# Photon quantum entanglement in the MeV regime and its application in PET
imaging

D.P. Watts1 J. Bordes1 J.R. Brown1 A. Cherlin2 R. Newton1 J. Allison3,4 M.
Bashkanov1 N. Efthimiou1,5 N.A. Zachariou1

###### Abstract

Positron Emission Tomography (PET) is a widely-used imaging modality for
medical research and clinical diagnosis. Here we demonstrate, through detailed
experiments and simulations, an exploration of the benefits of exploiting the
quantum entanglement of linear polarisation between the two positron
annihilation photons utilised in PET. A new simulation, which includes the
predicted influence of quantum entanglement on the interaction of MeV-scale
photons with matter, is validated by comparison with experimental data from a
cadmium zinc telluride (CZT) PET demonstrator apparatus. In addition, a
modified setup enabled the first experimental constraint on entanglement loss
for photons in the MeV regime. Quantum-entangled PET offers new methodologies
to address key challenges in next generation imaging. As an indication of the
potential benefits, we present a simple method to quantify and remove in-
patient scatter and random backgrounds using only the quantum entanglement
information in the PET events.

Department of Physics, University of York, Heslington, York, YO10 5DD, UK.

Kromek Group plc, Sedgefield, County Durham, TS21 3FD, UK.

Geant4 Associates International Ltd., Hebden Bridge, HX7 7BT, UK.

Department of Physics and Astronomy, University of Manchester, Manchester, M13
9PL, UK.

PET Research Centre, School of Health Sciences, University of Hull, Hull, HU6
7RX, UK.

##  1 Introduction

Positron Emission Tomography (PET) is a valuable imaging tool for exploring a
variety of cellular and molecular processes in vivo. PET can provide high
contrast and quantitative functional information about disease or therapy
response usually complemented by purely anatomical information provided by
other imaging modalities. A typical PET study involves the administration of a
radiotracer, a biologically active molecule which is labelled with a positron
($e^{+}$) emitting radionuclide to track metabolic activity. Subsequent
$e^{+}e^{-}$ annihilation produces two $0.511$ MeV $\gamma$-photons, moving in
approximately opposite directions. Their subsequent detection enables a line
of response (LOR) to be defined upon which the annihilation site is assumed to
be located. However, in addition to this spatial information, the two
annihilation $\gamma$ are described by a common entangled wavefunction which
results in correlations between their interaction processes even when
separated, the so called “spooky”[1, 2] action at a distance effect of quantum
mechanics. The potential imaging benefits from utilising quantum entanglement
information are more extensively investigated for optical ($\sim$eV) photons
[3, 4, 5], and more recently for X-ray photons ($\sim$10 keV)[6]. In this
work, we carry out a first simulation of the interaction of MeV scale photons
with matter in a PET-like system, which includes the effect of quantum
entanglement in their interactions. We explore the benefits of using this
quantum information in the analysis of PET data.

The most useful events collected to form a PET image (true events) are those
having a LOR which crosses the annihilation site (no interaction in the
patient). These true events are expected to maintain their quantum-entangled
nature. However, in PET these are recorded along with unwanted scatter and
random backgrounds. These have LORs displaced from the annihilation site(s)
which cause artefacts in the reconstructed image[7, 8], decrease the signal-
to-noise and distort the relationship between the image intensity and the
activity in the volume of interest. Scatter background arises when at least
one of the two annihilation $\gamma$ scatter prior to detection. In addition
to a displaced LOR, such decohering scatter reactions will lead to
entanglement loss for the subsequently detected photon pair. The scatter-to-
true ratios range from $\sim$0.2 for brain imaging to $\sim$2 for 3D abdominal
imaging[9]. The random background originates mainly from uncorrelated $\gamma$
pairs, producing LORs dispersed over the full image and which would, of
course, not be in an entangled state. Random-to-true ratios range from
$\sim$0.1 (brain imaging) to more than 1[9], influenced by the detector
properties (e.g the timing coincidence window for the accepted $\gamma$
photons) and the administered activity.

To investigate the benefits of exploiting quantum entanglement information in
PET, a new simulation accounting for the entangled interactions of
annihilation photons in matter was developed. This builds on the comprehensive
Geant4 simulation framework enabling full account of detector geometry,
experimental resolutions and backgrounds. Therefore, beyond the PET
application discussed here, the framework offers possibilities to improve the
achievable precision in measurements of entangled photons at the MeV scale,
such as precision tests of the predicted DCSc cross sections. To test its
validity, the new simulation is compared to high quality experimental data on
double Compton scattering (DCSc) of positron annihilation (Pa) gamma from a
22Na positron source. These new data were obtained using two state-of-the-art
semiconductor $\gamma$ detectors which, when placed back-to-back, provided a
PET-demonstrator apparatus. Additionally, simulated preclinical PET images
from a larger system of such detectors were simulated, to provide assessment
through simulation of the benefits of exploiting the new information.

In this first study, we used the linear polarisation of the $\gamma$ as the
experimental observable sensitive to the entangled nature of the photons, with
visibility achieved through observation of the DCSc process of the photons.
The two $\gamma$ from (ground state) para-positronium annihilation (anti-
parallel spins; $S=0$, $S_{z}=0$ where $S$ is the spin of the positronium)
have orthogonal linear polarisation with an entangled wave function which can
be expressed as:

$\ket{\Psi}=\frac{1}{\sqrt{2}}\>\Big{(}\ket{x}_{-}\ket{y}_{+}-\ket{y}_{-}\ket{x}_{+}\Big{)}$
(1)

where $\ket{x}_{-}$ and $\ket{y}_{-}$ represent a $\gamma$ propagating along
$-z$ with polarization in $x$ and $y$, respectively. $\ket{x}_{+}$ and
$\ket{y}_{+}$ have equivalent definitions but for the $+z$ direction. This
entangled Bell state is the only allowed state following annihilation of
ground state positronium to two photons[10, 11, 2].

The differential cross section for DCSc of the entangled annihilation gamma in
the allowed Bell state of equation 1, is given by[12, 11]:

$\frac{d^{2}\sigma_{double}}{d\Omega_{1}d\Omega_{2}}=\frac{r_{0}^{4}}{16}\>\Big{(}K_{a}(\theta_{1},\theta_{2})-K_{b}(\theta_{1},\theta_{2})\cdot{}\text{cos}(2\Delta\phi)\Big{)}$
(2)

where ${d\Omega_{1,2}}$ and $\theta_{1,2}$ are the solid angles and polar
scattering angles for $\gamma_{1}$, $\gamma_{2}$, respectively, $r_{0}$ is the
classical electron radius, $K_{a}$ and $K_{b}$ are kinematic
factors111$K_{a}=\frac{[(1-cos\theta_{1})^{3}+2]\cdot{}[(1-cos\theta_{2})^{3}+2]}{(2-cos\theta_{1})^{3}\cdot{}(2-cos\theta_{2})^{3}}$,
$K_{b}=\frac{sin^{2}\theta_{1}\cdot{}sin^{2}\theta_{2}}{(2-cos\theta_{1})^{2}\cdot{}(2-cos\theta_{2})^{2}}$,
and $\Delta\phi=\phi_{1}-\phi_{2}$ is the relative azimuthal scattering angle
(see Fig. 1 for definition of scatter angles). The form of equation 2 is
presented in the publication of Pryce and Ward[12] (also independently derived
by Snyder et al.[10]). In these early works, consistent results were obtained
when employing time dependent perturbation theory or a Klein-Nishina[13] based
approach, as outlined in the PhD thesis of Ward[11]. Subsequently the same
form has been derived in DCSc calculations in a matrix formalism[14] and
employing Kraus operators[15].

The DCSc cross section is modulated by the cos$(2\Delta\phi)$ term, resulting
in an enhancement ratio ($R$) between the maximum ($\Delta\phi=\pm
90^{\circ}$) and minimum ($\Delta\phi=0^{\circ}$) scattering probabilities of
$R=2.85$, achieved when both $\theta_{1}$ and $\theta_{2}$ are equal to
$81.7^{\circ}$ [10, 12, 11]. Bohm and Aharonov[2] were the first to recognize
that the ($\Delta\phi$) correlations between the DCSc annihilation photons was
an example of the kind of entanglement discussed by Einstein, Podolsky and
Rosen[16]. They also derived[2] the upper limit for a (hypothetical)
orthogonally polarised, but non-entangled (separable) state as $R=1.63$,
establishing that measured values above this limit are a witnesses of the
entanglement[2, 15, 14]. Experimental measurements of this amplitude [17, 18,
19, 20, 21, 22, 23, 24] focused on restricted kinematics of $\theta_{1}$,
$\theta_{2}$ around maximum visibility, and yielded $R$ values well beyond the
upper limit for a non-entangled state. Clear statistical agreement with the
entangled theory (equation 2) was established (with analytic corrections for
experimental aspects of the measurement).222The most precise measurements were
obtained by Langhoff[18] and Kasday et al.[21] with measurements $R=2.47\pm
0.07$ and $R=2.33\pm 0.10$ respectively, in agreement with equation 2 when
geometrical effects are accounted for. The consistency of the measured $R$ was
established[23, 22] for separation distances up to $2.5$ m, greatly exceeding
the coherence length associated with Pa ($0.12$ m[22]) and larger than the
diameter of a typical clinical PET scanner (0.8-1.3 m). The DCSc entanglement
witness was also exploited in early investigations of Bell’s inequalities,
providing evidence against (although not fully ruling out) hidden-variable
theories in quantum mechanics[24, 23, 21]. Subsequently, optical EPR
measurements proved more amenable due to the availability of more efficient
linear polarisation analysers, which enabled tests with less auxiliary
assumptions[25]. However, we remark that utilisation of the (experimentally
verified) correlation in DCSc scatter planes in this work (equation 2), does
not rely on the existence of loop-hole free EPR tests. The ability of the
entangled cross section (equation 2) to describe the $\Delta\phi$ distribution
($R$) for all previous data in positron annihilation (and for the current data
- see next section) gives confidence in its implementation in PET
simulation.333The small contribution of annihilation in flight, in which the
positron annihilates before thermalisation, is neglected as assumed for
previous Pa measurements[17, 18, 21, 22, 23, 24, 2] and only contributes in
tissue at the 2% level e.g[26]. We neglect three-$\gamma$ decays from ortho-
positronium ($S=1$) as they have a yield only 0.5% of the two-$\gamma$ yield
in PET[26], and could be further rejected by photon energy cuts. Initial
ortho-positronium formation may lead to two-$\gamma$ final states through
singlet ($S=0$) pickup annihilation with an (external) electron in the
surrounding medium e.g.[27]. We take such photons as having the correlated
wavefunction of equation 1 (and DCSc cross section of equation 2) as implicit
in previous measurements. This assumption is supported by the observation
that, despite a range of positron thermalising media being employed, no
deviation from equation 2 is observed in any previous experimental work on
Pa[17, 18, 21, 22, 23, 24, 2] (or the current work - see next section).

Figure 1: Scattering angle definition. Schematic figure showing the definition
of the scattering angles $\theta$ and $\phi$ for the DCSc of the two entangled
$\gamma$ photons. The direction vectors for the scattered photons are shown by
the red lines, with the corresponding scatter planes indicated by the shaded
rectangles. The z-axis is aligned with the $\gamma$ direction while the x-axis
is defined with respect to the detector in the laboratory frame. (Arbitrary)
mutually perpendicular orientations of the polarisation vectors of the
$\gamma$ are also shown.

Establishing the utility of quantum entangled PET (QE-PET) required the
implementation of the predicted effects of quantum entanglement into a
simulation of PET-photon interactions in matter. We incorporated entanglement
(as described by equation 2) in the leading Monte Carlo simulation package,
Geant4[28, 29], by developing new routines which enable communication between
the individual particle-tracking processes to reproduce the scattering cross
sections for entangled $\gamma$ pairs (equation 2). The new version, which we
refer to as QE-Geant4, will be included in future releases of the code. Any
further interactions of $\gamma$ after the initial Compton scatters reverted
to the standard Geant4 polarised Compton scattering routines, i.e. a Compton
interaction was assumed to completely collapse the entangled state shown in
equation 1.

The QE-Geant4 simulation was validated by comparing to measurements of DCSc of
Pa photons in a PET-demonstrator. The system was developed by Kromek Group
based on the DMatrix detector system[30] and comprised two 10 mm cubic
semiconducting cadmium zinc telluride (CZT) crystals, placed back-to-back and
separated by 55 mm. A segmented anode divided each crystal into 121 $0.8\times
0.8$ mm2 pixels, with depth information accessible from the anode drift
time[31]. Readout of the anode and cathode signals employed a bespoke ASIC.
The coincidence timing window of the system was set to 1 $\mu$s due to the
charge sweep-out time of a few tenths of $\mu$s. A 170 kBq 22Na source,
comprised of a 1mm diameter active bead housed in a thin plastic was placed
equidistant from each crystal face providing a source of annihilation
$\gamma$.

Measurements of $\Delta\phi$ were accessible for events where both
annihilation $\gamma$ interacted through a Compton scattering and a subsequent
photoelectric absorption. Such events produced, in both CZT modules, two
clusters of pixels with a total energy in the range 480 - 530 keV. Charge
sharing events, produced by a single hit between adjacent pixels, were
rejected by requiring a gap of at least one pixel between clusters. The polar
scattering angle was determined using the Compton scattering formula, assuming
the largest energy signal as the first interaction444Simulations indicate that
this assignment will be correct in $\sim 60\%$ of cases. Incorrect assignment
will result in a calculated polar scattering angle
$\theta_{calc.}=180^{\circ}-\theta_{real}$, but not affecting $\Delta\phi$ due
to the symmetry about $0^{\circ}$.. To enhance the amplitude of the
cos($2\Delta\phi)$ distribution (equation 2), coincidences were only retained
if the polar scattering angle was within $70^{\circ}{}\leq\theta\leq
110^{\circ}{}$. The azimuthal scattering angle was simply determined from the
energy-weighted centre-of-gravity of each cluster and only one- or two-pixel
clusters were considered. The azimuthal angular resolution depended on the
distance between the clusters. It ranged from $2.9^{\circ}$ (for the most
distant pixels) to $20.4^{\circ}$ for the closest pixels considered in this
analysis. The energy and $\theta$ cuts reduce the data to 29.2% and 5.4% of
the total two-cluster events respectively, and when combined retain 2.6% of
the total yield. The normalised (see caption) coincidence count rate as a
function of $\Delta\phi$ for these events is shown as the data points in Fig.
2. A strong cos($2\Delta\phi)$ modulation is seen with a measured enhancement
factor of $R=1.85\pm 0.04$ for the event yield selected by the analysis cuts.
555Selecting a tighter polar angle acceptance ($\theta_{1,2}=93-103^{\circ}$)
gave a measured enhancement factor of $R=1.95\pm 0.07$. We note that future
higher statistics data could further increase the visible enhancement by
selecting sub-set of events with larger intercluster separations and
correspondingly improved angular resolution.

The experimental setup was simulated with QE-Geant4. The simulated energy
deposits were matched to the CZT experimental detector response by accounting
for effects of charge transport, diffusion and self-repulsion[32], and smeared
to match the experimental energy resolution of each detector module (3.8 and
5.3% FWHM at 662 keV). The resulting simulated data were then passed through
the same analysis code as the experimental data.

Figure 2: Comparison of experimental and simulated scattering probabilities.
Black data points show the experimentally determined (normalised) coincidence
count rate as a function of $\Delta\phi$, obtained using the CZT detector
apparatus (see text). The prediction from the quantum-entangled simulation
(QE-Geant4) is shown as the blue line. Non-entangled predictions for
orthogonally polarised $\gamma$ pairs are shown as the red line. The
prediction for unpolarised annihilation $\gamma$ is shown by the green line.
The statistical uncertainties for the simulations are indicated by the line
widths. For comparison of the $\Delta\phi$ distributions, the experimental and
simulated data are normalised to unity for the data around the minima at
$\Delta\phi=0^{\circ}$ and $\pm 180^{\circ}$, specifically the mean yields
from bins 1, 9, 10 and 18.

The predictions from QE-Geant4 (Fig. 2 \- blue line) show a good agreement
with the measured $\Delta\phi$ distribution ($\chi^{2}/\nu=1.87$), providing
validation of QE-Geant4 simulation as well as the underlying entanglement
theory employed (the agreement between the measured $\Delta\phi$ distribution
and QE-Geant4 is also quantified on a bin-by-bin basis in the supplementary
materials). To better quantify the effects of entanglement on the measured
$\Delta\phi$ amplitude, a further simulation of (hypothetical) non-entangled,
but orthogonally polarised, annihilation $\gamma$ is also presented (red
line). The DCSc for this case is modelled by the standard (non-entangled)
polarised Compton scattering classes in Geant4, which model the scatter of
each $\gamma$ as an independent (separable) process 666As shown in the
supplementary materials the simulation results for both the entangled and non-
entangled systems agree with the analytical DCSc formulae derived for each
hypothesis by Bohm and Aharonov[2].. The size of the predicted $\Delta\phi$
modulation is clearly reduced if entanglement is neglected, and the
predictions are in clear disagreement with the experimental data. The results
illustrate the need for an account of quantum entanglement in PET simulation
to correctly describe the observed distribution of Compton scatter
planes777Earlier works simulating (non-entangled) polarisation effects in PET
using standard Geant4 classes[33, 34] could be further developed using this
new entangled framework.. In order to assess the influence of detector
acceptance on the measured $\Delta\phi$ distribution, a further simulation for
unpolarised annihilation $\gamma$ was also carried out (green line). The
intrinsic $R$ for such events is equal to unity, in agreement with analytical
calculations for mixed states in Bohm and Aharonov[2] and Caradonna et
al.[14].888We note that a very recent work[15] contradicted the original
theory of Bohm and Aharonov[2] and the interpretation of all previous
experimental works[23, 22, 2, 10, 12, 24, 21, 19, 18, 20, 17], by inferring a
contribution to $R$ from a hypothetical mixed state in Pa (in which the
$\gamma$ need not originate from the same annihilation event). The conclusion
has already been refuted by Caradonna et al.[14] who re-derived $R=1$ for
mixed states in a matrix formalism in agreement with Bohm and Aharonov[2]. We
note the unpolarised simulation presented here also supports that such events
are consistent with $R=1$. We therefore adopt the accepted theoretical
interpretation[2] in this work. The $\Delta\phi$ distribution for unpolarised
events in the detector acceptance (green line) is rather uniform albeit with a
small acceptance related enhancement near to the centre of the distribution,
which is also evidenced in the experimental data, and appears well modelled by
the simulation. It is clear that the measured $\Delta\phi$ distribution only
has modest influence from detector acceptance effects.

###  1.1 Investigation of entanglement loss when an annihilation gamma
scatters prior to measurement

The results in Fig 2, and previous measurements in more limited kinematics
from a range of different Pa sources, show that the azimuthal correlation of
the Compton scatter planes in DCSc of Pa photons is in agreement with the
entangled theory (equation 2) and has a correlation ($R$) beyond the upper
limit of a separable non-entangled state. We remark that the data in Fig 2 is
dominated by photons for which their first interaction is the (linear
polarisation analysing) Compton scatter reaction in the CZT detectors. We are
not aware of any previous measurement of the $\Delta\phi$ correlation for
photons which have undergone an identified decohering process prior to the
measurement.

To achieve this we adapted the experimental setup as shown in Fig. 3. Events
where one of the photons has undergone a Compton scattering process prior to
measurement of the $\Delta\phi$ correlation ( prior-scatter events) were
obtained by inclusion of a scattering medium (nylon) in the path of one of the
annihilation $\gamma$, with the corresponding CZT module rotated through 33∘
relative to the centre of the scatterer. The measured energy of the scattered
$\gamma$ in the CZT (obtained from the sum of the two energy deposits) matched
that expected from the reaction kinematics ($\sim$440 keV) and was well
separated from backgrounds.

The DCSc $\Delta\phi$ distributions measured by the CZT detectors for such
prior-scatter events are shown by the black data points in Fig. 4. For
comparison, the red data points (red line) show the measured (simulated)
$\Delta\phi$ correlations for back-to-back unscattered $\gamma$ respectively,
with the same polar angle cuts for the scattered photons as used for the
prior-scatter data999To increase the event yield both data sets have a wider
polar angle acceptance for the CZT scatters than employed for Fig. 2. (see
caption). The measured $\Delta\phi$ correlation for the prior-scatter events
is clearly diminished compared to the unscattered case. The QE-Geant4
prediction (blue line) also exhibits a diminished $\Delta\phi$ correlation, in
statistical agreement with the experimental data. In the QE-Geant4 modelling a
complete loss of entanglement is assumed following the first DCSc
(predominantly in nylon and CZT for this event sample). Any subsequent gamma
interactions are modelled as for polarised, independent $\gamma$ (i.e. a
separable state).

The consistency between the prior-scatter data and the corresponding
simulation predictions indicates the assumption of wavefunction collapse
between Pa $\gamma$ following a scatter process is consistent with the data.
The magnitude of the $\Delta\phi$ correlation for prior-scatter events is also
clearly diminished compared to the unscattered case. As far as we are aware
the world’s current data on this process is contained in Fig 4, so clearly a
future measurement programme with improved statistical accuracy and a wider
range of scattering kinematics are a clear next step for the field. Such
measurements (although planned) are currently out of reach of the small
demonstrator system used here101010The scatter data in Fig. 4 required over a
month of data taking.

Figure 3: Geant4 visualisation of the experimental setup for the scattering
measurement. The CZT crystals (red) are shown along with their support
structures (grey) and the nylon scattering medium (purple). The topology of a
typical scatter event are shown by the solid lines. The initial back-to-back
trajectory of the two annihilation $\gamma$ from a single Pa event can be seen
as the green lines originating at the source position. One of the photons
Compton scatters in the nylon scattering medium (purple). The subsequent
Compton scatters of both $\gamma$ within the CZT crystals, from which the
$\Delta\phi$ correlation is obtained, are evident from the kinks in the photon
trajectories. Figure 4: $\Delta\phi$ distributions for events with a prior-
scatter process The black data points were obtained with the setup of Fig 3
and show the measured $\Delta\phi$ distribution when one of the entangled
$\gamma$ has scattered through a polar angle of $\sim$33∘ prior to the
detection in the CZT. QE-Geant4 predictions for this setup are shown as the
blue line. The red data points show the measured $\Delta\phi$ distribution
without scatterer, in a back-to-back configuration, for the same polar scatter
angle range. The red line shows the QE-Geant4 predictions corresponding to
this latter setup. All simulations and data employ a selection of polar
scatter angles (within the CZT) in the range
$60^{\circ}\leq{}\theta\leq{}140^{\circ}$.

###  1.2 Quantum entangled PET

The ability of the new QE-Geant4 to accurately describe the observed
$\Delta\phi$ correlations for DCSc of Pa photons offers new possibilities to
separate the true (entangled) PET events from the backgrounds arising from
scatter and random coincidences, exploiting the differences in the
$\Delta\phi$ correlations to identify their contribution to the image on a
statistical basis (note the correlations do not offer the possibility to
identify contributions on an event-by-event basis). To illustrate this
potential, PET images were produced from a QE-Geant4 simulation of a
preclinical PET acquisition. A scanner composed of four rings of CZT (with the
same pixel size as the demonstrator) was defined, along with a standard
preclinical mouse phantom NEMA-NU4 (National Electrical Manufacturers
Association)[35] (Fig. 5). The phantom consisted of a cylinder of tissue
equivalent poly(methyl methacrylate) (PMMA) with five capillaries (1-5 mm in
diameter) filled with a solution of water and an $e^{+}$ source.

Figure 5: Geant4 visualisation of the simulated preclinical PET scanner. The
scanner consisted of four rings of 10 mm cubic CZT detector crystals (orange).
The NEMA-NU4 phantom consisted of a cylinder of PMMA (white) and five rods
which were filled with a solution of liquid water and $e^{+}$ source (blue).
The simulated trajectories of 20 pairs of annihilation $\gamma$ are shown in
green.

As a first study of QE-PET imaging benefits, two-dimensional PET images were
reconstructed from the simulated data using a simple filtered back-projection
(FBP) algorithm[36] and are shown in Figs. 6a and 7a, corresponding to
simulations with scatter or random backgrounds respectively. Only coincidences
where both $\gamma$ have polar scattering angles in the CZT within
$67^{\circ}\leq\theta\leq 97^{\circ}$ were retained, as a compromise between
keeping the enhancement ratio high and maintaining good statistics. The FBP
images clearly exhibit the structure of the NEMA-NU4 phantom, showing activity
from the 5 capillaries. The intensity profiles for a region of interest
crossing the two lowest capillaries (seen in yellow in Figs. 6a and 7a) were
extracted for two bins of $|\Delta\phi|$, $0^{\circ}\leq|\Delta\phi|\leq
20^{\circ}$ and $80^{\circ}\leq|\Delta\phi|\leq 100^{\circ}$. The bins
correspond (respectively) to regions where low and high fractions of true
events are expected (cf. Fig. 2). The profile for the true events can be
obtained using a simple subtraction of the profiles from each $|\Delta\phi|$
bin, scaled with factors obtained from the QE-Geant4 simulation (see
supplementary materials). The results of this subtraction for data sets with
scatter (random) backgrounds are shown by the red lines in Fig. 6d (Fig. 7d).
Good agreement is observed with the “actual” true profiles (blue lines), which
are reconstructed exclusively from the true coincidences identified using _a
piori_ knowledge from the simulations. An equivalent methodology can be used
to extract the scatter and random distributions in isolation (Figs. 6c and
7c). The overall magnitude and shape of the QE-PET and “actual” profiles are
in good agreement for both scatter and random background scenarios. The
fluctuations in the extracted profile (more prominent in Fig. 6c) are not
statistical in nature and appear to be caused by artefacts in the FBP imaging
produced by incomplete acceptance of the array[37] 111111The general features
of these fluctuations are mirrored in the (statistically independent) profiles
from each $\Delta\phi$ bin and are therefore not dominated by statistical
effects. Such fluctuations have been observed previously in FBP[37] and post-
processing methodologies have been attempted[38, 39, 40]. Note that for these
initial assessments, additional processing of the FBP image was not applied..
In each case, a $4^{th}$ order polynomial fit to the profile is additionally
included to enable the average trend to be compared with the “actual”
distribution.

Even this simple illustrative method, which only uses a fraction of the
available data from two of the $\Delta\phi$ bins, along with information from
the entangled simulation, indicates the feasibility of quantitative assessment
of the shape and magnitude of the image backgrounds with QE-PET121212The
previous simulation studies of (non-entangled) polarised PET[34] can be
improved further using a quantum entangled simulation as described here..

##  2 Discussion

We should remark that the information from QE-PET, as illustrated above, would
be obtained in addition to the single-pixel yield routinely analysed to
produce PET images. The QE-PET derived information on the backgrounds would
equally well apply to these standard PET events. The results (Fig. 6 and Fig.
7) were obtained from a simulation of 1012 $e^{+}e^{-}$ annihilations, which
is of the same order of magnitude as the radiotracer cumulated activity in a
patient during a typical clinical PET scan (a few 100 MBq activity integrated
over an acquisition time of 30 minutes). The accuracy in the extracted
background profiles are therefore indicative of what may be achieved in a PET
scan for this simulated detector geometry131313In a real PET scan scenario, a
combination of random and scatter backgrounds are present in the data. In
applying QE-PET, their combined contribution would be extracted
simultaneously. However, the random coincidence rate can be determined with
the delayed window method[41], hence the contributions of scatter and random
coincidences could be separated if required.. We should also note that these
first proof-of-principle results employed restrictive $\theta$ and
$\Delta\phi$ cuts, thereby only using a fraction of the available data. In
future work, these cuts will be optimised to further improve the achievable
accuracy by accounting for the interplay between enhancement magnitude and
event yield, as explored previously[34, 42]. However, ultimately we view the
optimal use of the new information would be within the framework of more
sophisticated imaging methodologies such as the forward model of a MLEM
(maximum-likelihood expectation-maximization)[43] image reconstruction
algorithm. Currently such approaches model scatter-backgrounds employing
either: scatter simulation algorithms[44, 45], which require analytical
modelling of the scanner; or CPU intensive Monte Carlo methods. Both
approaches require detailed anatomical information from a computed tomography
(CT) scan and rely on estimates of the underlying activity biodistribution,
which is a priori unknown and necessitates iterative approaches[46, 8, 47].
Implementation of the new QE-PET information in such iterative imaging
methods[48] would be an important next step.141414Time-of-flight (TOF) PET
methodologies using fast photon detectors have recently been explored to
address backgrounds in PET[49]. The QE-PET method can be employed in parallel
with TOF information where available. However, for systems where such timing
information is unavailable or of insufficient resolution (e.g. semiconductor
detectors as studied here), then QE-PET would offer unique opportunities.

Figure 6: Extraction of true and scatter contributions. (a) FBP two-
dimensional PET image of the NEMA-NU4 phantom for true events with a scatter
background. (b) Intensity profiles through the region indicated by the yellow
rectangle for different $\Delta\phi$ cuts, i.e. $0^{\circ}\leq|\Delta\phi|\leq
20^{\circ}$ (green), and $80^{\circ}\leq|\Delta\phi|\leq 100^{\circ}$ (blue).
(c) QE-PET profile for scatter background events extracted from a scaled
subtraction of the two $\Delta\phi$ cut profiles (red line) (see text). The
blue line shows the profile from the “actual” scatter events in isolation
using _a priori_ information from the simulation. The dashed line is a
$4^{th}$ order polynomial fit to the extracted scatter profile. (d) Profile
extracted for true events with QE-PET (red line) compared to the profile of
“actual” true events in (blue line). Figure 7: Extraction of true and random
contributions. As for Fig. 6 but for true events with a random background.

##  3 Conclusion

We have simulated the predicted effects of quantum entanglement in the
interaction of positron annihilation photons with matter, building on the
Geant4 simulation framework. The new simulation predictions were validated by
comparison with precision experimental data on double Compton scattering of
positron annihilation photons, obtained with a CZT pixelated semiconductor PET
demonstrator apparatus. The inclusion of quantum entanglement in the
simulation for the reaction process was shown to be crucial to correctly
describe the measured correlation between the Compton scatter planes. As well
as underpinning the QE-PET developments, the new simulation will enable
improved accuracy in any future simulations of PET imaging systems in medical
imaging, medical research and industrial applications.

A modified apparatus, incorporating an additional scattering medium, enabled a
first measurement of the Compton scatter plane correlation when one of the
(initially entangled) annihilation photons underwent a Compton scatter process
prior to measurement. The experimental data are compatible with simulation
predictions which assume the collapse of the entangled wavefunction following
the scatter process.

The new simulation was also used to model a CZT based preclinical scanner and
obtain a first simulation study of entangled PET imaging. A method to quantify
(and remove) spatially-resolved backgrounds from both in-patient scatter and
random coincidences is presented, suggesting entangled PET provides new
possibilities to address key challenges for next-generation PET imaging.

We acknowledge input from D. Jenkins to the manuscript. Simulation work was
undertaken on the Viking Cluster, which is a high performance computing
facility provided by the University of York. We are grateful for computational
support from the University of York High Performance Computing service, Viking
and the Research Computing team. The work has been supported by funding from
Innovate UK EP/P034276/1 and the UK Science and technology Facilities Council
(STFC) ST/K002937/1

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##  4 Supplementary materials

###  4.1 Verification of QE-Geant4 by comparison with analytical theories of
quantum entanglement

We verified that the simulated effects of quantum-entanglement, implemented in
the QE-Geant4 simulation, were in agreement with the theoretical calculations.
This was assessed by comparing the normalised coincidence count rate as a
function of $\Delta\phi$ for the case of a perfect detector having an
effectively infinite detecting medium ($e^{+}$ source placed at the centre of
a $10\times 10\times 10$ m3 cube of CZT) and using the exact $\gamma$ Compton
scattering angles from the simulation. In addition, we compared the analytical
calculations from a non-entangled hypothesis with the standard Geant4
predictions. The entangled and polarised simulations were carried out with QE-
Geant4 and standard Geant4 10.5, respectively. For consistency with the
analysis presented in the paper, the same polar scatter angle range for the
$\gamma$ was adopted ($67^{\circ}\leq\theta\leq 97^{\circ}$). The obtained
$\Delta\phi$ distributions for the entangled (non-entangled) scenarios are
shown by the blue (red) data points in Fig. 8.

The corresponding theoretical predictions were obtained as follows. From
equation 2, the probability of two entangled $\gamma$ Compton scattering with
a relative azimuthal angle $\Delta\phi$ takes the form:

$P(\theta_{1},\theta_{2})=A(\theta_{1},\theta_{2})\cdot{}cos(2\Delta\phi)+B(\theta_{1},\theta_{1}).$
(3)

For clarity, we focus on $\theta_{1}=\theta_{2}=\theta$ annihilations and
define the enhancement factor ($Enh$) as the ratio of the perpendicular
($\Delta\phi=\pm 90^{\circ}$) and parallel ($\Delta\phi=0^{\circ}$ and
$\Delta\phi=\pm 180^{\circ}$) scattering probabilities:

$Enh(\theta)=\frac{P_{\perp}(\theta)}{P_{\parallel}(\theta)}.$ (4)

In table 1 (A) of Bohm and Aharonov[2], the parallel and perpendicular
scattering probabilities for entangled $\gamma$ are given by:

$P_{\parallel}(\theta)=2\gamma(\gamma-2sin^{2}\theta)\quad\mathrm{and}\quad
P_{\perp}(\theta)=(\gamma-2sin^{2}\theta)^{2}+\gamma^{2}$ (5)

where $\gamma=(k_{0}/k)+(k/k_{0})$, with $k_{0}$ and $k$ the wave numbers of
the incident and scattered photons, respectively. The resultant enhancement
factor is plotted as the blue line in Fig. 8. Note that Pryce and Ward[12] and
Snyder _et al._[10] determined the enhancement factor with different
formalisms yielding strictly the same values.

Figure 8: Theoretical enhancement factors. Calculated using the formalism of
Bohm and Aharonov[2] for entangled $\gamma$ (blue) and for orthogonally
polarised non-entangled $\gamma$ (red).

Similarly, line $B_{2}$ of Table 1 of Bohm and Aharonov calculates the
parallel and perpendicular scattering probabilities for orthogonally polarised
non-entangled $\gamma$ as:

$P_{\parallel}(\theta)=2\gamma^{2}-4\gamma{}sin^{2}\theta+sin^{4}\theta\quad\mathrm{and}\quad
P_{\perp}(\theta)=2\gamma^{2}-4\gamma{}sin^{2}\theta+3sin^{4}\theta.~{}$ (6)

The non-entangled analytical prediction of the enhancement is plotted as the
red line in Fig. 8. The widely reported maximum of 2.85 at 81.7∘ is only
achieved if the fully entangled treatment is employed. Neglecting the
entanglement significantly reduces the maximum enhancement to 1.63.

In order to calculate the normalised scattering probability as a function of
$\Delta\phi$ for both entangled and orthogonally polarised non-entangled
$\gamma$, we combine equations 3 and 4 and solve for $\Delta\phi=0^{\circ}$
and $\Delta\phi=90^{\circ}$ yielding:

$Enh(\theta)=\frac{B(\theta)-A(\theta)}{B(\theta)+A(\theta)}$ (7)

which rearranges to:

$\frac{A(\theta)}{B(\theta)}=\frac{1-Enh(\theta)}{1+Enh(\theta)}.$ (8)

Thus, if we integrate equation 4 to obtain the enhancement factor over our
range of interest ($Enh_{\Delta\theta}$) and take the normalisation $1/B$, we
can calculate the relative double-scattering probability as:

$\frac{P_{\Delta\theta}}{B_{\Delta\theta}}=\frac{1-Enh_{\Delta\theta}}{1+Enh_{\Delta\theta}}\cdot{}cos(2\Delta\phi)+1$
(9)

The results of these calculations for our two cases are plotted as the solid
lines in Fig. 9. Satisfactory agreement is observed between the simulation and
the theoretical predictions for each case, demonstrating that our
implementation of quantum mechanical entanglement into Geant4 is consistent
with theory, and that standard Geant4 does not fully account for this.

Figure 9: Check of the consistency of the predicted $\Delta\phi$ correlation
from simulation and the underlying quantum theory. Data points and curves show
the normalised scattering probabilities as a function of the relative
azimuthal scattering angle, $\Delta\phi$, between the two $\gamma$. The
analytical theoretical predictions (solid lines) are shown for entangled
(blue) and non-entangled orthogonally polarised (red) annihilation $\gamma$,
calculated according to the formulae described in Table 1 of the publication
of Bohm and Aharonov[2]. These are compared to QE-Geant4 simulation results
for the entangled $\gamma$ (blue data points) and using standard Geant4 for
the non-entangled $\gamma$ (red) scenarios. See text for details of the
simulation.

###  4.2 Further statistical analysis of the agreement between experimental
data and simulations

In Fig. 10, we present a bin-by-bin determination of the magnitude of the
difference (the residual) between the experimental data and the simulation
predictions presented in Fig. 2. The QE-Geant4 simulation (blue data points)
matches the shape of the experimental data well over all $\Delta\phi$, with
residuals distributed about zero. The standard Geant4 simulation (red data
points) however produces large residuals, failing to reproduce the amplitude
of the experimentally observed $cos(2\Delta\phi)$ distribution. Furthermore,
we have calculated $\chi^{2}/\nu$ values of 1.87 and 42.8 for the (expt. data
- QE-Geant4) and the (expt. data - Geant4), respectively.

Figure 10: Residuals plot comparing the experimental data with simulations.
The bin-by-bin residuals (see text) obtained from a comparison of the
experimental data with the simulation predictions (all analysis criteria as in
Fig. 2). The residuals for Geant4 (red) and QE-Geant4 (blue) simulations are
both presented. The error bars are the quadrature sums of the statistical
error of the experimental data and simulation for each bin.

###  4.3 The adopted PET imaging method

A scanner consisting of an extension of our experimental PET-demonstrator
apparatus was implemented in QE-Geant4 (Fig. 5). To match the real detector
position resolution of the experimental setup, the coordinates of the
simulated energy deposits were smeared according to a Gaussian distribution
with $\sigma=0.8$ mm$/\sqrt{12}$ . QE-PET event data, including $\theta$ and
$\Delta\phi$, were stored in a list-mode file. They were sorted into three
sets: the first with no condition on $|\Delta\phi|$, the second and the third
contained only events within $0^{\circ}\leq|\Delta\phi|\leq 20^{\circ}$ or
$80^{\circ}\leq|\Delta\phi|\leq 100^{\circ}$ windows, respectively. Image
reconstruction was then performed, based on the methodology implemented in
GAMOS (Geant4-based Architecture for Medicine-Oriented Simulations)[50]. For
each set, data were histogrammed into sinograms with the “lm2pd” utility.
Images were reconstructed with the implementation of the single-slice
rebinning FBP (SSRB-FBP2D) algorithm[51]. A ramp filter was applied. Pixel
size was set to $0.6\times 0.6$ mm2. Images were processed with the NucMed
plugin of ImageJ[52], as recommended by the GAMOS users guide.

###  4.4 QE-PET method used to extract scatter and random contributions

The coincidence count rate varies as a function of the relative azimuthal
scattering angle $\Delta\phi$ (see for example, equation 2 and Fig. 2). For a
given $\Delta\phi$, the total number of coincidences can be considered as the
sum of true and scattered coincidences, i.e. $T(\Delta\phi)+S(\Delta\phi)$, as
illustrated in Fig. 11. For a given range of $\Delta\phi$, the total number of
coincidences can be expressed as a linear sum of the mean number of _trues_
and _scatters_ , weighted by appropriate scaling factors, e.g.

$P_{90}=t_{90}\cdot{}\bar{T}+s_{90}\cdot{}\bar{S}\quad\mathrm{and}\quad
P_{0}=t_{0}\cdot{}\bar{T}+s_{0}\cdot{}\bar{S}$ (10)

where $P_{90}$ and $P_{0}$ are the total number of coincidences in the
$|\Delta\phi|$ windows centered around $90^{\circ}$ and $0^{\circ}$,
respectively. $t_{90}=T(90^{\circ})/\bar{T}$ and $s_{0}=S(0^{\circ})/\bar{S}$
etc. These latter terms account for the small residual $\Delta\phi$ dependence
for the scatter coincidences161616From our fits to simulation, the residual
enhancement for scatter coincidences is $Enh=1.207$, compared to 2.541 for the
true coincidences. arising from any remaining correlation in the polarisation
planes of the detected $\gamma$171717We should remark that the size of this
residual enhancement is modelled assuming the expected collapse of the
entangled state, as supported by the results presented in Fig 4. If future
developments in our understanding of entanglement breaking at the MeV scale
result in changes in the residual $\Delta\phi$ dependence then scaling factors
can be extracted using the same method but with a simulation accounting for
such effects. Combining the two equations of 10, we can express the mean
scatter contribution, $\bar{S}$, purely in terms of a linear combination of
$P_{90}$ and $P_{0}$, i.e.

$\bar{S}=\frac{t_{90}\cdot{}P_{0}-t_{0}\cdot{}P_{90}}{t_{90}\cdot{}s_{0}-s_{90}\cdot{}t_{0}}.$
(11)

Figure 11: Illustration of the method to extract the scatter contribution.
Illustration of the different contributions to the total measured coincidences
(green), from true coincidences (blue) and scatter events (red). The y-scale
is arbitrary, and the relative number of scatter events and the amplitude of
their $cos(2\Delta\phi)$ dependence have been exaggerated for illustrative
purposes. Relevant features used to extract the scatter contribution from a
combination of two $\Delta\phi$ slices are indicated (see text).

The coefficients $s_{0},s_{90},t_{0}$ and $t_{90}$ are extracted from fits to
simulated QE-Geant4 data. If for $P_{0}$ and $P_{90}$ we use intensity
profiles extracted from images generated with different cuts on
$|\Delta\phi|$, we can use this methodology to extract a profile containing
only the scatter contribution to the profile. A similar method can be applied
to extract the _true_ coincidences. The process for extraction of random
background is identical, except that the coefficients $s_{0}$ and $s_{90}$ are
set to unity as the random coincidence count rate is independent of
$\Delta\phi$.

Equation 11 highlights that this method is independent of the relative numbers
of true and scatter coincidences; this information is contained within the two
$\Delta\phi$ slices. The only input required is the dependence of these
components on $\Delta\phi$. For random events, this is constant while for
scattered events there is a small residual $cos(2\Delta\phi)$ modulation. In
the results, we adopted _global_ coefficients integrated over the full
phantom. To test this assumption, we compared the $\Delta\phi$ dependence in
different regions of the phantom and found that the amplitude remained
consistent within statistical uncertainties ($\sim 5\%$). Further checks were
obtained from the extracted true and scatter contributions for profiles from
different regions of the phantom, while using a single set of _global_
coefficients for the residual scatter profile. The method was found to perform
satisfactorily in all cases studied. As an example, in Fig. 12 we present
profiles through the middle two capillaries. As was observed in Fig. 6, the
agreement between the extracted and “actual” true profiles is excellent. The
underlying shape of the scatter profile is different to that seen in Fig. 6c
and in good agreement with the “actual” scatter profile.

Figure 12: Extraction of true and scatter contributions from middle
capillaries. (a) FBP two-dimensional PET image of the NEMA-NU4 phantom for
true events with a scatter background. (b) Intensity profiles through the
region indicated by the yellow rectangle for different $\Delta\phi$ cuts, i.e.
$0^{\circ}\leq|\Delta\phi|\leq 20^{\circ}$ (green), and
$80^{\circ}\leq|\Delta\phi|\leq 100^{\circ}$ (blue). (c) QE-PET profile for
scatter background events extracted from a scaled subtraction of the two
$\Delta\phi$ cut profiles (red line). The blue line shows the profile from the
scatter events in isolation using the information from QE-Geant4. The dashed
line is a $4^{th}$ order polynomial fit to the extracted scatter profile. (d)
Profile extracted for true events with QE-PET (red line) compared to the
profile of true events in isolation using QE-Geant4 (blue line).