1.4.4. Quantum information with Gaussian states

1.4.4.1. Introduction

1.4.4.1.1. Phase space representation and definition of Gaussian states

Consider the single-mode field. We have photon-number operator, N̂ . Any Fock state $|l\rangle$ is an eigen-state of photon-number operator, i.e., $\hat{N} |l\rangle = l|l\rangle$.Also, we have creation operator $a^†$ and annihilation operator $a$ and $\hat{N} = a^† a$ $$a^\dagger a|l\rangle=l|l\rangle$$ $$a^\dagger|l\rangle =\sqrt{l+1}|l+1\rangle$$ $$a|l\rangle=\sqrt{l}|l-1\rangle$$

$$[a,a^\dagger]=aa^\dagger-a^\dagger a=1$$

$$|l\rangle=\frac{a^{\dagger^l}}{\sqrt{l!}}|0\rangle$$

$$|\chi\rangle=f(a^\dagger)|0\rangle$$

Let $\hat{x}_k$ and $\hat{p}k$ denote the ‘position’ and ‘momentum’ operators associated with the $k$th mode, respectively (k = 1, 2, · · · , n). $$\begin{aligned} \hat{x}_{k} &=\sqrt{\frac{1}{2 \omega_{k}}}\left(a_{k}+a_{k}^{\dagger}\right) \\ \hat{p}_{k} &=-i \sqrt{\frac{\omega_{k}}{2}}\left(a_{k}-a_{k}^{\dagger}\right) \end{aligned}$$ Defing $R=\left(R_{1}, R_{2}, \cdots, R_{2 n}\right)^{T}=\left(\omega_{1}^{1 / 2} \hat{x}_{1}, \omega_{1}^{-1 / 2} \hat{p}_{1}, \cdots, \omega_{n}^{1 / 2} \hat{x}_{n}, \omega_{n}^{-1 / 2} \hat{p}_{n}\right)^{T}$

these canonical commutation relations (CCRs) can be written compactly as $[R_j,R_k]=iJ_{jk}$.Here,$J=\oplus^n_{j=1}J_1$ with $$J_{1}=\left(\begin{array}{cc}{0} & {1} \\ {-1} & {0}\end{array}\right)$$ An n−mode density operator ρ is defined on the phase space that is a 2n-dimensional real vector space. The characteristic function is $$\chi(\xi)=\operatorname{Tr}[\rho \mathcal{W}(\xi)]$$ Here,$\mathcal{W}(\xi)=\exp \left(i \xi^{T} R\right)$is called Weyl operator. The density operator of any quantum state in Fock space can always be written in terms of its characteristic function and Weyl operators as follows $$\rho=\frac{1}{(2 \pi)^{m}} \int d^{2 m} \xi \chi(J \xi) \mathcal{W}(-J \xi)$$

A Gaussian state is defined as such a state that its characteristic function is Gaussian: $$\chi(\xi)=\exp \left[-\frac{1}{4} \xi^{T} \gamma \xi+i d^{T} \xi\right]$$ Here, $γ > 0$ is a real symmetric matrix and $g ∈ \mathbb{R} ^{2n}$

As shown below, the quantum vacuum state, coherent states, squeezed states, and thermal states are typical Gaussian states