## Appendix B Computations and Wigner Functions
### The Wigner Function for a Partially Entangled Double-Slit Experiment
\begin{array}{rl}\tilde{F}_{\theta}\left(p_{1},p_{2}\right):=&f_{p_{1}p_{2}\theta }\left(p_{1},p_{2}\right)-f_{p_{1}\theta}\left(p_{1}\right)f_{p_{2}\theta}\left( p_{2}\right)+\frac{B_{\theta}}{B_{\theta-0}}f_{p_{1}\theta=0}\left(p_{1} \right)f_{p_{2}\theta=0}\left(p_{2}\right)=\\ &=&+\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left[e^{4\text {i}h^{2}}+\cos\left(2 hp_{1}\right)\cos\left(2 hp_{2}\right)+\cos^{2}\left(2 \theta\right)\right]+\\ &+\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left[\sin^{2} \left(2\theta\right)e^{2\text{i}h^{2}}\left[\cos\left(2 hp_{1}\right)+\cos \left(2 hp_{2}\right)\right]-\cos\left(2\theta\right)\sin\left(2 hp_{1}\right) \sin\left(2 hp_{2}\right)\right]+\\ &+\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left[\cos^{2} \left(2\theta\right)e^{4\text{i}h^{2}}\cos\left(2 hp_{1}\right)\cos\left(2 hp_{ 2}\right)-\cos\left(2\theta\right)e^{4\text{i}h^{2}}\sin\left(2 hp_{1}\right) \sin\left(2 hp_{2}\right)\right]+\\ &-\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\sin\left(4 \theta\right)e^{2\text{i}h^{2}}\sin\left(2 hp_{1}\right)\sin\left(2 hp_{2} \right)=\\ &=&+\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left(e^{4 \text{i}h^{2}}+\cos^{2}\left(2\theta\right)+\frac{1}{2}\left[1+\cos^{2}\left( 2\theta\right)e^{4\text{i}h^{2}}\right]\left[\cos\left(2 hp_{+}\right)+\cos \left(2 hp_{-}\right)\right]\right)+\\ &+2\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\sin^{2}\left( 2\theta\right)e^{2\text{i}h^{2}}\cos\left(hp_{+}\right)\cos\left(hp_{-}\right) +\\ &-\frac{1}{2}\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\cos \left(2\theta\right)\left[1+e^{4\text{i}h^{2}}+2\sin\left(2\theta\right) \right]\left[\cos\left(2 hp_{-}\right)-\cos\left(2 hp_{+}\right)\right]=\\ &=&+\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left[e^{4 \text{i}h^{2}}+\cos^{2}\left(2\theta\right)+2\sin^{2}\left(2\theta\right)e^{2 \text{i}h^{2}}\cos\left(hp_{+}\right)\cos\left(hp_{-}\right)\right]+\\ &+\frac{1}{2}\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left( 1+\cos^{2}\left(2\theta\right)e^{4\text{i}h^{2}}+\cos\left(2\theta\right) \left[1+e^{4\text{i}h^{2}}+2\sin\left(2\theta\right)\right]\right)\cos\left(2 hp_{+}\right)+\\ &+\frac{1}{2}\frac{e^{-\frac{p_{1}^{2}+p_{2}^{2}}{2}}}{\sum_{\theta}^{2}}\left( 1+\cos^{2}\left(2\theta\right)e^{4\text{i}h^{2}}-\cos\left(2\theta\right) \left[1+e^{4\text{i}h^{2}}+2\sin\left(2\theta\right)\right]\right)\cos\left(2 hp_{-}\right).\end{array}  \tag{19}\]  
Substituting \(p_{-}=0\), we obtain  
\[ \begin{array}{rl}\tilde{F}_{\theta}\left(p_{+},p_{-}=0\right)&=\frac{e^{- \frac{p_{1}^{2}}{24}}}{\sum_{\theta}^{2}}\left[e^{4\text{i}h^{2}}+\cos^{2} \left(2\theta\right)+2\sin^{2}\left(2\theta\right)e^{2\text{i}h^{2}}\cos\left( hp_{+}\right)\right]+\\ &+2\frac{e^{-\frac{p_{1}^{2}}{42}}}{\sum_{\theta}^{2}}\left(1+\cos^{2}\left(2 \theta\right)e^{4\text{i}h^{2}}+\cos\left(2\theta\right)\left[1+e^{4\text{i}h^{ 2}}+2\sin\left(2\theta\right)\right]\right)\cos\left(2 hp_{+}\right)+\\ &+2\frac{e^{-\frac{p_{1}^{2}}{44}}}{\sum_{\theta}^{2}}\left(1+\cos^{2}\left(2 \theta\right)e^{4\text{i}h^{2}}-\cos\left(2\theta\right)\left[1+e^{4\text{i}h^{ 2}}+2\sin\left(2\theta\right)\right]\right).\end{array}  \tag{20}\]