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  • What is the difference between phase space representations (with real x & p as variables) and coherent state representation (with complex $\alpha$ as a variable) of $\rho$, W, P and Q functions in quantum optics?

  • How do first quantization and second quantization relate to different representations?

  • Are these two representations equivalent?

  • When do we use one or the other?

  • How and where does the Fermionic or Bosonic nature of the things we are describing using these tools come in?

  • Why do we need W, P and Q functions if they are derived from $\rho$?

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    $\begingroup$ Related. Yes, they are all equivalent, just as different coordinate systems are equivalent, best suited for different problems. The WP article hints at contexts. Alternate equivalent formalisms furnish insight and technical convenience in solving concrete problems. A good quantum optics texts may well illuminate all this. $\endgroup$ – Cosmas Zachos Aug 2 '18 at 21:49
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With a quantum communication engineer background, I'll try to ask you and expand Cosmas's answer as best as possible. I'm sorry if I can't give you further physical insights.

The difference between the $x-p$ plane, with $x,p\in\mathbb{R}$, and the $\alpha$ plane, with $\alpha\in\mathbb{C}$ is just a change of variable (it is equivalent to see $\mathbb{C}$ as an $\mathbb{R}^2$ plane). You can try by yourself, e.g., starting from the definition of the symmetrically ordered characteristic function (the computations are long and boring): $$\chi(u,v)=\text{Tr}\left\{\rho e^{-i(uP+vX)}\right\}$$

And making a change of variable by using this relationship: $$a=\frac{\hat{x}+i\hat{p}}{\sqrt{2}}$$ Where $\hat{x}=\sqrt{\frac{\omega}{\hbar}}X$ and $\hat{p}=\frac{1}{\sqrt{\hbar\omega}}P$. You arrive at the form: $$\chi(\xi)=\text{Tr}\left\{\rho e^{\xi a^{\dagger}-\xi^*a}\right\}$$ Doing an analogue change of variable with the Wigner function (i.e., the Fourier transform of $\chi(u,v)$) leads you from $W(x,p)$ to $W(\alpha)$. This time you need to express $x,p$ as a function of $\alpha$ and $\alpha^*$. The same reasoning applies to the Glauber-Sudarshan $P$ function and Husimi $Q$ function.

The different phase-space representations ($W$, $P$ and $Q$) comes out from the so called operator-ordering problem. This problem (and its solution) is purely mathematical and it has no physical justification, as the three function represents exactly the same state. As a consequence of the operator ordering problem, you can use the different representations to compute, respectively, the simmetrically/normally/anti normally ordered moments of the state. This is a first reason in unsing different, but equivalent, phase space representations. Indeed, is easy to show that you can use the Wigner function to calculate the simmetrically ordered moments, i.e., $$\text{Tr}\left\{ \rho\mathcal{S}\left[a^n(a^{\dagger})^m\right] \right\}= \int\alpha^n(\alpha^*)^mW(\alpha)d^2\alpha$$ While the $P$ representation can be used to calculate the normally ordered moments, i.e., $$\text{Tr}\left\{ \rho (a^{\dagger})^ma^n \right\}= \int\alpha^n(\alpha^*)^mP(\alpha)d^2\alpha$$ Furthermore, the $P$ function is useful to represent a state in the overcomplete basis of coherent states, i.e., $$\rho = \int P(\alpha)|\alpha\rangle\langle\alpha|d^2\alpha$$

I can't answer the other more physical questions, but I hope to have clarified the most theoretical issues.

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