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khfrekek92

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I'm trying to calculate a Wigner Function of an entangled state, and I'm not quite sure how to proceed. I have created this state by sending in vacuum and a squeezed state into a 50/50 BS, where the output state has a density operator:

$$\rho_{34}=S_{3}S_{4}S_{34}|00\rangle_{34}\langle 00|_{34}S_{34}^{\dagger}S_{4}^{\dagger}S_{3}^{\dagger}$$

Where $S_{3}$, $S_{4}$ and $S_{34}$ are the usual single and two mode squeezing operators.

Now I'm kind of stuck on how to calculate the Wigner function from here. I'm assuming I need to trace out one of the states as follows:

$$

\begin{aligned}

\rho_{3}&=Tr_{4}(\rho_{34})\\

&=S_{3}Tr_{4}[S_{4}S_{34}|00\rangle_{34}\langle 00|_{34}S_{34}^{\dagger}S_{4}^{\dagger}]S_{3}^{\dagger}\\

&=S_{3}Tr_{4}[S_{34}|00\rangle_{34}\langle 00|_{34}S_{34}^{\dagger}]S_{3}^{\dagger}\\

&=S_{3}Tr_{4}[|TMSV\rangle_{34}\langle TMSV|_{34}]S_{3}^{\dagger}\\

&=S_{3}\sum_{n=0}^{\infty}[_{4}\langle n|TMSV\rangle_{34}._{34}\langle TMSV|n\rangle_{4}]S_{3}^{\dagger}\\

&=\frac{S_{3}}{\cosh^{2}(r)}\sum_{m=0}^{\infty}[\tanh^{2m}(r)|m\rangle_{4}\langle m|_{4}]S_{3}^{\dagger}

\end{aligned}

$$Then to calculate the Wigner function, we use:

$$W_{3}=\frac{1}{2\pi{h}}\int_{-inf}^{inf}._{3}<q-\frac{y}{2}|\frac{S_{3}}{cosh^{2}(r)}\sum_{m=0}^{\infty}tanh^{2m}(r)|m>_{3}<m|S_{3}^{\dagger}|q+\frac{y}{2}>e^{\frac{ipy}{h}}dy$$

And then I get stuck. I have no idea how I would calculate this. Did I miss anything? Does anyone have anything that could maybe point me in the right direction? Thanks so much in advance!

$$\rho_{34}=S_{3}S_{4}S_{34}|00\rangle_{34}\langle 00|_{34}S_{34}^{\dagger}S_{4}^{\dagger}S_{3}^{\dagger}$$

Where $S_{3}$, $S_{4}$ and $S_{34}$ are the usual single and two mode squeezing operators.

Now I'm kind of stuck on how to calculate the Wigner function from here. I'm assuming I need to trace out one of the states as follows:

$$

\begin{aligned}

\rho_{3}&=Tr_{4}(\rho_{34})\\

&=S_{3}Tr_{4}[S_{4}S_{34}|00\rangle_{34}\langle 00|_{34}S_{34}^{\dagger}S_{4}^{\dagger}]S_{3}^{\dagger}\\

&=S_{3}Tr_{4}[S_{34}|00\rangle_{34}\langle 00|_{34}S_{34}^{\dagger}]S_{3}^{\dagger}\\

&=S_{3}Tr_{4}[|TMSV\rangle_{34}\langle TMSV|_{34}]S_{3}^{\dagger}\\

&=S_{3}\sum_{n=0}^{\infty}[_{4}\langle n|TMSV\rangle_{34}._{34}\langle TMSV|n\rangle_{4}]S_{3}^{\dagger}\\

&=\frac{S_{3}}{\cosh^{2}(r)}\sum_{m=0}^{\infty}[\tanh^{2m}(r)|m\rangle_{4}\langle m|_{4}]S_{3}^{\dagger}

\end{aligned}

$$Then to calculate the Wigner function, we use:

$$W_{3}=\frac{1}{2\pi{h}}\int_{-inf}^{inf}._{3}<q-\frac{y}{2}|\frac{S_{3}}{cosh^{2}(r)}\sum_{m=0}^{\infty}tanh^{2m}(r)|m>_{3}<m|S_{3}^{\dagger}|q+\frac{y}{2}>e^{\frac{ipy}{h}}dy$$

And then I get stuck. I have no idea how I would calculate this. Did I miss anything? Does anyone have anything that could maybe point me in the right direction? Thanks so much in advance!

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