## 4 Model for Asymmetric Entangled Double-Slit Experiment  
In this section, we consider a generalization of (4). Recalling the aforementioned analogy between the double-double slit and two measurement pointers measuring the entangled pair of particles, we note that [PERSON]'s system might be different from [PERSON]'s system in several aspects. [PERSON]'s pointer state could be implemented by an electron [22], cold ion [23] or any other quantum system. Moreover, his measuring system could correspond to variable strength coupling and could even have many degrees of freedom. However, we are only interested in one continuous degree of freedom out of the above giving rise to a pair of operators denoted by \(x_{2}\), \(p_{2}\) (not necessarily position and momentum), obeying the canonical commutation relations \(\left[x_{2},p_{2}\right]=i\). We therefore consider now asymmetric interactions corresponding to different measurements on the slits. Applying these interactions results in the following state  
\[\psi\left(x_{1},x_{2}\right)=M\left(e^{-a\left(x_{1}-h_{1}\right)^{2}}e^{-b \left(x_{2}-h_{2}\right)^{2}}+e^{-a\left(x_{1}+h_{1}\right)^{2}}e^{-b\left(x_ {2}+h_{2}\right)^{2}}\right), \tag{20}\]  
that is, [PERSON]'s two slits are not in the same distance as [PERSON]'s two slits, and his Gaussians have different widths than [PERSON]'s (\(M\) is a normalization constant). The Wigner function of this state appears in Appendix B.2. We begin by writing down the wavefunction in momentum space:  
\[\psi\left(p_{1},p_{2}\right)=\frac{M}{2\sqrt{ab}}\left(e^{-\frac{1}{4a}\left( p_{1}+2 ah_{1}i\right)^{2}-ah_{1}^{2}}e^{-\frac{1}{4b}\left(p_{2}+2 hh_{2}i \right)^{2}-bh_{2}^{2}}+e^{-\frac{1}{4a}\left(p_{1}-2 ah_{1}i\right)^{2}-ah_{1 }^{2}}e^{-\frac{1}{4b}\left(p_{2}-2 hh_{2}i\right)^{2}-bh_{2}^{2}}\right), \tag{21}\]  
and computing the probability density of \(p_{1}\):  
\[f_{p_{1}}\left(p_{1}\right)=\pi G\sqrt{\frac{2\pi}{a}}e^{-\frac{p_{1}^{2}}{2a }}\left[1+e^{-2 bh_{2}^{2}}\cos\left(2h_{1}p_{1}\right)\right]. \tag{22}\]  
As before, (22) yields an expression for the upper (lower) envelope by substituting \(\cos\left(2h_{1}p_{1}\right)=+1\left(-1\right)\)  
\[env_{asymmetric}^{\pm}=\pi G\sqrt{\frac{2\pi}{a}}e^{-\frac{p_{1}^{2}}{2a}} \left[1\pm e^{-2 bh_{2}^{2}}\right]. \tag{23}\]  
As in the previous section, these are approximations, valid for \(ah_{1}^{2}\gg 1\); however, we need not assume anything about \(b,h_{2}\). Again, we obtain a constant ratio  
\[r_{asymmetric}=\frac{1+e^{-2 bh_{2}^{2}}}{1-e^{-2 bh_{2}^{2}}}=\coth\left(bh_{2} ^{2}\right), \tag{24}\]  
and constant visibility  
\[V_{asymmetric}=e^{-2 bh_{2}^{2}}. \tag{25}\]  
We note that because of the asymmetry, the one-particle visibility for particle 1 (which is the one defined above) differs from the one-particle visibility for particle 2. Comparing (25) and (11), we wish to find a relation between the parameters determining the one-particle interference visibility. In the \"asymmetric\" wavefunction, it is determined by \(b,h_{2}\), while in the theta-wavefunction it is determined by \(a,h_{1},\sin\left(2\theta\right)\). Taking the two expressions to be equal, we obtain  
\[\frac{e^{-2 ah_{1}^{2}}+\sin\left(2\theta\right)}{1+\sin\left(2\theta\right)e^{-2 ah_{1}^{2}}}=e^{-2 bh_{2}^{2}}. \tag{26}\]  
Solving for \(\sin\left(2\theta\right)\) yields  
\[\sin\left(2\theta\right)=\frac{e^{-2 bh_{2}^{2}}-e^{-2 ah_{1}^{2}}}{1-e^{-2 ah_{ 1}^{2}}e^{-2 bh_{2}^{2}}}=\frac{V_{asymmetric}-e^{-2 ah_{1}^{2}}}{1-e^{-2 ah_{1}^{2} }V_{asymmetric}}. \tag{27}\]  
This also allows a simple way for [PERSON] to generate a _purification_ of her state, using only the parameters of her system and the interference visibility she measures. To put it more precisely, the state of the form (5) where \(\theta\) is given by (27), yields the same single-particle Wigner function for [PERSON] as the state (20) considered throughout this section (a proof appears in the Appendix B.3).