## Introduction

Positron emission tomography (PET) is a valuable tool for the in-vivo imaging of cellular and molecular processes. PET can provide high sensitivity and quantitative information on disease development and, more recently, therapy response. The PET information is usually complemented by purely anatomical information obtained using 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 γ-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 γ are predicted to be in 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. Such quantum-entangled correlations have recently shown imaging benefits for optical (~eV) photons3,4,5 and X-ray photons (~10 keV)6.

For the current study we use the linear polarisation of the γ as the experimental observable sensitive to the entangled nature of the PET photons, with visibility achieved through observation of the double Compton scattering (DCSc) process. The two γ from (ground state) para-positronium annihilation (anti-parallel spins; S = 0, Sz = 0 where S is the spin of the positronium) have orthogonal linear polarisation with an entangled wavefunction which can be expressed as

$$\left|{{\Psi }}\right\rangle =\frac{1}{\sqrt{2}}\ \left({\left|x\right\rangle }_{-}{\left|y\right\rangle }_{+}-{\left|y\right\rangle }_{-}{\left|x\right\rangle }_{+}\right),$$
(1)

where $${\left|x\right\rangle }_{-}$$ and $${\left|y\right\rangle }_{-}$$ represent a γ propagating along − z with polarisation in x and y, respectively. $${\left|x\right\rangle }_{+}$$ and $${\left|y\right\rangle }_{+}$$ 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 photons2,7,8.

The DCSc differential cross section of the entangled annihilation gamma in the allowed Bell state of Eq. (1), is given by8,9

$$\frac{{{\rm{d}}}^{2}{\sigma }_{\text{double}}}{{\rm{d}}{{{\Omega }}}_{1}{\rm{d}}{{{\Omega }}}_{2}}=\frac{{r}_{0}^{4}}{16}\ \left({K}_{a}({\theta }_{1},{\theta }_{2})-{K}_{b}({\theta }_{1},{\theta }_{2})\cdot \,{\text{cos}}\,(2{{\Delta }}\phi )\right),$$
(2)

where dΩ1,2 and θ1,2 are the solid angles and polar scattering angles for γ1 and γ2, respectively, r0 is the classical electron radius, Ka and Kb are kinematic factors (see Supplementary Note 1) and Δϕ = ϕ1 − ϕ2 is the relative azimuthal scattering angle (see Fig. 1 for the definition of scatter angles). The form of Eq. (2) is presented in the publication of Pryce and Ward9 (also independently derived by Snyder et al.7). In these early works, consistent results were obtained when employing time-dependent perturbation theory or a Klein-Nishina10 based approach, as outlined in the PhD thesis of Ward8. Subsequently, the same form has been derived in DCSc calculations in a matrix formalism11 and employing Kraus operators12. The DCSc cross section is modulated by the cos(2Δϕ) term, resulting in an enhancement ratio (R) between the maximum (Δϕ = ± 90) and minimum (Δϕ = 0) scattering probabilities of R = 2.85, achieved when both θ1 and θ2 are equal to 81. 77,8,9. Bohm and Aharonov2 were the first to recognise that the (Δϕ) correlations between the DCSc annihilation photons were an example of the kind of entanglement discussed by Einstein, Podolsky and Rosen13. They also derived2 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 witness of the entanglement2,11,12.

Previous measurements of the Δϕ amplitude in DCSc of positron annihilation (Pa) photons14,15,16,17,18,19,20,21 focused on restricted kinematics of θ1, θ2 around maximum visibility, and yielded R values well beyond the upper limit for a non-entangled state. Clear statistical agreement with the entangled theory (eq. (2)) was established (with analytic corrections for experimental aspects of the measurement). The most precise experiments were carried out by Langhoff15 and Kasday et al.18 giving measured enhancement factors of R = 2.47 ± 0.07 and R = 2.33 ± 0.10 respectively, in agreement with eq. (2) when geometrical effects are accounted for. The consistency of the measured R was established19,20 for separation distances up to 2.5 m, greatly exceeding the coherence length associated with Pa19 (0.12 m) and larger than the diameter of a typical clinical PET scanner (0.8–1.3 m). Further discussion of the previous measurements and the appropriateness of the entanglement formalism adopted here is presented in Supplementary Note 3.

The ability of the entangled cross section (eq. (2)) to describe the Δϕ 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. The most useful events collected to form a PET image (true events) have a LOR which crosses the annihilation site (no interaction in the patient). These true events are expected to maintain their quantum-entangled nature. However, 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 image22,23, decrease the signal-to-noise ratio 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 γ scatter prior to detection. In addition to a displaced LOR, such decohering scatter will lead to entanglement loss for the subsequently detected photon pair. The scatter-to-true ratios range from ~0.2 for brain imaging to ~2 for 3D abdominal imaging24. The random background originates mainly from uncorrelated γ 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 ~0.1 (brain imaging) to more than 124, influenced by the detector properties (e.g the timing coincidence window for the accepted γ photons) and the administered activity.

In this work, we implement the entangled description of the interactions of Pa photons into the comprehensive Geant425,26 particle simulation framework. The new simulation (QE-Geant4) is tested through a comparison with double Compton scattering of 22Na Pa gamma, measured by two state-of-the-art cadmium zinc telluride (CZT) semiconductor γ detectors, placed in a back-to-back PET configuration. We also provide a measurement of the diminished DCSc correlation for the case where one photon in the pair has scattered prior to measurement, achieved using a modified (non-back-to-back) setup. The benchmarked QE-Geant4 simulation is used to model a preclinical PET imaging apparatus, formed from an array of CZT detectors. This quantum-entangled PET (QE-PET) simulation indicates a spatially resolved extraction of the shape and magnitude of scatter and random contributions to PET images is achievable, using only the entanglement information contained in the data and information from the QE-Geant4 simulation.

## Results

### Comparison of simulated and measured Δϕ distributions

We implemented the entangled description of the interactions of Pa photons into the comprehensive Geant425,26 particle simulation framework (see methods), which enables full account of detector geometry, experimental resolutions and backgrounds. The simulation (QE-Geant4) was tested by comparing to measurements of DCSc of Pa photons in a PET-demonstrator apparatus. The system was developed by the Kromek Group based on the DMatrix detector system27 and comprised two 10 mm cubic semiconducting CZT crystals, placed back-to-back and separated by 87 mm. A segmented anode divided each crystal into 121 0.8 × 0.8 mm2 pixels, with depth information accessible from the anode drift time28. A 170 kBq 22Na source, comprised of a 1 mm diameter active bead housed in a thin plastic housing, was placed equidistant from each crystal face providing a source of positron annihilation γ.

Events in which the two annihilation γ undergo DCSc in the CZT were identified from the yield having two hit clusters in each head (see methods). The normalised (see caption) coincidence count rate for DCSc as a function of Δϕ for the event sample is shown as the data points in Fig. 2. A strong cos(2Δϕ) modulation is seen with a measured enhancement factor of R = 1.85 ± 0.04 for the event yield selected by the analysis cuts viz. polar scatter angles 70θ≤110 and summed energy 480−530 keV. Employing a tighter restriction on polar scattering angles (θ1,2 = 93−103) gave a measured enhancement factor R = 1.95 ± 0.07.

To directly compare the QE-Geant4 simulation with experimental data the predicted energy deposits in the CZT detector heads were matched to the experimentally determined CZT resolutions (see methods) and then passed through the same data analysis code and cuts as the experimental data. The predictions from the QE-Geant4 simulation are shown by the blue line in Fig. 2 and show a good agreement with the measured Δϕ distribution (χ2/ν = 1.87). The agreement provides a validation of the QE-Geant4 simulation for describing DCSc and confirms that the entangled theory (Eq. (2)) is consistent with the experimental data (the agreement on a bin-by-bin basis is presented in Supplementary Note 4).

To better quantify the predicted effects of entanglement on the measured Δϕ amplitude, a further simulation of (hypothetical) non-entangled, but orthogonally polarised, annihilation γ is also presented (red line). The DCSc for this case is modelled by the standard Geant4 polarised Compton scattering classes (the consistency between standard Geant4 and the theoretical predictions2 for such a (hypothetical) non-entangled state is shown in Supplementary Note 3). The size of the predicted Δϕ modulation is clearly reduced if entanglement is neglected, and the predictions are in clear disagreement with the experimental data. We remark that earlier standard Geant4 simulations of polarisation effects in PET29,30, could be further developed with this QE-Geant4 framework.

A further simulation for unpolarised annihilation γ 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 Aharonov2 and Caradonna et al.11. The predicted Δϕ 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 Δϕ distribution only has modest influence from detector acceptance effects.

The results in Fig. 2, and previous measurements in more limited kinematics from a range of different Pa sources2,7,9,14,15,16,17,18,19,20,21, show that the azimuthal correlation of the Compton scatter planes in DCSc of Pa photons is in agreement with the entangled theory (Eq. (2)) and has a measured enhancement factor (R) beyond the upper limit of a separable non-entangled state.

### Investigation of entanglement loss

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 Δϕ 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 Δϕ correlation (prior-scatter events) were obtained by inclusion of a scattering medium (nylon) in the path of one of the annihilation γ, with the corresponding CZT module rotated through 33 relative to the centre of the scatterer. The measured energy of the scattered γ in the CZT (obtained from the sum of the two energy deposits) matched that expected from the reaction kinematics (~440 keV) and was well separated from backgrounds.

The DCSc Δϕ 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) Δϕ correlations for back-to-back unscattered γ respectively, with the same binning as used for the prior-scatter data. The measured Δϕ correlation for the prior-scatter events is clearly diminished compared to the unscattered case. The QE-Geant4 prediction (blue line) also exhibits a diminished Δϕ correlation, in statistical agreement with the experimental data. In the QE-Geant4 modelling (see Methods) a complete loss of entanglement is assumed following the first DCSc (predominantly in nylon and CZT for this event sample) and any subsequent gamma interactions are modelled as for polarised, independent γ (i.e. a separable state). As far as we are aware the world’s current data on this process for positron annihilation is contained in Fig. 4. A future measurement programme with improved statistical accuracy and a wider range of scattering kinematics would be a clear next step for the field. Such measurements (although planned) are currently out of reach of the small PET demonstrator system used here (the data in Fig. 4 required over a month of data taking).

### Quantum-entangled PET

The ability of QE-Geant4 to accurately describe the observed Δϕ correlations in DCSc offers possibilities to separate the true (assumed entangled) PET events from backgrounds of scatter and random events. The method exploits the differences in the Δϕ correlations to identify their relative 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 crystal size as the demonstrator) was defined, along with a standard preclinical mouse phantom NEMA-NU4 (National Electrical Manufacturers Association)31 (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.

To study the QE-PET imaging benefits, two-dimensional PET images were reconstructed from the simulated data using a simple filtered back-projection (FBP) algorithm32 and are shown in Figs. 6a and 7a, corresponding to simulations with scatter or random backgrounds respectively. Only coincidences where both γ have polar scattering angles in the CZT within 67θ≤97 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, 7a) were extracted for two bins of Δϕ, 0Δϕ≤20 and 80Δϕ≤100. 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 Δϕ bin, scaled with factors obtained from the QE-Geant4 simulation (see Supplementary Note 5). 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 priori knowledge from the simulations. An equivalent methodology can be used to extract the scatter and random distributions in isolation (Figs. 6c, 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 array33. In each case, a 4th 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 Δϕ bins, along with information from the entangled simulation, indicates the feasibility of quantitative assessment of both the shape and magnitude of the image backgrounds with QE-PET. Further QE-PET analysis of the extracted profiles through the middle capillaries of the phantom is presented in Supplementary Fig. 5.

## 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 (Figs. 6, 7) were obtained from a simulation of 1012e+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 is therefore indicative of what may be achieved in a PET scan for this simulated detector geometry. We should also note that these proof-of-principle results employed restrictive θ and Δϕ 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 previously30,34. However, ultimately we view the optimal use of the information would be within the framework of more sophisticated imaging methodologies such as the forward model of a MLEM (maximum-likelihood expectation-maximization)35 image reconstruction algorithm. Currently such approaches model scatter backgrounds employing either: scatter simulation algorithms36,37, 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 approaches23,38,39. Implementation of the QE-PET information in such iterative imaging methods40 would be an important next step. Time-of-flight (TOF) PET methodologies using fast photon detectors have recently been explored to address backgrounds in PET41. 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 or compact PET), then QE-PET would offer unique opportunities. The PET study presented here corresponds to a preclinical PET scanner with a small-animal equivalent phantom. The scatter probability from the small-animal phantom, obtained from the QE-Geant4 simulation, is 15% providing a test of the QE-PET technique for the challenging scenario of small scatter backgrounds. Human PET, as referenced in the introduction, provides larger scatter contributions of up to 67% and should be more amenable to the method proposed.

In summary, 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 QE-Geant4 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 gave a good description of the measured correlation between the Compton scatter planes, while predictions based on a (hypothetical) non-entangled state could not describe this correlation. As well as underpinning the quantum-entangled PET (QE-PET) developments, the simulation will enable improved accuracy in any future simulations of standard PET for medical imaging, medical research or industrial applications. Additionally, the framework offers possibilities for further fundamental tests of entanglement at the MeV scale. 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 in statistical agreement with simulation predictions which assume the collapse of the entangled wavefunction following the scatter process. The QE-Geant4 simulation was also used to model a CZT based preclinical scanner and obtain a simulation study of QE-PET imaging. A method to quantify and remove backgrounds from both scatter and random coincidences is presented, suggesting quantum-entangled PET provides possibilities to address key challenges for next-generation PET imaging.

## Methods

### Implementation of entanglement in Geant4 simulation

We incorporated entanglement (as described by Eq. (2)) into Geant4, by developing routines which enable communication between the individual particle-tracking processes to reproduce the scattering cross sections for entangled γ pairs. The simulation framework, which we refer to as QE-Geant4, will be included in future releases of the code. For the current study QE-Geant4 was implemented into Geant4 version 10.5. The non-entangled and unpolarised predictions were obtained using an unmodified version 10.5.

The implementation of the entanglement is as follows. The standard physics routines in Geant4 for modelling Compton scattering of polarised gamma use the polarised Klein-Nishina theory embedded within the “Livermore” physics package42,43. In QE-Geant4 this modelling is adopted for all photon tracking processes, other than the first DCSc for which the entangled formalism is applied according to Eq. (2). The two Pa gammas (γ1, γ2) are processed sequentially: γ1 is tracked until it is destroyed using the standard Geant4 processes, but the kinematics of its first Compton scatter are stored and made accessible to the tracking processes of other particles. For γ2 the modelling of its (first) Compton scatter is chosen according to the distribution of Eq. (2), using the stored Compton scatter information from γ1. Any subsequent Compton scatter processes for either gamma revert to standard Geant4, i.e. the first DCSc process is assumed to completely collapse the entangled state of Eq. (1), with any subsequent interactions of the γ modelled as independent and separable photons. Supplementary Fig. 6 shows the entanglement implementation (discussed above) as a flow diagram.

A check of the QE-Geant4 simulation predictions through a comparison with the underlying analytic entangled theory7,9 is presented in Supplementary note 2. The consistency between the standard Geant4 simulation predictions and the analytic theory for DCSc of a (hypothetical) non-entangled state2 is also presented.

### The CZT PET demonstrator apparatus

A photo of the experimental setup is shown in Fig. 8. The properties of the CZT directly relevant to the analysis are included in the main text. The readout of the anode and cathode signals from the CZT employed a bespoke Application Specific Integrated Circuit (ASIC). The coincidence timing window for the system was set to 1 μs due to the charge sweep-out time of a few tenths of μs.

### Identification of double Compton scatter events

Measurements of Δϕ were accessible for events where both annihilation γ 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 interaction. Simulations indicate that this assignment will be correct in ~ 60% of cases. Incorrect assignment will result in a calculated polar scattering angle θcalc. = 180 − θreal, but not affecting Δϕ due to the symmetry about 0. To enhance the amplitude of the cos(2Δϕ) distribution (Eq. (2)), coincidences were only retained if the polar scattering angle was within 70θ≤110. 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 (for the most distant pixels) to 20.4 for the closest pixels considered in this analysis. The energy and θ 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.

We remark that future higher statistics data would enable the selection of a sub-set of Compton scatter events with improved azimuthal angular resolution (larger inter-cluster separations) than the data included in Fig. 2. This would be expected to increase the visible enhancement factor R accessible from the CZT apparatus.

### Implementing experimental resolutions in the simulation

The CZT experimental setup was simulated with QE-Geant4. To allow a direct comparison of simulated and experimental data the simulated energy deposits were matched to the CZT experimental detector response by accounting for effects of charge transport, diffusion and self-repulsion44, and smeared to match the experimental energy resolution of each detector module (3.8 and 5.3% FWHM at 662 keV). The location of the QE-Geant4 predicted energy deposits were convoluted with the position resolution of the CZT detectors, with coordinates of the simulated energy deposits smeared according to a Gaussian distribution with $$\sigma =\frac{0.8\,{\rm{mm}}}{\sqrt{12}}$$. The resulting simulated data were then passed through the same analysis code and cuts as for the experimental data.

### Details of the PET image extraction method

A scanner, comprising of an array of the CZT detectors used for the PET-demonstrator apparatus (see Fig. 5) was implemented in QE-Geant4. The predicted energy deposits were matched to the experimentally observed resolutions of the CZT detectors as described above. The simulated QE-PET event data were stored in a list-mode file, including the θ1,2 and Δϕ information for each event. The PET events were sorted into three groupings according to their event type: the first group had no condition on Δϕ, the second and the third contained only events having 0Δϕ≤20 or 80Δϕ≤100, respectively.

Image reconstruction was then performed for each data group, using the Filtered Back Projection (FBP) methodology implemented in GAMOS (Geant4-based Architecture for Medicine-Oriented Simulations)45. For each data group, the PET events were histogrammed into sinograms with the “lm2pd” utility. Images were reconstructed with the implementation of the single-slice rebinning FBP (SSRB-FBP2D) algorithm46. A ramp filter was applied. Pixel size was set to 0.6 × 0.6 mm2. Images were processed with the NucMed plugin of ImageJ47.

We note that the fluctuations in the extracted profiles (evident in Figs. 6c, 7c) have general features which are mirrored in the (statistically independent) profiles extracted for each Δϕ bin (before they are subtracted to extract the profile). We therefore take these fluctuations not to be dominated by statistical effects. Such fluctuations have been observed previously in FBP33 and post-processing methodologies have been attempted48,49,50. Note that for the initial assessments presented in the current work, no additional processing of the FBP image was applied.