 ref. [18]). We use the values of S for helium calculated by Brown [19], since the total ground state energy of the wave functions used to derive S only differ by 0.02% from the most accurate ground state energy values. For incident energies above the K-shell ionization threshold, around 90 percent of scattering occurs from K-shell electrons [18]. If both electrons of each scatterer is ejected from the K-shell, it is reported that the dependence of their angular distribution on the relative azimuth η is proportional to sin 2 η, which is maximum for perpendicular azimuths and zero for coplanar azimuths. Therefore, it is expected that some 65 percent of coincidences will correspond to both electrons originating in the K-shell. One may therefore expect to also observe these azimuthal correlations in Compton scattering experiments [20].

Theoretical precision of angular distributions of hard X-rays
The scattering function S is based on the impulse approximation (IA) model [21]. In this model, bound electrons are replaced by free electrons with a momentum distribution determined by the bound electron wave functions [18]. The validity of the IA approximation can be understood in terms of the uncertainty relation in which it is difficult to distinguish between bound and free electrons due to the short time scale of a Compton interaction, which is no greater than 10 −11 seconds [18,22]. The electron binding energy E b of an electronic shell is included in the IA model only insofar that the binding energy places a kinematic limit on the lowest incident photon energy that can inelastically scatter from a bound electron [18]. The binding energy E b also places a limit on the highest scattered photon energy E max that can be observed for a given photon incident energy E o. This limit is E max = E o -E b [18,23]. Taking into account electron binding energy, the scattered photon energy E is defined by the Compton formula where all energies are defined in units of 511 keV (i.e., mc 2 ). For a single photon beam scattering off electrons from a many electron system, it has been shown that the IA model is accurate to order of [23,24]. The fractional precision of the theoretical incoherent cross section, defined here as A(E oi, E os, where the energies of the scattered photons E i and E s are defined using Eq. (8) such Therefore, the incoherent cross sections for Ψ ± l and Φ ± l which takes into account the theoretical precision are expressed as follows where the theoretical uncertainty of the cross section ∆σ (IA) for the states Ψ ± l and Φ ± l are defined, respectively, as (12b) The theoretical precision is improved as ∆σ (IA) → 0, which can happen in one of two ways: If one is constrained to use a particular scattering medium, then the incident photon energy is chosen such that E o >> E b. Or, if one is constrained to a given photon energy E o, then a scattering medium with sufficiently low binding energy is chosen such that E b << E o.

The counting rates N ⊥ and N
Compton scattering of polarization entangled photons has been studied in the past by measuring a quantity known as the azimuthal ratio. This involves a coincidence measurement of both photons scattered through the same angle, i.e., θ s = θ i = θ [20,25]. The geometrical arrangement to measure the ratio of cross-polarized Bell states when the wave vectors are not colinear is illustrated schematically in Fig. (2). In this arrangement, the photon counter of the scattered idler is fixed at φ i = 0 o, such that only idler photons scattered in the trajectory plane are detected. Using Eqn. (5), this implies η = φ s. The quantity N ⊥ is defined as the coincidence count rate when the photon counter of the signal is set at φ s = 90 o, such that η = η (max) = 90 o. The quantity N is the coincidence count rate when the signal photon counter is set to φ s = 0 o, and therefore η = η (min) = 0 o, and the ratio of these quantities is then determined. The count rates N ⊥ and N are proportional to the incoherent cross sections, which can be expressed as follows and Before evaluating the ratios, which is done in the next two sections, it is instructive to plot Eqn. (13a) and (13b) and examine both their structure and the theoretical precision as a function of θ. These cross sections, normalized over the classical electron  (4) and (11) it can be shown that and  The angle ∆ϑ is = ϑ i − ϑ s is the difference in phase-matching angles, which were introduced in Sec. (2). As shown in Fig. (2), ∆ϑ is is the angle subtended between k s and k i. An energy only parameterized azimuthal ratio is defined for coincidence measurements in which η is independent of ∆ϑ is. The setup shown in Fig. (2) satisfies this definition. Therefore, the expected count rates N ⊥ and N as a function of θ will be invariant to changes in ∆ϑ is, which implies that the ratio of these quantities will be invariant as well. The expected energy only dependent ratio for cross-polarization entangled states that takes into account the theoretical precision is defined as follows

Energy parameterized azimuthal ratios
where we have applied error propagation methods to determine the theoretical precision. The subscript nd in R denotes 'nondegenerate' energies. Using Eqn. (3a) and Eqn. (4a) evaluated at η (max) = 90 o and η (min) = 0 o, R nd is given by where the scattering function S cancels due to the definition of the incoherent cross section given in Eqn. (3). In the case for degenerate energies, i.e., E os = E oi = E o, Eqn. (16) simplifies to where the subscript 'd' denotes 'degenerate' energy. Using equations of (14), the energy dependent ratios for parallel-polarization entangled Bell states are related to the ratios for cross polarization entangled states as follows The ratio R for E o = (12.5 keV)/(511 keV) = 0.0245 has been calculated and is shown in Fig. 4(a). The thickness of the curve gives an estimate of the theoretical precision. The ratios R (Ψ ± l ) nd and R (Φ ± l ) nd have also been calculated for the case when E os = 1.2E o (i.e., E os = 15 keV) and E oi = 0.8E o (i.e., E oi = 10 keV), and are shown in Fig. 4(b). As can be seen, both Fig. 4(a) and 4(b) show a significant asymmetry in the coincidence counting rates at θ ≈ 89.98 degrees, where the degenerate and nondegenerate ratios reach a peak value of 1713.04 and 1650.26, respectively. These peak values are approximately 600 times larger compared to the value of 2.85 predicted for 511 keV cross-polarized Bell states [20,25].

Azimuthal ratios parameterized by the incident energies and phase-matching angles
We now propose a coincidence measurement of the azimuthal ratio parameterized in terms of incident energies E os and E oi, and the difference in phase matching angles ∆ϑ is = ϑ i − ϑ s. (The phase matching angles used here are obtained from [13]). The geometrical arrangement is illustrated schematically in Fig. (5). As can be seen, the photon counter measuring the scattered signal photons is set at the same two positions as in Fig. (2). However, in this scheme, photon counter (1) is now fixed at right angles to the trajectory plane (i.e., φ i = 90 o ), rather than parallel to it, which was the case for energy parameterized ratios discussed in Sec. (6). Using Eqn. (5), we find that cos η = cos ∆ϑ is sin φ s. When photon counter (2) is at position (1), (φ s = 0 o ), the two scattering planes of the signal is perpendicular to that of the idler, and therefore, (η = η (max) = 90 o ). When the photon counter (2) is set at position (2), (φ s = 90 o ), it can be shown that η (min) = ∆ϑ is. We now define ρ nd (θ; E oi, E os, ∆ϑ is ) as the energy and phase-matching parameterized azimuthal ratio for the state Ψ ± l, such that and applying the equalities found in Eqns. (14), we can define the ratio for the state Equations (19) has been calculated and plotted in Fig. (6

Conclusion
Technology can now produce all 4 Bell states with sufficient incident energies required for Compton scattering experiments. Relative to 511 keV cross-polarized photons, and (c): Log-log plot of the inchorent differential cross sections used to calcuate the ratios in Fig. (a). (Photon energies and phase-matching angles are obtained from reference [13].) hard X-ray Bell states with degenerate or non-degenerate incident energies show a significant amplification in polarization correlations, making them amenable to experimental verification, provided a suitable scattering medium is selected. When all 4 Bell states are considered, relationships in the cross sections and ratios are found between cross and parallel-polarization entangled states. In addition, the azimuthal ratio can be further parameterized in terms of the phase matching angles of Bell state photons generated in parametric down-conversion processes. In principle, these mathematical relationships and phase-matching constraints each