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

Quantum Information With Gaussian States

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

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