# Transceiver designs to attain the entanglement assisted communications
capacity

Ali Cox  Quntao Zhuang  Christos N. Gagatsos  Boulat Bash  Saikat Guha
College of Optical Sciences, University of Arizona, Tucson AZ 85721
Department of Electrical Engineering, University of Arizona, Tucson AZ 85721

###### Abstract

Pre-shared entanglement can significantly boost communication rates in the
high thermal noise and low-brightness transmitter regime. In this regime, for
a lossy-bosonic channel with additive thermal noise, the ratio between the
entanglement-assisted capacity and the Holevo capacity—the maximum reliable-
communications rate permitted by quantum mechanics without any pre-shared
entanglement—scales as $\log(1/{\bar{N}}_{\rm S})$, where the mean transmitted
photon number per mode, ${\bar{N}}_{\rm S}\ll 1$. Thus, pre-shared
entanglement, e.g., distributed by the quantum internet or a satellite-
assisted quantum link, promises to significantly improve low-power radio-
frequency communications. In this paper, we propose a pair of structured
quantum transceiver designs that leverage continuous-variable pre-shared
entanglement generated, e.g., from a downconversion source, binary phase
modulation, and non-Gaussian joint detection over a codeword block, to achieve
this scaling law of capacity enhancement. Further, we describe a modification
to the aforesaid receiver using a front-end that uses sum-frequency generation
sandwiched with dynamically-programmable in-line two-mode squeezers, and a
receiver back-end that takes full advantage of the output of the receiver’s
front-end by employing a non-destructive multimode vacuum-or-not measurement
to achieve the entanglement-assisted classical communications capacity.

##  I Introduction

An emerging focus has been drawn to architecting the quantum internet [2, 3]:
a global network built using quantum repeaters [4, 5] and satellites [6] to
distribute entanglement at high rates to multiple distant users on-demand [7,
9, 8]. There are several well-known applications of shared entanglement, a new
information currency: distributed quantum computing [10], communications with
physics-based security [11], provably-secure access to quantum computers on
the cloud [12], and entanglement-enhanced distributed sensing [13, 14, 15,
16]. In this paper, we design a system for another impactful application of
shared entanglement: substantially improving classical communication rates in
certain regimes.

Transmission of electromagnetic (EM) waves in linear media, such as optical
fiber, atmosphere, and vacuum, can be described as propagation of a set of
mutually-orthogonal spatio-temporal-polarization modes over the single-mode
lossy bosonic channel with additive thermal-noise ${\cal
N}_{\eta}^{{\bar{N}}_{\rm B}}$, described by the Heisenberg evolution
${\hat{a}}_{\rm R}=\sqrt{\eta}\,{\hat{a}}_{\rm
S}+\sqrt{1-\eta}\,{\hat{a}}_{\rm E}$, where $\eta\in(0,1]$ is the modal
(power) transmissivity, and the environment ${\hat{a}}_{\rm E}$ is excited in
a zero-mean thermal state of mean photon number per mode ${\bar{N}}_{\rm B}$.
Alice encodes classical information by modulating the state of the
${\hat{a}}_{\rm S}$ modes, with the constraint of ${\bar{N}}_{\rm S}$ mean
photons transmitted per mode. The quantum limit of the classical
communications capacity, known as the Holevo capacity [17, 18], in bits per
transmitted mode, is given by:

$C(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm B})=g({\bar{N}}_{\rm
S}^{\prime})-g((1-\eta){\bar{N}}_{\rm B}),$ (1)

where ${\bar{N}}_{\rm S}^{\prime}\equiv\eta{\bar{N}}_{\rm
S}+(1-\eta){\bar{N}}_{\rm B}$ is the mean photon number per the
${\hat{a}}_{\rm R}$ mode at the channel’s output received by Bob, and
$g(x)\equiv(1+x)\log(1+x)-x\log(x)$ is the von Neumann entropy of a zero-mean
single-mode thermal state with mean photon number $x$ [19, 20]. In this paper,
we denote base-$2$ and natural logarithms by $\log$ and $\ln$, respectively.

If Alice and Bob pre-share an unlimited amount of entanglement as a resource
for transmitting classical data over ${\cal N}_{\eta}^{{\bar{N}}_{\rm B}}$
with a transmit photon number constraint of ${\bar{N}}_{\rm S}$ photons per
mode, the capacity (in bits per mode) increases to: [21, 22, 23, 24, 25, 26]:

$C_{\rm E}(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm B})=g({\bar{N}}_{\rm
S})+g({\bar{N}}_{\rm S}^{\prime})-g(A_{+})-g(A_{-}),$ (2)

where $C_{\rm E}$ is the entanglement assisted classical capacity of the
quantum channel ${\cal N}_{\eta}^{{\bar{N}}_{\rm B}}$, and
$A_{\pm}=\frac{1}{2}(D-1\pm({\bar{N}}_{\rm S}^{\prime}-{\bar{N}}_{\rm S})),$
with $D=\sqrt{({\bar{N}}_{\rm S}+{\bar{N}}_{\rm
S}^{\prime}+1)^{2}-4\eta{\bar{N}}_{\rm S}({\bar{N}}_{\rm S}+1)}.$

In the regime of a low-brightness transmitter (${\bar{N}}_{\rm S}\ll 1$) and
high thermal noise (${\bar{N}}_{\rm B}\gg 1$),

${C_{\rm E}}/{C}\approx\ln\mathopen{}\mathclose{{}\left({1}/{{\bar{N}}_{\rm
S}}}\right),$ (3)

which tends to infinity as ${\bar{N}}_{\rm S}\to 0$ [27]. Physically, this
means that for a fixed number of channel uses, i.e., transmitted modes, a
receiver that has access to a quantum system entangled with the transmitted
modes can extract many more message bits reliably per mode, in the low-signal-
brightness high-thermal-noise regime, compared with a receiver that has no
such access. The practical implications are potentially revolutionary in
radio-frequency (RF) communications, since the condition ${\bar{N}}_{\rm B}\gg
1$ is naturally satisfied at a long center wavelength. The ${\bar{N}}_{S}\ll
1$ regime is of particular interest to covert communications, where the
transmitter tries to hide the presence of the communication attempt [28]. Pre-
shared entanglement between Alice and Bob—distributed, e.g., via the quantum
internet—allows for an order of magnitude or more enhancement in classical
communications rate, depending on the operational regime of loss, noise, and
transmit power (see Fig. 1).

Figure 1: The ratio $C_{\rm E}/C$ is plotted as a function of ${\bar{N}}_{\rm
S}$ and ${\bar{N}}_{\rm B}$ for channel transmissivity, $\eta=0.01$. Also
shown is a plot of $\ln(1/{\bar{N}}_{\rm S})$, which is the scaling of $C_{\rm
E}/C$ as ${\bar{N}}_{\rm S}\to 0$ and ${\bar{N}}_{\rm B}\to\infty$.

Despite this large capacity advantage attainable with pre-shared entanglement
having been known for decades, a structured transmitter-receiver design to
harness this enhancement has been elusive. Continuous-variable (CV) superdense
coding yields a factor-of-two capacity advantage in the noiseless case, but
does not provide any advantage in the noisy regime [29]. Phase-only encoding
on many copies of pre-shared two-mode squeezed vacuum (TMSV) states,
$|\psi\rangle_{\rm SI}=\sum_{j=0}^{\infty}\sqrt{{\bar{N}}_{\rm
S}^{j}/(1+{\bar{N}}_{\rm S})^{j+1}}|j\rangle_{\rm S}|j\rangle_{\rm I}$,
attains $C_{\rm E}$ in the ${\bar{N}}_{\rm S}\ll 1$, ${\bar{N}}_{\rm B}\gg 1$
regime [27], but with a receiver measurement whose structured optical design
is unknown. Receivers based on optical parametric amplification (OPA) [30] and
sum-frequency-generation (SFG) [31] yield at most a factor-of-$2$ improvement
over $C$, as shown in [27] and Appendix A of this paper.

In this paper, we take an important step in solving this long-standing
problem. We use insights from the SFG receiver proposed for the quantum-
illumination radar [31, 32] to map the task of jointly detecting long code
blocks consisting of binary phase-correlated signal-idler mode pairs onto a
task of jointly detecting code blocks of coherent states, which in turn can be
done using the linear-optical ‘Green Machine’ (GM) joint-detection receiver
which has been shown to attain superadditive communication capacity with phase
modulation of coherent states [33] 111Jet Propulsion Laboratory developed a
decoding algorithm for the first-order length-$n$ Reed Muller codes that
employed the fast Hadamard transform in a specialized circuit that used
$(n\log n)/2$ symmetric butterfly circuits, for sending images from Mars to
the Earth as part of the Mariner 1969 Mission. This circuit came to be known
as the Green Machine named after its JPL inventor. Guha developed an optical
version of the Green Machine decoding circuit, replacing the butterfly
elements by 50-50 beamsplitters, which he showed achieved superadditive
communication capacity with Hadamard-coded coherent-state BPSK modulation,
i.e., communication capacity in bits transmissible reliably per BPSK symbol
that is fundamentally higher than that is physically permissible with any
receiver that detects each BPSK modulated pulse one at a time [33]. This
paper’s joint detection receiver for entanglement assisted communications
leverages insights from that optical Green Machine.. In section IV, we propose
two structured designs, one improving upon the other, each consisting of a
transmitter, modulation format, and receiver. We also show later in section IV
that both designs achieve the $\ln(1/{\bar{N}}_{\rm S})$ scaling in
entanglement-assisted capacity gain over the Holevo capacity in Eq. (3).
Preliminary results on the first of these designs and the associated capacity
scaling were published in a conference proceedings article by a subset of the
co-authors [34]. These proposed receivers also set the stage for designing
structured receivers, which with binary-phase modulated pre-shared TMSV
entanglement, can achieve the entanglement-assisted capacity, $C_{E}$. We give
a possible design of a $C_{E}$-achieving transceiver based on an adaptive
multi-mode non-destructive vacuum-or-not measurement in Section VI. To close
the gap to $C_{E}$, this new receiver reveals that not only is it necessary to
choose an optimal classical JDR on the blocks of coherent states once the
received correlated states are mapped to the coherent states, but it is also
necessary to modify the front-end of the receiver to avoid losing information
in the mapping process.

##  II Quantum limits of unassisted classical communications

The front end of the receiver designs proposed in this paper convert classical
information encoded in the direction of correlation between the received
signal and the stored idler modes of a two-mode zero-mean Gaussian state into
the mean field of a single-mode displaced thermal state. At the back-end, the
displaced thermal states produced by the receiver front-end are passed to
joint-detection receivers (JDRs) designed for classical communication, i.e.,
using a coherent state modulation without entanglement assistance [47, 48].
Therefore in this section we review the problem setup and JDR methods for
coherent-state-encoded classical communications over a lossy bosonic channel
without pre-shared entanglement. In particular, section II.1 is a review of
the definition of the unassisted Holevo capacity for arbitrary modulation
formats, among which section II.2 covers binary modulation formats, section
II.3 reviews joint-detection receiver designs for BPSK modulation that exhibit
superadditivity to approach the Holevo capacity, and finally section II.4
discusses code and receiver design ideas to achieve the ultimate unassisted
Holevo capacity. Some results in this section were derived in an unpublished
memo from 2011 [49]. In subsection II.3 we also present a new JDR design for
classical communications, which we use later as a module to build a JDR for
entanglement-assisted communications.

Figure 2: Schematic of classical optical communications over a lossy, noisy
channel. Each transmitted symbol of a length-$L$ code word is a single bosonic
mode excited in a coherent state $|\alpha_{ji}\rangle$, $1\leq i\leq L$,
$1\leq j\leq 2^{nR}$, which results in a displaced thermal state at the
channel output: $\hat{\rho}_{\rm
th}(\sqrt{\eta}\alpha_{ji},(1-\eta){\bar{N}}_{B})$ of mean field
$\sqrt{\eta}\alpha_{ji}$ and mean thermal photon number
$(1-\eta){\bar{N}}_{B}$. The total mean photon number of each received mode is
${\bar{N}}_{S}^{\prime}=\eta{\bar{N}}_{S}+(1-\eta){\bar{N}}_{B}$. To achieve
the Holevo capacity, the ultimate limit to the reliable communication rate,
the receiver must perform a joint quantum measurement on a long code word
block. Such a receiver is called a joint-detection receiver (JDR).

###  II.1 The ultimate Holevo capacity

Modulation and encoding using coherent states (or ideal laser light pulses)
suffices to attain the ultimate (Holevo) capacity of classical communications
over the lossy-noisy bosonic channel ${\cal N}_{\eta}^{{\bar{N}}_{B}}$ [20].
If the modulation is unrestricted, i.e., the coherent-state amplitude
$\alpha_{i}$ of the $i$-th transmitted mode can be chosen over the entire
complex plane, subject to a mean photon number constraint
${\bar{N}}_{S}=\langle\alpha_{i}\rangle$, the Holevo capacity is given by Eq.
(1) To achieve communications at a rate approaching this capacity, the
transmitter picks a code book consisting of $2^{nR}$ length-$n$ (i.e.,
$n$-mode) coherent state code words
$|\psi_{j}\rangle=|\alpha_{j1},\alpha_{j2},\ldots,\alpha_{jn}\rangle$, where
each symbol of each code word $\alpha_{ji}$, $1\leq j\leq 2^{nR}$, $1\leq
i\leq n$ is chosen i.i.d. from a Gaussian distribution,
$p(\alpha)=(1/\pi{\bar{N}}_{S})e^{-|\alpha|^{2}/{\bar{N}}_{S}}$,
$\alpha\in\mathbb{C}$. If a coherent state $|\alpha\rangle$,
$\alpha\in{\mathbb{C}}$ is transmitted at the input of the channel, the output
state ${\hat{\rho}}_{\rm th}(\sqrt{\eta}\alpha,{\bar{N}}^{{}^{\prime}}_{\rm
B})$ is a single-mode displaced thermal state with mean field amplitude
$\sqrt{\eta}\alpha$, and thermal-noise mean photon number,
${\bar{N}}^{{}^{\prime}}_{\rm B}=(1-\eta){\bar{N}}_{B}$. Its density operator
is given by:

${\hat{\rho}}_{\rm th}(\sqrt{\eta}\alpha,{\bar{N}}_{\rm
B}^{{}^{\prime}})=\int_{\mathbb{C}}\frac{1}{\pi{\bar{N}}_{\rm
B}^{{}^{\prime}}}e^{-|\beta-\sqrt{\eta}\alpha|^{2}/{\bar{N}}_{\rm
B}^{{}^{\prime}}}|\beta\rangle\langle\beta|d^{2}\beta,$ (4)

where $|\beta\rangle,\beta\in{\mathbb{C}}$ is a coherent state. The receiver
performs a joint quantum measurement on the $n$-mode received code word, each
of whose $n$ symbols are displaced thermal states as above, to make a guess
$\hat{j}$ on which code word was transmitted. Fig. 2 depicts this encoding-
transmission-decoding process schematically. If the rate of the code is chosen
to satisfy $R<C(\eta,\bar{N}_{S},\bar{N}_{B})$ (bits per transmitted mode),
then if the transmitter picks a random code book as described above and the
receiver uses the square root measurement [18] (a structured optical
realization of which is not known), the receiver’s probability of error in
picking the correct transmitted code word $P_{e}^{(L)}\to 0$, as $n\to\infty$
[17]. The Holevo capacity expression in Eq. (1) can be derived as follows:

$\displaystyle
C(\eta,\bar{N}_{S},\bar{N}_{B})=\max_{p(\alpha),\alpha\in{\mathbb{C}}}\,\mathopen{}\mathclose{{}\left\\{S\mathopen{}\mathclose{{}\left({\bar{\hat{\rho}}}_{R}}\right)-{\bar{S}}}\right\\},$
(5)

where $S(\hat{\rho})=-{\rm Tr}(\hat{\rho}\log\hat{\rho})$ is the von Neumann
entropy of the state $\hat{\rho}$, ${\bar{\hat{\rho}}}_{R}=\int
p(\alpha){\hat{\rho}}_{\rm th}(\sqrt{\eta}\alpha,{\bar{N}}_{\rm
T})d^{2}\alpha$ is the average state of the output mode, ${\bar{S}}=\int
p(\alpha)S\mathopen{}\mathclose{{}\left({\hat{\rho}}_{\rm
th}(\sqrt{\eta}\alpha,{\bar{N}}_{\rm T})}\right)\,d^{2}\alpha$ is the average
entropy of the possible output states, and the maximum is taken over all
probability distribution functions $p(\alpha)$, with $\alpha\in{\mathbb{C}}$,
satisfying the transmit mean-photon number constraint $\int
p(\alpha)|\alpha|^{2}d^{2}\alpha={\bar{N}}_{S}$. It can be shown that no other
quantum states used at the transmitter can achieve a capacity higher than that
achieved using coherent states with a Gaussian prior: the Holevo capacity as
given in Eq. (1) [20]. The unrestricted-modulation Holevo capacity, in the
lossless ($\eta=1$), noiseless (${\bar{N}}_{B}=0$) case is given by
$C(1,{\bar{N}}_{S},0)=g({\bar{N}}_{S})$, which in the small-${\bar{N}}_{S}$
limit, can be expanded as [49]:

$C(1,{\bar{N}}_{S},0)=-{\bar{N}}_{S}\ln{\bar{N}}_{S}+{\bar{N}}_{S}+\frac{{\bar{N}}_{S}^{2}}{2}+{\text{h.o.t.}},$
(6)

where the unit used is nats per mode, with $\ln 2$ nats equalling 1 bit, and
h.o.t. refers to “higher order terms”.

###  II.2 Holevo capacity with binary modulation constellations

If the encoding is restricted to the binary phase-shift keying (BPSK)
alphabet, i.e., coherent states $|\pm\alpha\rangle$ for each transmitted mode,
$|\alpha|^{2}={\bar{N}}_{S}$, the maximum possible communication rate, in bits
per transmitted mode, is the Holevo capacity of the BPSK alphabet,

$\displaystyle\chi_{\text{BPSK}}(\eta,\bar{N}_{S},\bar{N}_{B})=\max_{p\in[0,1]}\,\mathopen{}\mathclose{{}\left\\{S\mathopen{}\mathclose{{}\left({\bar{\hat{\rho}}}_{R}}\right)-{\bar{S}}}\right\\},$
(7)

where ${\bar{\hat{\rho}}}_{R}=p\hat{\rho}_{\rm
th}(\sqrt{\eta}\alpha,{\bar{N}}_{T})+(1-p)\hat{\rho}_{\rm
th}(-\sqrt{\eta}\alpha,{\bar{N}}_{T}))$ is the average state of the output
mode, and ${\bar{S}}=pS(\hat{\rho}_{\rm
th}(\sqrt{\eta}\alpha,{\bar{N}}_{T}))+(1-p)S(\hat{\rho}_{\rm
th}(-\sqrt{\eta}\alpha,{\bar{N}}_{T}))$ is the average entropy of the two
possible output states. To achieve a rate approaching this capacity, a random
code book should be constructed as explained in Section II.1, but with each
symbol of each code word $\alpha_{ji}$ chosen i.i.d. with equal priors, i.e.,
$p=1/2$, from the two possible amplitudes $\\{\pm\alpha\\}$. In the lossless
($\eta=1$), noiseless case (${\bar{N}}_{B}=0$), the BPSK Holevo capacity is
given by:

$\chi_{\text{BPSK}}(1,\bar{N}_{S},0)=h_{2}\mathopen{}\mathclose{{}\left([1+e^{-2\bar{N}_{S}}]/2}\right),$
(8)

where $h_{2}(x)=-x\log x-(1-x)\log(1-x)$, $x\in(0,1)$, is the binary entropy
function. Expanding the expression in the small-${\bar{N}}_{S}$ limit, we get
[49]:

$\chi_{\text{BPSK}}(1,{\bar{N}}_{S},0)=-{\bar{N}}_{S}\ln{\bar{N}}_{S}+{\bar{N}}_{S}+{{\bar{N}}_{S}^{2}\ln{\bar{N}}_{S}}+{\text{h.o.t.}},$
(9)

in units of nats per transmitted mode.

If the encoding is restricted to an on-off-keyed (OOK) modulation alphabet,
i.e., coherent state $|\alpha\rangle$ or vacuum $|0\rangle$ for each
transmitted mode, the maximum communication rate, the OOK Holevo capacity is
given by:

$\displaystyle\chi_{\text{OOK}}(\eta,\bar{N}_{S},{\bar{N}}_{B})=\max_{p\in[0,1]}\,\mathopen{}\mathclose{{}\left\\{S\mathopen{}\mathclose{{}\left({\bar{\hat{\rho}}}_{R}}\right)-{\bar{S}}}\right\\},$
(10)

where ${\bar{\hat{\rho}}}_{R}=p\hat{\rho}_{\rm
th}(\sqrt{\eta}\alpha,{\bar{N}}_{T})+(1-p)\hat{\rho}_{\rm
th}(0,{\bar{N}}_{T}))$ is the average state of the output mode,
${\bar{S}}=pS(\hat{\rho}_{\rm
th}(\sqrt{\eta}\alpha,{\bar{N}}_{T}))+(1-p)S(\hat{\rho}_{\rm
th}(0,{\bar{N}}_{T}))$ is the average output entropy per mode,
${\bar{N}}_{S}=p|\alpha|^{2}$ is the mean transmitted photon number per mode,
and $p$ is the prior probability with which the “on” symbol ($|\alpha\rangle$)
is transmitted. For the lossless, noiseless case,

$\displaystyle\chi_{\text{OOK}}(1,\bar{N}_{S},0)=$
$\displaystyle\max_{0<p<1}h_{2}\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(1-\sqrt{(1-2p)^{2}+4e^{-\frac{\bar{N}_{S}}{p}}p(1-p)}}\right)}\right),$
(11)

where the optimal on-prior, $p\sim\sqrt{{\bar{N}}_{S}/2}$ decreases as
${\bar{N}}_{S}$ decreases [49], resulting in the mean photon number of the
(infrequent) transmitted on-pulse given by,
${\bar{N}}_{S}/p\sim\sqrt{{\bar{N}}_{S}}$, in the ${\bar{N}}_{S}\ll 1$ regime.
Expanding $\chi_{\text{OOK}}(1,\bar{N}_{S},0)$ in the small-${\bar{N}}_{S}$
regime, we get:

$\chi_{\text{OOK}}(1,{\bar{N}}_{S},0)=-{\bar{N}}_{S}\ln{\bar{N}}_{S}+{\bar{N}}_{S}+\sqrt{2}{{\bar{N}}_{S}^{3/2}\ln{\bar{N}}_{S}}+{\text{h.o.t.}},$
(12)

in units of nats per transmitted mode [49].

In Fig. LABEL:fig:classcaps, we plot the Photon Information Efficiency
(PIE)—the bits-per-mode capacity divided by the mean photon number transmitted
per mode ${\bar{N}}_{S}$—for the ultimate Holevo limit, i.e.,
$C(1,{\bar{N}}_{S},0)/{\bar{N}}_{S}$ (black fine-dashed line), the BPSK Holevo
capacity, i.e., $\chi_{\text{BPSK}}(1,{\bar{N}}_{S},0)/{\bar{N}}_{S}$ (dark-
red solid line), and the OOK Holevo capacity, i.e.,
$\chi_{\text{OOK}}(1,{\bar{N}}_{S},0)/{\bar{N}}_{S}$ (light blue solid line).
Since each of these PIEs scale as $-\ln{\bar{N}}_{S}+1$ (in nats per photon)
up to the second-order term in the small-${\bar{N}}_{S}$ limit, it is not
surprising that each of these three Holevo-capacity PIE plots converge to one
another as ${\bar{N}}_{S}\to 0$. Intuitively, this implies that when
${\bar{N}}_{S}\ll 1$, the BPSK constellation (and even the OOK constellation)
suffices to closely achieve the ultimate Holevo capacity. This is because in
that regime, the density operator of the average state for the equi-prior
coherent-state BPSK alphabet closely approximates that of the optimal
Gaussian-prior coherent-state alphabet. The PIE plots in Fig.
LABEL:fig:classcaps can be also interpreted as the bits per photon attained
for the pure-loss bosonic channel ($\eta\in(0,1]$), with the x-axis re-labeled
$\eta{\bar{N}}_{S}$. This is because a coherent state $|\alpha\rangle$
transmitted through the pure-loss channel ${\cal N}_{\eta}^{{\bar{N}}_{B}}$,
${\bar{N}}_{B}=0$, results in coherent state $|\sqrt{\eta}\alpha\rangle$ at
the channel’s output.

With the BPSK alphabet
$\mathopen{}\mathclose{{}\left\\{|\alpha\rangle,|-\alpha\rangle}\right\\}$,
and assuming $\eta=1$, ${\bar{N}}_{B}=0$, if the receiver uses the optimal
symbol-by-symbol detection scheme—the Helstrom measurement (realized by the
Dolinar receiver [35, 51]) that achieves the minimum probability of error of
discriminating between the BPSK coherent states
$\mathopen{}\mathclose{{}\left\\{|\alpha\rangle,|-\alpha\rangle}\right\\}$
given by $q=\frac{1}{2}[1-\sqrt{1-e^{-4{\bar{N}}_{S}}}]$—the receiver induces
a binary symmetric channel (BSC) for the detection of each received mode. This
results in a (Shannon) capacity $C_{1}(1,{\bar{N}}_{S},0)=1-h_{2}(q)$ bits per
mode. It is simple to see that $\lim_{{\bar{N}}_{S}\to
0}[C_{1}(1,{\bar{N}}_{S},0)]/{\bar{N}}_{S}=2/\ln 2\approx 2.89$ bits per
photon [37, 38]. This PIE is plotted by the black coarse-dashed line in Fig.
LABEL:fig:classcaps. Closing the PIE gap between this BPSK $C_{1}$ and the
BPSK Holevo limit $\chi_{\text{BPSK}}$ requires joint detection receivers
(JDRs) that act collectively on long BPSK-modulated code words. The fact that
a receiver acting collectively on multiple modulated symbols can achieve a
fundamentally higher per-symbol communication rate is often termed
superaddivity of capacity.

###  II.3 Joint detection receivers for superadditive capacity with BPSK
modulation

Figure 4: Schematic of a JDR for BPSK-modulated Hadamard and the 1st order
Reed Muller (RM) codes. The Green Machine (GM) receiver front-end transforms
$L$ Hadamard code words into $L$-ary pulse-position modulation (PPM) with a
single pulse of mean field $\sqrt{L\eta}\alpha$ at one output mode. Single-
photon detectors at each of the $L$ outputs of the GM are used to identify the
pulse-containing mode. If a RM code is used, the GM transforms the $2L$ RM
code words into binary-phase-coded PPM. Single photon detectors are employed
at each GM output to identify the pulse-containing mode. An electro-optic
switch directs the remaining pulse (in the pulse-containing mode), after the
arrival of the first photon click, to a Kennedy receiver [41] to identify the
binary phase of that pulse.

Two JDR designs for BPSK modulation were proposed [33, 47], depicted
collectively in Fig. 4, employed the Hadamard code [33] and the first-order
Reed Muller (RM) code [47] respectively, with the receiver employing a linear-
optical front end known as the Green Machine (GM). There are $L$ Hadamard code
words each $L$-mode long, whereas there are $2L$ RM code words of the same
length. Assuming $\bar{N}_{B}=0$,The GM, a linear-optical circuit made out of
$(L\log L)/2$ 50-50 beamsplitters sketched in Fig. 4, turns the $L$-mode BPSK
Hadamard code book of mean field $\sqrt{\eta{\bar{N}}_{S}}$ per mode, into an
$L$-ary pulse-position modulation (PPM) with a $\sqrt{L\eta{\bar{N}}_{S}}$
mean field displaced thermal state pulse
${\hat{\rho}}_{\text{th}}(\sqrt{L}\eta{\bar{N}}_{S},\bar{N}_{B}^{{}^{\prime}})$
in one of $L$ output modes of the GM (and a zero-mean thermal state of mean
photon number $\bar{N}_{B}^{{}^{\prime}}=(1-\eta)N_{B}$ in the remaining $L-1$
modes). Using shot-noise-limited single photon detectors (PD in legend of Fig.
4) at each of the $L$ output modes induces an $L$-input $L+1$-output symmetric
erasure channel, with the erasure probability $p_{e}=e^{-L{\bar{N}}_{S}}$. The
capacity achieved by this JDR, when $\eta=1$, in bits per mode, is the Shannon
capacity of this channel, $(1-p_{e})\log(L)$, divided by $L$. The envelope of
the PIE attained by this BPSK code-JDR combination, for $\bar{N}_{B}=0$,
$L=2,2^{2},\ldots,2^{10}$, is plotted by the light brown dotted line in Fig.
LABEL:fig:classcaps. This PIE exceeds $C_{1}$, and hence exhibits super-
additive capacity [33].

If the RM code is used, at the output of the GM one obtains a binary-phase-
modulated PPM, i.e., one of the $L$ modes carrying one of two displaced
thermal states
$\hat{\rho}_{\text{th}}(\pm\sqrt{L\eta{\bar{N}}_{S}},\bar{N}_{B}^{{}^{\prime}})$
and the rest carrying thermal states
$\hat{\rho}_{\text{th}}(0,\bar{N}_{B}^{{}^{\prime}})$. Building upon the
receiver design in [47], we propose and analyze the following receiver for the
BPSK RM code. Each of the $L$ outputs of the GM is detected by a single photon
detector. The first detected click in any of those detectors is used to
identify the ‘pulse-containing’ mode. If no click occurs, it is deemed an
erasure. If a click is detected, the remainder of the (displaced thermal
state) pulse in this identified pulse-containing mode has mean field 0,
$\beta$ or $-\beta$ where
$\beta\in\mathopen{}\mathclose{{}\left[0,\sqrt{L{\bar{N}}_{S}}}\right)$ is
calculated based on the actual arrival time of the first photon click. This
remaining pulse is fed into a Kennedy receiver [41], which applies a coherent
displacement of $\beta$ prior to detecting it, to identify the binary phase of
the pulse in that mode. A click of the Kennedy receiver after nulling is more
likely to occur if the pulse prior to the nulling was $\beta$, hence the
receiver decides on $(+,j)$ if a click occurs, where $j$ is the index of the
pulse-carrying mode. The notation $(\pm,l)$ for $1\leq l\leq L$ labels the RM
code words such that $l$ is the index of associated Hadamard code word and
$\pm$ determines the parity of the code word (since each Reed-Muller code is
either a Hadamard code word or the negative of one, as depicted in the top
left box of Fig. 4). The Shannon capacity of the $2L$-input $2L+1$ output
channel thereby induced, divided by $L$, is the bits-per-mode capacity
attained by the above BPSK code-JDR pair, whose PIE we plot for $\eta=1$ and
$\bar{N}_{B}=0$ using the interweaving red curves in Fig. LABEL:fig:classcaps,
for $L=2,2^{2},\ldots,2^{10}$ 222This capacity calculation of the coherent-
state RM-code along with GM plus Kennedy-received based JDR emerges as a
special case (i.e., with the thermal noise mean photon numbers set to zero) of
the entanglement-assisted capacity calculation of the TMSV-assisted RM binary
phase code modulated JDR2 that we describe in detail in Section IV.3.. Even
though the envelope of these PIE plots is higher than that of the Hadamard
code GM-JDR combination, their scaling is inferior to that of the BPSK Holevo
limit.

###  II.4 Code and receiver designs to achieve the ultimate Holevo capacity

In Ref. [38], it was proven that no matter which coherent-state (amplitude
and/or phase) modulation format is employed, and no matter what error
correction code is used, if the JDR is built with passive linear optics,
coherent-state local oscillators, shot-noise-limited photon detectors, and
electro-optic feedforward, in the small-${\bar{N}}_{S}$ limit, assuming
$\eta=1$ and ${\bar{N}}_{B}=0$, the capacity achieved by such a linear-optical
JDR must scale as:

$C_{\text{LO-
JDR}}({\bar{N}}_{S})=-{\bar{N}}_{S}\ln{\bar{N}}_{S}-\ln\ln(1/{\bar{N}}_{S})+{\text{h.o.t.}},$
(13)

in units of nats per transmitted mode. The difference between the second term
in Eq. (13), i.e., $-\ln\ln(1/{\bar{N}}_{S})$, and the second terms in Eqs.
(6), (9), (11), i.e., $+{\bar{N}}_{S}$ captures the scaling gap between the
PIE plots of the GM-based JDRs and the Holevo-limit PIEs in Fig.
LABEL:fig:classcaps.

In order to close this gap to the Holevo limit, one would need the JDR to
employ quantum transformations that go beyond the linear-optical photon-
detection based JDRs considered in [38]. Referring back to Fig. 2 and its
caption, when the code length $L$ is large, and the code rate $R<C$, where $C$
is the Holevo capacity (of the unrestricted, or of the specific modulation
format in question), the quantum states of the $2^{LR}$ coherent-state code
words—either of the random code as described in Section II.1 or a structured
good code—are close to being mutually orthogonal. For this reason, there are
multiple JDR measurements that can tell them apart with an error rate that
goes to $0$ as $L\to\infty$, thereby achieving reliable communications at a
rate approaching $C$.

Examples of such JDR prescriptions are as follows:

  1. 1.

Square root measurement—The square-root measurement (also known as the
‘pretty-good measurement’) acts on the $L$-mode code word of a random code
book [17, 18] either in the optical domain, or in a matter-qubit processor,
after faithfully transducing the optical quantum state of the received code
word into a qubit register.

  2. 2.

Sequential decoding—This receiver acts on the received $L$-mode code word of a
random code book, with a coherent displacement (realized using an array of
highly-transmissive beamsplitters and strong coherent-state local oscillators)
matched to the negative amplitude of the $j$-th code word, followed by
applying an $L$-mode ‘vacuum-or-not’ (VON)—a non-destructive binary projective
measurement described by POVM elements
$\mathopen{}\mathclose{{}\left\\{|0\rangle^{\otimes L}\,{}^{\otimes L}\langle
0|,{\hat{I}}-|0\rangle^{\otimes L}\,{}^{\otimes L}\langle
0|}\right\\}$—stopping with decision ${\hat{j}}=j$ if it gets the ‘vacuum’
outcome, and if not, undoes the displacement on the $L$-mode post-measurement
output state, increments $j\to j+1$ and repeats the process [42].

  3. 3.

Polar code with successive cancellation—Encodes the information using a
classical-quantum Polar code [52, 53], and implements—all-optically or in a
matter-qubit domain—the successive cancellation decoder to decode the $R$ bits
successively.

  4. 4.

Unambiguous state discrimination—Implements the $2^{LR}+1$ outcome unambiguous
state discrimination (USD) measurement on the received $L$-mode code word of a
random code, which unambiguously identifies the transmitted code word or
produces an erasure outcome [54].

  5. 5.

Belief propagation with quantum messages—Couples the coherent state in each
received mode (of a $N$-ary constellation) of the $L$-mode code word to the
state of $\log N$ qubits of a quantum computer, while preserving the relative
inner-products of the quantum states of the modulation constellation, followed
by acting upon that $L\log N$ qubit register with a quantum belief-propagation
decoder [48, 50].

##  III Entanglement enhanced classical communications

In parallel with section II, we now describe the task of classical
communication over a lossy noisy bosonic channel with the added resource of
pre-shared entanglement. In subsection III.1 we review how the Holevo capacity
is defined when the receiver has access to a mode entangled with the
transmitted signal for each transmitted symbol. Then subsection III.2
restricts to the case of binary modulation and TMSV entanglement, and
subsection III.3 describes the working principle for the transceivers
introduced in detail in section IV.

###  III.1 Entanglement-assisted capacity of the lossy thermal-noise bosonic
channel

If the transmitter Alice and the receiver Bob pre-share an unlimited amount of
entanglement, with ${\bar{N}}_{\rm S}$ still denoting the mean number of
photons transmitted over the channel ${\cal N}_{\eta}^{{\bar{N}}_{\rm B}}$ per
mode, the capacity for sending classical data (in bits per mode) increases
above the Holevo capacity $C(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm B})$ in Eq.
(1) to [21, 22, 23, 24, 25, 26] Eq. (2), scaling like Eq. (3) in the regime of
a low-brightness transmitter (${\bar{N}}_{\rm S}\ll 1$) and high thermal noise
(${\bar{N}}_{\rm B}\gg 1$), which tends to infinity as ${\bar{N}}_{\rm S}\to
0$ [27]. The goal of this paper is to explore transmitter-receiver designs
that can achieve this $\ln\mathopen{}\mathclose{{}\left({1}/{{\bar{N}}_{\rm
S}}}\right)$ enhancement in capacity over the Holevo limit.

Intuitively, the scaling ${C_{\rm
E}}/{C}\sim\ln\mathopen{}\mathclose{{}\left({1}/{{\bar{N}}_{\rm S}}}\right)$
follows from the dominant term in the expression for $C_{\rm E}$ as
${\bar{N}}_{\rm S}\to 0$, being $-{\bar{N}}_{\rm S}\log{\bar{N}}_{\rm S}$ for
any constant ${\bar{N}}_{\rm B}>0$, while the Taylor series expansion of $C$
at ${\bar{N}}_{\rm S}=0$ yields $C={\bar{N}}_{\rm
S}\log\mathopen{}\mathclose{{}\left(1+((1-\eta){\bar{N}}_{\rm
B})^{-1}}\right)+o({\bar{N}}_{\rm S})$. Formally, one can use L’Hôpital’s rule
to obtain the following limit:

$\displaystyle\lim_{{\bar{N}}_{\rm S}\to 0}\frac{C_{\rm
E}}{C\ln\mathopen{}\mathclose{{}\left(\frac{1}{{\bar{N}}_{\rm S}}}\right)}$
$\displaystyle=\frac{1}{(1+(1-\eta){\bar{N}}_{\rm
B})\ln\mathopen{}\mathclose{{}\left(1+\frac{1}{(1-\eta){\bar{N}}_{\rm
B}}}\right)},$ (14)

which yields the scaling. Note that the right hand side (RHS) of (14) is zero
when ${\bar{N}}_{\rm B}=0$. This is consistent with the known fact that the
ratio $C_{\rm E}/C\leq 2$ in the noiseless (${\bar{N}}_{\rm B}=0$) regime.

The plot of $C_{\rm E}/C$ as a function of ${\bar{N}}_{\rm S}$ and
${\bar{N}}_{\rm B}$, shown in Fig. 1 for $\eta=0.01$ yields further insight.
At optical frequencies, at 300K, the Planck-Law-limited thermal mean photon
number per mode ${\bar{N}}_{\rm B}$ ranges between $10^{-5}$ to $10^{-6}$. At
such small ${\bar{N}}_{\rm B}$, despite the scaling in (3), the actual
capacity ratio is essentially at or below $2$ ($C_{\rm E}/C$ is at most $2$
when ${\bar{N}}_{\rm B}=0$) over the range of ${\bar{N}}_{\rm S}$: $10^{-6}$
to $10^{2}$; and hence the entanglement enhancement is significant only for
the meaninglessly-small values of ${\bar{N}}_{\rm S}$. However, at
${\bar{N}}_{\rm B}=100$ corresponding to the thermal noise at microwave
wavelengths, $\eta=10^{-3}$, and ${\bar{N}}_{\rm S}=10^{-3}$, reasonably
achieved with a spontaneous-parametric downconversion (SPDC) based
entanglement source employed by our proposed transceiver, $C_{\rm E}/C\approx
7$. This is a substantial improvement over the highest capacity achievable
without leveraging pre-shared entanglement.

It was shown in [22] that, in order to achieve $C_{\rm E}$ in Eq. (2), it
suffices for Alice and Bob to pre-share many copies of the following entangled
state:

$|\psi\rangle_{\rm
SI}=\sum_{n=0}^{\infty}\sqrt{\frac{{\bar{N}}_{S}^{n}}{(1+{\bar{N}}_{S})^{1+n}}}\,|n\rangle_{\rm
S}|n\rangle_{\rm I},$ (15)

known as the two-mode squeezed vacuum (TMSV). Let us consider a pulsed SPDC
source that produces entangled signal and idler pulses, each of duration $T$
seconds and optical bandwidth $W$ (typically $1$-$2$ THz) around their
respective center frequencies. The quantum description of the pair of signal-
idler entangled pulses is $|\psi\rangle_{\rm SI}^{\otimes M}$, where $M\approx
WT$ is the number of mutually-orthogonal temporal modes in each of the signal
and idler pulses.

###  III.2 Entanglement-assisted capacity with TMSV entanglement and binary
modulation constellations

In Ref. [13], it was shown that not only does pre-sharing TMSV entangled
states suffice to attain $C_{\rm E}$, but that phase modulation of the signal
mode (at Alice’s end) and transmitting that to Bob suffices to attain $C_{E}$
in the regime of low signal-to-noise (SNR) ratio (${\bar{N}}_{S}\ll 1$ and
${\bar{N}}_{B}\gg 1$). This low-SNR is the regime where the scaling ${C_{\rm
E}}/{C}\sim\ln\mathopen{}\mathclose{{}\left({1}/{{\bar{N}}_{\rm S}}}\right)$
holds true, and hence of interest here.

(a) Transmission of a $L$-length phase code word, where each phase is
modulated on to an $M$-temporal-mode signal pulse on Alice’s end of the pre-
shared SPDC-generated entanglement. The mean photon number per mode of each
received mode $R$, $\bar{N}_{S}^{\prime}=\eta\bar{N}_{S}+(1-\eta)\bar{N}_{B}$.
Bob’s JDR acts on $2ML$ modes.

(b) An equivalent parallel representation of the channel, explicitly showing
that Bob’s received code word comprises of $L$ ‘symbols’
$(\hat{\rho}_{1}^{{\rm RI}},\ldots,\hat{\rho}_{L}^{{\rm RI}})$, where each
symbol is uncorrelated and un-entangled with the others. The $M$ received-
idler mode pairs within each symbol hold phase-sensitive cross-correlations,
i.e., $\langle{\hat{a}}_{R}{\hat{a}}_{I}\rangle\neq 0$, which encode
information about the phase modulated by Alice on the $M$ modes of that
transmitted symbol.

Figure 5: Diagram of entanglement-assisted communication of classical data
over a lossy, noisy quantum channel. The S, I, E and R modes are labels,
respectively, for: the initial signal mode (to be sent over the channel ${\cal
N}_{\eta}^{{\bar{N}}_{B}}$ to the receiver), the idler mode retained at the
receiver (entangled with the corresponding S mode), the environment mode in a
thermal state of mean photon number $\bar{N}_{B}$, and the received mode at
the front end of the receiver.

Pursuant to phase modulation of the signal modes of pre-shared TMSV states
generated using an SPDC source, a schematic system diagram for entanglement-
assisted communications is depicted in Fig. 5. Alice maps her message to a
code book $\mathcal{C}$ (with $|\mathcal{C}|\leq 2^{Lr}$), each of whose code
words are $L$ phases, e.g., $(\theta_{1},\ldots,\theta_{L})$. Alice modulates
an SPDC-generated signal pulse, i.e., $M$ temporal modes, whose entangled
idler-mode counterparts are assumed pre-shared with Bob, with a single phase
symbol of that code word. So, transmission of one code word consumes $ML$ uses
of the single-mode lossy thermal-noise bosonic channel ${\cal
N}_{\eta}^{{\bar{N}}_{B}}$. The receiver Bob acts on $2ML$ modes—$ML$ modes of
Alice’s code word, corrupted by channel noise, that he receives, and the
corresponding $ML$ pre-shared idler modes—with a joint detection receiver
(JDR), to produce a guess of the transmitted code word. Bob’s received code
word comprises of $L$ ‘symbols’ $(\hat{\rho}_{1}^{{\rm
RI}},\ldots,\hat{\rho}_{L}^{{\rm RI}})$, where each symbol is uncorrelated and
un-entangled with the other symbols. Further, it is instructive to note that
even though the signal-idler modes were entangled, there is no entanglement
left amid the received-idler mode pairs within each symbol, due to the
entanglement-breaking lossy-noisy channel ${\cal N}_{\eta}^{{\bar{N}}_{B}}$.
However, the received-idler mode pairs within each symbol hold phase-sensitive
cross-correlations, i.e., $\langle{\hat{a}}_{R}{\hat{a}}_{I}\rangle$, which
encode information about the phase value modulated by Alice on that symbol at
the transmitter. If Bob is able to faithfully recover the transmitted message,
let the reliable-communications rate achieved by the above scheme be $r$ modes
per transmitted symbol. Then since each transmitted symbol comprises of $M$
temporal modes, the rate in bits per mode, $R=r/M$.

Figure 6: Ratios of the entanglement-assisted capacities for specific binary
modulation formats and the ultimate classical (Holevo) capacity without pre-
shared entanglement $C$, plotted as a function of $\bar{N}_{S}$ for $\eta=0.1$
and $\bar{N}_{B}=1$. $M=1$ is assumed for all the $C_{E}$ plots.

In Fig. 6, we further show that restricting the aforesaid phase modulation to
binary phase shift keying (BPSK), i.e., each of the $L$ symbols in Alice’s
code words comprising only of two phases
$\mathopen{}\mathclose{{}\left\\{0,\pi}\right\\}$, suffices to attain $C_{E}$
in the low SNR limit. This observation is reminiscent of the coherent state
BPSK modulation closely approaching the Holevo limit for unassisted classical
communications, as shown in Fig. LABEL:fig:classcaps. We further evaluate the
performance of an on-off keying (OOK) modulation format, wherein Alice’s
transmitted symbols are either an ‘on’ pulse, which is an $M$-temporal-mode
signal pulse of her pre-shared SPDC-entanglement, or an ‘off’ pulse, which is
Alice staying silent for an entire pulse duration. To evaluate the
entanglement-assisted communication rates using the BPSK and the OOK
modulation formats, $C_{\rm E}^{({\text{TMSV-BPSK}})}$ and $C_{\rm
E}^{({\text{TMSV-OOK}})}$ respectively, we calculate the (unassisted) Holevo
capacities of the modulated received-idler code words at Bob’s end. In other
words, for BPSK modulation, the maximum entanglement-assisted rate is:

$C_{\rm E}^{({\text{TMSV-
BPSK}})}=\max_{p\in[0,1]}\frac{\mathopen{}\mathclose{{}\left\\{S({\bar{\hat{\rho}}}_{RI})-{\bar{S}}}\right\\}}{M}\,{\text{bits
per mode}},$ (16)

with
${\bar{\hat{\rho}}}_{RI}=p\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{(0)}}\right)^{\otimes
M}+(1-p)\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{(\pi)}}\right)^{\otimes
M}$, where ${\hat{\rho}}_{RI}^{(\theta)}$ is a two-mode Gaussian state
obtained by applying a single-mode phase
$e^{i\theta{\hat{a}}_{S}^{\dagger}{\hat{a}}_{S}}$ on the signal mode of a two-
mode TMSV state of mean photon number per mode ${\bar{N}}_{S}$, followed by
the transmission of the signal mode through the channel ${\cal
N}_{\eta}^{{\bar{N}}_{B}}$. Further,
${\bar{S}}=pMS\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{(0)}}\right)+(1-p)MS\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{(\pi)}}\right)$.
For on-off-keying (OOK) modulation,

$C_{\rm E}^{({\text{TMSV-
OOK}})}=\max_{p\in[0,1]}\frac{\mathopen{}\mathclose{{}\left\\{S({\bar{\hat{\rho}}}_{RI})-{\bar{S}}}\right\\}}{M}\,{\text{bits
per mode}},$ (17)

with
${\bar{\hat{\rho}}}_{RI}=p\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{({\rm
on})}}\right)^{\otimes
M}+(1-p)\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{({\rm
off})}}\right)^{\otimes M}$, where ${\hat{\rho}}_{RI}^{({\rm on})}$ is a two-
mode Gaussian state obtained by the transmission of the signal mode of a two-
mode TMSV state of mean photon number per mode ${\bar{N}}_{S}/p$ through the
channel ${\cal N}_{\eta}^{{\bar{N}}_{B}}$, and ${\hat{\rho}}_{RI}^{({\rm
off})}=\hat{\rho}_{\rm th}(0,{\bar{N}}_{T})\otimes\hat{\rho}_{\rm
th}(0,{\bar{N}}_{S}/p)$ is a product of two zero-mean thermal states, with
${\bar{N}}_{T}=(1-\eta){\bar{N}}_{B}$. Further, the average output entropy,
${\bar{S}}=pMS\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{({\rm
on})}}\right)+(1-p)MS\mathopen{}\mathclose{{}\left({\hat{\rho}}_{RI}^{({\rm
off})}}\right)$.

The optimal prior $p$ for the BPSK modulation is $1/2$, whereas the optimal
on-prior $p$ for the OOK modulation (evaluated numerically) comes out to be
much smaller than $1/2$ for ${\bar{N}}_{S}\ll 1$, similar to what was seen to
be the case for coherent-state OOK-based classical communications, discussed
in Section II.2. There is however an important difference between coherent-
state OOK modulation for unassisted communications and TMSV-OOK modulation for
entanglement-assisted communications. In the latter, even for the ‘off’
symbols where Alice stays silent (transmits vacuum), she must still consume
pre-shared entanglement, i.e., discard the signal pulses during off-symbol
transmissions. Since the capacity calculations are done with a fixed mean
transmitted photon number per mode ${\bar{N}}_{S}$, and since the off
symbols—despite consuming pre-shared entanglement—do not transmit any photons
over the channel, the pre-shared entangled TMSV states need to have
${\bar{N}}_{S}/p$ mean photon number per mode. Since the amount of
entanglement (in ebits/mode) in a TMSV of mean photon number per mode
${\bar{N}}$ is given by $g({\bar{N}})$, which monotonically increases with
$\bar{N}$, the OOK modulation format for entanglement assisted communications
consumes more pre-shared entanglement compared to the BPSK method, and
achieves a capacity inferior to that of BPSK, as shown in Fig. 6. The
advantage the OOK modulation format enjoys however is that no phase modulation
is necessary at Alice’s end. Further, the sparse on-symbol transmission allows
Alice to use a pulse-position modulation (PPM) modulation-code over the OOK
alphabet, which allows her to use good outer (classical) error correction
codes to achieve the Shannon capacity of the transmitter-JDR pair in question,
as described later in Section V.1.

All the plots in Fig. 6, except for that of $C_{\rm E}/C$, were generated
numerically: by numerical evaluation of the capacity expressions in Eqs. (16)
and (17), and a Mathematica-generated best-fit line for $C_{\rm
E}^{({\text{TMSV-OOK}})}$ in the semi-logarithmic scale, to assess the scaling
of the capacity ratio $C_{\rm E}^{({\text{TMSV-OOK}})}/C$. It was found
(numerically) that $C_{\rm E}^{({\text{TMSV-OOK}})}/C\sim\ln(1/{\bar{N}}_{S})$
in the ${\bar{N}}_{S}\ll 1$, ${\bar{N}}_{B}\gg 1$ regime, confirming that even
an OOK modulation of pre-shared TMSV states can achieve the optimal scaling of
the ratio to the unassisted Holevo capacity $C$.

Each of the plots in Fig. 6 assume $M=1$, i.e., each individual temporal mode
of the SPDC-generated entanglement is individually modulated. This is hard to
achieve in practice as doing this will require THz-class electro-optic
modulators (EOMs). For the performance evaluation of the joint-detection
receivers we present in Section IV, we will use more reasonable values of
modes-per-symbol $M$, e.g., $M=10^{5}$ used for the rate plots in Fig. 10,
which for an SPDC optical bandwidth (modes per second) $W=10^{12}$ Hz,
translates to a much more reasonable EOM modulation bandwidth (modulated
symbols per second) of $10^{7}$, i.e., $10$ MHz.

###  III.3 Receiver design principles

Although the entanglement-assisted capacity of ${\cal
N}_{\eta}^{{\bar{N}}_{B}}$ can be saturated by encoding information in the
phase of the signal modes of pre-shared signal-idler SPDC mode pairs, it is
not immediately clear how to design a structured receiver to extract this
information on the receiver end. Initial guesses at designing a receiver could
be to employ either an optical parametric amplifier (OPA) or the phase-
conjugate receiver (PCR) proposed in [30] in the context of building receivers
for a quantum-illumination radar, to convert the information-bearing phase-
sensitive cross correlations $\langle{\hat{a}}_{R}{\hat{a}}_{I}\rangle$ into a
photon-number signature. Such a receiver acts jointly upon each received
noisy-modulated symbol ($M$ temporal modes) and the corresponding $M$-mode
idler pulse, either with an OPA or a PCR [27], followed by performing a
maximum-likelihood determination of the phase encoded in that symbol. However,
the bits-per-mode capacity attained by such a measurement that acts on one
modulated symbol at a time as they are received, cannot exceed twice the
ultimate classical Holevo capacity $C$ in the low $\bar{N}_{S}$, high
$\bar{N}_{B}$ regime (see Appendix A for proof).

Figure 7: The big picture view of the receivers proposed in this paper. Joint
detection on multiple signal-idler mode pairs is made possible by using a
quantum map (denoted $\mathcal{M}$) that depletes the cross-correlation
between the received-idler mode pair, converting it to the coherent
displacement of an ancilla mode.

In order to design a receiver to achieve $C_{E}$, or for that matter to exceed
$2C$ bits per mode, we must design a joint detection receiver (JDR)—along the
lines of JDRs for superadditive classical communications, as described in
Sections II.3 and II.4, and illustrated in Fig. 5—that acts on multiple
modulated symbols collectively. The challenge is to use a code word block of
signal-idler mode pairs to produce a strong detectable signature (e.g., photon
number at the output of the receiver) to pinpoint which code word was
received. The way in which this is done in the receiver design proposed here
is to first convert phase-sensitive cross-correlations in the return-idler
mode pairs into the coherent displacement of a single bosonic mode using a
non-linear sum-frequency generation (SFG) module, and then use a coherent
state JDR as described in Sections II.3 and II.4 on the (code word of)
coherently-displaced modes, as depicted in Fig. 7. The next Section describes
how this works, in further detail.

##  IV Joint detection receiver design for entanglement-assisted
communications

###  IV.1 Structure of the transmitter and receiver

Figure 8: Our first transceiver design JDR1, which uses the BPSK Hadamard
code, and applies a Green-Machine at the end of each SFG cycle to the
$k^{\text{th}}$ set of sum-frequency modes $\\{b_{k}^{(i)}\\}_{i}$ and
classically adds the outcomes of the post-green machine photon detectors. Thin
and thick lines depict weak (e.g., ${\bar{N}}_{\rm S}\ll 1$) and strong (e.g.,
${\bar{N}}_{\rm B}\gg 1$) mean photon number signals respectively. In an
actual realization, only one $L$-mode Green Machine is needed as the sum-
frequency modes ${\hat{b}}_{k}^{(i)}$, $1\leq k\leq K$, for each
$i\in\mathopen{}\mathclose{{}\left\\{1,\ldots,L}\right\\}$, appear in a
temporal sequence.

Consider the transmitter-receiver structure in Fig. 8. Alice uses a binary
phase shift keying (BPSK) modulation with a Hadamard code of order $L$, on the
signal (S) modes of TMSV states $|\psi\rangle_{\rm SI}$ whose idler (I) modes
are pre-shared with Bob. Let us assume $L$ is an integer power of $2$ such
that a Hadamard code exists. The $l$-th ‘pulse’ (comprising $M$ orthogonal
temporal modes) of the signal output of a pulsed spontaneous parametric
downconversion (SPDC) source, an $M$-fold tensor product TMSV
$|\psi\rangle_{\rm SI}^{\otimes M}$, is modulated by Alice, with the binary
phase $\theta_{l}\in\mathopen{}\mathclose{{}\left\\{0,\pi}\right\\}$. The
transmission of an entire Hadamard code word thus consumes $L$ SPDC signal
pulses, modulated with phases $\theta_{l},1\leq l\leq L$, over $ML$ uses of
the single-mode channel ${\cal N}_{\eta}^{{\bar{N}}_{\rm B}}$. The
corresponding idler modes are assumed losslessly pre-shared with Bob, e.g.,
using a fault-tolerant quantum internet. Given that in our regime of interest:
$(1-\eta){\bar{N}}_{\rm B}>\eta$, the channel ${\cal N}_{\eta}^{{\bar{N}}_{\rm
B}}$ is entanglement-breaking. Alice’s phase modulation of the signal mode
${\hat{a}}_{\rm S}$ of a TMSV state, followed by its transmission through
${\cal N}_{\eta}^{{\bar{N}}_{\rm B}}$, results in an output mode
${\hat{a}}_{\rm R}$ received by Bob with a (large) mean photon number
${\bar{N}}_{\rm S}^{\prime}=\eta{\bar{N}}_{\rm S}+(1-\eta){\bar{N}}_{\rm B}$.
Mode ${\hat{a}}_{\rm R}$ and the (weak) idler mode of the TMSV ${\hat{a}}_{\rm
I}$ with mean photon number ${\bar{N}}_{\rm S}$ held by Bob, are individually
in zero-mean thermal states. However, their joint quantum state is a
classically-correlated zero-mean Gaussian state (no longer entangled), with a
phase sensitive cross correlation $\langle{\hat{a}}_{\rm R}{\hat{a}}_{\rm
I}\rangle=\pm\sqrt{\eta{\bar{N}}_{\rm S}({\bar{N}}_{\rm S}+1)}$, where the
sign depends on the phase ($0$ or $\pi$) modulated by Alice. Note that the
amount of cross correlation in the received state is proportional to the
amount of cross correlation in the initially generated state which, being
entangled, is correlated by an amount even beyond the maximum allowed by
classical physics. So the term ‘entanglement-assisted’ applies in spite of the
entanglement-breaking channel.

The receiver employs the SFG, a non-linear optical process that runs SPDC in
reverse per the Hamiltonian ${\hat{H}}_{\rm SFG}=\hbar
g\sum_{m=1}^{M}\mathopen{}\mathclose{{}\left({\hat{b}}^{\dagger}{\hat{a}}_{{\rm
S}_{m}}{\hat{a}}_{{\rm I}_{m}}+{\hat{b}}{\hat{a}}_{{\rm
S}_{m}}^{\dagger}{\hat{a}}_{{\rm I}_{m}}^{\dagger}}\right)$, with $\hbar$ the
reduced Planck constant, and $g$ the non-linear interaction strength. Signal-
idler photon pairs from the $M$ input mode pairs
$\mathopen{}\mathclose{{}\left\\{{\hat{a}}_{{\rm S}_{m}},{\hat{a}}_{{\rm
I}_{m}}}\right\\}$, $1\leq m\leq M$, are up-converted to a sum-frequency mode
$\hat{b}$. The phase-sensitive cross-correlation at the input of the SFG,
$\langle{\hat{a}}_{{\rm S}_{m}}{\hat{a}}_{{\rm I}_{m}}\rangle$, manifests as
the mean field amplitude of a thermal state of $\hat{b}$ [31].

A single-mode displaced thermal state with mean field amplitude
$\alpha\in{\mathbb{C}}$, and thermal-noise mean photon number, ${\bar{N}}>0$
has the following density operator:

$\hat{\rho}_{\rm
th}(\alpha,{\bar{N}})=\int_{\mathbb{C}}\frac{1}{\pi{\bar{N}}}e^{-|\beta-\alpha|^{2}/{\bar{N}}}|\beta\rangle\langle\beta|d^{2}\beta.$
(18)

For ${\bar{N}}=0$, it reduces to the pure coherent state $|\alpha\rangle$. The
photodetection statistics of $\hat{\rho}_{\rm th}(\alpha,{\bar{N}})$ is
Laguerre-distributed [35]. The probability that it produces zero clicks when
detected with an ideal photon detector is

$\langle 0|\hat{\rho}_{\rm
th}(\alpha,{\bar{N}})|0\rangle=(1/({\bar{N}}+1))e^{-|\alpha|^{2}/({\bar{N}}+1)}.$
(19)

As depicted at the top of Fig. 8, Bob inputs the received $M$ modes of each of
the $L$ phase-modulated blocks of the Hadamard code, along with the
corresponding $M$ idler modes (pre-shared with the transmitted block of $M$
signal modes), into a feed-forward (FF) SFG module. An FF-SFG module stacks
$K$ SFG stages, each unitary corresponding to applying ${\hat{H}}_{\rm SFG}$
for a duration of $\frac{\pi}{2\sqrt{M}g}$, with $K$ beamsplitters and
combiners of transmissivities $\kappa=1/K$ and $1-\kappa$ respectively, as
shown. The $K$-stage SFG ensures that the signal input of each SFG has much
less than one photon per mode, so that we can use the “qubit-approximation”
analysis of the SFG from [31]. ${\hat{b}}_{k}^{(l)}$ denotes the sum-frequency
mode of the $k$-th SFG, $1\leq k\leq K$, of the $l$-th FF-SFG module, $1\leq
l\leq L$.

In the $\kappa\ll 1/{\bar{N}}_{\rm B}$ limit, the sum-frequency mode
${\hat{b}}_{k}^{(l)}$ is in a displaced thermal state $\hat{\rho}_{\rm
th}(\pm\alpha_{k},{\bar{N}}_{\rm T})$ [31], where the $\pm$ sign depends on
whether the mode block $i$ is modulated with phase $0$ or $\pi$. The mean
$\alpha_{k}=\sqrt{M\kappa\eta{\bar{N}}_{\rm S}(1+{\bar{N}}_{\rm
S})\mu^{k-1}}$, with
$\mu=\mathopen{}\mathclose{{}\left(1-\kappa(1+{\bar{N}}_{\rm
S}^{\prime})}\right)^{2}$, and ${\bar{N}}_{\rm T}=\kappa{\bar{N}}_{\rm
S}{\bar{N}}_{\rm S}^{\prime}$ [31]. Let us also define
$\bar{N}_{k}=|\alpha_{k}|^{2}$.

For a fixed $k$, the $L$ modes ${\hat{b}}_{k}^{(l)}$, $1\leq l\leq L$,
produced by the $k$-th FF-SFG gates are in a product of displaced thermal
states with the same mean photon number ${\bar{N}}_{\rm T}$, but with mean
field amplitudes $\alpha_{k}$ or $-\alpha_{k}$ corresponding to which Hadamard
code word was transmitted. On the other hand, for a fixed $l$, the $K$ modes
$\hat{b}_{k}^{(l)}$, $1\leq k\leq K$ can be approximated as having maximally
correlated noise (see appendix C). Therefore, it is possible, for each $l$, to
interfere the $K$ sum-frequency modes ${\hat{b}}_{k}^{(l)}$, $1\leq k\leq K$
on an appropriately-tuned beam-splitter array to produce a displaced thermal
state with mean thermal photon number $\approx K\bar{N}_{T}/7.61$, where
$\bar{N}_{T}$ is the mean thermal photon number of each sum-frequency mode
${\hat{b}}_{k}^{(l)}$, $\forall k,l$, as described above (see Fig. 9 for a
schematic).

We will refer to the JDR design in Fig. 8, proposed in [34], as JDR1. For each
$k\in\\{1,...,K\\}$, the $L$ modes ${\hat{b}}_{k}^{(l)}$, $1\leq l\leq L$ are
input to an $L$-mode Green Machine (GM), a linear-optical circuit comprising
$L\log_{2}(L)/2$ $50$-$50$ beasmplitters, denoted GMk. The GM transforms the
$L$-mode BPSK-modulated coherent-state Hadamard code word, e.g.,
$|\alpha_{k},-\alpha_{k},\ldots,\alpha_{k}\rangle$ into one of the $L$ code
words of order-$L$ coherent-state pulse-position modulation (PPM), e.g.,
$|0,\ldots,\sqrt{L}\alpha_{k},\ldots,0\rangle$ [33]. The bottom of Fig. 8
shows an example binary-phase $L=8$ Hadamard code, and the circuit of an
$8$-mode GM. At the output of GMk, one of the $L$ output modes (based on the
input Hadamard code word) is in a displaced thermal state $\hat{\rho}_{\rm
th}(\sqrt{L}\,\alpha_{k},{\bar{N}}_{\rm T})$. We call this the “pulse-
containing output” (mode). The remaining $L-1$ output modes of GMk are in the
zero-mean thermal state $\hat{\rho}_{\rm th}(0,{\bar{N}}_{\rm T})$.

Figure 9: Our second (upgraded) transceiver design JDR2 that uses a more
information-dense Reed-Muller (RM) code for the phase encoding and combines
the $\\{\hat{b}_{k}^{(l)}\\}_{k}$ modes for each $l$ via an array of beam
splitters into a single mode $\hat{b}_{0}^{(i)}$ before applying a single
Green Machine stage and a Kennedy Receiver-augmented photon detection stage
needed to decode the extra bit of information of the Reed-Muller code book
(further details are contained in the text).

All GM outputs are detected by single photon detectors [33]. The electrical
outputs of the $l$-th detectors, $1\leq l\leq L$, of each GM are classically
combined—by integrating all photon clicks—into one single output that is
monitored for zero or more clicks during each ($M$-temporal-mode) SPDC pulse
interval. Since the $K$ modes ${\hat{b}}_{k}^{(l)}$, $1\leq k\leq K$ in the
$l$-th FF-SFG module are in a temporal sequence, we only need one $L$-mode GM
and $L$ detectors to realize this receiver.

In Fig. 9, we sketch a JDR variant, which we call JDR2. Here, the sum-
frequency modes are combined using an array of beamsplitters before entering a
single Green Machine. Namely, for each $l\in\\{1,...,L\\}$ the $K$ modes
${\hat{b}}_{k}^{(l)}$, $1\leq k\leq K$ are combined on a set of beam-splitters
with transitivities $\gamma_{k}$ chosen such that the modes
$\hat{b}^{(l)}_{0}$, $1\leq l\leq L$ entering the Green Machine (i.e., output
of the final beam splitters of transmissivity $\gamma_{K-1}$) are in displaced
thermal states $\hat{\rho}_{\text{th}}(\pm\alpha_{0},\bar{N}_{T0})$ where
$\bar{N}_{T0}$ is the thermal mean photon number of the modes
$\hat{b}_{0}^{(l)}$, and

$|\alpha_{0}|^{2}=\sum_{k=1}^{K}\bar{N}_{k},$

which, in the limit as $K\rightarrow\infty$ ($\kappa\rightarrow 0$), evaluates
to

$|\alpha_{0}|^{2}=M\eta\bar{N}_{S}(1+\bar{N}_{S})\frac{1-e^{-2(1+\bar{N}^{\prime}_{S})}}{2(1+\bar{N}^{\prime}_{S})}.$

Although the thermal noise in the modes ${\hat{b}}_{k}^{(l)}$, $1\leq k\leq K$
are maximal-phase-insensitive classically-correlated, choosing the
$\gamma$-beam splitters to combine the mean fields into a single mode, as

$\gamma_{k}=\frac{\sum_{i=1}^{k}\alpha_{i}^{2}}{\sum_{i=1}^{k+1}\alpha_{i}^{2}}=\frac{1-(1-\kappa(1+\bar{N}_{S}^{\prime}))^{2k}}{1-(1-\kappa(1+\bar{N}_{S}^{\prime}))^{2(k+1)}},$
(20)

is not the choice that combines the total thermal noise into a single mode.
Instead, it is shown (numerically) in Appendix D that when the
$\gamma_{k}$-values are tuned to maximally combine the mean fields (Eq. (20)),
the thermal noise on the output modes $\hat{b}_{0}^{(i)}$ is given by:

$\bar{N}_{T0}\approx K\bar{N}_{T}/7.61=\bar{N}_{S}\bar{N}^{\prime}_{S}/7.61.$

Although one could use JDR2 with a BPSK Hadamard code as discussed in the
context of JDR1, we will consider instead BPSK modulation with the 1st order
Reed-Muller (RM) code. As shown in the $L=4$ example in the top-left corner of
Fig. 9, the RM code of code word length $L$ has $2L$ code words, i.e., twice
as many code words as the Hadamard code. It has all the Hadamard code words,
and each of their bit-flipped versions in the code book. Therefore, with RM
code words and JDR2, the pulse-containing mode at the output of the Green
Machine has a phase of $\pm 1$ depending on which of the two $L$-code word
halves of the RM code book the transmitted code word belonged in. To decode
this extra bit of information, JDR2’s final measurement stage uses a Kennedy
receiver as described in Section II.3 [41] with ‘exact nulling’, conditioned
on detecting a photon.

###  IV.2 Performance evaluation of JDR1

Let us consider the case of the transmission of the $L$-symbol Hadamard phase
code along with the JDR1, as described in the previous subsection. The $2^{L}$
possible (click, no-click) patterns at the $L$ classically-combined detector
outputs are classified into $L+1$ receiver outcomes: a click at a given output
and no clicks elsewhere, or an erasure, which is either zero clicks at all $L$
outputs, or clicks at multiple outputs. Our scheme thus induces an $L$-input
$(L+1)$-output discrete memoryless channel between the $L$ Hadamard code words
and the $L+1$ outcomes, which is identical to that induced by coherent-state
pulse-position modulation (PPM) and single photon detection with non-zero
background (or dark) click probability. The Shannon capacity of this channel
[39] divided by $ML$ is the bits-per-mode entanglement-assisted capacity
attained by our design. In other words,

$\displaystyle R_{\rm E}^{(M,L)}$ $\displaystyle=$
$\displaystyle\frac{1}{ML}\mathopen{}\mathclose{{}\left({p_{\rm e}}\log
L+(L-1)p_{\rm d}\log\frac{Lp_{\rm d}}{p_{\rm e}}}\right.$ (21)
$\displaystyle-$ $\displaystyle\mathopen{}\mathclose{{}\left.\big{(}p_{\rm
e}+(L-1)p_{\rm d}\big{)}\log\mathopen{}\mathclose{{}\left[1+\frac{(L-1)p_{\rm
d}}{p_{\rm e}}}\right]}\right),$

where $p_{\rm d}=(1-p_{\rm c})p_{\rm b}(1-p_{\rm b})^{L-2}$, $p_{\rm e}=p_{\rm
c}(1-p_{\rm b})^{L-1}$, $p_{\rm c}$ is the click probability at the pulse-
containing output, and $p_{\rm b}$ is the click probability at the non-pulse-
containing output.

To simplify the analysis, we assume that the photodetection statistics of
$i$-th outputs of each of the $K$ GMs are statistically independent, so
$1-p_{\rm c}=\Pi_{k=1}^{K}(1-p_{\rm c}^{(k)})$, where $1-p_{\rm
c}^{(k)}=\frac{1}{{\bar{N}}_{\rm T}+1}e^{-L\bar{N}_{k}/({\bar{N}}_{\rm
T}+1)}$. The capacity evaluated under this assumption will be a lower bound to
the actual capacity because in reality the photodetection outcomes of the
pulse-containing modes are positively correlated, reducing the variance of the
sum of the outcomes.

The expressions for the click probabilities simplify to:

$\displaystyle p_{\rm c}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{(1+{\bar{N}}_{\rm
T})^{K}}e^{-A\mathopen{}\mathclose{{}\left(\frac{1-\mu^{K}}{1-\mu}}\right)},{\text{and}}$
(22) $\displaystyle p_{\rm b}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{(1+{\bar{N}}_{\rm T})^{K}},$ (23)

with $A=ML\kappa\eta{\bar{N}}_{\rm S}({\bar{N}}_{\rm S}+1)/({\bar{N}}_{\rm
T}+1)$.

Figure 10: The thin magenta lines are plots of $R_{\rm E}^{(M,L)}/C$ for
$L=2,4,8,\ldots,2^{20}$ and $M=10^{5}$. This shows that the capacity ratio
scales as $\log(1/{\bar{N}}_{\rm S})$, which tends to infinity as
${\bar{N}}_{\rm S}\to 0$, for any $M$. However, this scheme (BPSK modulation,
Hadamard code, and our proposed structured joint-detection receiver) does not
achieve $C_{\rm E}$. We assume $\eta=0.01$ and ${\bar{N}}_{\rm B}=10$ photons
per mode for all the plots in this figure.

In Fig. 10, we plot $C_{\rm E}/C$ as a function of ${\bar{N}}_{\rm S}$ in the
${\bar{N}}_{\rm S}\ll 1$ regime, for $\eta=0.01$ and ${\bar{N}}_{\rm B}=10$.
We also plot $R_{\rm E}^{(M,L)}/C$, with $M=10^{5}$,
$L\in\mathopen{}\mathclose{{}\left\\{2,2^{2},\ldots,2^{20}}\right\\}$ and
taking $\kappa=1/K$ with $K=100$, which we found to be sufficiently large such
that further increasing $K$ did not affect the rate significantly. The
envelope $R^{(M)}_{\rm E}={{\rm sup}_{L}R_{\rm E}^{(M,L)}}$ shows that our
transceiver achieves an entanglement-assisted capacity gain that exceeds $2$
as ${\bar{N}}_{\rm S}\to 0$, the best achievable ratio with an OPA [30] or FF-
SFG receiver [31, 27] (see Appendix A).

In the next subsection we apply the conditions:

$\eta{\bar{N}}_{\rm S}\ll{\bar{N}}_{\rm S}\ll 1\ll{\bar{N}}_{\rm B}\ll K,$
(24)

and prove that our system design attains the optimal scaling of entanglement-
assisted communications capacity, i.e., ${R_{\rm
E}^{(M)}}/{C}\sim\ln(1/{\bar{N}}_{\rm S})$. Despite ${R_{\rm E}^{(M)}}$ not
meeting $C_{\rm E}$, it achieves the infinite-fold capacity enhancement
leveraging pre-shared entanglement, but most importantly—using quantum optical
states, processes and detection schemes that are readily realizable.

####  IV.2.1 Capacity scaling analysis

Let us first consider an order-$L$ pulse position modulation (PPM) alphabet
over a channel with loss and noise. PPM encodes information by the position of
a pulse (e.g., a coherent state of light) in one of $L$ orthogonal modes
(e.g., time bins) at the input, which is direct-detected at the output (e.g.,
by a single photon detector). Loss attenuates the transmitted pulse amplitude,
and noise results in potential detection events in one or more bins. Ignoring
detection events in multiple bins (i.e., treating them as “erasures”), and
assuming an equiprobable selection over the $L$ inputs (which maximizes the
throughput), the Shannon mutual information—expressed in bits per mode—of the
induced $L$-input $(L+1)$-output discrete memoryless channel (DMC), is given
by [39, Eq. (16)]:

$\displaystyle I^{(L)}_{\text{PPM}}$ $\displaystyle=\frac{p_{\rm e}}{L}\log
L+\frac{(L-1)}{L}p_{\rm d}\log\frac{Lp_{\rm d}}{p_{\rm e}}$
$\displaystyle\phantom{=}-\mathopen{}\mathclose{{}\left[\frac{p_{\rm
e}+(L-1)p_{\rm
d}}{L}}\right]\log\mathopen{}\mathclose{{}\left[1+\frac{(L-1)p_{\rm d}}{p_{\rm
e}}}\right],$ (25)

where $p_{\rm e}$ is the probability of the detection event occurring
exclusively in the bin corresponding to the position of the pulse at the
input, and $p_{\rm d}$ is the probability that a detection event occurs in a
single bin that is different from the one containing the input pulse. Denoting
by $p_{\rm c}$ the probability of a detection event in the bin corresponding
to the input pulse and by $p_{\rm b}$ the probability of a detection event in
another bin [39, Sec. IV],

$\displaystyle p_{\rm e}$ $\displaystyle=p_{\rm c}(1-p_{\rm
b})^{L-1},\,{\text{and}}$ (26) $\displaystyle p_{\rm d}$
$\displaystyle=(1-p_{\rm c})p_{\rm b}(1-p_{\rm b})^{L-2}.$ (27)

We specialize the result by Jarzyna and Banaszek [39] to find the channel
capacity of the DMC induced by the modulation-code-channel-receiver described
in Fig. 8.

Let us recall that our scheme involves BPSK-modulation of the signal modes of
$M$ pre-shared two-mode-squeezed-vacuum (TMSV) states, repeating the above $L$
times, encoding an order-$L$ binary Hadamard code, and transmission of the
$ML$ modulated modes over $ML$ uses of the single-mode lossy-noisy bosonic
channel $\mathcal{N}_{\eta}^{{\bar{N}}_{\rm B}}$, followed by demodulation and
detection by our joint detection receiver (JDR). This scheme results in
detection events that are statistically identical to demodulating PPM in the
presence of noise. Thus, we seek:

$\displaystyle R_{\rm E}^{(M)}$ $\displaystyle=\max_{L}\frac{1}{M}I^{(L)}_{\rm
PPM},$ (28)

where we determine $p_{\rm e}$ and $p_{\rm d}$ as follows. First, let’s recall
the definitions. The mean number of photons per mode in the signal modes of
the TMSV transmitted by Alice is ${\bar{N}}_{\rm S}$, and the mean photon
number of the thermal noise background per transmitted mode is ${\bar{N}}_{\rm
B}$. The modal power transmissivity of the bosonic channel is $\eta\in(0,1]$,
which implies that Bob’s received mean number of photons per mode is
${\bar{N}}_{\rm S}^{\prime}=\eta{\bar{N}}_{\rm S}+(1-\eta){\bar{N}}_{\rm B}$.
To calculate $p_{\rm c}$ and $p_{\rm b}$, we assume the photodetection
statistics of the $i$-th outputs of each of the $K$ Green Machines in the JDR
are statistically independent, and $K\gg{\bar{N}}_{\rm B}$. Thus, $1-p_{\rm
c}=1-\prod_{k=1}^{K}(1-p_{\rm c}^{(k)})$, where $1-p_{\rm
c}^{(k)}=\frac{1}{{\bar{N}}_{\rm T}+1}e^{-L{\bar{N}}_{k}/({\bar{N}}_{\rm
T}+1)}$ with ${\bar{N}}_{\rm T}={\bar{N}}_{\rm S}{\bar{N}}_{\rm
S}^{\prime}/K$, ${\bar{N}}_{k}=\frac{M\eta{\bar{N}}_{\rm S}(1+{\bar{N}}_{\rm
S})\mu^{k-1}}{K}$, and
$\mu=\mathopen{}\mathclose{{}\left[1-\frac{1+{\bar{N}}_{\rm
S}^{\prime}}{K}}\right]^{2}$. Thus:

$\displaystyle p_{\rm c}$ $\displaystyle=1-\frac{1}{(1+{\bar{N}}_{\rm
T})^{K}}e^{-A\mathopen{}\mathclose{{}\left(\frac{1-\mu^{K}}{1-\mu}}\right)},{\text{and}}$
(29) $\displaystyle p_{\rm b}$ $\displaystyle=1-\frac{1}{(1+{\bar{N}}_{\rm
T})^{K}},$ (30)

with $A=\frac{ML\eta{\bar{N}}_{\rm S}({\bar{N}}_{\rm S}+1)}{K({\bar{N}}_{\rm
T}+1)}$. Using the conditions:

$\displaystyle{\bar{N}}_{\rm S}\ll 1\ll{\bar{N}}_{\rm B}\ll K,$ (31)

we can make the following approximations using the limits as ${\bar{N}}_{\rm
S}\to 0$ and $K\to\infty$:

$\displaystyle{\bar{N}}_{\rm S}^{\prime}$
$\displaystyle\approx(1-\eta){\bar{N}}_{\rm B},$ (32)
$\displaystyle(1+{\bar{N}}_{\rm T})^{-K}$ $\displaystyle\approx
e^{-{\bar{N}}_{\rm S}(1-\eta){\bar{N}}_{\rm B}},\,{\text{and}}$ (33)
$\displaystyle\frac{A}{1-\mu}$ $\displaystyle\approx\frac{ML\eta{\bar{N}}_{\rm
S}}{2(1+(1-\eta){\bar{N}}_{\rm B})}.$ (34)

These lead to the following approximations for $p_{\rm c}$ and $p_{\rm b}$:

$\displaystyle p_{\rm c}$ $\displaystyle\approx
1-\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}\mathopen{}\mathclose{{}\left(\frac{ML\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})}+(1-\eta){\bar{N}}_{\rm B}}\right)}\right]$ (35) $\displaystyle p_{\rm b}$
$\displaystyle\approx 1-\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}(1-\eta){\bar{N}}_{\rm B}}\right],$ (36)

where $\gamma=1-e^{-2\mathopen{}\mathclose{{}\left(1+(1-\eta){\bar{N}}_{\rm
B}}\right)}$. Substitution of approximations in (35) and (36) into (26) and
(27) yields:

$\displaystyle p_{\rm e}$
$\displaystyle\approx\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}(L-1)(1-\eta){\bar{N}}_{\rm B}}\right]$
$\displaystyle\phantom{\approx}-\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}L\mathopen{}\mathclose{{}\left(\frac{M\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})}+(1-\eta){\bar{N}}_{\rm B}}\right)}\right]$ (37)
$\displaystyle\approx\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}L(1-\eta){\bar{N}}_{\rm B}}\right]$
$\displaystyle\phantom{\approx}-\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}L\mathopen{}\mathclose{{}\left(\frac{M\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})}+(1-\eta){\bar{N}}_{\rm B}}\right)}\right],$ (38) $\displaystyle p_{\rm
d}$ $\displaystyle\approx\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}L\mathopen{}\mathclose{{}\left(\frac{M\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})}+(1-\eta){\bar{N}}_{\rm B}}\right)}\right]$
$\displaystyle\phantom{\approx}-\exp\mathopen{}\mathclose{{}\left[-{\bar{N}}_{\rm
S}\mathopen{}\mathclose{{}\left(\frac{ML\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})}+(1-\eta){\bar{N}}_{\rm B}(L+1)}\right)}\right],$ (39)

where we assume $L\gg 1$ so that $L-1\approx L$ for the approximation in (38).
When ${\bar{N}}_{\rm S}\to 0$, we can approximate $p_{\rm e}$ and $p_{\rm d}$
by the Taylor series expansions at ${\bar{N}}_{\rm S}=0$ of (38) and (39),
respectively:

$\displaystyle p_{\rm e}$ $\displaystyle\approx\frac{{\bar{N}}_{\rm
S}ML\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm B})},$ (40) $\displaystyle p_{\rm
d}$ $\displaystyle\approx{\bar{N}}_{\rm S}(1-\eta){\bar{N}}_{\rm B}.$ (41)

Substituting (40) and (41) into the last two terms of (25), and approximating
$\frac{L-1}{L}\approx 1$, reveals that only the first term of (25) has a
significant dependence on $L$ in our regime of interest. Thus, for the optimal
order, we need:

$\displaystyle L^{*}$ $\displaystyle=\operatorname*{arg\,max}_{L}\frac{p_{\rm
e}}{L}\log L.$ (42)

The linear approximation in (41) is insufficient to find $L^{*}$. We follow
the methodology in [39] by substituting in (42) the quadratic Taylor series
expansion at ${\bar{N}}_{\rm S}=0$,

$\displaystyle p_{\rm e}$ $\displaystyle\approx\frac{L{\bar{N}}_{\rm
S}M\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm B})}$
$\displaystyle\phantom{\approx}-\frac{L^{2}{\bar{N}}_{\rm
S}^{2}M\eta\gamma\mathopen{}\mathclose{{}\left(M\eta\gamma+4(1-\eta){\bar{N}}_{\rm
B}(1+(1-\eta){\bar{N}}_{\rm B})}\right)}{8(1+(1-\eta){\bar{N}}_{\rm B})^{2}}.$

Let $v\equiv\frac{{\bar{N}}_{\rm
S}^{2}M\eta\gamma\mathopen{}\mathclose{{}\left(M\eta\gamma+4(1-\eta){\bar{N}}_{\rm
B}(1+(1-\eta){\bar{N}}_{\rm B})}\right)}{8(1+(1-\eta){\bar{N}}_{\rm B})^{2}\ln
2}$ and $u\equiv\frac{{\bar{N}}_{\rm S}M\eta\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})\ln 2}$. This reduces the problem in (42) to finding the location of the
extremal values of $f(L)=(u+vL)\ln L$ by solving

$\displaystyle\frac{\mathop{}\\!\mathrm{d}f(L)}{\mathop{}\\!\mathrm{d}L}$
$\displaystyle=\frac{u}{vL}-1-\ln L=0$ (43)

for $L$, which involves the principal branch of the Lambert $W$-function [61,
Sec. 4.13]:

$\displaystyle L^{*}$
$\displaystyle=\frac{u}{v}\mathopen{}\mathclose{{}\left[W\mathopen{}\mathclose{{}\left(\frac{u}{v}e}\right)}\right]^{-1},$
(44)

where $W\mathopen{}\mathclose{{}\left(xe^{x}}\right)=x$ for $x\geq-1$. Using
equality $\ln W(x)=\ln(x)-W(x)$ for $x>0$ [61, Eq. (4.13.3)] and asymptotic
expansion $W(x)=\ln(x)-\ln\ln(x)+o(1)$ as $\ln(x)\to\infty$ [61, Eq.
(4.13.10)] in our regime of interest ${\bar{N}}_{\rm S}\to 0$, we have:

$\log(L^{*})\approx\log\mathopen{}\mathclose{{}\left(\frac{w}{{\bar{N}}_{\rm
S}}}\right)-\log\mathopen{}\mathclose{{}\left(\ln\mathopen{}\mathclose{{}\left[\frac{we}{{\bar{N}}_{\rm
S}}}\right]}\right),$ (45)

where $w=\frac{4(1+(1-\eta){\bar{N}}_{\rm
B})}{M\eta\gamma+4(1-\eta){\bar{N}}_{\rm B}(1+(1-\eta){\bar{N}}_{\rm B})}$.
Substituting (40) and (45) into (28), we obtain:

$\displaystyle R_{\rm E}^{(M)}$ $\displaystyle\approx\frac{\eta{\bar{N}}_{\rm
S}\gamma}{2(1+(1-\eta){\bar{N}}_{\rm
B})}\mathopen{}\mathclose{{}\left[\log\mathopen{}\mathclose{{}\left[\frac{w}{{\bar{N}}_{\rm
S}}}\right]-\log\mathopen{}\mathclose{{}\left[\ln\mathopen{}\mathclose{{}\left[\frac{we}{{\bar{N}}_{\rm
S}}}\right]}\right]-g\mathopen{}\mathclose{{}\left[\frac{2(1-\eta){\bar{N}}_{\rm
B}(1+(1-\eta){\bar{N}}_{\rm B})}{M\eta\gamma}}\right]}\right],$ (46)

where $g(x)=(x+1)\log(x+1)-x\log x$. As ${\bar{N}}_{\rm S}\to 0$, the
logarithmic term dominates (46), and we obtain the scaling:

$R_{\rm E}^{(M)}=O\mathopen{}\mathclose{{}\left({\bar{N}}_{\rm
S}\log\mathopen{}\mathclose{{}\left(\frac{1}{{\bar{N}}_{\rm
S}}}\right)}\right).$ (47)

In Appendix B, we consider a cruder approximation of $R_{\rm E}^{(M)}$,
providing an alternative proof of the scaling in (47), but one that lets us
establish a connection with a problem that was studied by Wang and Wornell in
the context of coherent-state PPM modulation, where the dark click probability
per mode $\lambda$ is proportional to the mean photon number per mode $\cal E$
[60].

####  IV.2.2 Numerical rate calculations

In Figs. 11, 12, and 13, we compare the two approximations for ${R^{(M)}_{\rm
E}}$: the one we derived by modifying the Jarzyna-Banaszek analysis of PPM
applied to our problem, shown in Eq. (46) and labeled “our approx.” in Figs.
11, 12, 13, and the one we obtained from the Wang-Wornell PPM analysis, shown
in Appendix B, Eq. (65). It is seen that the former is closer to the true
envelope, especially for smaller values of $M$.

In Fig. 15 we plot (the exact) $R^{(M)}_{\rm E}$ as a function of
${\bar{N}}_{\rm S}$ for $M=10,10^{2},\ldots,10^{6}$. For the assumed values of
$\eta=0.01$ and ${\bar{N}}_{\rm B}=10$ photons per mode used for plots in this
figure, the highest capacity occurs at around $M\sim 10^{5}$ for JDR1 and
$M\sim 10^{4}$ for JDR2. The existence of such an optimum value of $M$ can be
explained by the negative sign of the $M$-dependent second-order term in (46).

For $\eta=0.01$ and ${\bar{N}}_{\rm B}=10$ photons per mode, our scheme
achieves the maximum rate at the modulation-block length $M\approx 10^{5}$.
For a typical SPDC entanglement source of optical bandwidth $W\sim 1$ THz, and
$M\approx WT$, $M=10^{5}$ modes in a signal pulse translates to a pulse
duration of $T\sim 100$ ns. This means the BPSK phase-modulation bandwidth
necessary would be $\sim 10$ MHz, which is readily realizable with commercial-
grade electro-optical modulators (EOMs) at $1550$ nm.

Figure 11: Plot of ${R^{(M,L)}_{\rm E}}/C$ with $M=10^{3}$, for
$L\in\mathopen{}\mathclose{{}\left\\{2,2^{2},\ldots,2^{20}}\right\\}$. We
assume $\eta=0.01$ and ${\bar{N}}_{\rm B}=10$ photons per mode, for all the
plots. Figure 12: Plot of ${R^{(M,L)}_{\rm E}}/C$ with $M=10^{4}$, for
$L\in\mathopen{}\mathclose{{}\left\\{2,2^{2},\ldots,2^{20}}\right\\}$. We
assume $\eta=0.01$ and ${\bar{N}}_{\rm B}=10$ photons per mode, for all the
plots. Figure 13: Plot of ${R^{(M,L)}_{\rm E}}/C$ with $M=10^{5}$, for
$L\in\mathopen{}\mathclose{{}\left\\{2,2^{2},\ldots,2^{20}}\right\\}$. We
assume $\eta=0.01$ and ${\bar{N}}_{\rm B}=10$ photons per mode, for all the
plots.

###  IV.3 Performance evaluation of JDR2

Let us now consider the RM code ($2L$ binary-phase code words each of length
$L$) and the JDR2, which together induce a 2$L$-input $2L+1$ discrete
memoryless channel. The additional output is the erasure outcome in which no
photons are detected. The transition matrix $\mathbf{X}$ is defined to be the
matrix of conditional probabilities such that $\mathbf{X}_{ji}$ is the
probability that the receiver decides on outcome $i$ given that code word $j$
was transmitted. In this case, $i$ runs from 1 to $2L+1$, indexing the
possible outcomes of the receiver and $j$ runs from 1 to $2L$, indexing the
code words of the Reed-Muller code. If the Reed-Muller code book is seen as a
Hadamard code book appended with its sign-flipped copy with the original copy
consisting of ‘plus-words’ and the sign-flipped copy consisting of ‘minus-
words’, even indices of $\mathbf{X}$ correspond to minus-words and odd indices
to plus-words. Because of symmetry, we observe that $\mathbf{X}$ has $6$
independent entries $\mathbf{X}_{11}$, $\mathbf{X}_{22}$, $\mathbf{X}_{12}$,
$\mathbf{X}_{21}$, $\mathbf{X}_{14}$, $\mathbf{X}_{1,2L+1}$, corresponding
respectively to the transition probabilities of: a plus-outcome on the pulse-
containing mode of the GM-output given that a plus-word was transmitted, a
minus-outcome on the pulse-containing mode given that a minus-word was
transmitted, a minus-outcome on the pulse-containing mode given that a plus-
word was transmitted, a plus-outcome on the pulse-containing mode given that a
minus-word was transmitted, a minus-outcome on a non-pulse-carrier given that
a plus-word was transmitted, and erasure (no clicks on any mode). Moreover, by
the same symmetry, the plus-words can be assumed to have equal priors
$p_{+}/L$, and similarly the minus-words can be assumed to have equal priors
$p_{-}/L=(1-p_{+})/L$ where $0\leq p_{+}\leq 1$. The entirety of $\mathbf{X}$
can then be written in terms of the above values, as:

$\displaystyle\mathopen{}\mathclose{{}\left(\begin{array}[]{cc}\mathbf{X}_{2i-1,2i-1}&\mathbf{X}_{2i-1,2i}\\\
\mathbf{X}_{2i,2i-1}&\mathbf{X}_{2i,2i}\end{array}}\right)=$
$\displaystyle\mathopen{}\mathclose{{}\left(\begin{array}[]{cc}\mathbf{X}_{11}&\mathbf{X}_{12}\\\
\mathbf{X}_{21}&\mathbf{X}_{22}\end{array}}\right)$ (48e)
$\displaystyle\mathbf{X}_{2j,2i}=\mathbf{X}_{2j-1,2i}\;=$
$\displaystyle\;\mathbf{X}_{14}$ (48f) $\displaystyle\mathbf{X}_{k,2n+1}\;=$
$\displaystyle\;\mathbf{X}_{1,2L+1}$ (48g)
$\displaystyle\mathbf{X}_{2j,2i-1}=$ $\displaystyle\mathbf{X}_{2j-1,2i-1}$
$\displaystyle=\;\frac{1}{L-1}(1-\mathbf{X}_{1,2L+1}-$
$\displaystyle\mathbf{X}_{11}-\mathbf{X}_{12})-\mathbf{X}_{14}$ (48h)

for $k=1,2,\cdots,2n$; $i,j=1,2,\cdots,L$; but $i\neq j$. Eq. (48h) says that
entries $\mathbf{X}_{2j,2i-1}$ and $\mathbf{X}_{2j-1,2i-1}$ are obtained for
free as result of the normalization condition of the rows of $\mathbf{X}$.

The information rate in bits per mode is obtained by dividing the mutual
information $I(\bar{N}_{S},\bar{N}_{B},\eta,p_{+})$ associated with the
transition matrix by $ML$, the total number of modes transmitted to send one
code word. Therefore,

$\displaystyle R_{E}^{(M,L)}(\bar{N}_{S},\bar{N}_{B},\eta,p_{+})=$
$\displaystyle\frac{1}{LM}I(\bar{N}_{S},\bar{N}_{B},\eta,p_{+})$
$\displaystyle=$
$\displaystyle\frac{1}{LM}\Bigl{(}\;\log_{2}(L)\bigl{(}(L-1)\mathbf{X}_{13}+(L-1)\mathbf{X}_{14}+p_{-}(\mathbf{X}_{21}+\mathbf{X}_{22})+p_{+}(\mathbf{X}_{11}+\mathbf{X}_{12})\bigr{)}$
$\displaystyle-((L-1)\mathbf{X}_{13}+p_{-}\mathbf{X}_{21}+p_{+}\mathbf{X}_{11})\log_{2}((L-1)\mathbf{X}_{13}+p_{-}\mathbf{X}_{21}+p_{+}\mathbf{X}_{11})$
$\displaystyle-((L-1)\mathbf{X}_{14}+p_{-}\mathbf{X}_{22}+p_{+}\mathbf{X}_{12})\log_{2}((L-1)\mathbf{X}_{14}+p_{-}\mathbf{X}_{22}+p_{+}\mathbf{X}_{12})$
$\displaystyle+(L-1)\mathbf{X}_{13}\log_{2}(\mathbf{X}_{13})+(L-1)\mathbf{X}_{14}\log_{2}(\mathbf{X}_{14})$
$\displaystyle+p_{-}\mathbf{X}_{21}\log_{2}(\mathbf{X}_{21})+p_{-}\mathbf{X}_{22}\log_{2}(\mathbf{X}_{22})+p_{+}\mathbf{X}_{11}\log_{2}(\mathbf{X}_{11})+p_{+}\mathbf{X}_{12}\log_{2}(\mathbf{X}_{12})\;\Bigr{)},$
(49)

where the dependence of the transition probability matrix elements on
$\bar{N}_{S}$, $L$, $M$, and $\bar{N}_{B}$ are given in Appendix F. The
optimal $p_{+}$ is solved numerically to satisfy

$\frac{\partial}{\partial p_{+}}I(\bar{N}_{S},\bar{N}_{B},\eta,p_{+})=0.$

In Fig. 14, we plot the bits-per-mode rates of JDR2 and JDR1 as a ratio over
the Holevo capacity $C$.

Figure 14: The capacity of JDR2 (green) and JDR1 (magenta) as functions of
$\bar{N}_{S}$ plotted as ratios over the ultimate unassisted Holevo capacity
$C$ assuming $\bar{N}_{B}=10$ and $\eta=0.01$. The individual green and pink
lines correspond to different values of $L$, the code word length. As
$\bar{N}_{S}$ decreases, the optimal value of $L$ increases. Also included in
the plot is the envelope of the capacity of JDR2 when code words are drawn
from a Hadamard code book instead of Reed-Muller code book (dotted brown)

.

Note that Eq. (49) subsumes the rate associated with using a Hadamard code
book, which can be obtained by setting $p_{-}=0$ and dividing the mutual
information by $M(L-1)$ instead of $ML$, since in the Hadamard code, all code
words share the same initial symbol ($\theta=0$), which can hence be appended
at the receiver-end instead of consuming a channel use. The resulting capacity
ratio is only slightly less than using the full Reed-Muller code, as Fig. 14
shows.

Fig. 15 shows that $M\approx 10^{4}$ is the new optimal $M$ for JDR2 as
opposed to $M\approx 10^{5}$ for JDR1. To understand this effect, note first
that repeating modes of the trasmitted code words $M$-fold effectively
increases the clarity of the received code words at the expense of consuming
$M$-times many channel uses. Of course, once $M$ is high enough, the code
words are almost perfectly distinguishable and further increasing $M$ only
hurts the communication rate by consuming unnecessary channel uses, but when
the receiver’s distinguishing power between two symbols is low (e.g. because
of the shot-noise associated with the particular design), adding clarity to
the code words is worth the extra channel uses, hence there is an optimal
value of $M>1$. Depending on the signal-to-noise (SNR) ratio of the output of
the receiver’s front-end, this optimal value of $M$ will be higher or lower.
Note that evaluating the Holevo information of the recieved ensemble
$\hat{\rho}^{(RI)}$ for $\theta\in\\{0,\pi\\}$, corresponding to $M=1$, in
section III.2 resulted in $C_{E}$. Thus a lower value for the optimal $M$ is
indicative of a receiver being closer to optimal. By combining the sum
frequency modes, JDR2 effectively filters out a significant portion of the
noise (into the unused outputs of the $\gamma$-beam splitters) and reduces the
SNR, which brings down the optimal value of $M$. Since this corresponds to
less channel uses, JDR2 achieves a higher rate than JDR1.

(a) $R_{E}^{(M,L)}/C$ envelopes of JDR1.

(b) $R_{E}^{(M,L)}/C$ envelopes of JDR2.

Figure 15: Here we plot the envelopes of $R_{E}^{(M,L)}/C$ (taken over
$L=2,4,8,...,2^{2}1$), for $M=10,100,...,10^{6}$. This shows that an optimum
performance occurs at $M\sim 10^{5}$ for JDR1 and $M\sim 10^{4}$ for JDR2. We
assume $\eta=0.01$ and $\bar{N}_{B}=10$ photons per mode, for all the plots in
these figures.

##  V PPM and OOK modulation formats for Entanglement-Assisted Communications

In Section III.2, we discussed the entanglement assisted capacity achievable
with on-off based modulation formats, while leveraging continuous-variable
SPDC-based pre-shared entanglement, and showed that the $\log(1/{\bar{N}}_{\rm
S})$ capacity-ratio improvement over the Holevo capacity is attainable with
such modulation formats. Despite the capacity not being as good as phase-only
modulation formats, and the need for more pre-shared entanglement, the on-off
modulation formats are easier to realize experimentally. Further, the pulse-
position modulation (PPM) scheme, a modulation-code over the on-off keying
(OOK) alphabet, has a close connection to quantum ranging, which also
saturates the $\log(1/{\bar{N}}_{\rm S})$ scaling [32]. The optimal receiver
design for OOK and PPM modulation formats are not known. In this section, we
discuss these two on-off based modulation formats for entanglement-assisted
communications, in further detail.

###  V.1 Pulse position modulation (PPM)

At the $L$ output modes of the $K$ Green Machine (GM) circuits of JDR1, shown
in Fig. 8, the state of the $LK$ output modes resembles pulse-position
modulation (PPM): One block of $K$ modes carries displaced thermal states
$\hat{\rho}_{\rm th}(\sqrt{L{\bar{N}}_{k}},{\bar{N}}_{\rm T})$, where
${\bar{N}}_{k}=M\kappa\eta{\bar{N}}_{\rm S}(1+{\bar{N}}_{\rm S})\mu^{k-1}$,
$1\leq k\leq K$, with $\mu=(1-\kappa(1+{\bar{N}}_{\rm S}^{\prime}))^{2}$,
${\bar{N}}_{\rm T}=\kappa{\bar{N}}_{\rm S}{\bar{N}}_{\rm S}^{\prime}$,
${\bar{N}}_{\rm S}^{\prime}=\eta{\bar{N}}_{\rm S}+(1-\eta){\bar{N}}_{\rm B}$.
The remainder $L-1$ of the $K$-mode blocks are excited in zero-mean thermal
states $\hat{\rho}_{\rm th}(0,{\bar{N}}_{\rm T})$.

An alternative to this scheme described in Section IV is for Alice to directly
modulate PPM code words. Alice and Bob need to pre-share (brighter) SPDC
signal-idler mode pairs of mean photon number per mode $L{\bar{N}}_{\rm S}$,
and Alice sends an $M$-temporal mode signal pulse (of mean photon number
$L{\bar{N}}_{\rm S}$) and nothing (vacuum) in $L-1$ pulse slots. Thus, only
$M$ modes are occupied by signal pulses out of each $ML$ transmitted modes.
FF-SFG stages are used to demodulate, as before, but no GM stages are needed.
The state of the $LK$ output modes of the $L$ $K$-stage FF-SFG modules are
identical to the above: one block of $K$ modes carries displaced thermal
states $\hat{\rho}_{\rm th}(\sqrt{L{\bar{N}}_{k}},{\bar{N}}_{\rm T})$, and the
remainder $L-1$ of the $K$-mode blocks will be excited in zero-mean thermal
states $\hat{\rho}_{\rm th}(0,{\bar{N}}_{\rm T})$.

The mean transmit photon number of both schemes are identical. The discrete-
memoryless channel (DMC) induced by the modulation-code-receiver combination
for both schemes are identical. Hence, the capacities achieved by the two
schemes are identical. The optimal PPM order for the second scheme is the
optimal Hadamard-code length for the first scheme. That optimal PPM-order (or
Hadamard code length) is given by: $L\approx({\cal E}\log(1/{\cal E}))^{-1}$
with ${\cal E}=M\eta{\bar{N}}_{\rm S}/(2{\bar{N}}_{\rm B})$, which translates
to $L{\bar{N}}_{\rm
S}\approx\frac{{\bar{N}}_{0}}{\log({\bar{N}}_{0}/{\bar{N}}_{\rm S})}$ with
${\bar{N}}_{0}=2{\bar{N}}_{\rm B}/(M\eta)$. For the parameters in Fig. 15,
i.e., $\eta=0.01$, ${\bar{N}}_{\rm B}=10$, $M=10^{4}$, we get
${\bar{N}}_{0}=0.2$, and optimal $L\approx 7$. This implies that that, for
${\bar{N}}_{\rm S}<0.01$, $L{\bar{N}}_{\rm S}\lesssim 0.07$, and that the
idler pulses are still in the regime that the implicit “qubit approximation”
analysis of the SFG [31] is valid.

There are key operational differences, however, between the two schemes, which
are described below:

  1. 1.

Peak power usage—Even though the mean photon number that is transmitted over
the channel is identical for both schemes, the peak power is not. The PPM
scheme uses $L$ times more peak power than the BPSK scheme. For the
aforementioned parameters, the optimal PPM order $L\approx 7$, which implies
the peak power is $7$ times that of BPSK. However, the BPSK scheme is slightly
more restrictive since Hadamard codes exist only for $L$ that is an integer
power of $2$. However, it is possible to redesign the BPSK scheme with
complex-valued Hadamard codes that would work for all integer $L$.

  2. 2.

Entanglement consumption—More important than the peak power advantage the BPSK
scheme enjoys is that its entanglement consumption is lower. Despite the fact
that the mean photon number per transmitted mode is ${\bar{N}}_{\rm S}$ for
both schemes, in the PPM scheme, every $M$-mode SPDC pulse that needs to be
pre-shared must have $L{\bar{N}}_{\rm S}$ photons per mode. This is true, even
though $(L-1)/L$ fraction of the signal pulses of the pre-shared entangled
states will never be transmitted in the PPM scheme. This is a major drawback
for this scheme.

  3. 3.

Receiver complexity—The BPSK scheme needs the $K$ Green Machine circuits in
JDR1 and one in JDR2 in addition to the FF-SFG modules. That is an added
receiver complexity for the BPSK scheme over the PPM scheme.

  4. 4.

Using the noise modes of FF-SFG stages—In the BPSK scheme described in Section
IV, we ignore the $LK$ “noise modes” labeled ${\hat{e}}_{k}^{(i)}$ in Fig. 8.
In our operational regime, for both the BPSK and PPM schemes, the state of
${\hat{e}}_{k}^{(i)}$ is close to zero-mean thermal state of the same mean
photon number as that of the corresponding sum-frequency mode,
${\hat{b}}_{k}^{(i)}$. The capacity analyses (for both BPSK and PPM) above
ignores ${\hat{e}}_{k}^{(i)}$ modes. There is information about the
transmitted code word in them, which can only increase the achievable
capacity. For the PPM scheme, one can simply do photon counting on all the
${\hat{e}}_{k}^{(i)}$ modes. For the pulse-containing block of $K$ noise modes
${\hat{e}}_{k}^{(i)}$, $1\leq k\leq K$, on-off direct detection of those modes
effectively doubles the energy of the “on” PPM pulse, causing the capacity-
ratio plots to shift right by $\log_{10}2$. This is a small improvement, but
one that only needs additional single-photon detectors. A similar capacity
improvement for the BPSK scheme leveraging the ${\hat{e}}_{k}^{(i)}$ modes
requires a feedback-based scheme like in [31], where, based on photon-
detection events at the noise modes, one adaptively applies two-mode squeezing
before and after each of the SFG stages within the FF-SFG modules.

###  V.2 On-off keying (OOK)

Finally, PPM can be thought of as a modulation code over an on-off keying
(OOK) alphabet, and hence its capacity is strictly inferior to that of OOK,
although it is very close to OOK when ${\cal E}\ll 1$. This means that an OOK
version of our modulation format also attains the $\log(1/{\bar{N}}_{\rm S})$
capacity ratio. The “on” symbol (transmission of the $M$-mode signal pulse) is
associated with a prior probability $p$ and the “off” symbol (no signal
transmission) with a prior probability $1-p$, with $p\sim{\cal E}\log(1/{\cal
E})$ assuming the role of the inverse-order $1/L$ of PPM, except that there is
now no restriction that there must be exactly one “on” pulse in every
$L$-pulse block.

As described in Section III.2, the entanglement assisted capacity attainable
with PPM and OOK based modulation (with TMSV pre-shared entanglement) is
strictly inferior to that attainable with BPSK modulation on TMSV pre-shared
entanglement. However, BPSK modulation on TMSV, paired with the GM-based JDR1
described in Section IV, attains the same entanglement assisted capacity
achievable with PPM with TMSV but without the GM. Despite this, the BPSK-based
JDRs from Section IV may be more practical in the near term compared to PPM
and OOK formats, which require Alice and Bob to pre-share more entanglement
(i.e., the pre-shared signal-idler mode pairs need a higher mean photon number
per mode). High-rate fault-tolerant entanglement distribution to pre-share the
resource necessary for supporting entanglement-assisted communications is
likely to be the most expensive process in a future implementation.

##  VI Receiver designs to attain $C_{E}$

As discussed in section III, BPSK modulation on TMSV pre-shared entanglement
suffices to closely attain $C_{E}$ in the ${\bar{N}}_{S}\ll 1$,
${\bar{N}}_{B}\gg 1$ regime (the regime where the most entanglement assisted
gain $C_{E}/C\sim\ln(1/{\bar{N}}_{S})$) holds. In Section IV, we developed and
analyzed two receivers for BPSK modulation on TMSV, the JDR1 and JDR2, both of
which attain entanglement-assisted communication rates that exceed the $2C$
limit ($C$ being the unassisted Holevo capacity) associated with symbol-by-
symbol measurements. However, these JDRs are not optimal, as there is still a
large gap to $C_{E}$. We refer to the JDR2 design depicted in Fig. 9 for
notation, in this section.

There are two distinct sources of information inefficiencies that must be
addressed to close this capacity gap:

Figure 16: Plot (red dots) of the Holevo capacity of the effective output
ensemble
$\\{\hat{\rho}_{\text{th}}(-\alpha_{0},\bar{N}_{T0}),\hat{\rho}_{\text{th}}(\alpha_{0},\bar{N}_{T0})\\}$
of a single SFG-module of JDR2 (at the output of the beam splitter
$\gamma_{K-1}$ of Fig. 9), as a ratio to the ultimate unassisted capacity $C$.
For comparison, the plot also includes (blue diamonds) the Holevo capacity of
the ensemble
$\\{\hat{\rho}_{\text{th}}(0,\bar{N}_{T0}),\hat{\rho}_{\text{th}}(\alpha_{0},\bar{N}_{T0})\\}$
(corresponding to OOK modulation rather than BPSK), as a ratio to $C$. Both
plots are shown overlaid on Fig. 14 and drawn assuming $\bar{N}_{B}=10$ and
$\eta=0.01$. The thick dashed black line is $C_{E}/C$.

###  VI.1 Receiver back-end that achieves the Holevo-capacity of unassisted
communications

Even if the FF-SFG receiver front-end was designed to optimally convert the
modulated phase in a symbol to the displacement of a single BPSK coherent
state without losing any Holevo information content, receivers such as JDR1
and JDR2 that are based on passive linear optics, coherent-state local
oscillators and shot-noise-limited photon number detection, are known to be
insufficient to achieve the Holevo capacity of the induced phase-modulated
coherent-state constellation [38], and hence fall short of achieving $C_{E}$.
The various receiver designs we discussed in Section II.4 —in the context of
achieving the quantum (Holevo) limit of classical communications—can be used
as a back-end for an entanglement-assisted receiver design to close the gap
between the green envelope and the red dots in Fig. 16.

Note that the envelope of the JDR2 capacity in Fig. 16 falls short of the
Holevo capacity of the FF-SFG output ensemble (red dots) in the same way that
the Reed-muller-GM-Dolinar receiver of [47] falls short of the BPSK Holevo
capacity for coherent-state modulated communication (see Fig.
LABEL:fig:classcaps). To close the gap, one could implement a quantum belief-
propagation receiver, recently proposed in [48] as the back-end of JDR2. The
Reed-muller code would get replaced by an LDPC code, and the demodulation
phase would require a photon-to-qubit conversion step before running a quantum
message-passing circuit to decode the code words, exemplified by a photon-to-
ion transduction in Fig. 17.

Figure 17: An alternative to JDR2 to attain the FF-SFG BPSK Holevo capacity
(red dots in Fig. 16). See [50] for definitions of the example circuit
components.

Just as JDR2, a BPQM-based receiver for EA communications has the added
complication that the detection front-end has nonzero thermal noise compared
to noiseless classical communication. The analysis is left as future work.

###  VI.2 Receiver front-end that extracts information from the noise modes

The red dots in Fig. 16 reveal that the Holevo capacity of the ensemble
$\\{\hat{\rho}_{\text{th}}(\pm\alpha_{0},\bar{N}_{T0})\\}$ at the output of
the beam splitter $\gamma_{K-1}$ combining the outputs of the sum-frequency
modes does not attain $C_{E}$, either in value or in scaling. Yet the input
ensemble does attain $C_{E}$, as shown in section III (Fig. 6). Given that the
$\kappa$-beam splitters and SFG operations are unitary, and hence do not
decrease the information content of the input, the information missing in the
combined sum-frequency modes warrants tracking down.

One candidate for this missing information are the noise modes ${\hat{e}}$ in
Fig. 9. But, they are in a zero-mean thermal state that has no information
about the modulated binary phase on the symbol. We claim that sandwiching each
SFG gate with two-mode-squeezing gates $S(r)$ and $S(-r)$, whose action on the
joint state of the signal and idler modes is described by the unitary

$S(r):=e^{\text{sinh}^{-1}(r)(\hat{a}_{S}^{\dagger}\hat{a}_{I}^{\dagger}-\hat{a}_{S}\hat{a}_{I})},$

with the associated Heisenberg mode transformation given by
$\hat{a}_{S}\longrightarrow\sqrt{1+r^{2}}\,\hat{a}_{S}+r\,\hat{a}_{I}^{\dagger}$
(swapping $S$ for $I$ for the idler mode transformation), enables the
extraction of the information missing in the sum frequency modes from the
photon detection statistics of the noise mode $\hat{e}$. To see how, we
consider the effect of a single sandwiched SFG gate on the received signal and
idler in the qubit approximation as depicted in Fig. 18, ignoring the sum-
frequency mode.

Figure 18: Diagram of the effective action of an SFG gate on the signal and
idler modes (ignoring the sum-frequency mode) in the qubit approximation,
sandwiched by TMS operations with squeezing amount $r$.

Let $\hat{a}_{S,1},\hat{a}_{S,2}$ and $\hat{e}$ be associated with modes as
depicted in Fig. 18. It was shown in [31] that if the input signal and idler
are dim enough, the evolution time $t$ can be chosen appropriately, in
particular $gt=\pi/2$, such that the SFG gate effectively acts as a two-mode
squeezer whose squeezing coefficient $-\xi_{\theta,r}$ depends on $\theta$ and
$r$, being equal to the negative of the phase-dependent cross correlation
$\langle\hat{a}_{S,1}\hat{a}_{I}\rangle$ at the input of the SFG gate. Up to
first order in $\bar{N}_{S}$ and $r$,

$\xi_{\theta,r}=e^{i\theta}\sqrt{\kappa\eta\bar{N}_{S}(\bar{N}_{S}+1)}+r.$
(50)

Note that even though all of the operations in Fig. 18 including the beam
splitters are unitary, the information content of the ensemble

$\\{\hat{\rho}_{\theta}^{\text{(SIE)}}\,\mid\,\theta\in\\{0,\pi\\}\\}$

of the joint state of the signal (S), idler (I) and noise (E) modes (a
Gaussian state in the qubit approximation for the SFG gate) after the gates
and beam splitters is less than that of the input ensemble. The reason is that
in order to ignore the sum-frequency mode, the SFG gate has been reduced to a
two-mode unitary whose action is a function of $\theta$, the very parameter in
which the information is encoded. If the full SFG unitary is considered as a
$\theta$-independent operation then one cannot neglect the sum-frequency mode
when evaluating the information content at the output, at which point one
finds that some of the information in the received state has been funneled
into the sum-frequency mode, as evaluated in detail in section IV.

Figure 19: Holevo information of the BPSK ensemble
$\hat{\rho}_{\theta}^{\text{(SIE)}}$ for $\theta\in\\{0,\pi\\}$ (in
bits/channel use), plotted as a function of $r$, where
$\hat{\rho}_{\theta}^{\text{(SIE)}}(r)$ is the state of the signal, idler and
noise modes after the operations depicted in Fig.18. $\chi$ was evaluated
numerically by diagonalizing
$\frac{1}{2}(\hat{\rho}_{0}^{(SIE)}+\hat{\rho}_{\pi}^{(SIE)})$ in the Fock
basis with cutoff $n=18$.

Moreover, it is not obvious that the information content of
$\hat{\rho}_{\theta}^{(\text{SIE})}$ is independent of $r$. Since the
effective SFG action is also a function of $r$, the amount of information
removed via the $\theta$-dependence of the effective SFG could in principle be
parameterized by $r$. However, when the Holevo capacity of
$\hat{\rho}_{\theta}^{(\text{SIE})}$ is evaluated as a function of $r$ (Fig.
19), it is seen to be independent of $r$.

Consider the case when $r=0$ in Fig.18. If increasing $r$ to a nonzero value
does in fact activate the noise mode $\hat{e}$ so that the value of $\theta$
is reflected in the $\hat{e}$ mode’s photon detection statistics, which we
have yet to show, Fig. 19 raises the question of how the $\theta$-information
is embedded in the joint state of the modes $\hat{a}_{S,1},\hat{a}_{I}$ and
$\hat{e}$ before the post-SFG TMS gate. The answer is that the $\theta$
dependence is in the phase-sensitive cross-correlation
$\langle\hat{a}_{S,2}\hat{a}_{I}\rangle$, whose expression for any value of
$r$ is given by

$\begin{split}\langle\hat{a}_{S,2}\hat{a}_{I}\rangle=\pm(1-\kappa(\frac{1}{2}+\bar{N}_{B}))\sqrt{\eta\bar{N}_{S}}\\\
+\mathcal{O}(\bar{N}_{S}^{3/2})+\mathcal{O}(\kappa
r^{2})+\mathcal{O}(\kappa^{3/2})\end{split}$ (51)

where $\pm$ corresponds to $\theta=0$ and $\theta=\pi$.

Note that neither the $\hat{a}_{S,1}$-mode nor the $\hat{a}_{S,2}$-mode’s mean
photon number depends on $\theta$, so when the modes $\hat{a}_{S,2}$ and
$\hat{a}_{S,1}$ recombine on the $1-\kappa$ beam splitter to produce the
$\hat{e}$-mode, a $\theta$-dependent mean photon number on the $\hat{e}$-mode
can only result if the $\hat{a}_{S,1}$ and $\hat{a}_{S,2}$-modes are
classically correlated by an amount that depends on $\theta$. At the input of
the post-SFG TMS gate in Fig. 18, the actual expression for this classical
correlation is

$\displaystyle\langle\hat{a}_{S,2}^{\dagger}\hat{a}_{S,1}\rangle=$
$\displaystyle\sqrt{\kappa}\bar{N}_{B}+\mathcal{O}(\kappa\bar{N}_{S}^{3/2})$
$\displaystyle+\mathcal{O}(\kappa^{3/2})+\mathcal{O}(\sqrt{\kappa}(\bar{N}_{S}^{2}+r^{2})),$
(52)

which is clearly independent of $\theta$.

Eq.s (52) and (51) reveal the crucial role of the post-SFG two-mode squeezing
gate depicted in Fig. 18. By linearly mixing the quadratures
$\hat{a}_{S,1}^{\dagger}$ and $\hat{a}_{I}$, the post-SFG TMS gate transfers
$\theta$-dependence from the correlation
$\langle\hat{a}_{S,2}\hat{a}_{I}\rangle$ onto the correlation
$\langle\hat{a}_{S,2}^{\dagger}\hat{a}_{S,1}\rangle$ so that upon recombining
on the $1-\kappa$ beam splitter, the mean photon number of the $\hat{e}$ mode
is given by

$\begin{split}\langle\hat{e}^{\dagger}\hat{e}\rangle=\pm
2r\sqrt{\eta\kappa\bar{N}_{S}}\\\ +\kappa\eta\bar{N}_{S}+r^{2}\\\
+\mathcal{O}((\bar{N}_{S}-\kappa)r^{2}),\end{split}$ (53)

where $\pm$ corresponds to $\theta=0$ and $\theta=\pi$. This matches the
coherent contribution to the mean photon number of the sum-frequency mode
whose coherent displacement, after the pre-SFG squeezing, is approximately
$\langle\hat{b}\rangle\approx r\pm\sqrt{\kappa\eta\bar{N}_{S}}$. The
distillation of the phase of the signal-idler cross-correlation into the sum-
frequency and noise modes by the squeezer-sandwiched SFG operation is sketched
out in phase-space in Fig. 20

Figure 20: Sandwiching the SFG gate with squeezers allows for extraction of
information from the environment modes as well as the sum-frequency modes. The
effect of the phase of the cross-correlation between the signal and idler
modes on the SFG output modes $\hat{e}$ and $\hat{b}$ when the squeezing
parameters $r_{k}$ are adjusted to null one of the BPSK symbols is depicted by
cartoons of the Wigner function representation of the states.

Without the squeezers sandwiching the SFG gates, the information bridging the
gap between $C_{E}$ and the Holevo capacity of the ensemble of the combined
sum-frequency modes (red dots in Fig. 16) would be trapped in the cross-
correlations $\langle\hat{e}^{(i)}_{k}\hat{a}_{I}\rangle$ between the noise
modes and the idler mode, undetectable by photon counters on the
$\hat{e}^{(i)}_{k}$ modes for any $i$ and $k$.

With the squeezing parameters $r_{k}^{(i)}=\pm\alpha_{k}^{(i)}$ chosen to null
one of the hypotheses, the effective output state of an FF-SFG module (a
multimode state consisting of $\hat{b}_{0}^{(i)}$ and i.i.d.
$\hat{e}_{k}^{(i)}$-modes for $k$ ranging from $1$ to $K$) is
$\hat{\rho}_{\text{th}}(\pm\sqrt{2}\alpha_{0},\bar{N}_{T0})$, as shown in
Appendix E. The Holevo capacity of this ensemble achieves $C_{E}$, as Fig. 21
shows. A receiver back-end design that actually achieves the full $C_{E}$
latent in the front-end output however, must be a JDR that is not only Holevo-
capacity-attaining for BPSK modulated coherent states (with a small amount of
additive thermal noise), but also one that is able to see the effective output
of the front-end as
$\hat{\rho}_{\text{th}}(\pm\sqrt{2}\alpha_{0},\bar{N}_{T0})$ when the noise
modes are included.

Figure 21: Plot (red dots) of the Holevo capacity of the effective output
ensemble
$\\{\hat{\rho}_{\text{th}}(-\sqrt{2}\alpha_{0},\bar{N}_{T0}),\hat{\rho}_{\text{th}}(\sqrt{2}\alpha_{0},\bar{N}_{T0})\\}$
of a single SFG-module of JDR2 (capturing both the output of $\gamma_{K-1}$
and the noise modes $\\{\hat{e}^{(i)}_{k}\\}_{k=1}^{K}\\}$), as a ratio over
the ultimate classical capacity $C$. The plot also includes the Holevo
capacity (blue triangles) of the ensemble
$\\{\hat{\rho}_{\text{th}}(0,\bar{N}_{T0}),\hat{\rho}_{\text{th}}(\sqrt{2}\alpha_{0},\bar{N}_{T0})\\}$
(corresponding to OOK modulation).

One possible complete structured receiver design that could potentially attain
$C_{E}$ is inspired by the Holevo-capacity attaining vacuum-or-not JDR for
coherent-state modulation proposed in Ref. [42]. The receiver would
successively null the patterns of cross-correlations on the encoded mode-pair-
blocks using the appropriate squeezing amounts on the sandwiching squeezers,
then perform a multi-mode vacuum-or-not (VON) measurement on the joint output
state of the SFG-Green-Machine module, as pictured in Fig. 22.

Figure 22: Vacuum-or-not joint receiver design idea for entanglement-assisted
communications to attain the entanglement-assisted capacity $C_{E}$.

The analysis of such a receiver is complicated by the fact that correct
nulling of the code words does not result in perfect vacuum, as for the case
of noiseless coherent state encoded communication [42]. Moreover, while the
idea of combining an SFG-based JDR with a VON measurement possibly attains
$C_{E}$ in principle, it will be difficult to construct in practice. Two major
experimental challenges associated with building such a receiver are the
successful implementation of a multi-mode vacuum-or-not measurement, e.g.,
leveraging the V-STIRAP interaction in a light-atom interaction [43], and the
ability to push the entire multi-mode output state back through the SFG in
reverse losslessly after each round of the vacuum-or-not measurement.

In Appendix E, we provide strong evidence to this vacuum-or-not (VON) receiver
back-end achieving $C_{E}$. We calculate the block vacuum-probability with
correct and incorrect codeword nulling, showing that these probabilities are
exactly those of the VON receiver of [42] applied on the noisy BPSK coherent
state codewords with symbols
$\hat{\rho}_{\text{th}}(\pm\sqrt{2}\alpha_{0},\bar{N}_{T0})$.

Notably, the sandwiching squeezers were an essential component in [31] for
closing the 6dB gap to the quantum Chernoff exponent for the illuminated
target-detection problem so the TMSV-sandwiched SFG gates are in a sense,
natural to consider as a component for the task of communication. Even so, the
optimal receiver designs for the tasks of sensing and communication are
different, the former implementing a feed-forward structure paralleling the
Dolinar receiver for optimal symbol-by-symbol coherent-state discrimination,
and the latter implementing a joint multi-mode non-destructive vacuum-or-not
module described above.

##  VII Conclusion

Bridging the gap to $C_{\rm E}$ will require better codes and more complex
quantum joint detection receivers. Ideas for follow on works include (1)
developing theoretical work on receiver performance classification based on
constituent optical elements as in Ref. [38], (2) using information in the
extra modes ${\hat{e}}_{k}^{(l)}$’s in Fig. 8 as in [31], and (3) FF-SFG
modules consisting of SFG gates sandwiched by two-mode-squeezing stages as in
[31] to successively null the mean field amplitudes of the thermal states of
the sum-frequency modes and the mean thermal photon numbers of the environment
modes unused by JDR1 and JDR2. In addition, we would like to note that as
${\bar{N}}_{\rm B}\to\infty$, $C_{\rm E}/(C\ln{\bar{N}}_{\rm S})\to 1$, while
$R_{\rm E}^{(M)}/(C\ln{\bar{N}}_{\rm S})\to 1/2$ [46], which indicates a
possible check, to see if the entanglement-assisted capacity attained by an
improved receiver design improves this ratio from $1/2$ to $1$. Finally, as an
alternative to the originally-proposed random-coding method to achieve $C_{\rm
E}$ [21], “position based encoding” was proposed to achieve $C_{\rm E}$ over a
general quantum channel [55, 56, 57, 58, 32]. However, a structured optical
receiver of the associated receiver’s joint-detection measurement is unknown,
and is an interesting topic for future work.

Pre-shared entanglement improves capacity when the transmitted power is low
and thermal-noise mean photon number is high, despite the entanglement not
surviving these conditions. While typically uncommon, this regime corresponds
to _covert communications_ , a security modality where the mere attempt to
transmit data must be hidden. Covert transmission using $n$ total modes is
constrained to per-mode photon number ${\bar{N}}_{\rm S}=c/\sqrt{n}$ for a
constant $c$ that depends on ${\bar{N}}_{\rm B}$, $\eta$, and the desired
stringency of covertness [44, 45]. Thus, one can transmit covertly and
reliably only ${O}(\sqrt{n})$ bits using $n$ modes without entanglement
assistance, however, pre-shared entanglement breaks this _square root law_ ,
allowing transmitting $\mathcal{O}(\sqrt{n}\log n)$ bits covertly [46].

## Acknowledgments

SG, CNG, and AC acknowledge General Dynamics Mission Systems (GDMS) for
supporting this research. AC and BB was partially funded by the National
Science Foundation (NSF) grant CCF-2006679. QZ and BB further acknowledge the
Army Research Office (ARO) Grant Numbers W911NF-19-1-0418 and
W911NF-19-1-0412, respectively. The authors acknowledge William Clark of GDMS,
as well as Michael Bullock and Zheshen Zhang of University of Arizona for many
valuable discussions.

S.G. has outside interests in Guha LLC, Xanadu Quantum Technologies, Quantum
Network Technologies, and SensorQ Technologies. These have been disclosed to
the University of Arizona and reviewed in accordance with its conflict of
interest policies. Any resulting conflicts of interest from these interests
will be managed by The University of Arizona in accordance with its policies.

##  Appendix A OPA receiver analysis

In the low photon number regime (${\bar{N}}_{\rm S}\ll 1$) the communication
capacities are well-approximated by the Taylor series expansion around
${\bar{N}}_{\rm S}=0$. For example, the Holevo capacity $C(\eta,{\bar{N}}_{\rm
S},{\bar{N}}_{\rm B})$ is:

$C(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm B})=\eta{\bar{N}}_{\rm
S}\log\mathopen{}\mathclose{{}\left(1+\frac{1}{(1-\eta){\bar{N}}_{\rm
B}}}\right)+o({\bar{N}}_{\rm S}).$ (54)

Here we derive the Taylor series expansion of the entanglement-assisted
communication capacity with an SPDC source, BPSK modulation, and the OPA
receiver [30] of gain $G$. We use it to evaluate the entanglement-assisted
capacity gain achieved by an OPA receiver over the Holevo capacity. This
channel’s capacity is the classical mutual information between the random
binary phase input $\theta\in\\{0,\pi\\}$, $P(\theta=0)=q$, modulating the
block of $M$ transmitted symbols (i.e., $M$-fold tensor product of TMSV
states) and the photon-count output $L$ of Bob’s detector, optimized over the
probability distribution of the input defined by $q$:

$\displaystyle C_{\text{EA-OPA}}(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm
B})=\max_{q}I(\theta;{\bar{N}}_{\rm S}).$ (55)

The probability that the photon counter records $k$ photons over $M$ modes is:

$\displaystyle P(k|\theta;M)=\frac{1}{(1+{\bar{N}}_{\theta})^{M}}{k+M-1\choose
k}\mathopen{}\mathclose{{}\left(\frac{{\bar{N}}_{\theta}}{1+{\bar{N}}_{\theta}}}\right)^{k}.$
(56)

When phase $\theta$ is transmitted, the mean received photon number per mode
is:

$\displaystyle{\bar{N}}_{\theta}=G{\bar{N}}_{\rm S}+(G-1){\bar{N}}_{\rm
S}^{\prime}+2C_{p}\sqrt{G(G-1)}\cos(\theta),$ (57)

where ${\bar{N}}_{\rm S}$ is the mean photon number in each signal and idler
mode, ${\bar{N}}_{\rm B}$ is the mean thermal noise injected by the
environment, $\eta$ is the channel transmissivity, ${\bar{N}}_{\rm
S}^{\prime}\equiv\eta{\bar{N}}_{\rm S}+(1-\eta){\bar{N}}_{\rm B}+1$, $G$ is
the gain of the OPA, and $C_{p}\equiv\sqrt{\eta{\bar{N}}_{\rm
S}({\bar{N}}_{\rm S}+1)}$.

The Taylor series of mutual information $I(\theta;{\bar{N}}_{\rm S})$ at
${\bar{N}}_{\rm S}=0$ is:

$\displaystyle I(\theta;{\bar{N}}_{\rm S})=-{\bar{N}}_{\rm
S}\sum_{k=0}^{\infty}\sum_{\theta\in\\{0,\pi\\}}\mathopen{}\mathclose{{}\left.Q_{\theta}(k,{\bar{N}}_{\rm
S})}\right|_{{\bar{N}}_{\rm S}=0}+o({\bar{N}}_{\rm S}),$

where

$\displaystyle Q_{\theta}(k,{\bar{N}}_{\rm
S})=\begin{cases}q\frac{\mathrm{d}P(k|0;M)}{\mathrm{d}{{\bar{N}}_{\rm
S}}}\log\mathopen{}\mathclose{{}\left(q+(1-q)\frac{P(k|\pi;M)}{P(k|0;M)}}\right),&\theta=0\\\
(1-q)\frac{\mathrm{d}P(k|\pi;M)}{\mathrm{d}{{\bar{N}}_{\rm
S}}}\log\mathopen{}\mathclose{{}\left((1-q)+q\frac{P(k|0;M)}{P(k|\pi;M)}}\right),&\theta=\pi\end{cases}.$
(58)

Substitution of (56) and evaluation of $Q_{\theta}(k,{\bar{N}}_{\rm
S})\big{\rvert}_{{\bar{N}}_{\rm S}=0}$ by taking the limit
$\lim_{{\bar{N}}_{\rm S}\to 0}Q_{\theta}(k,{\bar{N}}_{\rm S})$ yields:

$\displaystyle I(\theta;{\bar{N}}_{\rm S})={\bar{N}}_{\rm S}8q(1-q)\eta$
$\displaystyle\times\sum_{k=0}^{\infty}G(G-1)^{k-1}({\bar{N}}_{\rm
B}^{\prime})^{k-2}(G+(1-\eta)(G-1){\bar{N}}_{\rm
B})^{k-M-2}(k+(G-1)M({\bar{N}}_{\rm B}^{\prime})^{2}{k+M-1\choose k}$
$\displaystyle+o({\bar{N}}_{\rm S}),$ (59)

where ${\bar{N}}_{\rm B}^{\prime}\equiv 1+(1-\eta){\bar{N}}_{\rm B}$. Well-
known results for the moments of binomial distribution are used to evaluate
the sum in (59). Maximizing over $q$ yields:

$\displaystyle C_{\text{EA-OPA}}(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm B})=$
$\displaystyle\frac{2{\eta}GM{\bar{N}}_{\rm S}}{{\bar{N}}_{\rm
B}^{\prime}(G+(1-\eta)(G-1){\bar{N}}_{\rm B})}$
$\displaystyle+o({\bar{N}}_{\rm S}).$ (60)

The maximum gain from using the SPDC source, BPSK modulation and the OPA
receiver over the Holevo capacity when ${\bar{N}}_{\rm S}\ll 1$ and
${\bar{N}}_{\rm B}\gg 0$ is thus:

$\displaystyle\lim_{G\downarrow 1}\lim_{{\bar{N}}_{\rm
B}\to\infty}\frac{C_{\text{EA-OPA}}(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm
B})}{M\times C(\eta,{\bar{N}}_{\rm S},{\bar{N}}_{\rm B})}=2,$ (61)

where $\lim{G\downarrow 1}$ indicates a one-sided limit taken from above, and
we normalize the denominator by $M$ to account for employing block encoding of
$M$ symbols. We note that, with such normalization, the gain does not depend
on $M$. There is also no dependence on the transmissivity $\eta$.

##  Appendix B Connection with PPM with dark-click rate proportional to mean
energy per slot

In this Appendix, we consider a cruder approximation of $R_{\rm E}^{(M)}$,
providing an alternative proof of the scaling in (47), but one that lets us
establish a connection with a problem that was studied by Wang and Wornell in
the context of coherent-state PPM modulation, where the dark click probability
per mode $\lambda$ is proportional to the mean photon number per mode $\cal E$
[60].

Recall that $R^{(M)}_{\rm E}={{\rm sup}_{L}R_{\rm E}^{(M,L)}}$ is the envelope
of capacities attained by our scheme over all $L$, for a given $M$.
Approximations in (32)-(34) lead to the following simplified asymptotic
expressions: $1-p_{\rm c}\approx e^{-(L{\cal E}+\lambda)}$, and $1-p_{\rm
b}\approx e^{-\lambda}$, $\lambda=c{\cal E}$, with

$\displaystyle{\cal E}$ $\displaystyle=\frac{M\eta{\bar{N}}_{\rm
S}}{2(1+(1-\eta){\bar{N}}_{\rm B})}$ (62) $\displaystyle c$
$\displaystyle=\frac{2(1-\eta){\bar{N}}_{\rm B}(1+(1-\eta){\bar{N}}_{\rm
B})}{M\eta}.$ (63)

This is exactly the setting of $L$-mode coherent-state PPM modulation and
direct detection, where the dark click probability per mode $\lambda$ is
proportional to the mean photon number per mode ${\cal E}$ [60]. The leading-
order terms of the optimal capacity for this setting, in the regime of ${\cal
E}\ll 1$, is given by:

$C_{\rm PPM}({\cal E})\approx{\cal E}\log\frac{1}{\cal E}-{\cal
E}\log\ln\frac{1}{\cal E}-{\cal E}\ln(1+c),$ (64)

with the optimal PPM order, $L=\lfloor\mathopen{}\mathclose{{}\left({{\cal
E}\log(1/{\cal E})}}\right)^{-1}\rfloor$ [60]. Applying this result to our
problem, we get

$\displaystyle{R^{(M)}_{\rm E}}$ $\displaystyle=$ $\displaystyle\frac{C_{\rm
PPM}({\cal E})}{M},$ (65)

with ${R^{(M)}_{\rm E}}\approx(\eta{\bar{N}}_{\rm
S}/(2(1+(1-\eta){\bar{N}}_{\rm B})))\log(2(1+(1-\eta){\bar{N}}_{\rm
B})/(M\eta{\bar{N}}_{\rm S}))$ in the leading order. In the same regime as
above, $\kappa{\bar{N}}_{\rm S}\ll{\bar{N}}_{\rm S}\ll 1\ll{\bar{N}}_{\rm B}$,
the leading order term for the Holevo capacity (attained using coherent states
and Gaussian amplitude-and-phase modulation), $C\approx\eta{\bar{N}}_{\rm
S}/{\bar{N}}_{\rm B}$, and that of the entangled-assisted capacity (achieved
via an SPDC transmitter and phase-only modulation), $C_{\rm
E}\approx(\eta{\bar{N}}_{\rm S}/{\bar{N}}_{\rm B})\log(1/{\bar{N}}_{\rm S})$
[27]. It therefore follows that,

$\frac{R^{(M)}_{\rm
E}}{C}\sim\log\mathopen{}\mathclose{{}\left(\frac{1}{{\bar{N}}_{\rm
S}}}\right),\,\forall M,$ (66)

proving that our transmitter-receiver structure attains the optimal capacity
scaling.

(a) $\kappa=1$

(b) $\kappa=.05$

Figure 23: The Holevo capacity (ratio to ulimate classical Holevo capacity) at
the output of the first sum-frequency mode. Notice that when $\kappa=1$, i.e.
when a single SFG-gate is fed all of the signal mode without splitting off a
part of it using a beam splitter, the optimal SFG evolution time is not
$gt=\pi/2$ but less. Moreover even for the optimal value of $gt\approx 1.13$,
the capacity does not attain $C_{E}$, as the nonlinearity of the SFG gate
prevents complete conversion of the signal-idler cross-correlation into the
sum-frequency mode mean field. Only in the regime in which the qubit
approximation is valid does the optimal value of $gt$ approach $\pi/2$. Plots
are drawn for $\bar{N}_{B}=0.8$ using Fock-basis cutoff $n=3$.

##  Appendix C Analysis of the sum-frequency modes produced by the SFG-
operations

Figure 24: Setup for the evaluation of the joint sum-frequency state

The analysis of JDR2 relies on the key assumptions that 1) each sum-frequency
output mode is individually a displaced thermal state and 2) two successive
sum-frequency output states are in a maximally classically correlated joint
Gaussian state with covariance matrix given by

$\displaystyle\sigma_{th2}$
$\displaystyle=\frac{1}{2}\langle\\{\overset{\longrightarrow}{\Delta\hat{a}},\overset{\longrightarrow}{\text{\raisebox{0.0pt}[6.2pt]{$\Delta\hat{a}^{\dagger}$}}}\\}\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left(\begin{smallmatrix}\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}+\frac{1}{2}&\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}&0&0\\\
\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}&\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}+\frac{1}{2}&0&0\\\
0&0&\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}+\frac{1}{2}&\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}\\\
0&0&\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}&\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}+\frac{1}{2},\end{smallmatrix}}\right)$
(67)

and mean field given by

$\mu_{th2}=\langle(\hat{a}_{1},\hat{a}_{2},\hat{a}_{1}^{\dagger},\hat{a}_{2}^{\dagger})\rangle=(\alpha_{1},\alpha_{2},\alpha_{1}*,\alpha_{2}*)$
(68)

where $\alpha_{1}=\sqrt{\kappa\eta\bar{N}_{S}(1+\bar{N}_{S})}$ and
$\alpha_{2}=\sqrt{\kappa\eta\bar{N}_{S}(1+\bar{N}_{S})(1-\kappa(1+\bar{N}_{S}^{\prime}))^{2}}$.

In order to justify these assumptions independently from [31], we compute the
fidelity of the actual sum-frequency output gate with the state whose
covariance matrix is given in Eq. (67) and compute the ratio
$\frac{1}{2}\langle\\{\hat{b}_{1}\hat{b}_{2}^{\dagger}\rangle\\}/\sigma_{\text{max}}$
where
$\sigma_{\text{max}}=(\sigma_{th2})_{12}=\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}$
between the actual cross-correlation and the maximal classical cross-
correlation. The ratio is plotted in Fig. 27

The evolution of the signal, idler and sum-frequency modes through the sum-
frequency generation is done in the Schrödinger picture according to

$\tilde{\hat{\rho}}_{t}=e^{-it\hat{H}_{SFG}}\tilde{\hat{\rho}}_{0}e^{it\hat{H}_{SFG}}.$
(69)

where $\tilde{\hat{\rho}}$ is the state consisting of the signal, idler, and
sum-frequency modes represented in the Fock-basis with cutoff at $n=6$. The
SFG Hamiltonian $\hat{H}_{SFG}$ (see section IV for definition) in Eq. (69) is
also represented in the Fock basis as a matrix.

Excluding the initial vacuum sum-frequency mode, the state $\hat{\rho}_{0}$ at
the input of the SFG receiver indicated by the dashed line in Fig. 24 is a
3-mode zero-mean Gaussian state with covariance matrix
$\sigma:=\frac{1}{2}\langle\\{\Delta\vec{\hat{r}},\Delta\vec{\hat{r}}\\}\rangle$
given by

(
A0B0Ccos(θ)-κCsin(θ)0A0B-κCsin(θ)-κCcos(θ)B0D01-κCcos(θ)-1-κCsin(θ)0B0D-1-κCsinθ-1-κCcos(θ)κCcos(θ)-κCsin(θ)1-κCcos(θ)-1-κCsin(θ)¯N_S+120-κCsin(θ)-κC
cos(θ)-1-κCsin(θ)-1-κCcos(θ)0¯N_S+12 ) (70)

where

$\displaystyle\Delta\vec{\hat{r}}:=($
$\displaystyle\hat{q}_{S}-\langle\hat{q}_{S}\rangle,\hat{p}_{S}-\langle\hat{p}_{S}\rangle,$
$\displaystyle\hat{q}_{V}-\langle\hat{q}_{V}\rangle,\hat{p}_{V}-\langle\hat{p}_{V}\rangle,$
$\displaystyle\hat{q}_{I}-\langle\hat{q}_{I}\rangle,\hat{p}_{I}-\langle\hat{q}_{I}\rangle).$

and

$\displaystyle A:=$
$\displaystyle\bar{N}_{B}(\kappa-\eta\kappa)+\eta\kappa\bar{N}_{S}+\frac{1}{2}$
$\displaystyle B:=$
$\displaystyle\sqrt{\kappa(1-\kappa)}((1-\eta)\bar{N}_{B}+\eta\bar{N}_{S})$
$\displaystyle C:=$
$\displaystyle\sqrt{\eta}\sqrt{\bar{N}_{S}(\bar{N}_{S}+1)}$ $\displaystyle
D:=$
$\displaystyle(1-\eta)(1-\kappa)\bar{N}_{B}+\bar{N}_{S}(\eta-\eta\kappa)+\frac{1}{2}.$

To evolve $\hat{\rho}_{0}$ through two iterations of the SFG cycle, as
diagrammed in Fig. 24, we first represent $\hat{\rho}_{0}$ in the Fock-basis
with Fock cutoff $n=6$ using the formula from [62]:

$\langle\mathbf{m}|\hat{\rho}_{0}|\mathbf{n}\rangle=T\times\text{lhaf}(\bar{A})$
(71)

where $|\mathbf{n}\rangle$ is a photon number product state represented by
photon number vector $\mathbf{n}$. $T$ is a scalar factor computed from the
displacement and the covariance matrix of $\hat{\rho}_{0}$ (see [62] for
expression). $\bar{A}$ is derived from the covariance matrix by repeating rows
and columns of $\sigma$ and entries of mean $\mathbf{\mu}$ according to
$\mathbf{m}$ and $\mathbf{n}$.

To compute the loop-Hafnian lhaf in Eq. (71) we use a function from Python
library the-walrus, based on multi-dimensional Hermite polynomials [63].

$\hat{\rho}_{0}$ is upgraded to a 4-mode state via Kronecker product with a
vacuum initial sum-frequency mode. The state is then successively evolved via
the SFG unitary with evolution coefficient $gt=\pi/2$, a beam-splitter unitary
of transmissivity $(1-\kappa)$, a beam-splitter unitary of transmissivity
$\kappa$, and a final SFG unitary of evolution coefficient $gt=\pi/2$. The
state is partially traced out and Kronecker-multiplied with vacuum at each
step appropriately to maintain a 4-mode form throughout the evolution. Two of
the four modes are traced out at the end to produce the final joint-sum-
frequency state, which are plotted as matrices in the Fock basis
representation in Fig. 25 for $\bar{N}_{S}=.5$, $\bar{N}_{B}=1$, $\theta=\pi$,
$\eta=.01$, $\kappa=.8$, and $gt=\pi/2$.

$\pi/2$ is the value of $gt$ which corresponds to all cross-correlation
between signal and idler modes having been converted to a displacement of the
sum-frequency mode in the qubit approximation. This is verified numerically by
Fig. 23(b), which plots the Holevo capacity of the BPSK ensemble at the first
sum-frequency mode.

Figure 25: Plots of the actual state (top row) at the output of the first two
sum-frequency modes and the maximally classically correlated two-mode
displaced thermal state (bottom row) approximating the actual joint-sum-
frequency state with means
$\alpha_{1}=\sqrt{\kappa\eta\bar{N}_{S}(1+\bar{N}_{S})}$ and
$\alpha_{2}=\sqrt{\kappa\eta\bar{N}_{S}(1+\bar{N}_{S})(1-\kappa(1+\bar{N}_{S}^{\prime}))^{2}}$
and mean thermal noise $\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}$ on each mode,
where $\bar{N}_{S}^{\prime}=\eta\bar{N}_{S}+(1-\eta\bar{N}_{B})$. The fixed
channel parameters are $\bar{N}_{S}=.5$, $\bar{N}_{B}=1$, $\theta=\pi$,
$\eta=.01$, $\kappa=.8$, and $gt=\pi/2$. If the plots are seen as $7\times 7$
block-matrices, then the row and column indices of the blocks correspond to
the Fock-numbers of the mode $\hat{b}_{2}$, and the row and column indices
within each block correspond to the Fock-numbers of the mode $\hat{b}_{1}$.

The similarity between the actual joint state $\hat{\rho}_{b1,b2}$ and
maximally classically correlated two-mode displaced thermal state
$\hat{\rho}_{\text{th2}}$ with covariance matrix given by Eq. (67) and plotted
in Fig. 25 can be measured by the Uhlmann fidelity defined as
$\mathcal{F}(\hat{\rho}_{1},\hat{\rho}_{2}):=\text{Tr}(\sqrt{\sqrt{\hat{\rho}_{1}}\hat{\rho}_{2}\sqrt{\hat{\rho}_{1}}})^{2}$.
The fidelities between $\hat{\rho}_{b1,b2}$ and $\hat{\rho}_{\text{th2}}$ and
between single sfg-output $\hat{\rho}_{b_{1}}$ and $\hat{\rho}_{\text{th}}$ is
plotted in Fig. 26 as a function of $\kappa$ and $\bar{N}_{S}$.

Figure 26: Plots of the fidelity between output modes and the Gaussian
approximation assumed by [31]

Since both the approximation and the actual joint state are close to vacuum
for small $\kappa$ and $\bar{N}_{S}$, the fidelity might not capture the full
difference in the nature of the states. To check that the sum-frequency modes
are indeed maximally classically correlated, the ratio between the phase-
insensitive cross-correlation
$\langle\\{\hat{b}_{1},\hat{b}_{2}^{\dagger}\\}\rangle/2$ and the maximum
allowed phase-insensitive cross correlation
$\sigma_{\text{max}}=\kappa\bar{N}_{S}\bar{N}_{S}^{\prime}$ is plotted in Fig.
27. The phase-sensitive correlation of two successive sum-frequency modes is
also plotted in Fig. 28.

(a) Large scale plot

(b) Small scale plot zooming in on the region where $\bar{N}_{S}\ll 1$ and
$\kappa\ll 1$

Figure 27: Ratio between phase-insensitive cross-correlation of
$\hat{\rho}_{b1,b2}$ and maximally allowed phase-insensitive cross-correlation
$\sigma_{\text{max}}=\bar{N}_{B}\bar{N}_{S}(1-\eta)\kappa+\bar{N}_{S}^{2}\eta\kappa$.
Both plots use values $\bar{N}_{B}=1$, $\theta=\pi$ and Fock-basis cutoff
$n=6$.

Figure 28: Ratio between the phase-sensitive cross-correlation of
$\hat{\rho}_{b1,b2}$ and the maximally allowed phase-insensitive cross-