$$\left|\Phi_\text{Bell} \right> =\frac{1}{\sqrt{2}}\left(\left|H\right>_a\left|H\right>_b+\left|V\right>_a\left|V\right>_b\right)$$

$$p=\varepsilon_0\left( \chi_1 E+\chi_2 E^2+\chi_3 E^3+... \right)\\ E(t)= E_1e^{-i\omega_1t}+E_2e^{-i\omega_2t}+E_1^*e^{+i\omega_1t}+E_2^*e^{+i\omega_2t}\\ p^{(2)}(t)= \chi_2E^2(t)= \chi_2\left( {E_1}^2 e^{-2i\omega_1t}+{E_2}^2e^{-2i\omega_2t}+2E_1E_2 e^{-i(\overbrace{\omega_1}^\text{UV}+\overbrace{\omega_2}^{\text{Vakuum-Feld mit }\omega_2=\frac{1}2\omega_1})t }+2E_1E_2^*e^{-i(\omega_1-\omega_2)t}+c.c.\right)\\+2\chi_2\left( E_1E_1^*+E_2E_2^*\right) \\ \underbrace{\text{Spontaneous}}_\text{Vakuum}\ \underbrace{\text{parametric}}_\text{E-Feld}\ \left. \underbrace{\text{down-conversion}}_{\omega_\text{UV}=2\omega_\text{IR}}\right.$$

$$\left|\Phi^\pm\right>=\frac{1}{\sqrt{2}}\left(\left|H\right>\left|H\right>\pm\left|V\right>\left|V\right>\right)$$

$$\left|\Psi^\pm\right>=\frac{1}{\sqrt{2}}\left(\left|H\right>\left|V\right>\pm\left|V\right>\left|H\right>\right)$$

$$\left|\Phi_\text{pump}\right>= \cos(\vartheta)\left|V\right>_p+e^{i\phi}\sin(\vartheta) \left|H\right>_p$$

$$\left|\Phi_\text{pump}\right>\rightarrow \left|\Psi_\text{DC}\right>=\cos(\vartheta)\left|H\right>_s\left|H\right>_i+e^{2i\phi}\sin(\vartheta) \left|V\right>_s\left|V\right>_i\\\vartheta = \frac{\pi}{4}\Rightarrow 45\text{° polarisation pump, }\phi = 0\\ \Rightarrow \left|\Psi_\text{DC}\right>=\frac{1}{\sqrt{2}}\left(\left|H\right>_s\left|H\right>_i+ \left|V\right>_s\left|V\right>_i\right)$$

$$\left| V_\alpha \right>= \cos(\alpha)\left|V\right> -\sin(\alpha)\left|H\right>\\ \left| H_\alpha \right>= \cos(\alpha)\left|V\right> +\sin(\alpha)\left|H\right>$$

$$V_\alpha V_\beta,\ V_\alpha H_\beta,\ H_\alpha V_\beta,\ H_\alpha H_\beta\\ P_{VV}(\alpha,\beta)= \left|\left<V_\alpha\right|_s\left<V_\beta\right|_i\left|\Psi_\text{DC}\right> \right|^2=\left| \sin(\alpha)\sin(\beta) \cos(\vartheta) + e^{2i\phi}\cos(\alpha)\cos(\beta) \sin(\vartheta) \right|^2$$

$$E(\alpha,\beta)=P_{VV}(\alpha,\beta)+P_{HH}(\alpha, \beta)-P_{VH}(\alpha, \beta)-P_{HV}(\alpha,\beta)$$

$$S=E(a,b)-E(a,b')+E(a',b)+E(a'b')$$

$$p(\lambda)\geq 0,\ \int p(\lambda) \text{d}\lambda = 1$$

$$A(\lambda,\alpha)\rightarrow \begin{cases}+1\text{, V-polarisiert } \\-1\text{, H-polarisiert } \end{cases}\\ B(\lambda,\alpha)\rightarrow \begin{cases}+1\text{, V-polarisiert } \\-1\text{, H-polarisiert } \end{cases}\\ P_{VV}(\alpha,\beta)=\int\frac{1+A(\lambda,\alpha)}{2}\cdot \frac{1+B(\lambda,\beta)}{2}p(\lambda)\text{d}\lambda\\ P_{VH}(\alpha,\beta)=\int\frac{1+A(\lambda,\alpha)}{2}\cdot \frac{1-B(\lambda,\beta)}{2}p(\lambda)\text{d}\lambda\\ P_{HV}(\alpha,\beta)=\int\frac{1-A(\lambda,\alpha)}{2}\cdot \frac{1+B(\lambda,\beta)}{2}p(\lambda)\text{d}\lambda\\ P_{HH}(\alpha,\beta)=\int\frac{1-A(\lambda,\alpha)}{2}\cdot \frac{1-B(\lambda,\beta)}{2}p(\lambda)\text{d}\lambda\\ E(\alpha,\beta)=\int A(\lambda,\alpha)\cdot B(\lambda,\beta)P(\lambda)\text{d}\lambda\\ S= A(\lambda,a)B(\lambda,b)-A(\lambda,a)B(\lambda,b)+ A(\lambda,a')B(\lambda,b)+ A(\lambda,a')B(\lambda,b')\\= A(\lambda ,a)\left[ B(\lambda, b) -B(\lambda,b')\right]+A(\lambda ,a')\left[ B(\lambda, b) +B(\lambda,b')\right]\\ \left<S\right>=\int S(\lambda,a,a',b,b')p(\lambda)\text{d}\lambda=E(a,b)-E(a,b')+E(a',b)+E(a',b')=S\\ -2<s<+2\Rightarrow S\leq 2\\ a=0^\circ,\ a'=45^\circ,\ b=22,5^\circ,\ b'=67,5^\circ\\\text{QM}\Rightarrow S=2\sqrt{2}>2$$

$$P_{VV}=\frac{N(\alpha,\beta)}{N(\alpha,\beta)+N(\alpha_\perp,\beta)+N(\alpha,\beta_\perp)+N(\alpha_\perp,\beta_\perp)}\\ \alpha _\perp=a+90^\circ,\ \beta_\perp=\beta +90^\circ\\ E(a,b)=\frac{N(\alpha,\beta)+N(\alpha_\perp,\beta)-N(\alpha,\beta_\perp)-N(\alpha_\perp,\beta_\perp)}{N(\alpha,\beta)+N(\alpha_\perp,\beta)+N(\alpha,\beta_\perp)+N(\alpha_\perp,\beta_\perp)}\widehat{=}\text{ 4 Mesungen}$$

$$\alpha =0^\circ,45^\circ,90^\circ,135^\circ\text{ und }\beta =22,5^\circ,\ 67,5^\circ,\ 112,5^\circ,157,5^\circ$$