# A Weighted Autoencoder-Based Approach to Downlink NOMA Constellation Design
††thanks: This article/publication is based upon work from COST Action
INTERACT, CA20120, supported by COST (European Cooperation in Science and
Technology).
This paper has received funding from the European Union’s Horizon 2020
research and innovation programme under Grant Agreement number 856967.
This work was supported by Grant PID2021-128373OB-I00 (6G-AINA) funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe,”

Vukan Ninkovic, Dejan Vukobratovic  Faculty of Technical Sciences
University of Novi Sad
Novi Sad, Serbia
{ninkovic<EMAIL_ADDRESS>Adriano Pastore, Carles Antón-Haro  Centre
Tecnològic de Telecomunicacions de Catalunya (CTTC/iCERCA)
Parc Mediterrani de la Tecnologia (PMT), Av. Carl Friedrich Gauss 7, Building
B4
08860 Castelldefels, Spain
{adriano.pastore<EMAIL_ADDRESS>
###### Abstract

End-to-end design of communication systems using deep autoencoders (AEs) is
gaining attention due to its flexibility and excellent performance. Besides
single-user transmission, AE-based design is recently explored in multi-user
setup, e.g., for designing constellations for non-orthogonal multiple access
(NOMA). In this paper, we further advance the design of AE-based downlink NOMA
by introducing weighted loss function in the AE training. By changing the
weight coefficients, one can flexibly tune the constellation design to balance
error probability of different users, without relying on explicit information
about their channel quality. Combined with the SICNet decoder, we demonstrate
a significant improvement in achievable levels and flexible control of error
probability of different users using the proposed weighted AE-based framework.

###### Index Terms:

Deep autoencoders, Non-orthogonal multiple access, Successive interference
cancellation

##  I Introduction

Improving the resource usage efficiency of 5G and beyond-5G wireless systems
is essential to accommodate increasing user requirements. Non-orthogonal
multiple access (NOMA) is a recent addition to the wireless signal processing
toolset that promises further advances in achieving improved spectrum usage
under additional benefits of reduced latency, increased throughput, higher
connection density and improved fairness [1, 2]. The main idea behind NOMA is
that multiple users can be served by the same resources if appropriate joint
signal encoding (e.g., using superposition encoding principle) and efficient
signal decoding (e.g., using successive interference cancellation) is applied
[3, 4].

Recent trends see shifting the design of encoding and decoding procedures from
conventional to machine learning (ML)-based methods. The trend is initiated in
the domain of point-to-point communication systems [5, 6], but has since
expanded to multi-user NOMA setup [7, 8, 9, 10]. In this paper, we follow this
trend, and use a deep autoencoder (AE)-based approach to design encoding and
decoding solution for downlink NOMA. Building upon the work in [8], we apply:
1) a weighted loss function to control error probability balance across
different users, 2) SICNet architecture [7] to enhance deep AE-based decoding
capability. Using the proposed weighted AE approach, we are able to obtain
significant improvement and flexibility in the error rate performance, as
evidenced by simulation experiments.

The paper is organised as follows. In Sec. II, we provide background on
generic NOMA system, and recent ML-based methods we use in our work. In Sec.
III, we represent NOMA system as end-to-end AE-based system and introduce our
weighted AE approach combined with SICNet. Performance evaluation and
comparison with the baseline AE-based NOMA is presented in Sec. IV. The paper
is concluded in Sec. V.

##  II Background and System Model

###  II-A Downlink NOMA Transmission

We consider a problem of downlink NOMA transmission of $L$ (different)
messages from a base station (BS) to $L$ users. The BS jointly encodes user
messages into a signal $\mathbf{x}$ and transmits it over $n$ channel uses. We
assume a user message
$\mathbf{s}_{\ell}\in\mathbb{F}_{2}^{k_{\ell}},\ell\in\\{1,2,\ldots,L\\}$ is a
binary (information) sequence of length $k_{\ell}$ bits.

Encoding process for the $\ell^{\textrm{th}}$ user can be described as
$f_{\ell}:\mathbb{F}_{2}^{k_{\ell}}\rightarrow\mathbb{R}^{n}$. In other words,
the $\ell$-th user maps its information sequence $\mathbf{s}_{\ell}$ into a
signal $\mathbf{q}_{\ell}$ obtained as
$\mathbf{q}_{\ell}=f_{\ell}(\mathbf{s}_{\ell}),\mathbf{q}_{\ell}\in\mathbb{R}^{n},\ell\in\\{1,2,\ldots,L\\}$.
Based on the $L$ signals
($\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{\textrm{L}}$), the BS
generates the transmitted message $\mathbf{x}$ of length $n$
($\mathbf{x}\in\mathbb{R}^{n}$), usually by exploiting generic superposition
coding function $F(\cdot)$:

$\displaystyle\mathbf{x}=F(f_{1}(\mathbf{s}_{1}),\ldots,f_{\textrm{L}}(\mathbf{s}_{\textrm{L}}))=F(\mathbf{q}_{1},\ldots,\mathbf{q}_{\textrm{L}}),$
(1)

where in the case of conventional superposition coding the function $F(\cdot)$
represents a weighted linear combination [4]. At the output of the encoder,
$\mathbf{x}$ obeys an average power constraint, for which,
$\frac{1}{M_{c}}\sum_{i=1}^{i=M_{c}}\|\mathbf{x}_{i}\|_{2}^{2}=n$, where
$M_{c}=2^{k_{1}+k_{2}+\ldots+k_{\textrm{L}}}$.

Each of the $L$ users has distinct channel conditions, which transforms input
message $\mathbf{x}\in\mathbb{R}^{n}$ into $L$ output sequences
$\mathbf{y}_{\ell}\in\mathbb{R}^{n},\ell\in\\{1,2,\ldots,L\\}$ following the
probabilistic channel model $p(\mathbf{y}_{\ell}|\mathbf{x})$. In this paper,
we consider a set of independent additive white Gaussian noise (AWGN) channels
between the BS and each of the $L$ users characterised by the set of noise
variances $\\{\sigma^{2}_{\ell}\\}_{\ell=1,\ldots,L}$ [11]. This model is
suitable for downlink NOMA in OFDM-based systems, and it can be extended to a
set of independent block-fading channels [8].

At the receiver side, an estimated information sequence for the
$\ell^{\textrm{th}}$ user is obtained via the function
$g_{\ell}:\mathbb{R}^{n}\rightarrow\hat{\mathbf{s}}_{\ell}$, i.e., the
decoding process for each user can be defined as
$\hat{\mathbf{s}}_{\ell}=g_{\ell}(\mathbf{y}_{\ell}),\ell\in\\{1,2,\ldots,L\\}$.

The goal is to design a combination of encoders
$\\{\\{f_{\ell}\\}_{\ell=1,\ldots,L},F\\}$ and decoders
$\\{g_{\ell}\\}_{\ell=1,\ldots,L}$ that will optimally balance (in a Pareto
optimal sense) the set of message error probabilities
$\\{P_{\textrm{e}_{\ell}}\\}_{\ell=1,\ldots,L}$:

$\displaystyle
P_{\textrm{e}_{\ell}}=\frac{1}{2^{k_{\ell}}}\sum_{\mathbf{s}_{\ell}\in\mathbb{F}_{2}^{k_{\ell}}}\mathbb{P}\\{\hat{\mathbf{s}}_{\ell}\neq\mathbf{s}_{\ell}|\mathbf{s}_{\ell}\\}$
(2)

across the set of $L$ receivers.

###  II-B Power–Domain Downlink NOMA

In the conventional downlink NOMA scheme, the function $F(\cdot)$ defined in
Eq. 1 is a weighted linear combination of user signals obtained by allocating
different power coefficients as the weights of the linear combination [3, 2]:

$\displaystyle\mathbf{x}=F(\mathbf{q}_{1},\ldots,\mathbf{q}_{\textrm{L}})=\sum_{\ell=1}^{\ell=L}\sqrt{p_{\ell}}\mathbf{q}_{\ell},$
(3)

where $p_{\ell}$ represents power associated with user $\ell$. Without loss of
generality, we assume that the power is allocated to users in ascending order
$p_{1}<p_{2}<p_{3}<\ldots<p_{\textrm{L}}$ [1]. From the receiver perspective,
the $\ell^{\textrm{th}}$ user observes:

$\displaystyle\mathbf{y}_{\ell}=h_{\ell}\mathbf{x}+\mathbf{z}=h_{\ell}\sum_{\ell=1}^{\ell=L}\sqrt{p_{\ell}}\mathbf{q}_{\ell}+\mathbf{z}$
(4)

where $h_{\ell}$ represents complex channel coefficient between BS and user
$\ell$, while additive white Gaussian noise (AWGN) with variance
$\sigma^{2}_{\ell}$ is denoted as $\mathbf{z}$. For simplicity and without
loss of generality, we suppose that $h_{\ell}=1,\ell\in\\{1,2,\ldots,L\\}$,
i.e., we focus on the AWGN channel (Section II-A).

A combination of superposition coding at the BS and successive interference
cancellation (SIC) decoding at the receiver is commonly used in downlink NOMA
[1]. SIC exploits different powers allocated to different users at the
transmitter side. The user with the power coefficient $p_{\textrm{L}}$ (the
strongest power coefficient) will be decoded first (treating other users
signals as noise) and subtracted from $\mathbf{x}$. This procedure is
successively repeated for the remaining users until all user messages are
decoded [3, 4]. Note that the SIC performance is dependent on a perfect
knowledge of channel state information (CSI) and power allocation
coefficients, and may degrade if imperfect CSI is used [12].

###  II-C SICNet

In order to overcome the above-mentioned problem of the conventional SIC
receiver, the authors in [7] proposed a deep neural network (DNN)-based
decoding algorithm, entitled SICNet, that incorporates DNNs to recover user
messages. SICNet preserves original SIC decoder structure and algorithmic
flow, however, each block is replaced with a DNN that performs classification
task [7]. Output of each DNN can be used as a soft estimate (by using softmax
activation function) and concatenated with the input of the subsequent DNN. In
such a way, the input to the SICNet of $\ell^{\textrm{th}}$ user is its
received signal $\mathbf{y}_{\ell}$ concatenated with soft estimates of users
$\ell+1$ to $L$. This approach avoids the explicit need for prior CSI and
power allocation knowledge [7].

###  II-D Autoencoder–Based Communication Systems

The conventional point-to-point communication system with a transmitter and a
receiver has been redesigned from the deep learning perspective in the form of
an end-to-end autoencoder (AE) [5], as illustrated in Fig. 1a (blue blocks).

Figure 1: Communication system represented as a deep autoencoder: a)
Conventional system - Blue blocks [5] b) NOMA downlink system with SICNet [7]
\- Red blocks; Black blocks are used by both a) and b)

An input message $m$, which can be represented as a sequence of $k$ bits
$\mathbf{s}=(s_{1},s_{2},\ldots,s_{\textrm{k}})$ (where $k=\log_{2}{M}$), is
encoded as a one-hot vector
$\mathbf{u}=(u_{1},u_{2},\ldots,u_{\textrm{M}})\in\\{0,1\\}^{M}$. The
transmitter, described by $f_{\textrm{AE}}:m\rightarrow\mathbf{x}$, is
represented as a feed-forward neural network with $H$ hidden layers, followed
by a bottleneck layer of width $n$. At the output of transmitter, a batch
normalization layer ensures that the average power constraint on $\mathbf{x}$
are met (Section II-A).

The AWGN channel is constructed as an additive noise layer with output
$\mathbf{y}=\mathbf{x}+\mathbf{z}$ (as it is shown in Fig. 1, blue receiver),
where $\mathbf{z}$ contains $n$ independent and identically distributed
samples of a Gaussian random variable with zero mean and variance
$\sigma^{2}$.

The receiver, described as $g_{\textrm{AE}}:\mathbf{y}\rightarrow\hat{m}$, is
implemented as a feed-forward neural network with $H$ hidden layers and
softmax activation function at its output,
$\mathbf{b}=(b_{1},b_{2},\ldots,b_{\textrm{M}})\in(0,1)^{M}$,
$\|\mathbf{b}\|_{1}=1$. All other hyperparameters are the same as in the AE
transmitter part. The receiver takes the channel output $\mathbf{y}$ as its
input and produces a message estimate $\hat{m}$ as
$\hat{m}=\arg\max_{i}\\{b_{i}\\}$.

Except the output layer of the transmitter and receiver that have linear and
softmax activation function, respectively, all other layers are activated
using rectified linear units (ReLU).

The presented AE is trained in an end-to-end manner to minimize the message
error probability $P_{\textrm{e}}$ (Eq. 2), by using stochastic gradient
descent with Adam optimizer [13]. In other words, $f_{\textrm{AE}}$ and
$g_{\textrm{AE}}$ are jointly optimized. In order to optimize $P_{\textrm{e}}$
(and parameters of the AE), the minimization of categorical–cross entropy
between $\mathbf{u}$ and $\mathbf{b}$ is used as a surrogate for the message
error probability [5]:

$\displaystyle\ell(\mathbf{u},\mathbf{b})=-\sum_{i=1}^{M}u_{i}\log{b_{i}},$
(5)

##  III Autoencoder–Based Downlink (DL) NOMA

In this section, we present a flexible and efficient way to learn encoding and
decoding strategy for NOMA downlink transmission. Conventional NOMA downlink
communication system with SIC decoding, described in Sections II-A and II-B,
can be implemented in a DNN fashion as an extension of the end-to-end AE-based
scheme presented in Section II-D, as illustrated in Fig. 1b (red blocks)[8].

Starting from the superposition coding at the transmitter side, a composition
of the function $F(\cdot)$ and single-user encoding functions
$\\{f_{\ell}\\}_{\ell=1,2,\ldots,L}$ (Eq. 1), is replaced with a single
learning process. More precisely, an encoder is jointly optimized with the set
of $L$ decoders using an end-to-end AE-based approach, i.e., the function
pairs $(f_{\textrm{AE}},\\{g_{AE_{\ell}}\\}_{\ell=1,2,\ldots,L})$ are obtained
using an end-to-end training procedure. Individual users messages
$\mathbf{s}_{\ell}\in\mathbb{F}_{2}^{k_{\ell}},\ell\in\\{1,2,\ldots,L\\},$ are
represented as a message index from a set
$\mathcal{M}_{\ell},|\mathcal{M}_{\ell}|=M_{\ell}=2^{k_{\ell}},$ and one-hot
encoded, where $\mathbf{u}_{\ell}$ denotes one-hot encoding form of the
$\ell^{\textrm{th}}$ user message $\mathbf{s}_{\ell}$, and
$\mathbf{U}=\\{\mathbf{u}_{1},\mathbf{u}_{2},\ldots,\mathbf{u}_{\textrm{L}}\\}$.
AE input message $m$ is defined as a concatenation of individual user messages
in their bit representations, i.e.,
$m=[\mathbf{s}_{1};\mathbf{s}_{2};\ldots;\mathbf{s}_{\textrm{L}}]$ (Fig. 1b)
[8]. One-hot encoding of the AE message $m$ maps
$k_{C}=k_{1}+k_{2}+\ldots+k_{\textrm{L}}$ bits into a one-hot encoding
$\mathbf{u}_{C}=(u_{1},u_{2},\ldots,u_{M_{C}})\in\\{0,1\\}^{M_{C}}$ where
$M_{C}=2^{k_{C}}$. The rest of the transmitter is the same as in Section II-D
(Fig. 1, black blocks).

The transmitted message $\mathbf{x}=f_{\textrm{AE}}(m)$ is passed through $L$
different AWGN channels whose outputs
$\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{\textrm{L}}$ are available
at the respective receivers. Without loss of generality, we suppose that each
of the $L$ channels has different signal-to-noise ratio ($SNR$) and
$SNR_{1}>SNR_{2}>\ldots>SNR_{\textrm{L}}$.

At the receiver side, each of the $L$ users estimates the corresponding
message through the feed-forward neural network, described in Section II-D
(there are in total $L$ DNN receivers, one per each user). In order to
implement SICNet [7] (Section II-C), soft output of each preceding users’ DNN
is connected to the input of the next user (Fig. 1-red block receiver), where
the decoding process starts from the user with the most degraded channel
($\mathbf{y}_{\textrm{L}}$). The softmax output of all $L$ users is collected
into
$\mathbf{B}=\\{\mathbf{b}_{1},\mathbf{b}_{2},\ldots,\mathbf{b}_{\textrm{L}}\\}$,
where each $\mathbf{b}_{\ell}$ is of length
$M_{\ell}=2^{k_{\ell}},\ell\in\\{1,2,\ldots,L\\}$.

The goal is to learn a pair
($f_{\textrm{AE}},\\{g_{AE_{\ell}}\\}_{\ell=1,2,\ldots,L}$) that minimizes the
individual user message error probabilities
$\\{P_{\textrm{e}_{\ell}}\\}_{\ell=1,2,\ldots,L}$ (Eq. 2). As in Section II-D,
minimization of $P_{\textrm{e}_{\ell}}$ is replaced with the loss function
based on the categorical-cross entropy, defined for the $\ell^{\textrm{th}}$
user as:

$\displaystyle\ell_{\ell}(\mathbf{u}_{\ell},\mathbf{b}_{\ell})=-\sum_{i=1}^{M_{\ell}}u_{{\ell_{i}}}\log{b_{{\ell_{i}}}},$
(6)

The total loss function for the AE-based downlink NOMA is the sum of the
individual users’ loss contributions:

$\displaystyle
L(\mathbf{U},\mathbf{B})=\sum_{j=1}^{L}\ell_{j}(\mathbf{u}_{j},\mathbf{b}_{j})$
(7)

###  III-A Weighted Autoencoder-Based Design of Downlink NOMA

In order to train AE-based downlink NOMA communication system described in the
previous section, the SNR difference $\Delta SNR$ between subsequent receivers
is needed as one of the hyperparameters of the training process. As channel
conditions may change during the testing phase, this may lead to a potential
performance degradation.

In our previous work [14], we explored a novel class of autoencoders that
utilise a compound loss function in the context of single-user unequal error
protection (UEP) coding. Such AE method is able to flexibly balance between
error probabilities among the set of messages (message-wise UEP) or between
specific subblocks of different importance classes within a single message
(bit-wise UEP). Inspired with this approach, a similar weighted sum approach
can be used in the AE-based downlink NOMA by introducing weights associated to
$L$ different users (Eq. 7), leading to a weighted total loss function:

$\displaystyle
L(\mathbf{U},\mathbf{B})=\sum_{j=1}^{L}\lambda_{j}\ell_{j}(\mathbf{u}_{j},\mathbf{b}_{j}),$
(8)

where $\lambda_{j}$ represents a weight factor associated to the user $j$,
$\sum_{j=1}^{L}\lambda_{j}=1$, and $\lambda_{j}\geq 0$.

Loss function defined in Eq. 7 is a special case of the weighted sum presented
in Eq. 8, where $\lambda_{\ell}=\frac{1}{L},\ell\in\\{1,2,\ldots,L\\}$. By
incorporating the above mentioned compound loss function (Eq. 8), all users
can be trained assuming they experience the same SNR (thus disregarding
$\Delta SNR$ as a hyperparameter) while adjusting the desirable users’
performance using
$\mathbf{\lambda}=\\{\lambda_{1},\ldots,\lambda_{\textrm{L}}\\}$. Although a
simple alteration of loss function, the proposed approach introduces a single
“knob” (parameter $\lambda$) one can tune to design a family of NOMA
constellations that progressively balance the users’ error probabilities. The
influence of $\lambda$ on the system performance is elaborated in the next
section.

##  IV Performance Evaluation of the Weighted AE–Based Downlink NOMA

###  IV-A Training Procedure

For simplicity, and without loss of generality, the number of users in all
conducted experiments is restricted to two
($\mathbf{\lambda}=\\{\lambda_{1},\lambda_{2}\\}=\\{\lambda,1-\lambda\\}$),
and $k_{1}=k_{2}=k$ (number of messages is $M_{c}=2^{k_{1}+k_{2}}=2^{2k}$).
Apart from the introduction of the suitable loss function (Eq. 7 and 8), the
same training procedure as presented in [5] is preserved, i.e., AE-based NOMA
downlink system is optimized by using stohastic gradient descent with Adam
optimizer[13] (learning rate $\alpha=0.0009$, $\beta_{1}=0.9$,
$\beta_{2}=0.999$). Regarding the number of bits associated to the each user,
for both encoder and each of two decoders, single fully–connected hidden layer
is considered ($H=1$) with $M_{c}$ neurons. Batch size is set to 3000. When
Eq. 7 is used as a loss function, training is performed at $\Delta
SNR=SNR_{1}-SNR_{2}=9$ dB, where $SNR_{1}=15$ dB. With introduction of
$\lambda$ (Eq. 8) both decoders are trained on the same $SNR_{1}=SNR_{2}=6$
dB. During the testing phase, $\Delta SNR$ is preserved at $9$ dB for all
conducted experiments. Training and test data sets contain $10^{5}$ and
$2\times 10^{6}$ messages sampled at random from $\\{1,2,\ldots,M_{c}\\}$,
respectively.

$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$24$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$SNR_{1}$$P_{\textrm{e}}$[8]
\- $P_{\textrm{e}_{1}}$ ($\lambda=0.5$)[8] \- $P_{\textrm{e}_{2}}$
($\lambda=0.5$)[8] \+ SICNet \- $P_{\textrm{e}_{1}}$ ($\lambda=0.5$)[8] \+
SICNet \- $P_{\textrm{e}_{2}}$ ($\lambda=0.5$)[8] \+ SICNet \-
$P_{\textrm{e}_{1}}$ ($\lambda=0.55$)[8] \+ SICNet \- $P_{\textrm{e}_{2}}$
($\lambda=0.55$)[8] \+ SICNet \- $P_{\textrm{e}_{1}}$ ($\lambda=0.6$)[8] \+
SICNet \- $P_{\textrm{e}_{2}}$ ($\lambda=0.6$) Figure 2: Different
architectures $(P_{\mathrm{e_{1}}},P_{\mathrm{e_{2}}})$ versus $SNR$
performance comparison for $\lambda=\\{0.5,0.55,0.6\\}$ in NOMA downlink
transmission with two users ($k_{1}=2$, $k_{2}=2$, $n=2$, $\Delta SNR=9$ dB).

###  IV-B Comparison with the State-of-the-Art

####  IV-B1 Case 1 - $(k,n)=(2,2)$

In order to examine performances of the proposed approach, the two-user
scenario and corresponding architecture form [8] is recreated. More precisely,
BS jointly encodes $k_{1}=k_{2}=k=2$ bits per user ($M_{c}=2^{2k}=16$) and
sends it over $n=2$ channel uses.

In Fig. 2 we compare performances obtained with the proposed approaches and
architecture replicated from [8]. Channel conditions for the second user can
be obtained as $SNR_{2}=SNR_{1}-\Delta SNR$, where $SNR_{1}$ is presented on
the Fig. 2 $x$-axis. Significant performance improvement in terms of error
performance of both users can be observed with the introduction of the SICNet
[7], compared to the baseline results in [8]. Moreover, use of compound loss
function (Eq. 8) by incorporating non-uniform users weights in the loss
function (parameter $\lambda$) provides for a flexible system design, where a
desirable trade-off between the two users’ performance can be easily
controlled through manipulation of $\lambda$ value.

Influence of different $\lambda$ values, introduced during the training
process, on system performances is illustrated in Fig. 3, where we plot
($P_{\textrm{e}_{1}},P_{\textrm{e}_{2}}$) against
$\mathbf{\lambda}=\\{0,0.1,0.2,\ldots,0.9,1\\}$. As the value of $\lambda$
increases, the loss function (Eq. 8) begins to favour the first user. This is
reflected in a graceful improvement of $P_{\textrm{e}_{1}}$ on Fig. 3,
simultaneously with the graceful degradation of $P_{\textrm{e}_{2}}$. By
tuning $\lambda$, the system can be adapted to a different requirements and
variable channel conditions ($\Delta SNR$). The testing phase is done on
$SNR_{1}=12$ dB and $\Delta SNR=9$ dB, i.e., although the second user is
tested on $SNR_{2}=3$ dB and its slope is gentler, it still outperforms the
first user for $\lambda$ values below 0.3 (Fig. 3).

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\lambda$BLER$P_{\textrm{e}_{1}}$-User
1$P_{\textrm{e}_{2}}$-User 2 Figure 3:  $\lambda$ influence on
($P_{\textrm{e}_{1}}$, $P_{\textrm{e}_{2}}$) in two users downlink NOMA
transmission ($k_{1}=2$, $k_{2}=2$, $n=2$, $SNR_{1}=12$ dB, $\Delta SNR=9$
dB).

Learned messages $\mathbf{x}$ are two-dimensional real values
($\mathbf{x}\in\mathbb{R}^{2}$), thus they can be visualized as symbols (or
points) in 2D ”constellation” (Fig. 4). Fig. 4 illustrates the influence of
$\lambda$ on learned constellation. We observe that the distance between
messages associated to user 1 increases with increasing $\lambda$ (loss
function favors the first user), while the opposite happens to the second user
(shape of the constellation transforms from the square ($\lambda=0.5$) to a
rectangle ($\lambda=0.6$)).

$-2$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$2$$-2$$-1$$0$$1$$2$User 2User
1In–phaseQuadrature$\lambda=0.5$$\lambda=0.6$ Figure 4: $\lambda$ influence on
learned constellations for two users NOMA downlink transmission ($k_{1}=2$,
$k_{2}=2$, $n=2$, $M_{c}=2^{k_{1}+k_{2}}=16$).

####  IV-B2 Case 2 - $(k,n)=(4,4)$

Most of the existing works targeting AE-based downlink NOMA restrict their
attention to the case of two-bit user messages ($k=2$) [8, 15]. Exploring how
learned NOMA will behave in higher dimensions, we further expand the number of
bits per user to $k_{1}=k_{2}=k=4$ ($M_{c}=2^{2k}=256$) and the number of
channel uses to $n=4$. Although the number of messages grows exponentially,
the proposed approach significantly outperforms an architecture from [8] (Fig.
5). More precisely, from Fig. 5 we observe that SICNet receiver noticeably
improves the performance of the second user (when $\Delta SNR$ is known),
while the first user performance remains comparable to [8]. On the other hand,
with the introduction of the compound loss function ($\lambda$, Eq. 8), the
proposed approach significantly outperforms [8] in terms of the first user
performance (Fig. 5, green and cyan solid lines), with the price of slight
performance degradation for the second user.

We note that moving to higher constellation dimensions (i.e., from $n=2$ to
$n=4$), although incurs higher complexity ($M_{c}=256$), leads to a
considerable improvement of error performance of both users. It also leads to
faster saturation of the second user performance with the increase of
$\lambda$ (i.e., already for $\lambda>0.55$ on Fig. 5).

##  V Conclusion

In this paper, we presented a flexible and efficient weighted AE-based method
for design of downlink NOMA constellations. The method demonstrates promising
performance under simplicity in training and tuning to the desired error
probability balance between users. In the future work, we plan explore a
combination of outer low-density parity-check codes (LDPC) with inner weighted
AE-based dowlink NOMA constellations.

$10$$10.5$$11$$11.5$$12$$12.5$$13$$13.5$$14$$14.5$$15$$15.5$$16$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$SNR_{1}$$P_{\textrm{e}}$[8]
\- $P_{\textrm{e}_{1}}$ ($\lambda=0.5$)[8] \- $P_{\textrm{e}_{2}}$
($\lambda=0.5$)[8] \+ SICNet \- $P_{\textrm{e}_{1}}$ ($\lambda=0.5$)[8] \+
SICNet \- $P_{\textrm{e}_{2}}$ ($\lambda=0.5$)[8] \+ SICNet \-
$P_{\textrm{e}_{1}}$ ($\lambda=0.55$)[8] \+ SICNet \- $P_{\textrm{e}_{2}}$
($\lambda=0.55$)[8] \+ SICNet \- $P_{\textrm{e}_{1}}$ ($\lambda=0.6$)[8] \+
SICNet \- $P_{\textrm{e}_{2}}$ ($\lambda=0.6$) Figure 5: Different
architectures $(P_{\mathrm{e_{1}}},P_{\mathrm{e_{2}}})$ versus $SNR$
performance comparison for $\lambda=\\{0.5,0.55,0.6\\}$ in NOMA downlink
transmission with two users ($k_{1}=4$, $k_{2}=4$, $n=4$, $\Delta SNR=9$ dB).

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# Gravitationally Mediated Entanglement: Newtonian Field vs. Gravitons

Daine L. Danielson<EMAIL_ADDRESS>Gautam Satishchandran
<EMAIL_ADDRESS>Robert M. Wald<EMAIL_ADDRESS>Enrico Fermi
Institute and Department of Physics, The University of Chicago, 933 East 56th
Street, Chicago, Illinois 60637, USA

###### Abstract

We argue that if the Newtonian gravitational field of a body can mediate
entanglement with another body, then it should also be possible for the body
producing the Newtonian field to entangle directly with on-shell gravitons.
Our arguments are made by revisiting a gedankenexperiment previously analyzed
by Belenchia et al. [1, 2], which showed that a quantum superposition of a
massive body requires both quantized gravitational radiation and local vacuum
fluctuations of the spacetime metric in order to avoid contradictions with
complementarity and causality. We provide a precise and rigorous description
of the entanglement and decoherence effects occurring in this
gedankenexperiment, thereby significantly improving upon the back-of-the-
envelope estimates given by Belenchia et al. and also showing that their
conclusions are valid in much more general circumstances. As a by-product of
our analysis, we show that under the protocols of the gedankenexperiment,
there is no clear distinction between entanglement mediated by the Newtonian
gravitational field of a body and entanglement mediated by on-shell gravitons
emitted by the body. This suggests that Newtonian entanglement implies the
existence of graviton entanglement and supports the view that the experimental
discovery of Newtonian entanglement may be viewed as implying the existence of
the graviton.

##  1 Introduction

General relativity and quantum field theory are the two fundamental pillars of
modern physics. Their union in the form of a theory of quantum gravity remains
the most significant open issue in theoretical physics. Although one can
formulate an essentially satisfactory theory of linearized quantum gravity
perturbed off of some fixed background spacetime, severe difficulties arise in
formulating a nonperturbative theory of quantum gravity. While strong
arguments can be given that gravity should be quantized [3, 4, 5, 6, 7, 8],
these difficulties have led some to suggest that gravity may be fundamentally
classical, that the description of gravity with quantum mechanics requires a
radical modification of quantization [9, 10, 11], or that the question of
quantization is ill-posed [12]. Of central importance to this debate is the
prediction of quantized gravitational radiation in the form of gravitons, the
existence of which has not yet been verified experimentally.

As already noted by Feynman in the 1950’s [13, 14], some key issues regarding
the quantization of gravity can be explored by considering the gravitational
field sourced by a quantum superposition of a massive body. Due to recent
advances in maintaining coherent spatial superpositions,111Spatial
superpositions of masses on the scale of $10^{5}$ amu over distances of order
microns have been achieved [15, 16, 17, 18] and recent proposals have
suggested up to nanogram scale superpositions [19, 20]. many actual
experiments involving such superpositions have recently been proposed [21, 22,
23]. Given the rapid progress toward proposed “low-energy” tabletop
experiments [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39],
it is of interest to understand what such low-energy phenomena might teach us
about the fundamental nature of quantum gravity.

The analysis by Belenchia et al. [1, 2] of a gedankenexperiment originally
proposed by [40] provides strong evidence that low-energy experiments can
probe quantum field theoretic aspects of gravity. In this gedankenexperiment,
an experimenter, Alice, puts a massive body (hereinafter referred to as a
“particle”) into a quantum superposition at different spatial locations. At a
later time, she recombines the particle and determines its quantum coherence.
In the meantime—at a spacelike separation from the recombination portion of
Alice’s experiment—another experimenter, Bob, measures the Newtonian
gravitational field of Alice’s particle to try to determine its position. If
Bob succeeds, then by complementarity, Alice’s particle must be decohered.
But, if Bob influences the state of Alice’s particle, then causality would be
violated. The analysis by Belenchia et al. [1, 2] showed that, in order to
avoid contradictions with complementarity or causality, quantum gravity must
have fundamental features of a quantum field theory at low energies,
specifically the quantization of gravitational radiation (which decoheres
Alice’s particle without the presence of Bob) and local vacuum fluctuations
(which limits Bob’s ability to measure the position of Alice’s particle).

However, the analysis of [1, 2] made only back-of-the-envelope estimates for
the decoherence effects associated with Alice’s recombination and Bob’s
measurement. Furthermore, it considered only a particular type of measurement
that Bob might make. An important purpose of this paper is to reanalyze this
gedankenexperiment, allowing Bob to make any measurement whatsoever in the
region spacelike separated from Alice’s recombination region. We give a
precise analysis of the decoherence associated with radiation emitted by
Alice’s particle and the decoherence associated with Bob’s measurement. We
thereby confirm in a rigorous way the conclusions that had been drawn in [1,
2] from their back-of-the-envelope estimates.

Our analysis sheds additional light on the issue of whether tabletop
experiments probe only quantum properties of the Newtonian gravitational field
[41]. Since Bob sees only the Newtonian gravitational field of Alice’s
superposition during the time of his measurement, it is natural to view this
Newtonian field as mediating entanglement between Bob and Alice. Indeed, if
Alice decides to recombine her body at a much later time, the resulting
correlations between the state of Bob’s measuring apparatus and the state of
Alice’s particle must be viewed as having been mediated by the Newtonian field
of Alice’s particle. However, we will show that if Alice follows her protocol
and recombines her particle in a region spacelike separated from Bob’s
measurements, then it is much more natural to view Bob as having measured on-
shell gravitons that were emitted by Alice’s particle; i.e., although Bob may
believe that he is measuring a Newtonian gravitational field, he is actually
measuring long wavelength gravitons. This viewpoint makes it clear that if the
protocols of the gedankenexperiment are followed, then Bob is merely a
“bystander” and his measurements have no relevance to the decoherence of
Alice’s particle.

Thus, in the circumstances of our gedankenexperiment, there is no clear
distinction between entanglement of Alice’s particle with Bob’s apparatus that
is mediated by a Newtonian field and entanglement of Alice’s particle with
gravitons that then interact with Bob’s apparatus. This suggests that, in more
general circumstances, entanglement mediated by a Newtonian field is not fully
distinguishable from entanglement with gravitons and, hence, that the
experimental discovery of entanglement by a Newtonian field may be viewed as
evidence for existence of the graviton as a fundamental particle of
nature.222Additional arguments for this conclusion have been given in [42].
Furthermore, our analysis provides support for the conclusions of [2] that the
Newtonian field itself can store and transmit quantum information.

In Sec. 2, we review the gedankenexperiment of [40] and its analysis by [1].
In Sec. 3 we analyze the decoherence effects associated with the emission of
quantized radiation by Alice’s particle and the decoherence effects associated
with measurements made by Bob. In Sec. 4 we reanalyze the gedankenexperiment
in a more precise way and provide a proof that no violations of causality or
complementarity occur. Some further remarks and conclusions are given in Sec.
5.

Throughout the paper, we will work in Planck units where $G=c=\hbar=1$.

##  2 The gedankenexperiment of Mari et al. and its resolution by Belenchia et
al.

In this section we review the gedankenexperiment initially proposed by Mari et
al. [40] and its resolution given by Belenchia et al. [1]. There are
electromagnetic and gravitational versions of this gedankenexperiment. For
simplicity and definiteness, we shall first focus on the electromagnetic
version and then discuss the modifications to the analysis needed for the
gravitational case.

Figure 1: The setup for the gedankenexperiment of [40], as analyzed in [1].
Alice’s particle (in blue) is originally in the superposition state Eq. 2.1
with the two wave packets separated by distance $d$. Bob is at a distance
$D\gg d$ from Alice and, at a prearranged time, he releases a particle (in
orange) from a trap and attempts to gain information about which path Alice’s
particle took by determining the strength of the Coulomb/Newtonian field of
Alice’s particle. Meanwhile, at a corresponding prearranged time, Alice
recombines her particle and determines its coherence as described in the text.
Bob does his measurement within time $T_{B}<D$ and Alice recombines her
particle in time $T_{A}<D$, so their actions are performed in spacelike
separated regions.

The gedankenexperiment is illustrated in Fig. 1. At some time in the distant
past, Alice sent a charged particle with spin in the positive $x$ direction
through a Stern-Gerlach apparatus that is oriented in the $z$ direction. We
assume that this process was done sufficiently slowly so as to produce
negligible radiation and that Alice’s particle can be described by ordinary,
nonrelativistic quantum mechanics. After going through the Stern-Gerlach
apparatus, her particle is then in a superposition state of the form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}+\ket{\downarrow;A_{2}}\big{)}$
(2.1)

where $\ket{A_{1}}$ and $\ket{A_{2}}$ describe spatially separated wave
packets and $\ket{\uparrow}$ and $\ket{\downarrow}$ represent eigenstates of
$z$ spin. At a prearranged time, Bob attempts to determine which path Alice’s
particle followed by measuring the Coulomb field of Alice’s particle. One way
that Bob could do this is to release a charged particle from a trap at the
prearranged time; if Alice’s particle takes the right path in Fig. 1, the
Coulomb field near Bob will be stronger and the motion of Bob’s particle will
be influenced more, so by measuring the position of his particle at a later
time, Bob can obtain some “which-path” information about Alice’s particle. At
a corresponding, prearranged time, Alice recombines her particle by putting it
through a “reversing Stern-Gerlach apparatus” [40, 24]. Alice then determines
the coherence of her recombined particle by measuring its spin in the $x$
direction.333In the version of the gedankenexperiment discussed in [1], Alice
determines the coherence of her particle by performing an interference
experiment on the particle wave packets. An alternative resolution of that
version of the gedankenexperiment was proposed in [43], based upon postulating
fundamental limits to the ability to resolve interference fringes as
originally proposed by [44]. This alternative resolution would not be
applicable to the version of the gedankenexperiment being considered here. If
her had particle maintained perfect coherence, it would evolve back to an
eigenstate of spin in the positive $x$ direction. By contrast, if the
components of the original superposition Eq. 2.1 had completely decohered,
Alice would find that the spin is in the positive $x$ direction only $50\%$ of
the time. By repeating the gedankenexperiment as many times as necessary,
Alice can build up good statistics on the $x$ spin and thereby determine the
degree of decoherence of her particle. By the prearranged protocol, the
spacetime region in which Alice does the recombination and spin measurement is
spacelike separated from the region in which Bob does his measurements, as
illustrated in Fig. 1.

This gedankenexperiment appears to lead to a contradiction with
complementarity or causality. If Bob acquires any which-path information from
his measurement, the state of Bob’s particle must be correlated with Alice’s
to some degree. In that case, by complementarity, Alice’s particle cannot be
in a perfectly coherent superposition and she will find her particle to have
spin in the negative $x$ direction some of the time. On the other hand, since
Bob and Alice perform their actions in spacelike separated regions, by
causality, it is impossible for Bob’s measurements to have any effect on
Alice’s results, so the fact that he obtained some which-path information
cannot degrade the coherence of Alice’s particle. So, if Bob’s measurement
does not influence Alice’s spin measurement, we would appear to have a
violation of complementarity, whereas if Bob’s measurement does influence
Alice’s spin measurement, we have a clear violation of causality.

A resolution of this apparent paradox was given in [1]. This resolution is
based upon Bob’s limitations in acquiring which-path information due to vacuum
fluctuations and Alice’s limitations in maintaining coherence due to the
emission of entangling radiation. Bob’s limitations due to vacuum fluctuations
can be estimated as follows. In the electromagnetic case, the difference of
the Coulomb electric fields associated with the different paths of Alice’s
particle is given by

$E\sim\frac{\mathcal{D}_{A}}{D^{3}}$ (2.2)

where $D$ is the distance between Alice and Bob and $\mathcal{D}_{A}=q_{A}d$,
where $q_{A}$ is the charge of Alice’s particle and $d\ll D$ is the distance
between the two paths of Alice’s particle. If Bob must perform his measurement
in time $T_{B}$, the difference in the final position of his particle due to
the difference in the Coulomb fields of Alice’s particle is

$\delta x\sim\frac{q_{B}}{m_{B}}\frac{\mathcal{D}_{A}}{D^{3}}T^{2}_{B}$ (2.3)

where $q_{B}$ is the charge of Bob’s particle and $m_{B}$ is its mass. On the
other hand, vacuum fluctuations of the electromagnetic field produce
fluctuations in the position of Bob’s particle of order

$\Delta x\sim\frac{q_{B}}{m_{B}}.$ (2.4)

Thus, on account of the “noise” due to vacuum fluctuations, Bob can acquire
significant which-path information only if

$\frac{\mathcal{D}_{A}}{D}>\bigg{(}\frac{D}{T_{B}}\bigg{)}^{2}.$ (2.5)

In particular, if Bob abides by his protocol $T_{B}<D$, he can acquire
significant which-path information only when $\mathcal{D}_{A}>D$.

Alice’s limitations on maintaining coherence due to radiation can be estimated
as follows. When Alice recombines her particle over a time $T_{A}$, she
reduces the initial effective dipole $\mathcal{D}_{A}$ to zero. By the Larmor
formula, this should result in emission of entangling radiation corresponding
to an average energy flux $\sim(\mathcal{D}_{A}/T_{A}^{2})^{2}$. Thus the
total energy radiated should be $\sim\mathcal{D}^{2}_{A}/T_{A}^{3}$. This
radiation should be composed of photons of frequency $\sim 1/T_{A}$. Thus the
total number of entangling photons emitted when Alice recombines her particle
should be

$N\sim\frac{\mathcal{D}^{2}_{A}}{T^{2}_{A}}.$ (2.6)

If $N>1$, then Alice’s particle will undergo significant decoherence due to
entanglement with radiation, independent of what Bob does. In particular, if
Alice abides by her protocol $T_{A}<D$, she can maintain coherence only when
$\mathcal{D}_{A}<D$.

The above estimates allow one to provide the resolution given in [1]. If
$\mathcal{D}_{A}>D$, then Bob can acquire significant which-path information,
so by complementarity, Alice’s particle must correspondingly be significantly
decohered. However, in this case the radiation emitted when Alice does her
recombination will decohere her particle independently of what Bob does, so
there is no reason to believe that Bob’s measurement “caused” the decoherence,
i.e., there is no obvious violation of causality. On the other hand, if
$\mathcal{D}_{A}<D$, then Alice should be able to largely maintain the
coherence of her particle during the recombination. But in this case, Bob
cannot acquire significant which-path information, so complementarity does not
imply decoherence of Alice’s particle and, again, there is no obvious
violation of causality.

The analysis of the gravitational version of the gedankenexperiment within the
context of linearized quantum gravity is very similar, with the main
difference being the replacement of “dipole” by “quadrupole.” Alice’s original
separation of the particle into a superposition of two paths does not produce
an effective dipole on account of conservation of center of mass—her
laboratory must produce an equal and opposite compensating mass dipole. Thus,
Eq. 2.3 gets replaced by

$\delta x\sim\frac{\mathcal{Q}_{A}}{D^{4}}T^{2}_{B}$ (2.7)

where ${\mathcal{Q}}_{A}=m_{A}d^{2}$, where $m_{A}$ is the mass of Alice’s
particle. The replacement of Eq. 2.4 is the Planck length which, in our units,
is given by

$\Delta x\sim 1.$ (2.8)

Since Alice now dominantly would emit quadrupole radiation during her
recombination, the replacement of Eq. 2.6 is

$N\sim\frac{\mathcal{Q}^{2}_{A}}{T^{4}_{A}}.$ (2.9)

Suppose that Bob and Alice follow their protocols, so that $T_{B}<D$ and
$T_{A}<D$. Then if $\mathcal{Q}_{A}>D^{2}$, Bob can acquire significant which-
path information but Alice decoheres her particle with gravitational radiation
independent of what Bob does. Conversely, if $\mathcal{Q}_{A}<D^{2}$, then
Alice should be able to largely maintain the coherence of her particle during
the recombination, but Bob cannot acquire significant which-path information.
Thus, as in the electromagnetic case, there is no obvious contradiction with
complementarity or causality.

The above analysis of [1] resolves the apparent paradox posed by the
gedankenexperiment. Interestingly, this analysis shows that both quantized
radiation and vacuum fluctuations are essential for resolving the paradox.
Nevertheless, there are some unsatisfactory aspects of this analysis. In
particular, only back-of-the-envelope estimates of the various effects were
made, so only a rough, order of magnitude relation was obtained between the
decoherence due to radiation during Alice’s recombination and the decoherence
associated with Bob’s measurement. Furthermore, one might consider ways in
which Bob might improve his ability to obtain which-path information. For
example, suppose that Bob, together with $n-1$ assistants, sets up $n$
separate experiments like the one pictured in Fig. 1 to measure the
Coulomb/Newtonian field of Alice’s particle. Suppose that each of these $n$
experiments are done in regions that are spacelike separated from Alice’s
recombination region and spacelike separated from each other. If each of these
experiments could be treated as independent, one would obtain an improvement
of $1/\sqrt{n}$ in Bob’s ability to overcome the noise due to vacuum
fluctuations. Bob would then be able to obtain a corresponding improvement in
his acquisition of which-path information, so if $n$ could be taken to be
sufficiently large, we would again get a contradiction with complementarity or
causality. In fact, vacuum fluctuations over spacelike separated regions are
correlated, so it is not obvious that the $n$ experiments can be treated as
independent. But it also is not obvious that a scheme of this sort would not
work. Thus, while the analysis of [1] is satisfactory for indicating that
there are no obvious contradictions with complementarity or causality, it is
not adequate for conclusively showing that no such contradictions can ever
occur in this type of gedankenexperiment.

As already stated in the Introduction, an important purpose of this paper is
to improve the analysis of [1] by giving much more precise versions of the
above estimates. We will thereby show in a much more rigorous way that no
contradictions with complementarity or causality can occur in this type of
gedankenexperiment. As a very important by-product, we will also obtain
additional insights into how the state of Alice’s particle and the state of
Bob’s apparatus become correlated. Should this correlation be viewed as being
mediated by the Coulomb/Newtonian field of Alice’s particle or by on-shell
photons/gravitons emitted during the recombination process? We will show that
both viewpoints are correct, i.e., they are equivalent descriptions of the
same phenomena. We begin in the next section by giving precise descriptions of
the decoherence due to Alice and the decoherence due to Bob.

##  3 Decoherence due to Alice and Decoherence due to Bob

In this section, we give a more precise characterization of the decoherence of
Alice’s particle due to radiation emitted when she recombines her particle and
the decoherence associated with Bob’s measurements. These characterizations
will be used in the next section to reanalyze the gedankenexperiment. In this
section we will explicitly discuss the electromagnetic version of the
gedankenexperiment, since the language and concepts are more familiar in this
context. However, exactly the same discussion applies to the gravitational
case, with appropriate substitutions of “graviton” for “photon,” “Newtonian”
for “Coulomb,” etc.

###  1 Decoherence Due to Alice

We first consider the decoherence of Alice’s particle that would occur in the
absence of Bob or any other external influence.

Previously, we stated that after Alice sends her particle through a Stern-
Gerlach apparatus at an early time, the particle is in the superposition state
Eq. 2.1. However, this expression ignores the electromagnetic field, which is
in a different state depending upon the state of Alice’s particle.
Heuristically, the state of the total system should be of the form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}\otimes\ket{\psi_{1}}+\ket{\downarrow;A_{2}}\otimes\ket{\psi_{2}}\big{)}$
(3.1)

where states $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ formally correspond to
coherent states of the Coulomb field of Alice’s particle in states
$\ket{\uparrow;A_{1}}$ and $\ket{\downarrow;A_{2}}$ respectively. However,
this is only a formal expression because the “Coulomb states”
$\ket{\psi_{1}}$, $\ket{\psi_{2}}$ are not well defined—we would need to
define the state space of the full interacting quantum field theory to define
them. Nevertheless, formally, one could argue that these formal Coulomb states
should be orthogonal and that therefore Alice’s particle is already decohered
at the earliest time depicted in Fig. 1. However this decoherence is a “false
decoherence” in the sense of [45]. If Alice recombines her particle slowly
enough and if there are no external influences, she will be able to fully
restore the coherence of her particle.

As Alice recombines her particle and moves its components along noninertial
paths, formally the total state should continue to be of the form Eq. 3.1.
However, while the recombination process is occurring, there is no way to
meaningfully separate $\ket{\psi_{1}}$ or $\ket{\psi_{2}}$ into a “Coulomb
part” (which is not an independent degree of freedom and should cause only a
false decoherence of Alice’s particle) and a “radiation part” (which is a
state of the free electromagnetic field that should be responsible for a true
decoherence). Since we do not have a well-defined inner product between
$\ket{\psi_{1}}$ and $\ket{\psi_{2}}$, we cannot, in general, meaningfully say
how much true decoherence has occurred at any finite time during this process.

However, the situation improves considerably if we go to asymptotically late
times. At asymptotically late times, the electromagnetic field naturally
decomposes into a radiation field that propagates to null infinity and a
Coulomb field that follows Alice’s particle to timelike infinity. The
asymptotic Coulomb field is completely determined by the asymptotic state of
Alice’s particle and does not represent an independent degree of freedom (see
e.g. [46]). Thus, at asymptotically late times, the state of the total system
is of the form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}_{i^{+}}\otimes\ket{\Psi_{1}}_{\mathscr{I}^{+}}+\ket{\downarrow;A_{2}}_{i^{+}}\otimes\ket{\Psi_{2}}_{\mathscr{I}^{+}}\big{)}.$
(3.2)

Here $\ket{\uparrow;A_{1}}_{i^{+}}$ and $\ket{\downarrow;A_{2}}_{i^{+}}$
represent the asymptotically late time states of the components of Alice’s
recombined particle and $\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and
$\ket{\Psi_{2}}_{\mathscr{I}^{+}}$ represent the states of the radiation field
at null infinity that would arise if, over all time, the states of Alice’s
particle were $\ket{\uparrow;A_{1}(t)}$ and $\ket{\downarrow;A_{2}(t)}$,
respectively. Note that after recombination, the spatial wave packets
describing the “$1$” and “$2$” states coincide, so, in particular, we have
$\ket{A_{1}}_{i^{+}}=\ket{A_{2}}_{i^{+}}$, but we keep the $1$ and $2$
subscripts for notational clarity.

It is very important to recognize that—unlike Eq. 3.1—Eq. 3.2 is not merely a
formal expression. The states $\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and
$\ket{\Psi_{2}}_{\mathscr{I}^{+}}$ are well-defined Fock space states of the
“out” Hilbert space of the electromagnetic field and have a well-defined
description in terms of photons.444In a general scattering process, there will
be a nontrivial electromagnetic “memory effect,” resulting in infrared
divergences in the description of the quantum state (see e.g.
[Satishchandran_2021, 47, 48]). In that case, the electromagnetic “out” state
cannot be expressed as a state in the standard Fock space and cannot be given
a proper description in terms of photons. However, such infrared divergences
do not occur in cases where the charges are not relatively boosted at
asymptotically early and late times as we consider here, so such infrared
issues play no role in the analysis of this gedankenexperiment. Similar
divergences which arise due to the gravitational memory effect also play no
role in the (linearized) gravitational version of the gedankenexperiment. The
failure of $\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and
$\ket{\Psi_{2}}_{\mathscr{I}^{+}}$ to coincide implies a decoherence of
Alice’s particle. The degree of decoherence of the asymptotic state of Alice’s
particle is given by

${\mathscr{D}}_{\text{Alice}}=1-\left\lvert\braket{\Psi_{1}}{\Psi_{2}}_{\mathscr{I}^{+}}\right\rvert$
(3.3)

where $\braket{\Psi_{1}}{\Psi_{2}}_{\mathscr{I}^{+}}$ denotes the inner
product of the states $\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and
$\ket{\Psi_{2}}_{\mathscr{I}^{+}}$ on $\mathscr{I}^{+}$. This equation is a
precise and general version of the decoherence estimate given in Sec. 2 based
on the number of “entangling photons” that are emitted. If
$\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and $\ket{\Psi_{2}}_{\mathscr{I}^{+}}$
differ by more than one photon, they should be nearly orthogonal, and the
decoherence will be nearly complete.

Figure 2:  Alice recombines her particle at event $P$ and subsequently keeps
her recombined particle in inertial motion. $\Sigma$ is an arbitrary Cauchy
surface passing through $P$.

We are interested in the case depicted in Fig. 2 where Alice recombines her
particle as in the gedankenexperiment—but without the presence of Bob—and
after recombination, she keeps her combined particle in inertial motion at all
future times. Then, to the causal future of the recombination event $P$, the
electromagnetic field will correspond to the Coulomb field of the recombined
particle. Let $\Sigma$ be an arbitrary Cauchy surface passing through $P$.
Extend the Coulomb field of the recombined particle to the entire region to
the future, $I^{+}(\Sigma)$, of $\Sigma$ (i.e., not just the causal future of
$P$). Subtract this Coulomb field from the electromagnetic field in this
region. The electromagnetic field associated with $\ket{\uparrow;A_{1}}$ with
the final Coulomb field subtracted will thus correspond to a well-defined
state $\ket{\Psi_{1}}_{\Sigma}$ of the source-free electromagnetic field on
$\Sigma$. Similarly, the electromagnetic field associated with
$\ket{\downarrow;A_{2}}$ with the final Coulomb field subtracted will
correspond to a well-defined state $\ket{\Psi_{2}}_{\Sigma}$ on $\Sigma$. At
“time” $\Sigma$, the joint state of Alice’s particle and the electromagnetic
field is described by

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}_{\Sigma}\otimes\ket{\Psi_{1}}_{\Sigma}+\ket{\downarrow;A_{2}}_{\Sigma}\otimes\ket{\Psi_{2}}_{\Sigma}\big{)}.$
(3.4)

In contrast to Eq. 3.1, this is a completely meaningful expression;
$\ket{\Psi_{1}}_{\Sigma}$ and $\ket{\Psi_{2}}_{\Sigma}$ are well-defined
states of the source-free electromagnetic field. Under time evolution,
$\ket{\Psi_{1}}_{\Sigma}$ and $\ket{\Psi_{2}}_{\Sigma}$ evolve to
$\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and $\ket{\Psi_{2}}_{\mathscr{I}^{+}}$,
respectively. Since time evolution is unitary, we may express the decoherence
Eq. 3.3 of Alice’s particle as

${\mathscr{D}}_{\text{Alice}}=1-|\braket{\Psi_{1}}{\Psi_{2}}_{\Sigma}|.$ (3.5)

This is our desired expression for the decoherence due to Alice. It is clear
that if there are no time constraints on Alice’s recombination, then by doing
the recombination adiabatically—so that negligible radiation is emitted to
infinity—she can make the decoherence arbitrarily small.

###  2 Decoherence Due to Bob

We now consider the decoherence that would occur if Bob makes a measurement
that obtains some which-path information about Alice’s particle. We assume
that Alice recombines her particle adiabatically in the distant future—after
Bob has completed his measurements—in such a way that had Bob not been
present, no decoherence would have occurred. Thus, any decoherence in this
situation can be attributed to Bob. This situation corresponds to experimental
proposals such as [24].

Since Bob is now part of the system, heuristically, the state of the total
system after Alice has put her particle through the initial Stern-Gerlach
apparatus but before Bob has begun his measurements is now

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}\otimes\ket{\psi_{1}}+\ket{\downarrow;A_{2}}\otimes\ket{\psi_{2}}\big{)}\otimes\ket{B_{0}}$
(3.6)

where $\ket{B_{0}}$ is the initial state of Bob’s apparatus and again
$\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ are the formal Coulomb states of Alice’s
particle. We wish to consider a situation wherein Bob turns on his apparatus
for a time $T_{B}$ and makes a measurement of the Coulomb field of Alice’s
particle in order to try to obtain which-path information. We assume that Bob
carries out his measurement in such a way that he emits negligible radiation
to infinity. For example if Bob measures the motion of a charged particle
released from a trap as described in the previous section, the sensitivity of
his experiment will depend on $q_{B}/m_{B}$ but the emitted radiation will
vary as $q^{2}_{B}$, so by taking $q_{B}$ and $m_{B}$ sufficiently small, he
should be able to carry out his measurements with negligible emitted
radiation.555The assumption that Bob emits negligible radiation is being made
so as to make our discussion simpler and cleaner, but it is not essential for
the analysis. We allow Bob to make any field measurement whatsoever, i.e., we
do not restrict him to measuring the trajectory of a particle released from a
trap. For the analysis of this subsection, we do not place any limits on
$T_{B}$, i.e., we do not require $T_{B}<D$.

Since no radiation is emitted by Bob or Alice, at asymptotically late times,
the state of the electromagnetic field at null infinity will be
$\ket{0}_{\mathscr{I}^{+}}$ for either state of Alice’s superposition. Thus,
the final state of the electromagnetic field plays no role in entanglement and
we need only be concerned with the Alice-Bob system. The final state of the
Alice-Bob system will be of the form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}_{i^{+}}\otimes\ket{B_{1}}_{i^{+}}+\ket{\downarrow;A_{2}}_{i^{+}}\otimes\ket{B_{2}}_{i^{+}}\big{)}$
(3.7)

where $\ket{B_{1}}_{i^{+}}$ and $\ket{B_{2}}_{i^{+}}$ are the final states of
Bob’s apparatus for Alice’s states $\ket{\uparrow;A_{1}}$ and
$\ket{\downarrow;A_{2}}$, respectively. The failure of $\ket{B_{1}}_{i^{+}}$
and $\ket{B_{2}}_{i^{+}}$ to coincide corresponds to Bob having acquired
which-path information about Alice’s particle. The corresponding decoherence
of Alice’s particle is

${\mathscr{D}}_{\text{Bob}}=1-|\braket{B_{1}}{B_{2}}_{i^{+}}|.$ (3.8)

However, since Bob stops interacting at time $T_{B}$, we can equivalently
calculate the inner product at time $T_{B}$

${\mathscr{D}}_{\text{Bob}}=1-|\braket{B_{1}}{B_{2}}_{T_{B}}|.$ (3.9)

This gives the decoherence associated with Bob’s measurement. In the
circumstance considered here where Alice emits no radiation, it is clear that
this decoherence can be viewed as being caused by Bob. It also is clear that
in this circumstance, the decoherence should be viewed as being mediated by
the Coulomb field of Alice’s particle.

Equation 3.9 is a precise and general version of the decoherence estimate
given in Sec. 2 based upon Bob’s ability to get which-path information. The
amount of which-path information Bob can obtain is determined by the extent to
which Bob can design a measurement so that $\ket{B_{1}}_{T_{B}}$ is nearly
orthogonal to $\ket{B_{2}}_{T_{B}}$. The degree to which $\ket{B_{1}}_{T_{B}}$
is orthogonal to $\ket{B_{2}}_{T_{B}}$ determines how much decoherence of
Alice’s particle must occur.

##  4 Reanalysis of the gedankenexperiment

We now are in a position to reanalyze the gedankenexperiment of Sec. 2. We
will again explicitly consider the electromagnetic version of the
gedankenexperiment, but the exactly same discussion applies to the
gravitational case with the appropriate word substitutions. The spacetime
diagram of the gedankenexperiment is redrawn in Fig. 3 in order to show three
Cauchy surfaces, $\Sigma_{1}$, $\Sigma_{2}$, and $\Sigma_{3}$, that will play
an important role in our reanalysis.

Figure 3:  A spacetime diagram of the gedankenexperiment of Fig. 1 showing the
three Cauchy surfaces, $\Sigma_{1}$, $\Sigma_{2}$, and $\Sigma_{3}$. The
Cauchy surface $\Sigma_{1}$ passes through Alice’s region after recombination
but is such that the region in which Bob performs his measurements (shaded in
gray) lies to the future of $\Sigma_{1}$. (We have depicted Bob as releasing a
particle from a trap, but Bob is allowed to perform any measurement whatsoever
in the gray region.) The Cauchy surface $\Sigma_{2}$ is such that it passes
through Alice’s region before she starts the recombination process but is such
that Bob’s measurement lies to the past of $\Sigma_{2}$. The Cauchy surface
$\Sigma_{3}$ passes through Alice’s region after recombination and is such
that Bob’s measurement lies to the past of $\Sigma_{3}$.

We reanalyze the decoherence of Alice’s particle using the results of the
previous section as follows. First, consider the portion of the spacetime of
Fig. 3 that lies to the past of Cauchy surface $\Sigma_{1}$. At the time
represented by $\Sigma_{1}$, Alice has completed her recombination but Bob has
not yet begun performing his measurements. The portion of the spacetime lying
to the past of $\Sigma_{1}$ is identical to the portion of the spacetime of
Fig. 2 lying to the past of a corresponding Cauchy surface $\Sigma$. Thus, we
may apply the results of Sec. 3.1 to conclude that the decoherence of Alice’s
particle is given by

${\mathscr{D}}_{\text{Alice}}=1-\left\lvert\braket{\Psi_{1}}{\Psi_{2}}_{\Sigma_{1}}\right\rvert,$
(4.1)

where $\ket{\Psi_{1}}_{\Sigma_{1}}$ and $\ket{\Psi_{2}}_{\Sigma_{1}}$ are the
radiation states on $\Sigma_{1}$ obtained by subtracting the common Coulomb
field from the states of the electromagnetic field corresponding to Alice’s
particle being in states $\ket{\uparrow;A_{1}}$ and $\ket{\downarrow;A_{2}}$,
respectively. Since Alice’s recombination is complete at time $\Sigma_{1}$,
Eq. 4.1 should yield the exact expression for the decoherence of Alice’s
particle.

However, we also can analyze the decoherence of Alice’s particle by
considering the portion of the spacetime that lies to the past of the Cauchy
surface $\Sigma_{2}$. At time $\Sigma_{2}$, Alice has not yet started her
recombination, but Bob has completed his measurements. Thus, the situation
here is identical to the setup considered in Sec. 3.2. Hence, we may apply the
results of Sec. 3.2 to conclude that a decoherence of Alice’s particle given
by

${\mathscr{D}}_{\text{Bob}}=1-|\braket{B_{1}}{B_{2}}|$ (4.2)

must occur as a result of Bob’s measurements, where $\ket{B_{1}}$ and
$\ket{B_{2}}$ represent the states of Bob’s apparatus after completion of his
measurement. It is possible that more decoherence of Alice’s particle could
occur as Alice performs her recombination. However, since Bob has completed
his measurement and stops interacting after time $\Sigma_{2}$, it is
impossible for the decoherence of Alice’s particle to be less than this.

It follows that there would be a paradox if it were possible for Bob to do a
measurement in such a way that

$|\braket{B_{1}}{B_{2}}|<\left\lvert\braket{\Psi_{1}}{\Psi_{2}}_{\Sigma_{1}}\right\rvert,$
(4.3)

i.e., such that the decoherence associated with Bob’s measurement is greater
than the decoherence due to Alice. If Eq. 4.3 held, then Bob’s measurement
either would result in a violation of causality [if it induced an additional
decoherence of Alice’s particle beyond that given by Eq. 4.1], or it would
result in a violation of complementarity (if it did not induce such an
additional decoherence). Eq. 4.3 is a precise statement of the potential
paradox posed by the gedankenexperiment of Sec. 2.

However, it is now easy to see that no such paradox can ever arise. At time
$\Sigma_{1}$, the state of the joint Alice-field-Bob system is described by

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}\otimes\ket{\Psi_{1}}_{\Sigma_{1}}+\ket{\downarrow;A_{2}}\otimes\ket{\Psi_{2}}_{\Sigma_{1}}\big{)}\otimes\ket{B_{0}}$
(4.4)

where $\ket{\Psi_{1}}_{\Sigma_{1}}$ and $\ket{\Psi_{2}}_{\Sigma_{1}}$ are the
radiation states on $\Sigma_{1}$ (with the common Coulomb field subtracted),
and $\ket{B_{0}}$ is the initial state of Bob’s detector. We now consider the
evolution of this state to the Cauchy surface $\Sigma_{3}$. There is no
evolution of Alice’s state, since $\Sigma_{3}$ is the same time as
$\Sigma_{1}$ as far as Alice’s state is concerned. However, the radiation
interacts with Bob’s measuring apparatus. In the case where Alice’s state is
$\ket{\uparrow;A_{1}}$, Bob’s state evolves to $\ket{B_{1}}$, whereas if
Alice’s state is $\ket{\downarrow;A_{2}}$, Bob’s state evolves to
$\ket{B_{2}}$. It follows that the state Eq. 4.4 on $\Sigma_{1}$ must evolve
to the state on $\Sigma_{3}$ described by

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;A_{1}}\otimes\ket{\Psi^{\prime}_{1}}_{\Sigma_{3}}\otimes\ket{B_{1}}+\ket{\downarrow;A_{2}}\otimes\ket{\Psi^{\prime}_{2}}_{\Sigma_{3}}\otimes\ket{B_{2}}\big{)}.$
(4.5)

Here $\ket{\Psi^{\prime}_{1}}_{\Sigma_{3}}$ and
$\ket{\Psi^{\prime}_{2}}_{\Sigma_{3}}$ are the radiation states that arise
from $\ket{\Psi_{1}}_{\Sigma_{1}}$ and $\ket{\Psi_{2}}_{\Sigma_{1}}$,
respectively, after interaction with Bob. The states
$\ket{\Psi^{\prime}_{1}}_{\Sigma_{3}}$ and
$\ket{\Psi^{\prime}_{2}}_{\Sigma_{3}}$ depend on the interaction with Bob, so
they cannot be calculated without knowing exactly what Bob is measuring.
However, no matter what Bob does, the joint evolution from $\Sigma_{1}$ to
$\Sigma_{3}$ must be unitary. It follows that the norms of states are
preserved and that

$\displaystyle\braket{\Psi^{\prime}_{1}}{\Psi^{\prime}_{2}}_{\Sigma_{3}}\braket{B_{1}}{B_{2}}$
$\displaystyle=$
$\displaystyle\braket{\Psi_{1}}{\Psi_{2}}_{\Sigma_{1}}\braket{B_{0}}{B_{0}}$
(4.6) $\displaystyle=$
$\displaystyle\braket{\Psi_{1}}{\Psi_{2}}_{\Sigma_{1}}.$

It then follows immediately that

$|\braket{B_{1}}{B_{2}}|\geq\left\lvert\braket{\Psi_{1}}{\Psi_{2}}_{\Sigma_{1}}\right\rvert$
(4.7)

so the inequality Eq. 4.3 can never be satisfied. This is precisely what we
wished to show.

Although the above argument completes our proof that no contradiction with
causality or complementarity can ever arise in this gedankenexperiment—no
matter what Bob chooses to measure—it remains to give a more intuitive
explanation of our new resolution of the gedankenexperiment and connect it
with the discussion of Sec. 2.

The main new ingredient that we have added to the analysis is that we may view
Bob as measuring aspects of the radiation emitted by Alice’s particle. It may
seem strange to talk about “emitted radiation” that is present in a region
that is spacelike separated from the region where the emission is taking
place. Indeed, this may, by itself, appear to be a violation of causality!
However, this kind of phenomenon is a basic feature of quantum field theory,
with no violation of causality involved. The mode function of a particle in
quantum field theory is a positive frequency solution and cannot be sharply
localized. If a photon is emitted by a source in some localized region
$\mathcal{O}$, there always will be some amplitude for the photon to be
present in a region spacelike separated from $\mathcal{O}$. Indeed, as
discussed in detail in [49], there are cases where the emitted photon is
mostly localized in a spacelike separated region. This does not lead to a
violation of causality because an observer in the spacelike separated region
will not be able to tell whether she is observing a photon or a vacuum
fluctuation—she can tell the difference between these possibilities only when
she enters the causal future of $\mathcal{O}$. In the present case, the
electromagnetic field in Bob’s region can be viewed either as corresponding to
the superposition of the Coulomb fields of Alice’s particle with no
radiation—as would be natural to do if we view Bob’s region as lying to the
past of time $\Sigma_{2}$—or as the single Coulomb field of Alice’s combined
particle together with free radiation—as would be natural to do if we view
Bob’s region as lying to the future of time $\Sigma_{1}$. These viewpoints are
indistinguishable in Bob’s region.

The radiation viewpoint allows us to understand why Bob cannot produce any
additional decoherence beyond what Alice produces during her recombination.
Bob can obtain which-path information only by measuring (i.e., scattering
and/or absorbing) the entangling photons that “previously” were emitted by
Alice. Therefore, the state of his apparatus cannot become more correlated
with Alice’s particle than the radiation emitted by Alice, as we have proven
above in Eq. 4.7.

Note that, as we have just argued, in the gedankenexperiment, Bob is merely an
“innocent bystander” with regard to the decoherence of Alice’s particle, since
he is merely measuring the entangling radiation emitted by the particle that
was the true cause of the decoherence. However, suppose that Alice does not
follow the protocol assigned to her in the gedankenexperiment and instead
recombines her particle very slowly at a later time, so as not to produce any
radiation. Then, despite her attempts to keep perfect coherence, she will find
that her particle has decohered by the amount Eq. 4.2. In this case, Bob’s
measurement is the true cause of her particle’s decoherence [2].
Interestingly, when Bob performs his measurements, he has no way of knowing
whether he will turn out to be an “innocent bystander” or the cause of
decoherence of Alice’s particle.

Finally, we note that the analysis of the gedankenexperiment summarized in
Sec. 2 was based upon the limitations on Alice’s ability to maintain coherence
due to radiation and the limitations on Bob’s ability to get which-path
information due to vacuum fluctuations. The reanalysis of the
gedankenexperiment given above gave a more precise version of Alice’s
limitations on maintaining coherence due to radiation. However, we did not
mention “vacuum fluctuations” in the discussion of the decoherence associated
with Bob’s measurements, so it might appear that the reanalysis differs in
this respect. However, this is not the case: The radiation fields
$\ket{\Psi_{1}}_{\Sigma_{1}}$ and $\ket{\Psi_{2}}_{\Sigma_{1}}$ have different
expected values of the electromagnetic field. Their failure to be orthogonal
can be viewed as a manifestation of the same type of fluctuations in these
states as occurs in the vacuum state; if these states did not have such
fluctuations, they would be fully distinguishable and hence orthogonal. But,
as is evident from 4.7, it is the failure of $\ket{\Psi_{1}}_{\Sigma_{1}}$ and
$\ket{\Psi_{2}}_{\Sigma_{1}}$ to be orthogonal that limits Bob’s ability to
make $\ket{B_{1}}$ and $\ket{B_{2}}$ orthogonal. Thus, there is a direct
connection between vacuum fluctuations and the limitations on Bob’s ability to
obtain which-path information.

##  5 Summary and Conclusions

In this paper, we have reanalyzed the gedankenexperiment discussed in [1]. Our
reanalysis validates the arguments that had been made in [1] using only back-
of-the-envelope estimates, and it shows in a much more precise way—and under
completely general assumptions about the measurements that Bob makes—that no
violations of causality or complementarity can occur.

Perhaps the most interesting aspect of our reanalysis is the equivalence of
two viewpoints on how the state of Bob’s measuring apparatus becomes
correlated with the state of Alice’s particle. In the gravitational version of
the gedankenexperiment, one can say either that (i) Alice’s particle became
entangled with on-shell gravitons emitted during the recombination process and
Bob’s apparatus then interacted with these gravitons—thereby transferring some
of the entanglement present in these gravitons to his apparatus—or that (ii)
the Newtonian gravitational field of Alice’s particle mediated an entanglement
of Bob’s apparatus with Alice’s particle. If Alice follows her protocol but
Bob fails to make any measurement, then it is essential to take viewpoint (i)
to understand why Bob’s inaction has no effect whatsoever on the decoherence
of Alice’s particle. Conversely, if Bob follows his protocol but Alice
recombines her particle adiabatically at a later time, one must take viewpoint
(ii) to understand how Bob’s measuring apparatus becomes correlated with the
Alice’s particle [2]. But if Alice and Bob each follow the protocols of the
gedankenexperiment, then both (i) and (ii) provide a valid description of the
process that occurs.

Indeed, it is essential that both (i) and (ii)—or, alternatively, neither (i)
nor (ii)—be valid descriptions of the process. To see this, suppose that (i)
fails, i.e., Alice’s particle does not emit entangling gravitons, but suppose
that (ii) holds, i.e., Bob’s apparatus is able to entangle with Alice’s
particle via its Newtonian gravitational field. Then Alice’s particle would
not decohere in the absence of Bob. It follows that if it decohered in the
presence of Bob we would have a violation of causality, whereas if it did not
decohere in the presence of Bob we would have a violation of complementarity.
Thus, it is not consistent for (i) to fail but (ii) to hold. Conversely,
suppose (i) holds, i.e., Alice’s particle emits quantized entangling
gravitational radiation, but suppose that (ii) fails, i.e., Bob’s apparatus is
unable to entangle with Alice’s particle via its Newtonian gravitational
field. Then, since, as we have seen, under the protocol of the
gedankenexperiment, the difference of the Newtonian fields of Alice’s particle
can be equivalently viewed as quantized radiation emitted by Alice’s particle,
this would imply that Bob is unable to interact with quantized gravitational
radiation in any way that results in entanglement. This would not make sense
in any theory where quantized gravitational radiation can be produced.666It
has been argued that it may be impossible, in principle, to measure the energy
of a single graviton [12]. Bob is not required here to resolve an individual
graviton but merely to become entangled, at least to some degree, with
gravitons.

These considerations show that there is a direct relationship between
Newtonian entanglement and the existence of gravitons. Our argument for such a
relationship is strictly valid only within the protocol of the
gedankenexperiment, where the measurement of the Newtonian field/gravitons is
carried out within a time span no longer than the light travel time to the
source. Nevertheless, these considerations yield strong support for the view
that any observation of entanglement mediated by a Newtonian field provides
evidence for the existence of the graviton.

###### Acknowledgements.

D.L.D. acknowledges support as a Fannie and John Hertz Foundation Fellow
holding the Barbara Ann Canavan Fellowship and as an Eckhardt Graduate Scholar
in the Physical Sciences Division at the University of Chicago. This research
was supported in part by NSF Grant No. 21-05878 to the University of Chicago.

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