d}}$. These parameters
will be fixed to be values shown in Table I (with the exception of $L$ and
$r_{\mathrm{d}}$). For focus, we assume a horizontal channel in the results we
show here; however, other configurations can be easily accommodated via small
changes in the simulations [31]. A total of 10 uniformly distributed phase
screens are used, and the grid size of complex numbers is $1500{\times}1500$.
A typical PDF from our calculations can be seen in Fig. 2.

Beyond turbulence, other real-world system issues can affect the channel
transmissivity. Most notable are the pointing and tracking errors between the
transmitter and receiver that create a jitter and/or deterministic offsets of
the beam direction [23, 24]. In our channel model, we will account for these
effects by the addition of erasures into the PDF of the transmissivity. That
is, for each optical beam sent from the transmitter to the receiver, we
consider that there is a probability $p_{\mathrm{e}}$ of it entirely missing
the receiver (corresponding to $T_{j}=0$). In other words, for a given
communications setup (e.g., the transceiver apertures, propagation distance,
and wavelength) with corresponding PDF $\mathcal{P}$, and an erasure
probability, $p_{\mathrm{e}}$, the values of $T_{j}$ are now given by the
following modified PDF,

$\displaystyle T_{j}=\begin{cases}T_{j}\sim\mathcal{P}&\mathrm{with\
probability\ }(1-p_{e})\\\ 0&\mathrm{with\ probability\ }p_{e},\end{cases}$
(7)

where $X\sim Y$ indicates that a random variable $X$ is drawn from
distribution $Y$. This channel model represents a good approximation to the
real-world channel of interest to us - a channel where erasure probability is
high111In channels where $\mathcal{P}$ indicates a small probability for
$T\sim 0$, and where $p_{e}\sim 0$, we find that no improvement from the
protocol is likely - the transmission may as well be just direct. If _a
priori_ known that $p_{e}=0$, we have a Gaussian channel, and the introduction
of non-Gaussian states or measurements into the protocol is needed if any
advantage is to be forthcoming.. Also, the extension of the channel model via
a new probability, $p_{e}$, to parameterize the additional loss caused by
transceiver movement/offsets allows us to compare better with previous related
work.

TABLE I: Free-space communications parameters $\lambda$ [nm] |  $r_{t}$ [cm] |  $C_{n}^{2}$($\text{[m]}^{-2/3}$) |  $l_{0}$ [mm] |  $L_{0}$ [m]
---|---|---|---|---
1550 | 2.5 | $2.47\times 10^{-13}$ | 7.5 | 1.57
Figure 2: Transmissivities obtained from the model of the free-space channel
following the parameters in Table I. The shaded area represents the standard
deviation of the curves.

##  III Simulations

Here, we first outline the CFs of the main quantum resources we use and make
clear how we determine the fidelity - the key metric we use. We then discuss
the performance of the quantum and classical communication under the protocol,
detailing how the optimization of $g$ is determined for each simulation run.

###  III-A Quantum resources

The quantum signal is a coherent state $\ket{\alpha}$, whose CF is,

$\displaystyle\makebox{\large$\chi$}_{\ket{\alpha}}(\lambda)=\exp\left[-\frac{|\lambda|^{2}}{2}+(\lambda\alpha^{*}-\lambda^{*}\alpha)\right].$
(8)

The two-mode squeezed vacuum (TMSV) state $\ket{\psi_{\mathrm{AB}}}$ used in
the encoding has a CF,

$\displaystyle\makebox{\large$\chi$}_{\mathrm{TMSV}}(\lambda_{\mathrm{A}},\lambda_{\mathrm{B}})=\exp\left[-\frac{1}{2}\left(|\lambda_{\mathrm{A}}^{\prime}|^{2}+|\lambda_{\mathrm{B}}^{\prime}|^{2}\right)\right].$
(9)

In deriving this, the following transformation has been used[33],

$\displaystyle\hat{S}_{\mathrm{AB}}(\varrho)\hat{D}(\lambda_{\mathrm{A}})\hat{D}(\lambda_{\mathrm{B}})\hat{S}^{\dagger}_{\mathrm{AB}}(\varrho)=\hat{D}(\lambda_{\mathrm{A}}^{\prime})\hat{D}(\lambda_{\mathrm{B}}^{\prime}),$
(10)
$\displaystyle\lambda_{j}^{\prime}=\cosh(r)\lambda_{j}+e^{i\phi}\sinh(r)\lambda_{k}^{*}~{}~{}~{}~{}j,k=\\{\mathrm{A,B}\\};~{}~{}j\neq
k,$

with $\hat{S}_{\mathrm{AB}}$ the two-mode squeezing operator, and with
$\varrho=re^{i\phi}$. Without loss of generality, the value $\phi=\pi$ is set.
Finally, the CF of the vacuum state $\ket{0}$, that appears in Eq. 5 is,

$\displaystyle\makebox{\large$\chi$}_{\ket{0}}(\lambda)=\exp\left[-\frac{|\lambda|^{2}}{2}\right].$
(11)

Fidelity is used as the metric of the effectiveness of the protocol. The
fidelity represents the closeness between the quantum signal $\ket{\psi_{s}}$
and the output state $\ket{\psi_{\mathrm{out}}}$. In the CF formalism, the
fidelity is computed as,

$\displaystyle\mathcal{F}_{\alpha}=\frac{1}{\pi}\int
d^{2}\lambda\makebox{\large$\chi$}_{\mathrm{s}}(\lambda)\makebox{\large$\chi$}_{\mathrm{out}}(-\lambda).$
(12)

The fidelity as defined in Eq. 12 will depend on each input state’s value
$\alpha$. Therefore, the mean fidelity over an ensemble of coherent states
must be considered. The following Gaussian distribution specifies the
ensemble,

$\displaystyle
P_{\alpha}(\alpha)=\frac{1}{\sigma_{\alpha}\pi}\exp\left[-\frac{|\alpha|^{2}}{\sigma_{\alpha}}\right],$
(13)

with $\sigma_{\alpha}$ the variance. Therefore, the fidelity over the ensemble
of coherent states will be used, defined as

$\displaystyle\mathcal{F}=\int
d{\alpha}^{2}P_{\alpha}(\alpha)\mathcal{F}_{\alpha}.$ (14)

Ideally, we would like to compute the fidelity considering a uniform
distribution of coherent states (corresponding to $\sigma_{\alpha}\to\infty$);
however, this would be analytically intractable. An approximation to a
uninform distribution can be obtained by setting a large enough variance,
$\sigma_{\alpha}=10$ (a value we adopt through this work). For reference, the
so-called classical limit, $\mathcal{F}_{\mathrm{class}}=0.52$ sets the
baseline where fidelities of transmission below this value could be obtained
using purely classical communications[34].

Using Eq. 5 with Eq. 14 it becomes possible to find an analytical expression
of the fidelity for the ensemble of coherent states. This expression is
provided in Appendix C, Eq. 41. Additionally, the fidelity for direct
transmission for the ensemble of coherent states,
$\mathcal{F}^{\mathrm{dir}}$, is also presented in Appendix C, Eq. 43.

###  III-B Transmission of quantum information

In the following calculations, a homodyne measurement efficiency of
$\eta^{2}=0.9$ is set unless specified otherwise. Fluctuations of the free-
space channel produce several effects contributing to the excess noise [20,
21]. We have modeled the excess noise in our simulation as
$\epsilon=\epsilon_{\mathrm{ph}}+\epsilon_{\mathrm{det}}$. Here,
$\epsilon_{\mathrm{ph}}$ is related to noise introduced by the displacement
used in the encoding (see [11, 12]) and will be modified by the transmissivity
of the channel. The other term, $\epsilon_{\mathrm{det}}$ is detector excess
noise, which we set to $\epsilon_{\mathrm{det}}=0.013$ [21]. As an example,
for $T=0.5$, we have $\epsilon=0.018$. When computing the fidelity, every
protocol realization involves a different sample of the values $T_{j}$. Thus,
the parameter $g$ must be optimized for each protocol realization, as shown in
Fig. 3. The following procedure was followed to compute the mean fidelities.
First, using the numerical simulations described in Section II-B, 30000
samples of $T_{j}$ were obtained for a given propagation distance. Next,
fidelity was obtained for a specific realization of the channels using three
sampled values of $T_{j}$ in Eq. 41.

Figure 3: Fidelity as a function of $g$, each curve corresponds to a specific
set of values of $[T_{1},T_{2},T_{3}]$. The red dots mark the value of
$g=g_{\mathrm{opt}}$ corresponding to the maximum fidelity, and each label
denotes the values of the set $[T_{1},T_{2},T_{3}]$ corresponding to the
adjacent curve.

All possible transmissivity occurrences must be accounted for to compare the
protocol’s effectiveness to direct transmission. For direct transmission, the
total fidelity is calculated as

$\displaystyle\mathcal{F}^{\mathrm{dir}}_{\mathrm{total}}=(1-p_{\mathrm{e}})\mathcal{F}^{\mathrm{dir}}+p_{\mathrm{e}}\mathcal{F}_{\ket{0}}.$
(15)

Figure 4: Total fidelity for different values of erasure rate. Solid lines
represent transmission via the protocol, and dashed lines correspond to direct
transmission. The ancilla entangled state uses a squeezing value of $r=10$ dB.

The results comparing the mean total fidelity between the protocol and direct
transmission are shown in Fig. 4. The results indicate that the protocol
improves the transmission fidelity for values of $p_{\mathrm{e}}$ up to 0.35.
The advantage provided by the protocol decreases as the distance increases.
For values, $p_{\mathrm{e}}{>}0.35$, it is observed that the protocol does not
provide any advantage for the $r_{d}$ values shown.

###  III-C Transfer of classical information

Now the error rates in transmitting classical information via the protocol are
analyzed. An error during the transmission of classical information
corresponds to Bob misidentifying the symbol sent initially by Alice. The
excess noise added to the system increases the variance of the syndrome
measurements. Thus, if the noise is too high, there is a probability that the
syndrome measurements appear on a different partition from the one Alice meant
to encode. The error probability can be directly quantified by the ratio
between the variance of the syndrome measurements, $\sigma_{s}$, and the
magnitude of the displacement used during the classical encoding, $|\Delta|$.
For a given value of $|\Delta|$, a larger $\sigma_{s}$ means a higher Bit
Error Rate (BER). Alternatively, a more significant displacement can also be
used to increase the distance between partitions and offset the effects of
$\sigma_{s}$, increasing the Signal-to-Noise Ratio (SNR).

If all the states used during the protocol are Gaussian, then the measurement
results will follow a Gaussian distribution [35]. Solving the integration in
Eq. 32 in Appendix A, we see that the syndrome measurements have the mean
values given by

$\displaystyle\mu_{\tilde{x}}=\frac{1}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Re}(\Delta)$
$\displaystyle\mu_{\tilde{p}}=\frac{1}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Im}(\Delta).$
(16)

The variance $\sigma_{s}$ depends on the specific values $T_{j}$, the amount
of squeezing $r$, and the size of the ensemble of states being transmitted
using the protocol $\sigma_{\alpha}$. The expression of $\sigma_{s}$ is
presented in Appendix D.

Once we have determined the moments of the syndrome measurements, we can
compute the BER. For simplicity, we assume the classical information consists
of single bits encoded in Binary Phase-Shift Keying [5]. In this case, the BER
of the protocol will be,

$\displaystyle\mathrm{BER}=\frac{1}{2}\mathrm{erfc}\left(\frac{\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}|\Delta|}{\sqrt{2\sigma_{s}^{2}}}\right).$
(17)

A value of interest is the size of the displacement required to maintain the
BER to a tolerable level. For a BER of $10^{-9}$, we see in Fig. 5 the mean
values of $|\Delta|$ required for the special case when $T_{1}=0$ in mode $1$
(or similarly mode $2$). If instead mode $3$ possesses $T_{3}=0$, the
displacement requirements are approximately half of the values shown.

Figure 5: Mean magnitude of the displacement, in shot-noise units, required to
keep the error rate below $10^{-9}$ when the transmissivity is zero on mode
$1$. A squeezing value $r=10$ dB is considered in the TMSV state.

##  IV Enhancement via non-Gaussian operations

Non-Gaussian states represent a key element in CV quantum information, and
there has been great interest in their use to enhance quantum communications
protocols [36, 37, 38, 39, 40, 41]. Particularly, in CV quantum teleportation,
certain non-Gaussian operations have shown to enhance the fidelities of
transmitted states [42, 33, 43, 44]. Thus, it is natural to ask if the same
holds for this protocol.

One viable experimental way to construct non-Gaussian states shown in Fig. 6,
where a mode of a TMSV is combined with a BS with transmissivity $T_{\kappa}$
with a $M$ photon Fock state, and a photon discriminating detector is used to
detect $N$ photons. In the following, we consider a Photon Subtraction (PS),
setting $M=0,N=1$; Photon Addition (PA) with $M=1,N=0$; and Photon Catalysis
(PC) setting $M=1,N=1$. We also consider the successive application of the PS
and PA operations, PS-PA (order reversed in PA-PS). All the operations are
applied in mode $3^{\mathrm{o}}$ of the initial TMSV state as the first step
in our protocol.

Besides applying non-Gaussian operations on a TMSV, an additional non-Gaussian
entangled state is considered. This state is prepared at the start of the
protocol by the application of the two-mode squeezing operation to a Bell
state, the state known as the Squeezed Bell (SB) state, given by

$\displaystyle\ket{\psi_{\mathrm{SB}}}$
$\displaystyle=(\cos^{2}(\vartheta)+\sin^{2}(\vartheta))^{-1/2}$
$\displaystyle\times\hat{S}_{\mathrm{AB}}(\varrho)\Big{[}\cos(\vartheta)\ket{00}+\sin(\vartheta)\ket{11}\Big{]}.$
(18)

The implementation of the non-Gaussian states is achieved by replacing
$\ket{\psi_{\mathrm{AB}}}$ in Fig. 1 with the corresponding non-Gaussian
state. The CF of each non-Gaussian state must be computed first and used in
Eq. 5, where the non-Gaussian CF replaces
$\makebox{\large$\chi$}_{\mathrm{AB}}$. The CF of the states used in this work
have been calculated in previous works [43], and are listed in Appendix E for
completeness.

In the CF of each non-Gaussian state, a free parameter exists corresponding to
the transmissivity of the beam-splitter involved in each operation. This free
parameter is optimized to maximize the fidelity of transmission. In the case
of the SB state, the $\vartheta$ free parameter is optimized. For the PS-PA
and PA-PS, the exact value of $T_{\kappa}$ is used in both successive non-
Gaussian operations. Finally, it is assumed that a quantum memory is available
at the transmitter, such that the non-Gaussian states can be prepared and
stored to be used on demand during the protocol222Non-Gaussian operations have
a non-unity success probability. Without a quantum memory, this success
probability would need to be accounted for in the resulting fidelities..

Figure 6: Non-Gaussian operations used to generate a non-Gaussian resource
state to be used in the teleportation channel. The operations are applied on a
single mode of a TMSV state.

The results are presented in Fig. 7. To simplify the analysis, we evaluate the
fidelities for a simplified channel, where
$\mathcal{P}=\delta(T_{j}-T^{\prime})$ in Eq. 7, the excess noise is set as
$\epsilon=0$, and $\eta=1$. Of all the non-Gaussian resources used, only two
states represent an enhancement over the TMSV squeezed state: the PA-PS and
the SB states. Moreover, the gain in fidelity is only observed when the
squeezing in the entangled states is low, around $r=4.7$ dB. As the squeezing
increases, the fidelities obtained by the non-Gaussian states get closer to
the ones obtained by the TMSV state, for a value of $r=10$ dB, the difference
in fidelity is of the order of $10^{-2}$. We point out that there is an
agreement between the two non-Gaussian resources that represent an enhancement
between the protocol presented here and CV quantum teleportation [43].
Moreover, when mode $3$ is erased, the fidelities obtained using the non-
Gaussian states are equal to the ones obtained when the TMSV state is used.

Now, we address how the use of non-Gaussian states affects classical encoding.
When non-Gaussian states are used, the syndrome measurement results follow a
non-Gaussian PDF. Numerical integration over this non-Gaussian PDF indicates
that the values of $|\Delta|$ required to keep low BER are considerably higher
when the non-Gaussian states are used. Approximately $80\%$ larger $|\Delta|$
is required in the SB and PA-PS states when mode $1$ is erased333A case where
the application of the PA-PS operation is applied to mode $2^{\mathrm{o}}$
instead of mode $3^{\mathrm{o}}$ during the preparation of the non-Gaussian
state was also tested. The results showed no difference in the displacement
requirements between the two cases..

Figure 7: Fidelities obtained for a loss channel for the different non-
Gaussian states. In each case, the squeezing of the TMSV state before the non-
Gaussian operation is fixed at a value $r=4.7$ dB.

##  V Discussion

Here, we discuss our new results in relation to previous similar work, some
alternative encoding schemes, and some additional input states.

###  V-A Relation to other work

Other works [25, 26, 27, 28, 29] have previously looked at similar encoding
schemes. In [25, 26], a three-mode encoding scheme was studied in the context
of a quantum state sharing scheme. Although not aimed at the communication
scenario, the nature of this scheme was similar to the encoding required for
an erasure channel in the communication context. In [28], a four-mode encoding
scheme used to transfer two coherent input states over an erasure channel was
investigated. In [29], a three-mode encoding scheme similar to that used in
the context of state sharing was analyzed for the communication scenario, with
extensions to include non-Gaussian operations. This latter work was again only
in the scenario of an erasure channel. Channel errors beyond a simple erasure
were considered in [27] \- namely, a displacement error.

The results presented here represent outcomes different from all these
previous works in several aspects. First, we consider a channel model more
appropriate to that anticipated for free-space atmospheric communications in
which one transceiver is untethered. Simulations of this channel are then used
to analyze a modified protocol deployed over a three-mode channel. The key
modification in the protocol is an optimization phase in which the
transmissivity measured for each channel is used as an input. This allows for
encoding and decoding in the more general case where transmissivities in all
channels are non-zero. Importantly, our protocol leads to improved results
relative to the situation where any loss is simply considered a complete
erasure. Second, we have included additional displacements within the encoding
phase to include classical signaling. Third, we have investigated non-Gaussian
operations within the modified protocol.

###  V-B Alternative encodings and post selection

Thus far, we have considered combined classical-quantum communication embedded
in the same signal. However, other possibilities exist. For example, the
reference pulses could embed classical information – simple on-off keying
being one scenario. In this scenario, the lack of any LO sent in a channel
could indicate a zero and its presence, a one. The quantum signal, in this
case, would not need an additional displacement, but the trade-off would be
that the quantum information rate would be reduced. Exactly how much reduction
would occur would depend on the coherence time of the channel and the pulse
rate of the source, but for anticipated timescales of order 1ms and available
pulse rates of 100MHz, this reduction would be minimal (since the channel is
stable for thousands of sent LOs - these can be re-used). This trade-off of a
reduced quantum information rate would have the benefit of a reduction in the
excess noise of ${\approx}0.02$ and would yield an increase of fidelity
(${\approx}0.05$). An added benefit would be a reduction of the complexity of
deployment (one less displacement operation) and potential power savings
(fewer LOs sent). On-off keying could also be applied to the quantum signals
(with a vacuum detection being mapped to a zero). However, such schemes would
need a modified digital modulation scheme (to avoid ‘on’ signals which have
small displacements).

Our results presented here have utilized an averaging over all channel
conditions. In practice, post-selection could be utilized to select the
channels such that, at most only one presents an erasure. Again, a trade-off
in performance vs. throughput would be in play here. Post-selection would be
most beneficial under high erasure error rates. For example, post-selecting to
allow at most a single erasure at an error rate $p_{\mathrm{e}}=0.3$ would
have the benefit of the fidelity being increased by ${\approx}0.15$, at a
reduction in throughput of 0.8.

###  V-C Entanglement distribution

We are also interested in evaluating the protocol’s effectiveness in
distributing entangled states. Considering an input TMSV state,
$\ket{\psi^{\mathrm{in}}_{\mathrm{MN}}}$ with squeezing $r_{\mathrm{MN}}$, we
can evaluate the fidelity with the state obtained after mode $\mathrm{M}$ has
been transmitted via the protocol. Although not shown, our results show that
there are scenarios where an advantage over direct transmission can be found
similar to before. For equal squeezing values in the states
$\ket{\psi_{\mathrm{AB}}}$ and $\ket{\psi^{\mathrm{in}}_{\mathrm{MN}}}$, we
observe the use of the protocol does present an advantage over direct
transmission.

Ideally, for entangled input states, we would like to test the protocol’s
effectiveness using an application of quantum communications, such as
entanglement distribution. However, using only fidelity can be an incomplete
metric of the transmission of quantum information, especially for entangled
states [45]. Entanglement distribution is perhaps better measured by computing
the _Reverse Coherent Information_ (RCI) of the channel, as it represents a
lower bound on the distillable entanglement [46]. The RCI is defined as,

$\displaystyle\mathcal{R}=S(\rho_{\mathrm{N}})-S(\rho_{\mathrm{MN}}),$ (19)

with $S(\cdot)$ the Von Neumann entropy, and $\rho_{\mathrm{MN}}$ a maximally
entangled TMSV state ($r_{\mathrm{MN}}\to\infty$), where mode $\mathrm{M}$ has
been transmitted via the channel using the protocol.

For our protocol, an upper bound on the RCI can be found using the fact that
the entropy is concave,

$\displaystyle S(\rho_{\mathrm{MN}})\geq\sum_{k}p_{k}S(\rho_{\mathrm{k}}),$
(20)

where the index $k$ iterates over all possible erasure combinations on three
modes (no erasure, erasure on mode 1, erasure on mode 2, and so forth),
$\rho_{\mathrm{k}}$ is the corresponding state after error correction for
combination $k$, and $p_{k}$ the probability of each combination. In the
combinations when error correction is not possible, e.g. erasures on modes 1
and 2, $\rho_{k}=\text{Tr}_{\mathrm{M}}[\rho^{\mathrm{in}}_{\mathrm{MN}}]$.
Using the upper bounds of the RCI as a new metric, we are interested in
comparing the protocol with direct transmission. Direct transmission, in this
case, corresponds to transmitting mode $\mathrm{M}$ (of the state
$\ket{\psi_{\mathrm{MN}}}$) via the channel. We compute the upper bounds of
the RCI for the simplified channel used for Fig. 7.

The resulting upper bounds of RCI are presented in Fig. 8. The results show
that the upper bound on the RCI obtained from the protocol and direct
transmission is virtually equal for values $T^{\prime}$ close to 1; as
$T^{\prime}$ decreases in value, the protocol stops presenting an advantage,
and the RCI becomes lower than the one obtained by direct transmission. The
lack of an advantage by the protocol in this simulation can be understood by
the fact that loss in the ancilla modes (modes $2^{\prime}$ and $3^{\prime}$)
in the protocol propagates to the final state. Thus, while the transmission of
coherent states can be effectively enhanced by using the protocol presented
here, we warn that this may not be the case for applications that are
dependent on the distribution of entangled states.

Figure 8: RCI obtained using the protocol (solid lines), and by direct
transmission (dashed lines) over a fixed loss channel with a probability of
erasure $p_{\mathrm{e}}$. The values $\epsilon=\epsilon_{\mathrm{det}}$, and
$\eta=1$ are used. A maximally entangled TMSV is considered as
$\ket{\psi_{\mathrm{AB}}}$.

##  VI Conclusions

In this work, a new error correction protocol was introduced. Aimed at
deployment over realistic free-space optical channels where at least one
transceiver is untethered, the protocol was designed to encompass both
classical and quantum information on the encoded signal. We showed, relative
to non-encoded direct transmission, that the protocol improves the fidelity of
transmitted coherent states over a wide range of losses and erasure
probabilities. In addition, using ancillary non-Gaussian entangled bipartite
states in the signal encoding further increased the fidelity of transmission.
Variants on a theme for the protocol were discussed, including its application
to the transmission of components of a multi-mode entangled state. Our results
demonstrate that signal loss in atmospheric channels can be compensated for
significantly at the price of additional implementation complexity.

## Acknowledgements

We thank Ziqing Wang for the valuable discussions. The Australian Government
supported this research through the Australian Research Council’s Linkage
Projects funding scheme (project LP200100601). The views expressed herein are
those of the authors and are not necessarily those of the Australian
Government or the Australian Research Council. Approved for Public Release;
Distribution is Unlimited; #23-0238.

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##  Appendix A Output state CF derivation

We will derive the CF of the output quantum state obtained by the protocol. To
do so, we follow the diagram presented in Fig. 1. As the first step, consider
the application of a displacement operation on mode 2 of a bipartite state
$\hat{\rho}_{\mathrm{AB}}$,

$\displaystyle\rho^{\prime}=\hat{D}(\Delta)\hat{\rho}_{\mathrm{AB}}\hat{D}^{\dagger}(\Delta).$
(21)

Then the CF of the state $\rho^{\prime}$ is,

$\displaystyle\makebox{\large$\chi$}_{\rho^{\prime}}(\lambda_{1},\lambda_{2})$
$\displaystyle=\text{Tr}[\hat{D}(\Delta)\hat{\rho}_{\mathrm{AB}}\hat{D}^{\dagger}(\Delta)\hat{D}(\lambda_{1})\hat{D}(\lambda_{2})]$
(22)

Using the following properties of the displacement operator,

$\displaystyle\hat{D}(\alpha)\hat{D}(\beta)=e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}\hat{D}(\alpha+\beta);$
$\displaystyle\hat{D}^{\dagger}(\alpha)\hat{D}(\beta)\hat{D}(\alpha)=e^{(\alpha^{*}\beta-\alpha\beta^{*})}\hat{D}(\beta),$
(23)

the CF of $\hat{\rho}^{\prime}$ is then,

$\displaystyle\makebox{\large$\chi$}_{\hat{\rho}^{\prime}}(\lambda_{1},\lambda_{2})$
$\displaystyle=\text{Tr}[e^{(\Delta^{*}\lambda_{2}-\Delta\lambda_{2}^{*})}\hat{D}(\lambda_{1})\hat{D}(\lambda_{2})\hat{\rho}_{AB}]$
$\displaystyle=\makebox{\large$\chi$}_{\mathrm{AB}}(\lambda_{1},\lambda_{2})\mathrm{D}_{\Delta}(\lambda_{2})$
(24)

where we defined the function
$\mathrm{D}_{\Delta}(\lambda)=e^{(\Delta^{*}\lambda_{-}\Delta\lambda^{*})}$
and $\makebox{\large$\chi$}_{\mathrm{AB}}$ is the CF of $\hat{\rho}_{AB}$. Now
the CF of the initial state at the start of the protocol, considering the
displacement applied to the entangled state, is

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\lambda_{1}\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\lambda_{2},\lambda_{3}\right)\mathrm{D}_{\Delta}\left(\lambda_{2}\right).$
(25)

The effect of BS1 acting on modes 1 and 3 can be described as a change in
variables, such as

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{\lambda_{1}+\lambda_{2}}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{\lambda_{1}-\lambda_{2}}{\sqrt{2}},\lambda_{3}\right)\mathrm{D}_{\Delta}\left(\frac{\lambda_{1}-\lambda_{2}}{\sqrt{2}}\right).$
(26)

After that, the effect of the loss acting independently on every channel
transforms the CF to

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{\sqrt{T_{1}}\lambda_{1}+\sqrt{T_{2}}\lambda_{2}}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{\sqrt{T_{1}}\lambda_{1}-\sqrt{T_{2}}\lambda_{2}}{\sqrt{2}},\sqrt{T_{3}}\lambda_{3}\right)$
$\displaystyle\times\mathrm{D}_{\Delta}\left(\frac{\sqrt{T_{1}}\lambda_{1}-\sqrt{T_{2}}\lambda_{2}}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\ket{0}}\left(T_{1}^{*}\lambda_{1}\right)\makebox{\large$\chi$}_{\ket{0}}\left(T_{2}^{*}\lambda_{2}\right)$
$\displaystyle\times\makebox{\large$\chi$}_{\ket{0}}\left(T_{3}^{*}\lambda_{3}\right),$
(27)

with $T_{j}^{*}=\sqrt{1+\epsilon-T_{j}}$. Thereafter, BS2 is applied by Bob on
modes 1’ and 2’,

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{\sqrt{T_{1}}(\lambda_{1}+\lambda_{2})+\sqrt{T_{2}}(\lambda_{1}-\lambda_{2})}{2}\right)$
$\displaystyle\times\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{\sqrt{T_{1}}(\lambda_{1}+\lambda_{2})-\sqrt{T_{2}}(\lambda_{1}-\lambda_{2})}{2},\sqrt{T_{3}}\lambda_{3}\right)$
$\displaystyle\times\mathrm{D}_{\Delta}\left(\frac{\sqrt{T_{1}}(\lambda_{1}+\lambda_{2})-\sqrt{T_{2}}(\lambda_{1}-\lambda_{2})}{2}\right)$
$\displaystyle\times\makebox{\large$\chi$}_{\ket{0}}\left(\frac{T_{1}^{*}(\lambda_{1}+\lambda_{2})}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{T_{2}^{*}(\lambda_{1}-\lambda_{2})}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\ket{0}}\left(T_{3}^{*}\lambda_{3}\right).$
(28)

Finally, to perform a dual homodyne measurement, BS3 is applied, transforming
the state. The efficiency of the homodyne measurements can be modeled by
considering a BS with transmissivity $\eta^{2}$, with an extra vacuum mode
placed before the detectors. Then the CF of the state before homodyne
measurements is,

$\displaystyle\makebox{\large$\chi$}_{\mathrm{BS3}}(\lambda_{1},\lambda_{2},\lambda_{3})$
$\displaystyle=$
$\displaystyle\makebox{\large$\chi$}_{\mathrm{BS3}}^{\prime}(\lambda_{1},\lambda_{2},$
$\displaystyle\lambda_{3})\mathrm{D}_{\Delta}\left(\lambda_{1}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\frac{(\lambda_{2}+\lambda_{3})}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right),$ (29)

where

$\displaystyle\makebox{\large$\chi$}_{\mathrm{BS3}}^{\prime}(\lambda_{1},\lambda_{2},\lambda_{3})=$
$\displaystyle~{}\makebox{\large$\chi$}_{\mathrm{s}}\left(\lambda_{1}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\frac{\lambda_{2}+\lambda_{3}}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower 1.2571pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)\times$
$\displaystyle~{}\makebox{\large$\chi$}_{\mathrm{AB}}\left(\lambda_{1}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\frac{(\lambda_{2}+\lambda_{3})}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},\frac{\sqrt{T_{3}}\eta(\lambda_{2}-\lambda_{3})}{\sqrt{2}}\right)\times$
$\displaystyle~{}\makebox{\large$\chi$}_{\ket{0}}\left(T_{1}^{*}\left(\frac{\lambda_{1}}{\sqrt{2}}+\eta\frac{\lambda_{2}+\lambda_{3}}{2}\right)\right)\times$
$\displaystyle~{}\makebox{\large$\chi$}_{\ket{0}}\left(T_{2}^{*}\left(\frac{\lambda_{1}}{\sqrt{2}}-\eta\frac{\lambda_{2}+\lambda_{3}}{2}\right)\right)\times$
$\displaystyle~{}\makebox{\large$\chi$}_{\ket{0}}\left(T_{3}^{*}\eta\frac{\lambda_{2}-\lambda_{3}}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{1-\eta^{2}}\lambda_{1}\right)\times$
$\displaystyle~{}\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{1-\eta^{2}}\lambda_{2}\right).$
(30)

It is convenient to express the measurement results in the phase space
representation by transforming the complex arguments into two real numbers,
$x_{j}=\lambda_{j}+\lambda_{j}^{*}$ and
$p_{j}=i(\lambda_{j}^{*}-\lambda_{j})$. Then the homodyne measurements are
represented by integrating the measured modes.

$\displaystyle\makebox{\large$\chi$}_{\mathrm{m}}(x,p)$
$\displaystyle=\frac{\mathcal{P}(\tilde{x},\tilde{p})^{-1}}{(2\pi)^{2}}\int
dx_{2}dp_{3}\makebox{\large$\chi$}_{\mathrm{BS3}}\left(x,p,x_{2},0,0,p_{3}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
e^{-i\tilde{x}p_{3}+i\tilde{p}x_{2}},$ (31)

with $\tilde{\mathcal{P}}(\tilde{x},\tilde{p})$ the PDF of any pair of
measurement results, given by

$\displaystyle\tilde{\mathcal{P}}(\tilde{x},\tilde{p})=\frac{1}{(2\pi)^{2}}\int
dx_{2}dp_{3}\makebox{\large$\chi$}_{\mathrm{BS3}}\left(0,0,x_{2},0,0,p_{3}\right)$
$\displaystyle\times e^{-i\tilde{x}p_{3}+i\tilde{p}x_{2}}.$ (32)

Expanding the terms in $\mathrm{D}_{\Delta}$, we have,

$\displaystyle\mathrm{D}_{\Delta}$
$\displaystyle\left(x_{1}T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower 1.2571pt\hbox
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}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\frac{x_{2}}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},p_{1}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\frac{p_{3}}{\sqrt{2}}\eta
T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)=$
$\displaystyle\exp\Big{[}-ix_{1}T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower 1.2571pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Im}(\Delta)+ip_{1}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Re}(\Delta)\Big{]}\times$
(33) $\displaystyle\exp\Big{[}-ix_{2}\frac{\eta T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Im}(\Delta)}{\sqrt{2}}+ip_{3}\eta\frac{T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Re}(\Delta)}{\sqrt{2}}\Big{]}.$

Inserting this expression into Eq. 32 and manipulating the terms, it becomes
clear that the mean of the syndrome measurement results is displaced following
Eqs. III-C.

The additional exponential term in Eq. 31 indicates that the state after the
measurement requires a corrective displacement to be recovered. This
corrective displacement must also account for the additional displacement
induced by the displacement $\Delta$. The corrective displacement can be
implemented via,

$\displaystyle\hat{D}_{\mathrm{corr}}(\tilde{x},\tilde{p})=$
$\displaystyle\exp\Big{[}-ip\left(\sqrt{2}\tilde{x}^{\prime}g+T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Re}(\Delta)\right)+$
$\displaystyle
ix\left(\sqrt{2}\tilde{p}^{\prime}g+T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\text{Im}(\Delta)\right)\Big{]},$
(34)

with

$\displaystyle\tilde{x}^{\prime}=\tilde{x}-\mu_{\tilde{x}},$
$\displaystyle\tilde{p}^{\prime}=\tilde{p}+\mu_{\tilde{p}},$ (35)

where $\mu_{\bar{x}(\bar{p})}$ is defined in Eq. III-C. Here, the correction
includes a factor of $\sqrt{2}$ to compensate for the global factor that
appears in the arguments of $\makebox{\large$\chi$}_{\mathrm{BS3}}$ in Eq. 29.

Finally, since the output CF will depend on a specific set of measurement
results, the mean over all possible measurement outcomes, weighted by their
corresponding probability, must be considered, that is

$\displaystyle\makebox{\large$\chi$}_{\mathrm{out}}(x,p)$ $\displaystyle=\int
d\tilde{x}d\tilde{p}\mathcal{P}(\tilde{x},\tilde{p})\makebox{\large$\chi$}_{\mathrm{m}}(x,p)\hat{D}_{\mathrm{corr}}(\tilde{x},\tilde{p})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int
d\tilde{x}d\tilde{p}dx_{2}dp_{3}\makebox{\large$\chi$}_{\mathrm{BS3}}^{\prime}(x,p,x_{2},0,0,p_{3})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
e^{i\tilde{x}^{\prime}(p_{3}-\sqrt{2}pg)-i\tilde{p}^{\prime}(x_{2}-\sqrt{2}xg)}.$
(36)

At this point, the definition of the Dirac delta function,
$\frac{1}{2\pi}\iint e^{i\beta x-i\beta\alpha}f(\alpha)d\beta
d\alpha=\int\delta(x-\alpha)f(\alpha)d\alpha$, can be used twice to obtain

$\displaystyle\makebox{\large$\chi$}_{\mathrm{out}}(x,p)$ $\displaystyle=\int
dx_{2}dp_{3}\delta(p_{3}-\sqrt{2}p)\delta(x_{2}-\sqrt{2}x)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\makebox{\large$\chi$}_{\mathrm{BS3}}^{\prime}(x,p,x_{2},0,0,p_{3}).$
(37)

Removing the integrands using the properties of the $\delta$ function, the Eq.
5 presented above is recovered.

##  Appendix B Specific examples of error correction

To further understand the error correction, we outline the following three
cases, corresponding to different values of $g$.

####  B-1 $g=0$

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\lambda\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\lambda,0\right)\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{1+\epsilon-\frac{T_{1}+T_{2}}{2}}\lambda\right).$
(38)

In this case, the resulting state is independent of the value of $T_{3}$.
Moreover, the excess noise in channels $2^{\prime}$ and $3^{\prime}$ does not
propagate to the output state. However, when $T_{1}{\neq}T_{2}$ imperfect
destructive interference in BS2 will translate to additional excess noise
introduced by the entangled state, which will be proportional to the squeezing
of the state. When $T_{1}=T_{2}$, the resulting state is equivalent to the one
obtained if the signal state were to be transmitted directly through the
channel without using the protocol.

####  B-2 $g=\frac{1}{\eta}$

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\sqrt{T_{1}}\lambda\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\sqrt{T_{1}}\lambda,\sqrt{T_{3}}\lambda^{*}\right)\times$
$\displaystyle\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{3+3\epsilon+2\left(\frac{1}{\eta^{2}}-1\right)-2T_{1}-T_{3}}\lambda\right).$
(39)

The resulting state will be independent of the value of $T_{2}$. Unlike the
previous case, the resulting state is affected by the loss and excess noise
affecting modes $2^{\prime}$ and $3^{\prime}$. Consider the case when
$T_{1}=T_{3}$, then it becomes clear that the vacuum contribution appears in
the output state three times. On an ideal scenario, with $T_{1}=T_{3}=1$, the
quantum signal can be fully recovered in the limit $V\rightarrow\infty$ as in
this limit
$\makebox{\large$\chi$}_{\mathrm{AB}}(\lambda,\lambda^{*})\rightarrow 1$ (see
Eq. 9) [29].

####  B-3 $g=-\frac{1}{\eta}$

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\sqrt{T_{2}}\lambda\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(-\sqrt{T_{2}}\lambda,-\sqrt{T_{3}}\lambda^{*}\right)\times$
$\displaystyle\makebox{\large$\chi$}_{\ket{0}}\left(\sqrt{3+3\epsilon+2\left(\frac{1}{\eta^{2}}-1\right)-2T_{2}-T_{3}}\lambda\right).$
(40)

In this case, the resulting state is independent of $T_{1}$. Similarly to the
previous case, the loss and excess noise from the ancillary entangled state
propagates to the output state.

##  Appendix C Fidelity of Transmission

To compute the fidelity between the states $\ket{\phi_{\mathrm{in}}}$ and
$\ket{\phi_{\mathrm{out}}}$ first the CF of
$\makebox{\large$\chi$}_{\mathrm{out}}$ is obtained using Eq. 5. Next, an
analytical expression for a single input coherent state can be obtained by
solving the integration in Eq.12. Finally, this fidelity is averaged over the
ensemble of coherent states by solving the integral in Eq. 14. The expression
obtained is,

$\displaystyle\mathcal{F}=\frac{2}{2\sigma_{\alpha}\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\tilde{g}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}-1\right)^{2}+c_{1}+1},$
(41)

with

$\displaystyle c_{1}=$ $\displaystyle
V\left(\left(T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower 1.2571pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\tilde{g}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)^{2}+T_{3}\tilde{g}^{2}\right)-$
$\displaystyle 2\sqrt{V^{2}-1}\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\tilde{g}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)\sqrt{T_{3}}\tilde{g}+\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\tilde{g}T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right)^{2}+$
$\displaystyle(1+\epsilon)(2\tilde{g}^{2}+1)+g^{2}(1-\eta^{2})-T^{\prime}.$
(42)

Additionally, the analytical expression for the fidelity of direct
transmission can be obtained by following the same procedure as above, where
the CF of the state transmitted directly through the channel replaces
$\makebox{\large$\chi$}_{\mathrm{out}}$. The expression obtained is,

$\displaystyle\mathcal{\bar{F}}^{\mathrm{DT}}=\frac{2}{2\sigma_{\alpha}(\sqrt{T}-1)^{2}+2}.$
(43)

where $T$ is the transmissivity of the channel.

##  Appendix D Derivation of the SNR of the measurement results

The SNR ratio of the syndrome measurement is used to calculate the bit error
rate of the encoded classical communications. To find the SNR, we need to find
the variance of the syndrome measurement results, $\sigma_{\mathrm{s}}$. If
all the states involved in the protocol are Gaussian, then solving the
integrating in Eq. 32, we observe that $\tilde{\mathcal{P}}$ corresponds to a
Gaussian distribution,

$\displaystyle\tilde{\mathcal{P}}(\tilde{x},\tilde{p})=\frac{1}{2\sigma_{\mathrm{s}}\pi}\exp\left[-\frac{\tilde{x}^{\prime
2}+\tilde{p}^{\prime 2}}{2\sigma_{s}^{2}}\right],$ (44)

where

$\displaystyle\sigma_{s}^{2}=\eta^{2}\left(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower
1.2571pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{2}\sigma_{\alpha}+c_{2}\right)+1.$
(45)

with

$\displaystyle c_{2}=$ $\displaystyle
V\left(\frac{T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to4.31pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower-0.45pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{2}}{2}+\frac{T_{3}}{2}\right)-\sqrt{V^{2}-1}(T_{\mathord{\leavevmode\hbox
to4.31pt{\vbox to4.31pt{\pgfpicture\makeatletter\hbox{\hskip
0.45pt\lower-0.45pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.9pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{1.70709pt}{0.0pt}\pgfsys@lineto{1.70709pt}{3.41418pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{1.70709pt}\pgfsys@lineto{3.41418pt}{1.70709pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\sqrt{T3})+$
$\displaystyle\frac{T_{\mathord{\leavevmode\hbox to4.31pt{\vbox
to0.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.45pt\lower 1.2571pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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##  Appendix E CF of non-Gaussian states

####  E-1 Photon addition, photon subtraction, and photon catalysis

In the following, we apply the procedures of [32] for determining CFs. The
unnormalized CF of the state resulting from the application of the non-
Gaussian operation to a mode of the TMSV state after it has been transmitted
through the channel is,

$\displaystyle\makebox{\large$\chi$}^{\prime}_{\mathrm{PS}}($
$\displaystyle\xi_{\mathrm{A}},\xi_{\mathrm{B}})=\frac{T_{\kappa}-1}{T_{\kappa}}\exp{\left[-\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}$
(47)
$\displaystyle\times\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{[}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right]}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}})\bigg{]},$

where

$\displaystyle
f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}})=\int\frac{d\xi^{2}}{\pi(1-T_{\kappa})}\makebox{\large$\chi$}_{\mathrm{TMSV}}(\xi_{\mathrm{A}},\xi)$
(48)
$\displaystyle\times\exp{\Big{[}\frac{1+T_{\kappa}}{2(T_{\kappa}-1)}(|\xi|^{2}+|\xi_{\mathrm{B}}|^{2})}+\frac{\sqrt{T_{\kappa}}}{T_{\kappa}-1}(\xi_{\mathrm{B}}\xi^{*}+\xi_{\mathrm{B}}^{*}\xi)\Big{]}.$

Similarly, the unnormalized CF after the PA operator is applied is,

$\displaystyle\makebox{\large$\chi$}^{\prime}_{\mathrm{PA}}(\xi_{\mathrm{A}},$
$\displaystyle\xi_{\mathrm{B}})=(T_{\kappa}-1)\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}$
(49)
$\displaystyle\times\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{[}\exp{\left[-\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}})\bigg{]}.$

When the PC operator is applied, the unnormalized CF is,

$\displaystyle\makebox{\large$\chi$}^{\prime}_{\mathrm{PC}}(\xi_{\mathrm{A}},$
$\displaystyle\xi_{\mathrm{B}})=q^{2}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{\\{}\exp{\left[-|\xi_{\mathrm{B}}|^{2}\right[}$
(50)
$\displaystyle\times\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{[}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}})\bigg{]}\bigg{\\}}$
$\displaystyle-q\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}\frac{\partial}{\partial\xi_{\mathrm{B}}}\bigg{\\{}\exp{\left[-|\xi_{\mathrm{B}}|^{2}\right[}$
$\displaystyle\times\frac{\partial}{\partial\xi_{\mathrm{B}}^{*}}\bigg{[}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}})\bigg{]}\bigg{\\}}$
$\displaystyle-q\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}\frac{\partial}{\partial\xi_{\mathrm{B}}^{*}}\bigg{\\{}\exp{\left[-|\xi_{\mathrm{B}}|^{2}\right[}$
$\displaystyle\times\frac{\partial}{\partial\xi_{\mathrm{B}}}\bigg{[}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}})\bigg{]}\bigg{\\}}$
$\displaystyle+f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},\sqrt{T_{\kappa}}),$

where $q=\frac{T_{\kappa}-1}{T_{\kappa}}$.

For the sequential use of the PS and PA operators, the operations PS-PA and
PA-PS, we consider the two non-Gaussian operations to use the same beam-
splitter transmissivity, $T_{\kappa}$. For PS-PA, the unnormalized CF is,

$\displaystyle\makebox{\large$\chi$}^{\prime}_{\mathrm{PS-
PA}}(\xi_{\mathrm{A}},\xi_{\mathrm{B}})$ (51)
$\displaystyle\quad=q^{2}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{\\{}\exp{\left[-|\xi_{\mathrm{B}}|^{2}\right[}$
$\displaystyle\quad\times\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{[}\exp{\left[\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},{T_{\kappa}})\bigg{]}\bigg{\\}}.$

The unnormalized CF for PA-PS is,

$\displaystyle\makebox{\large$\chi$}^{\prime}_{\mathrm{PA-
PS}}(\xi_{\mathrm{A}},\xi_{\mathrm{B}})$ (52)
$\displaystyle\quad=(T_{\kappa}-1)^{2}\exp{\left[-\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{\\{}\exp{\left[|\xi_{\mathrm{B}}|^{2}\right[}$
$\displaystyle\quad\times\frac{\partial^{2}}{\partial\xi_{\mathrm{B}}\partial\xi_{\mathrm{B}}^{*}}\bigg{[}\exp{\left[-\frac{|\xi_{\mathrm{B}}|^{2}}{2}\right[}f(\xi_{\mathrm{A}},\xi_{\mathrm{B}},{T_{\kappa}})\bigg{]}\bigg{\\}}.$

####  E-2 Squeezed Bell-like states

The normalized CF of an SB state is [42],

$\displaystyle\makebox{\large$\chi$}_{\mathrm{SB}}(\xi_{\mathrm{A}},\xi_{\mathrm{B}})=(\cos^{2}(\vartheta)+\sin^{2}(\vartheta))^{-1/2}$
$\displaystyle\exp\Big{[}-\frac{1}{2}\left(|\xi_{\mathrm{A}}^{\prime}|^{2}+|\xi_{\mathrm{B}}^{\prime}|^{2}\right)\Big{]}\Big{[}\cos^{2}(\vartheta)+2\cos(\vartheta)\sin(\vartheta)$
$\displaystyle\times\Re\\{\xi_{\mathrm{A}}^{\prime}\xi_{\mathrm{B}}^{\prime}\\}+\sin^{2}(\vartheta)(1-|\xi_{\mathrm{A}}^{\prime}|^{2})(1-|\xi_{\mathrm{B}}^{\prime}|^{2})\Big{]},$
(53)

where $\Re\\{z\\}$ is the real part of $z$ and the transformation given by Eq.
10, is used.