## 3 A Partially Entangled Double-Slit Experiment  
In this section, we shall construct a model for a partially entangled double-slit experiment. Following Sec. III of [1], we start from (4), substitute \(h_{1}=h_{2}=h\), construct an \"opposite\" wavefunction and take a normalized superposition of the two, yielding  
\[\psi_{\theta}\left(x_{1},x_{2}\right)=\frac{A_{\theta}}{A}\left[\psi\left(x_{ 1},x_{2}\right)\cos\theta+\psi\left(x_{1},-x_{2}\right)\sin\theta\right], \tag{5}\]  
where \(\theta\in\left[0,\frac{\pi}{4}\right]\) is a parameter controlling the amount of entanglement (note the entanglement strength decreases as \(\theta\) increases), and  
\[\frac{1}{A_{\theta}^{2}}=\frac{\pi}{a}\left[1+2e^{-2 ah^{2}}\sin\left(2\theta \right)+e^{-4 ah^{2}}\right]. \tag{6}\]  
Propagation of a wave from the near to the far field transforms an initial state into its Fourier transform. Thus, the position wavefunction in some time \(t\) such that the wave has propagated to reach a screen located in the far field domain, is proportional to the initial momentum wavefunction. This implies the momentum wavefunction may be used to define and compute the interferometric quantities we wish to consider (i.e., one- and two-particle visibility). The momentum wavefunction and related derivations appear in Appendix B.1. The momentum probability density is  
\[f_{p_{1},\theta}\left(p_{1}\right)=\sqrt{\frac{2\pi}{a}}\pi B_{\theta}e^{- \frac{p_{1}^{2}}{2a}}\left(\left[1+e^{-2 ah^{2}}\cos\left(2 hp_{1}\right)\right] +\sin\left(2\theta\right)\left[e^{-2 ah^{2}}+\cos\left(2 hp_{1}\right)\right] \right). \tag{7}\]  
Taking a close look at (7), we notice that this probability density function is a sum of two parts: one which is independent of \(p_{1}\) (up to normalization), and an oscillating part, with amplitude proportional to \(\sin\left(2\theta\right)\). This allows us to write down expressions for the \"envelopes\" of these oscillations  
Figure 1: The proposed “double-double slit” setup: each particle of an entangled two-mode Gaussian state \(\psi_{\theta}\left(x_{1},x_{2}\right)\) (5) is sent through a double slit with a screen behind it. The amount of quantum entanglement between the two particles is controlled by a parameter \(\theta\in\left[0,\frac{\pi}{4}\right]\). The experiment is repeated many times with the same initial state, and the positions where the particles hit the screen are registered to infer the one-particle and two-particle probability distributions. Generally, the distance between each pair of slits may vary. The plotted probability distributions on the screens are for \(\theta=\frac{\pi}{4}\). \[env_{+}\left(p_{1}\right) =\pi\sqrt{\frac{2\pi}{a}}B_{\theta}e^{-\frac{P_{1}^{2}}{2a}}\left(1+ e^{-2 ah^{2}}\right)\left[1+\sin\left(2\theta\right)\right] \tag{8}\] \[env_{-}\left(p_{1}\right) =\pi\sqrt{\frac{2\pi}{a}}B_{\theta}e^{-\frac{P_{1}^{2}}{2a}}\left(1 -e^{-2 ah^{2}}\right)\left[1-\sin\left(2\theta\right)\right]. \tag{9}\]  
These were computed by substituting \(\pm 1\) for the cosine. However, it should be noted that because of the Gaussian, the extreme points of \(f_{p_{1},\theta}\) are not necessarily located in the extreme points of \(\cos\left(2 hp_{1}\right)\). Thus, the above envelopes are merely approximations. They are \"good\" approximations if the Gaussian changes much slower than the cosine, i.e., \(ah^{2}\gg 1\). Now we wish to construct measures for the strength of interference. First, we compute the ratio between the envelopes  
\[r_{\theta}\triangleq\frac{env_{+}\left(p_{1}\right)}{env_{-}\left(p_{1} \right)}=\frac{1+e^{-2 ah^{2}}}{1-e^{-2 ah^{2}}}\cdot\frac{1+\sin\left(2\theta \right)}{1-\sin\left(2\theta\right)}=\coth\left(ah^{2}\right)\frac{1+\sin \left(2\theta\right)}{1-\sin\left(2\theta\right)}, \tag{10}\]  
where \(\coth\) denotes the hyperbolic cotangent.