# Entanglement of annihilation photons

Alexander Ivashkin  Institute for Nuclear Research RAS, Moscow, 117312, Russia
<EMAIL_ADDRESS>Dzhonrid Abdurashitov  Institute for Nuclear Research RAS,
Moscow, 117312, Russia  Alexander Baranov  Institute for Nuclear Research RAS,
Moscow, 117312, Russia  National Research Nuclear University MEPhI, Moscow,
115409, Russia  Fedor Guber  Institute for Nuclear Research RAS, Moscow,
117312, Russia  Sergey Morozov  Institute for Nuclear Research RAS, Moscow,
117312, Russia  Sultan Musin  Institute for Nuclear Research RAS, Moscow,
117312, Russia  Moscow Institute of Physics and Technology, Moscow, 141701,
Russia  Alexander Strizhak  Institute for Nuclear Research RAS, Moscow,
117312, Russia  Moscow Institute of Physics and Technology, Moscow, 141701,
Russia  Igor Tkachev  Institute for Nuclear Research RAS, Moscow, 117312,
Russia

###### Abstract

We present a new experimental study of the quantum entanglement of photon
pairs produced in positron-electron annihilation at rest. Each annihilation
photon has an energy that is five orders of magnitude higher than the energy
of photons in optical experiments, which provides a unique opportunity for
controlled formation of decoherent states and successive measurements of the
same photon.The experimental setup includes a system of Compton polarimeters
to measure the Compton scattering of annihilation photons in entangled and
decoherent states. Decoherent states are prepared by pre-scattering of one of
the initial photons prior to measurements in polarimeters. For the first time,
a direct comparison of the polarization correlations of annihilation photons
in the entangled and thus prepared separable states has been carried out. The
angular distributions of scattered photons turned out to be the same in both
quantum states, which is an unexpected discovery for the quantum-entangled
positron emission tomography. Moreover, the correlation function in the Bell’s
inequality is also the same for entangled and separable states. It follows
that, despite numerous measurements in a series of experiments, there is still
no experimental proof of the entanglement of annihilation photons. These
results are in line with recent theoretical predictions of an identical
Compton scattering cross-section for entangled and specific mixed separable
quantum states and cast doubt on the universality of the Bell’s theorem for
testing the entanglement and nonlocality in quantum theory.

## Introduction

The entanglement of a quantum system is a simple consequence of the
superposition principle and means that the state of the system cannot be
represented as a product of the states of individual subsystems. Historically,
however, this term was introduced by Schrödinger, referring to the
entanglement of our knowledge of quantum systems [1, 2]. Such an
interpretation of entanglement is quite relevant for the current situation
with a system of two photons formed by positron-electron annihilation at rest.
The description of this system has been the subject of pioneering work on
quantum entanglement in the past century and, as we will see below, is still
unclear in some respects.

The study of this two-photon system has a long and remarkable history, which
can be divided into several stages. Initially, the idea of measuring a pair of
annihilation photons was proposed in 1946 by Wheeler [3]. He considered at
that time a hypothetical bound system of an electron and a positron with an
orbital moment equal to one or zero. In the latter case, due to the
conservation of angular momentum and parity, the photons of the pair formed
during positron-electron annihilation have mutually perpendicular
polarization. In the same article, in order to test the predicted correlation
of photon polarization, J. Wheeler proposed an experimental scheme with two
Compton scatterers and detectors of scattered photons, which has already
become a classic. Since photons are scattered predominantly perpendicular to
the polarization plane, the dependence of the number of registered photons on
the angle between the scattering planes should have a maximum or minimum at
$90^{\circ}$ or $0^{\circ}$, respectively.

Almost simultaneously, two theoretical papers by Pryce and Ward [4] and by
Snyder et al. [5] predicted the behavior of the angular distribution of
scattered annihilation gamma rays. These distributions were obtained on the
basis of the Klein-Nishina formula [6] with the following initial state

${\Psi}=\frac{1}{\sqrt{2}}(\ket{{H_{1}}{V_{2}}}+\ket{{V_{1}}{H_{2}}}),$ (1)

where H(V) represent the horizontal (vertical) linear polarization of the
first or second photon. In this state, photons do not have a definite
polarization, albeit their polarizations are mutually orthogonal. It was shown
that the ratio $R={N_{\perp}}/{N_{\parallel}}$ of the number of counts for
perpendicular and parallel orientations of scattering planes reaches a maximum
of $R=2.85$ in the case when photons are scattered at an angle of $82^{0}$ to
their initial momenta. These predictions were brilliantly confirmed in a
pioneering experiment by Wu and Shaknov [7]. Their experimental ratio
$R=2.04\pm 0.08$ was consistent with the theory, taking into account the
finite solid angles of the detectors.

Seven years later, the results obtained were seriously rethought in the paper
by Bohm and Aharonov [8], devoted to the experimental verification of the
famous EPR paradox by Einstein, Podolsky and Rosen [9]. The authors emphasized
that positron-electron annihilation with zero orbital angular momentum
produces two photons described by the entangled wave function, Eq. (1), and
therefore provides a particular entangled system for studying the EPR paradox.
The authors calculated ratio $R$ and the angular distributions of Compton
scattered photons for several types of quantum states. They confirmed
$R\approx 2.85$ for the state Eq. (1) and concluded that $R=1$ for mixed
separable state of annihilation photons described by the density matrix:

$\rho_{\perp}=\frac{1}{2}(\ket{{H_{1}}{V_{2}}}\bra{{H_{1}}{V_{2}}}+\ket{{V_{1}}{H_{2}}}\bra{{V_{1}}{H_{2}}}).$
(2)

According to the author’s statement, the measured ratio $R=2.0$ [7] is
experimental evidence of distant correlations leading to the EPR-paradox. It
is worth noting that the experimental verification of entanglement was first
discussed seven years before the appearance of famous Bell’s theorem [10, 11],
which is basic instrument of modern experimental tests of entanglement.

In subsequent years a series of experiments [12, 13, 14, 15, 16, 17] was
performed to measure the ratio $R$ with better accuracy. The most precise
results were obtained by Langhoff [12] ($R=2.47\pm 0.07$) and by Kasday et al.
[14] ($R=2.33\pm 0.10$). The geometrical corrections that account for the
finite solid angles of detectors provided the results consistent with the
theoretically predicted value $R=2.85$.

Quite recently Caradonna et al. [18] derived the cross-sections for Compton
scattering of annihilation photons in several (entangled and mixed separable)
states using the matrix representation of the Klein-Nishina formula. Their
results confirmed the credibility of Bohm-Aharonov finding and, consequently,
the validity of relevant experimental tests.

Strong polarization correlations for annihilation photons in entangled states
and the theoretically predicted absence of correlations for separable states
motivated the development a new generation of Positron Emission Tomography
(PET), the so-called quantum-entangled QE-PET with Compton scattering
reconstruction [19],[20]. Attention is paid to the kinematic restrictions on
the angular distributions of scattered gammas in attempts to suppress the
scatter and random backgrounds and improve the quality of PET images. In
recent years, significant efforts have been made to create working prototypes
of PET using the reconstruction of the Compton scattering kinematics. More
recently, the research Jagiellonian Positron Emission Tomograph (J-PET) [21,
22] was built of plastic scintillators. Identification of double Compton
scattering of photons in J-PET plastic bars makes it possible to determine the
linear polarization of primary photons and to measure the angular
correlations.

The verification of entanglement mentioned above relies exclusively on the
angular correlations of scattered annihilation photons. One might wonder why
Bell’s theorem [10, 11], or the Clauser- Horne-Shimony-Holt (CHSH) inequality
[23] as a particular practical case, has not been applied here. The
explanation is quite simple and is related to the low analyzing power of
Compton polarimeters, which are the only tool for measuring the polarization
of high-energy gamma rays. The analyzing power is strongly dependent on the
energy of gamma-ray and is less than 0.7 for annihilation photons with an
energy of 511 keV, even for an ideal polarimeter. This issue has been
discussed at length by Clauser and Shimoni [24], who pointed out that
nonlocality in positron-electron annihilation can only be demonstrated if the
analyzing power is greater than 0.83. In this case, the $S$-function in the
CHSH inequality, reduced to the product of the analyzing powers of two Compton
polarimeters, would exceed the required limit of 2. However, in reality, for
annihilation photons, this S-function can only reach a maximum of 1.4 even for
an ideal polarimeter.

Nevertheless, the $S$-function was measured in 1996 by Osuch et al. [25]. They
counted the number of scattered photons in Compton polarimeters located at
different azimuthal angles. The correlation coefficients were obtained from
the coincidence of the count rates between two sets of polarimeters installed
on opposite sides of the annihilation photon source. The $S$-function
constructed from these coefficients perfectly reproduces the theoretically
predicted behavior:

$S=-p_{0}(3\cos(2\phi)-\cos(6\phi)),$ (3)

where $\phi$ is the azimuthal angle between the polarimeters, and $p_{0}$ is
the product of the analyzing powers of two opposite Compton polarimeters. Of
course, the resulting maximum value of $S$ was well below the theoretical
limit of $2$. Nevertheless, subsequent corrections by the measured values of
the analyzing power of the polarimeters allowed the authors to state that the
CHSH inequality is violated by twelve standard deviations.

The agreement of numerous experimental results with the predictions of quantum
theory gives the impression of a complete understanding of the behaviour of a
system of two annihilation photons and confidence in the entanglement of their
initial state [18]. However, the situation became rather uncertain in 2019
with the appearance of the theoretical work of Hiesmayr and Moskal [26], who
applied an open quantum formalism to the Klein-Nishin formula and obtained the
same Compton scattering cross section for both entangled and mixed separable
states of annihilation photons. Their results fundamentally contradict
previous theoretical considerations and, therefore, claim the incompleteness
of existing experimental studies, which are based on the assumed difference in
the Compton scattering of entangled and separable photon pairs. Moreover, as
we will show below, the identical Compton scattering cross sections lead to
the same $S$-functions in the CHSH inequality for entangled and separable
mixed states. This raises the question of the relevance of Bell’s theorem for
testing quantum entanglement.

The results in ref.[26] obviously affect the very possibility of using quantum
entanglement in the development of PET imaging. To resolve apparent
theoretical contradictions, Watts et al. [27] built a prototype PET with
modern semiconductor gamma detectors and a passive Compton scatterer for one
of the annihilation photons. They measured the angular correlation of
scattered photons for two types of events: entangled photons without a passive
scatterer and pre-scattered events with a passive scatterer placed in the path
of one of the annihilation photons. In the latter case, the decohering process
in the passive scatterer leads to a separable state with lost entanglement.
Unfortunately, the low sensitivity to the measured polarization and large
statistical measurement errors did not allow to draw an unambiguous
conclusion. New dedicated measurements with intact and prescattered initial
states of annihilation photons are required to solve the theoretical puzzle.
For this purpose, we have developed a setup [28] of Compton polarimeters with
a large solid angle and an active intermediate Compton scatterer, which has
been operating for about a year and a half at INR RAS. In this paper, we
report our rather unexpected findings on polarization correlations in Compton
scattering of entangled and decoherent pairs of annihilation photons.

## Methods

Each annihilation photon has an energy equal to the electron mass (511 keV),
which is five orders of magnitude higher than the energy of optical photons.
Such a sharp difference has both advantages and disadvantages in measuring
polarization states. The need to use Compton polarimeters with a relatively
low analyzing power has already been discussed and makes it problematic to use
the CHSH inequality to test photon entanglement. At the same time, the high
energy of photons makes it possible to create a separable, similar to that
described by the Eq.2, mixed state through the Compton scattering of one of
the photons. The selection of events with a sufficiently low energy release in
a Compton scatterer would ensure the decohering process without noticeable
distortion of the initial two-photon kinematics. Thus, annihilation photons
provide a unique opportunity for direct comparison of both initially entangled
and prepared separable mixed states.

The principles for measuring the polarization correlations of these two types
of quantum states are illustrated in Fig. 1.

Figure 1:  Principal scheme for measuring polarization correlations of
initially entangled and prepared decoherent states of annihilation photons. It
includes two Compton polarimeters and an intermediate scatterer. Each Compton
polarimeter consists of a scatterer and two orthogonal detectors of scattered
gammas. $\varepsilon_{1}$ ($\varepsilon_{2}$) and $\varepsilon_{1}^{\prime}$
($\varepsilon_{2}^{\prime}$) are the polarization vectors of the initial and
scattered gammas, respectively. $N_{\parallel}$ and $N_{\perp}$ denote
detectors parallel (perpendicular) to the initial polarization vector.

Two Compton polarimeters are required for measuring both photons with opposite
momenta. Each polarimeter consists of a Compton scatterer and two detectors of
scattered gammas arranged orthogonally. An intermediate scatterer is placed in
the path of one of the initial photons to create a tagged subset of pre-
scattered events. Due to the decohering process during the interaction,
entanglement is lost for the corresponding pairs of tagged photons.

The measurements are based on the dependence of the differential cross section
of Compton scattering on the polarization direction, which is given by the
well-known Klein-Nishina formula [6]:

$\frac{d\sigma}{d\Omega}=\frac{(r_{e}\epsilon)^{2}}{2}\cdot\left(\epsilon+\frac{1}{\epsilon}-2\sin^{2}{\theta}\cdot\cos^{2}{\phi}\right),$
(4)

where ${r_{e}}$ is the classical electron radius, $\epsilon\equiv E_{1}/E$,
$E$ ($E_{1}$) is the energy of the incident (scattered) photon, $\theta$ is
the scattering angle and $\phi$ is the angle between the scattering plane and
the direction of polarization of the incident photon. The scattered photon
energy is ${E_{1}=E\cdot m_{e}/(m_{e}+E\cdot(1-\cos{\theta})})$, where $m_{e}$
is the electron mass. As follows from Eq.4, photons scatter predominantly
orthogonally to the polarization plane.

Like optical polarimeters, the Compton polarimeter has two main components.
Namely, the Compton scatterer measures the polarization of photons in place of
a conventional polaroid, and the detectors of scattered photons play the role
of photodetectors.

The main characteristic of a Compton polarimeter, which determines the
sensitivity to the measured polarization, is the analyzing power:
$A(\theta)=\frac{N_{\perp}-N_{\parallel}}{N_{\perp}+N_{\parallel}}$, where
${N_{\perp}}$ $({N_{\parallel}})$ denotes the number of registered events in
counters located perpendicular (parallel) to the polarization of the incident
photons. Equivalently, at large statistics,

$A(\theta)=\frac{\frac{d\sigma}{d\Omega}(\theta,\phi=90^{\circ})-\frac{d\sigma}{d\Omega}(\theta,\phi=0^{\circ})}{\frac{d\sigma}{d\Omega}(\theta,\phi=90^{\circ})+\frac{d\sigma}{d\Omega}(\theta,\phi=0^{\circ})}.$
(5)

Using Eq. 4, the analyzing power is obtained as [29]:

$A(\theta)=\frac{\sin^{2}{\theta}}{E_{1}/E+E/E_{1}-\sin^{2}{\theta}}.$ (6)

As follows from Eq. 6, for a given energy of the initial gamma, the analyzing
power depends on the scattering angle. For completely polarized photons with
an energy of 511 keV, the analyzing power reaches a maximum of $A=0.69$ at
$\theta=82^{\circ}$.

In the case of an entangled pair of photons with mutually orthogonal
polarizations, the probability of Compton scattering at scattering angles
$\theta_{1},\theta_{2}$ is given by the following expression [5],[18], [26] :

$P(\Delta\phi)={k_{1}k_{2}\left(1-\alpha(\theta_{1})\alpha(\theta_{2})\cos(2\Delta\phi)\right)},$
(7)

where $k_{1}$, $k_{2}$ are the kinematic factors for the first and second
scattered photons, $\Delta\phi$ is the angle between scattering planes, the
parameters $\alpha(\theta_{1})$ and $\alpha(\theta_{2})$ coincide with the
analyzing power of the corresponding Compton polarimeter. The product of the
analyzing powers is equal to the modulation factor $\mu$ [30] which determines
the sensitivity of a setup to the measured polarization. Therefore, Eq.7,
together with the dependence of photon counts on the azimuthal angle, also
gives the modulation factor $\mu$ of the experimental setup.

## Experimental setup

Based on the above principles an experimental setup [28] was constructed to
measure the polarization correlation of annihilation photons in both entangled
and separable states. The setup comprises two equivalent arms of Compton
polarimeters and a 22Na positron source placed between these arms, as shown in
Fig.2.

Figure 2:  Scheme of a two-arm experimental setup. Each arm consists of a
plastic scatterer on the setup axis and 16 NaI(Tl) counters orthogonal to the
axis. The 22Na source of positrons is placed in a lead collimator between the
arms closer to intermediate GAGG scatterer for preparing annihilation photons
in decoherent states.

A source of positrons with an activity of $\sim 50$ MBq was fabricated by
irradiating a 1 mm thick aluminum plate with 130 MeV protons at the INR RAS
isotope facility [31]. The source is located in a horizontal hole in a lead
cube providing collimated beams of annihilation photons in opposite directions
from the source. Each arm of the setup consists of a plastic scintillation
scatterer and a ring of 16 NaI(Tl) scintillation detectors of scattered
photons with an azimuthal angle between adjacent detectors of $22.5^{\circ}$.
NaI(Tl) counters detect photons scattered at an angle of about $90^{\circ}$.
Each pair of orthogonal NaI(Tl) counters and a plastic scatterer of the same
arm form an elementary Compton polarimeter. Due to the azimuthal symmetry of
the setup, each NaI(Tl) counter registers the vertical or horizontal
polarization component depending on the orientation of the scattering plane.
The chosen layout leads to a compensation for possible systematic errors
caused by different efficiencies and inaccuracies in the positions of the
NaI(Tl) counters.

The distance between the plastic scatterers is about 70 cm. To break down the
entanglement of annihilation photons in a subset of events, an intermediate
thin scatterer, a gadolinium-aluminum-gallium garnet (GAGG) scintillator, is
placed next to one of the plastic scatterers. The 22Na positron source is
located 10 cm closer to the arm with the GAGG scintillator to ensure that the
first photon interaction occurs in the intermediate scatterer.

The GAGG scatterer is the key element that separates events in entangled or
decoherent states. An interaction in the GAGG scintillator means that a pair
of initially entangled annihilation photons has undergone a decoherence
process. Therefore, the reliability of interaction identification in the
intermediate scatterer is the most important feature of the setup.

Figure 3:  Left - time coincidence spectra between signals in intermediate
GAGG and plastic scatterers. Red and black lines correspond to events with
energy release in GAGG in the ranges of 2–40 keV and 40–120 keV, respectively.
The numbers indicate the time resolution for these two cases. Right - the
energy spectra in GAGG scatterer for events within the true time coincidence
peak. Here, events are selected that hit the NaI(Tl) counters. Insert shows
the extended GAGG energy spectrum for all events, regardless of the hits in
NaI(Tl) counters. The prominent peak at 170.5 keV corresponds to photons
backscattered by the adjacent plastic scatterer and absorbed by GAGG. This
peak is used for energy calibration of the intermediate scatterer.

Energy deposition and signal timing are measured in all scintillators, NaI(Tl)
detectors, intermediate GAGG and plastic scatterers. The timing and amplitude
of the signals used to identify the interaction in the GAGG scintillator are
shown in Fig.3.

The time coincidence spectra of signals from intermediate and plastic
scatterers are presented for two cases, with low and high energy deposition in
the GAGG. In the first case, the scattering angles in GAGG are small and the
influence on the momentum of the initial annihilation photon is minimal. In
these events, the noise of the photodetector and readout electronics stronger
affects the time resolution compared to events with higher energy deposition.

Fig.3, on the right, shows the energy spectra in the intermediate scatterer
for events in true time coincidence window with hits in the NaI(Tl) counters.
The high light yield of GAGG [32, 33] makes it possible to detect recoil
electrons with an energy threshold of several keV. The permanent energy
calibration of GAGG is performed by detecting 170.5 keV energy deposition from
photons backscattered by an adjacent plastic scatterer and absorbed in GAGG.

We define the tagging of events of various types analyzed in polarimeters as
follows. A pair of initial photons is considered entangled if no interaction
is observed in the intermediate GAGG scatterer. Otherwise, the detected energy
in the GAGG scatterer means that the photons have undergone decoherence.

The sensitivity of the setup to polarization measurements was studied by the
Monte Carlo simulation using the latest version of Geant4 [34, 35] particle
simulation framework, where the theoretical formula Eq.7 is implemented for
initial annihilation photons. Since the NaI(Tl) detectors are located
orthogonally to the Compton scatterers, the theoretical ratio of counts for
perpendicular and parallel orientations of the scattering planes can be
$R=2.6$ which is less than the maximum $R=2.85$ for the optimal $82^{\circ}$
scattering angle. In a setup with a realistic detector geometry, photons are
detected in the range of scattering angles of $80^{\circ}-100^{\circ}$, which
leads to an additional reduction to $R=2.40$. The modulation factor for
entangled photons equals to $\mu=0.41$, which is about 15% less than the
maximum theoretical value $\mu=0.48$ for the optimal scattering geometry.

Finally, Monte Carlo simulations estimated the contribution of systematic
errors caused by possible inaccuracies in the positions of the detectors and
the positron source. Due to the azimuthal location of the detectors,
systematic errors turned out to be almost an order of magnitude smaller than
the statistical errors achieved and were not taken into account.

## Results

### Experimental spectra

Identification of the Compton scattering kinematics is carried out by
monitoring the energy release in all active elements of Compton polarimeters,
namely, in NaI(Tl) detectors of scattered photons, plastic scatterers, and the
intermediate GAGG scatterer. Scattering of the initial annihilation photons at
$90^{\circ}$ releases on average an equivalent energy of 255 keV both in the
plastic scatterer and in the NaI(Tl) counter. According to Monte Carlo
simulations, in a setup with a realistic detector geometry, photons are
detected in the range of scattering angles of $80^{\circ}-100^{\circ}$ with
energy deposition in NaI(Tl) from 235 keV to 280 keV. This is illustrated in
Fig.4, which shows energy deposition spectra in NaI(Tl) detectors for various
Compton scattering kinematics.

Figure 4: Left - energy spectra in NaI(Tl) counters for events in entangled
states without energy deposition in the intermediate GAGG scatterer (blue
line) and in decoherent states with energy deposition in the intermediate GAGG
scatterer below 30 keV (red line). Right - the energy spectra in NaI(Tl)
counters for decoherent photons with energy deposition in the intermediate
GAGG scatterer between 30 keV and 110 keV. The blue line is the experimental
data, the red line is the result of the Monte Carlo simulation.

The relatively narrow NaI(Tl) energy peak for entangled states reflects the
range of scattering angles of registered photons. The situation is completely
different for decoherent states after the interaction of photons with an
intermediate scatterer. Even a few percent energy loss (below 30 keV) of the
original 511 keV photons in the GAGG scintillator leads to a significant
distortion of the energy spectrum. This is the effect of double Compton
scattering of the initial annihilation photon in both GAGG and plastic
scintillators. After the first scattering in GAGG, the photon deviates within
a few degrees from its original direction. Subsequent interaction in a plastic
scintillator increases the ranges of scattering angles and energy deposition
of photons registered in NaI(Tl).

Events with a higher energy deposition in the intermediate scatterer (up to
110 keV) form a complex structure in the NaI(Tl) spectrum. Visible peaks in
this energy spectrum correspond to particular cases of the kinematics of
Compton scattering. The Monte Carlo simulation of the double Compton
scattering of 511 keV photons confirmed that the observed structure in the
NaI(Tl) spectrum reflects the kinematics of the first scattering in the GAGG
scintillator, which leads to distinct groups of events. These particular cases
of Compton scattering and the correlation between energy depositions in the
NaI(Tl) and GAGG scintillators are shown in Fig. 5.

Figure 5:  Correlation between energy depositions in the intermediate GAGG
scatterer and in NaI(Tl) counters. Several groups of events in black boxes
marked as "a"," b", "c" and "d" can be distinguished. They correspond to
different Compton scattering kinematics in the GAGG scintillator, shown in the
diagrams to the left/right of the correlation plot. The brown, red and blue
boxes in the diagrams mark the intermediate GAGG scatterer, the plastic
scatterer and the NaI(Tl) counter, respectively.

In this figure, several distinct groups of events can be identified, labeled
as classes "a", "b", "c", and "d". The double Compton scattering kinematics
for these events can be determined using a simple analytical calculation or
Monte Carlo simulation. Group "a" represents scattering processes with an
electron recoil energy in the GAGG scintillator below 30 keV and, accordingly,
with the smallest scattering angles.

The range of scattering angles by the GAGG in the "b" and "c" regions is
identical and it is larger as compared to the "a"-events. However, events in
the "b” region have maximum energy deposition in NaI(Tl) detectors from two
sequential Compton scatterings with the first scattering in the GAGG in the
direction of the NaI(Tl) counter. In contrast to these events, the "c" group
represents a double Compton scattering with the sum of scattering angles
exceeding $90^{\circ}$, since the first scattering in the GAGG scintillator
deflects the photon in the opposite direction from the NaI(Tl) counter.
Therefore, more energy is left in the plastic scintillator and less for the
NaI(Tl) detector remains.

The most complex kinematics of double Compton scattering is observed for the
“d" group of events with the lowest energy registered by NaI(Tl). The original
annihilation photons in these events pass through the GAGG scintillator
without interaction and are backscattered in the plastic scatterer. The second
scattering at about $90^{\circ}$ occurs in the GAGG scintillator, followed by
photon registration in NaI(Tl) counters. Unlike other classes of events with
relatively small scattering angles in GAGG, the "d" group represents Compton
scattering at maximum angles about $180^{\circ}$ and with maximum energy
losses in plastic scatterer.

### Angular distributions of scattered photons.

Previous experiments with entangled annihilation photons have mainly measured
the dependence of the number of counts of scattered photon detectors on the
azimuthal angle between these detectors. This dependence is described by Eq.7
and has a cosine-like behaviour. As follows from the formula, the ratio of
counts $R$ for perpendicular and parallel orientations of the scattering
planes reaches a maximum of $R=2.85$ if both photons are scattered at an angle
of $82^{0}$ to their initial momenta. According to Eq. 7, the number of counts
in detectors of scattered photons can be approximated as:

$N(\phi)=A-B\cos(2\Delta\phi),$ (8)

where $\Delta\phi$ is the angle between scattering planes, A and B are the fit
constants. As follows from Eq. 7 and Eq. 8, the ratio $R$ equals to
$R=({A+B})/({A-B})$, while the modulation factor is $\mu=B/A$.

Figure 6:  Dependence of coincidence counts in NaI(Tl) detectors on the
azimuthal angle between these detectors for entangled (left) and decoherent
(right) initial states of photon pairs. The solid line corresponds to the
fitting function Eq. 8.

In our experiment, the data set without energy deposition in intermediate GAGG
scatterer is associated with entangled photons, Eq. 1. In this case, the
modulation factor coincides with the product of the analyzing powers of the
corresponding Compton polarimeters,
$\mu=\alpha({\theta_{1}})\cdot\alpha({\theta_{2}})$, see Eq. 7. Other events
with interaction in the GAGG scatterer correspond to decoherent photon pairs
in the initial state. Experimental angular distributions for these two types
of events with different initial quantum states are shown in Fig. 6. The data
were approximated by the function Eq. 8. For the entangled photons, the
fitting parameters give $R=2.435\pm 0.018$, which is consistent with the
Geant4 Monte Carlo simulation, taking into account that the NaI(Tl) detectors
are set orthogonal to the symmetry axis and register photons scattered in
range $80^{\circ}-100^{\circ}$. The modulation factor equals to
$\mu=A^{2}=0.418\pm 0.003$, which is about 15% less than the maximum
theoretical value of 0.48 for ideal Compton polarimeters.

The second plot in Fig. 6 shows the angular distributions for the entire set
(classes "a", "b", "c" in Fig. 5) of decoherent events, excluding
backscattered ones. Surprisingly, decoherent events exhibit almost the same
azimuthal behavior with a ratio $R=2.41\pm 0.10$ and a modulation factor
$\mu=0.414\pm 0.017$.

Interaction in GAGG is equivalent to quantum mechanical measurement. After the
measurement, the participating photon must be in the “H" or “V" state with
equal probability. The other photon must collapse into a state of opposite
polarization if there is no exchange of angular momentum with the electron,
which is true at low energies $E\ll m_{e}$ or forward scattering
$\theta^{2}\ll m_{e}/E$, Eqs. (87.20-21) in ref.[36]. Therefore, after
scattering in GAGG, under one of these conditions, decoherent photons must be
in a mixed separable state Eq. 2. Events of classes "a", "b", "c" satisfy the
forward scattering condition. Then the observed cosine-like behavior
contradicts earlier theoretical findings [8], see also recent ref. [18]. On
the contrary, our experimental results are in agreement with the recent
prediction [26] of the equivalent Compton scattering cross section for photons
in entangled and mixed separable states, Eq. 1 and Eq. 2 respectively.

To get more information and insight, angular distributions were constructed
for all classes of decoherent events classified in Fig. 5. The dependencies of
the number of scattered photons registered in the NaI(Tl) counters on the
azimuthal angle between these photons are shown in Fig. 7. From the fitting
parameters presented in these plots, one can calculate the ratio $R$ of the
numbers of counts for the perpendicular and parallel orientations of the
scattering planes, which are very close to each other, $R_{a}=2.43\pm 0.15$,
$R_{b}=2.52\pm 0.21$, $R_{c}=2.35\pm 0.17$ for classes "a"," b", "c",
respectively.

Figure 7:  Dependence of coincidence counts in NaI(Tl) detectors on the
azimuthal angle between these detectors for four classes ( "a", "b", "c" and
"d" in Fig. 5) of decoherent events. The solid line corresponds to the fitting
function Eq. 8. The numbers in the blue area on the graphs indicate the $R$
ratio for the corresponding class of events.

The smallest ratio $R_{d}=1.48\pm 0.13$ is observed for class "d" events. This
is due to the partial depolarization of backscattered photons, as follows from
the analysis of the Klein-Nishina formula for linearly polarized gamma rays
[36]. Namely, according to Eq. (87.12) in ref. [36], with $E=m_{e}$ and
backscattering by $180^{\circ}$, one fifth of photons change from vertical to
horizontal polarization and vice versa. Note, that in [26] the possibility of
such transitions is missed. Adding the corresponding fraction of

$\rho_{\parallel}=\frac{1}{2}(\ket{{H_{1}}{H_{2}}}\bra{{H_{1}}{H_{2}}}+\ket{{V_{1}}{V_{2}}}\bra{{V_{1}}{V_{2}}})$

to the density matrix Eq. 2 we find that the modulation factor reduces by a
factor of $0.6$. We should expect $R\approx 1.66$ for such a depolarized
state, which is consistent with the experimental result, Fig. 7. Note, that
$R=1$ if $\rho=\frac{1}{2}(\rho_{\perp}+\rho_{\parallel})$.

Since the angular distributions of scattered photons turned out to be the same
for entangled and separable initial states, our results show that these
dependences cannot be used to experimentally test the entanglement of
annihilation photons.

###  $S$-function in CHSH inequality.

The CHSH inequality as a particular variant of Bell’s theorem is used in
classical experiments with pairs of entangled optical photons [37, 38, 39]. A
suitable setup includes two dual-channel polarimeters separating orthogonal
linear polarization parallel and perpendicular to some arbitrary $\vec{a}$ and
$\vec{b}$ directions for the first and second photons, respectively. By
measuring the coincidence rate $R$ of photons with different spin
orientations, the correlation coefficients can be constructed as:

$E(\vec{a},\vec{b})=\frac{R_{a_{\parallel}b_{\parallel}}+R_{a_{\perp}b_{\perp}}-R_{a_{\parallel}b_{\perp}}-R_{a_{\perp}b_{\parallel}}}{R_{a_{\parallel}b_{\parallel}}+R_{a_{\perp}b_{\perp}}+R_{a_{\parallel}b_{\perp}}+R_{a_{\perp}b_{\parallel}}},$
(9)

where the indexes $a_{\parallel}$, $a_{\perp}$, $b_{\parallel}$ and
$b_{\perp}$ denote the photons with parallel or perpendicular polarization for
the direction $\vec{a}$ or $\vec{b}$, respectively.

By measuring the correlation coefficients for four different polarimeter
orientations, $\vec{a},\vec{b},\vec{a}^{\prime},\vec{b}^{\prime}$, the
following function can be composed:

$S=E(\vec{a},\vec{b})-E(\vec{a},\vec{b}^{\prime})+E(\vec{a}^{\prime},\vec{b})+E(\vec{a}^{\prime},\vec{b}^{\prime}).$
(10)

In the case of entangled photons, $S$-function reaches a maximum of $\lvert
S\rvert=2\sqrt{2}=2.83$ for certain optimal polarimeter orientations:

$\displaystyle(\vec{a},\vec{b})=(\vec{a}^{\prime},\vec{b}^{\prime})=(\vec{a}^{\prime},\vec{b})=\phi_{opt}$
$\displaystyle(\vec{a},\vec{b}^{\prime})=(\vec{a},\vec{b})+(\vec{a}^{\prime},\vec{b}^{\prime})+(\vec{a}^{\prime},\vec{b}),$

where $(\vec{a},\vec{b})$ denotes an angle between vectors $\vec{a}$ and
$\vec{b}$, and the optimal azimuthal angles $\phi_{opt}$ are multiples of
$\phi=22.5^{\circ}$. This clearly violates the CHSH inequality $\lvert
S\rvert\leq 2$ that follows from Bell’s theorem if underlying hidden local
variables would exist.

As discussed by Clauser and Shimony [24], the CHSH inequality is never
violated for annihilation photons due to the low efficiency (analyzing power)
of Compton polarimeters. To map the real-world experiments with non-ideal
polarimeters to the ideal case, the $S-$function is usually normalized to the
product of corresponding efficiencies. Moreover, even without such a
normalisation, measurements of annihilation photons in entangled and
decoherent states on the same setup allow a direct comparison of the
$S$-function for these two types of events. One should expect a fundamentally
different behavior of this function for entangled and separable states, since
in the latter case there are no nonlocal correlations.

Since in our setup the angle between adjacent detectors of scattered photons
is $22.5^{\circ}$, the procedure for measuring the correlation coefficients
and the $S$-function is straightforward (see also Osuch et al. [25]). The
elementary Compton polarimeter consists of a scatterer and two orthogonal
detectors of scattered photons, as shown in Fig 1. To measure the correlation
coefficient $E(\vec{a},\vec{b})$, two elementary Compton polarimeters are
requested on opposite sides of the setup with orientations $\vec{a}$ and
$\vec{b}$. As an example, Fig. 1 shows a scheme for measuring the correlation
coefficient $E(90^{\circ})$. The $S$-function was determined using four
polarimeters with orientations $\vec{a}$, $\vec{a}^{\prime}$ on one side and
$\vec{b}$, $\vec{b}^{\prime}$ on the other. There are 16 Compton polarimeters
on each side of the setup. And the count rates for a given correlation
coefficient are summed for all relevant combinations of polarimeters.

The measured $S-$functions for entangled and decoherent annihilation photons
are presented in Fig. 8. In the latter case, class “a" events with an energy
deposition below 30 keV in the intermediate GAGG scatterer were selected to
ensure the minimum difference from the kinematics of entangled photons. The
fit of the experimental points by the theoretical function Eq. 3 is shown by
the solid line. The normalizing factor $p0$ is equal to the product of the
analyzing powers of two polarimeters. Surprisingly, the behavior of both
$S-$functions is almost identical regardless of the quantum state of the
photons. Moreover, the normalizing factors are practically identical and
coincide with the modulation factors estimated from the angular dependencies
of the coincidence counts in the detectors of scattered photons, see Fig. 6.

Figure 8:  Dependence of the S-function on the relative azimuthal angle
between polarimeters for entangled (left) and decoherent photons of class "a"
(right). The solid line corresponds to the fitting function Eq. 3

The identity of the $S$-functions indicates that the CHSH inequality, even
corrected for the efficiency of Compton polarimeters, cannot be used to test
the entanglement of annihilation photons.

## Discussion

In this work, we utilized two methods for studying entanglement in the Compton
scattering of annihilation photons. The first method is based on the discovery
by Bohm and Aharonov [8] of the relationship between strong angular
correlations of scattered photons and entanglement. According to them, a large
ratio $R>2$ of the number of photon counts for the perpendicular and parallel
orientations of the scattering planes indicates non-local correlations of
annihilation photons.

The second approach involves the CHSH inequality, which is directly related to
Bell’s theorem. Osuch et al. [25] measured the $S$-function in this inequality
and found that it agrees with quantum theory. Their correction of the
$S$-function for the efficiency of Compton polarimeters confirmed the
violation of the CHSH inequality.

The agreement between the theoretical and experimental results led to the
conclusion that the system of two annihilation photons is entangled [18].
However, a recent paper by Hiesmayr and Moskal [26] has shaken the belief.
They found that the cross section for Compton scattering of annihilation
photons, described by the Eq. 7, is identical for entangled and mixed
(separable) state Eq. 2. Since the cross section is of fundamental importance
and determines all dependencies between scattered photons, the angular
distributions and values of the $S$-function in the CHSH inequality must be
the same for both quantum states. Nevertheless, Caradonna et al. [18]
disagreed with Hiesmayr and Moskal [26], confirming the results of Bohm and
Aharonov [8].

In our experiment, we compared Compton scattering of initial annihilation
photons and decoherent pairs prepared by preliminary forward scattering of one
of the initial photons before measurements in a polarimeter. The new results
have revealed the same behavior of the angular distributions and $S$-functions
for both quantum states. (The correlations of a backscattered photon with its
pair are smaller, but we came to the conclusion that this state cannot be
approximated by the Eq. 2.) Our measurements reveal that the entanglement of
annihilation photons is still awaiting experimental confirmation using more
advanced techniques.

Also, according to new experimental results, the similarity of angular
correlations of scattered photons limits the use of quantum entanglement in
the planned next generation positron emission tomography (QE-PET). Most
scattered background events cannot be rejected by the constraints on the
angular distribution of scattered photon pairs. At the same time, the random
background obviously has no angular correlations and can be suppressed by
applying appropriate kinematic cuts.

Perhaps the most intriguing discovery is the identity of the $S$-function in
the CHSH inequality for the initial and prepared separable state of photon
pairs, Eq. 2. However, this is a consequence of the same Compton scattering
cross section for both quantum states, as stated in ref. [26]. The equivalence
of these cross sections raises the question of the universality of Bell’s
theorem for testing the entanglement and nonlocality of quantum theory. This
issue has been theoretically debated for a long time [40, 41]. For example, a
very recent discussion [42, 43, 44] predicts a violation of CHSH inequality in
a completely local system of neon ${}^{21}{\rm Ne}$ atoms with spin 3/2. The
results presented provide an experimental basis for these theoretical
discussions.

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## Acknowledgements

The authors are grateful to B. Zhuikov for fabricating the ${}^{22}{\rm Na}$
radioactive source of positrons and to D. Levkov, M. Libanov, and A. Panin for
useful discussions.