# The definition of Gaussian State

Could you clarify to me what is a Gaussian state? I know what is a Gaussian function and Gaussian distribution, but I don't know how to respond to other when they ask me to provide the definition of a Gaussian state.

Thank you.

• On $L^2(\mathbb{R}^d)$, $\psi(x)=\frac{1}{(\sqrt{\pi})^d}e^{-\lvert x\rvert^2}$ is a normalized gaussian state (vector of the Hilbert space). – yuggib Nov 4 '15 at 11:03
• @TBBT Are you referring to QFT or QM? – Valter Moretti Nov 4 '15 at 11:15
• I am referring to G.S in Quantum Information Processing. – TBBT Nov 5 '15 at 4:11

A Gaussian state is a ground or thermal state of a (bosonic or fermionic) Hamiltonian which is quadratic in the creation and annihiliation operators. Those states are fully characterized by expectation values of quadratic operators, and thus $4N^2$ parameters for $N$ fermions or bosons.
The most general description of a quantum system is given by a density matrix $\rho$. It has dimensions of $N \times N$, where $N$ is the number of degrees of freedom of the system: 2 for a 2 level quantum system (qubit), 3 for 3-level etc. But often we deal with the systems that have infinite number of degrees of freedom. Such systems are quantum harmonic oscillators, modes of light and other systems with a certain Hamiltonian as was pointed out in a previous answer. The density matrix approach is not the most convenient for such systems as you have to deal with infinite matrices and moreover if you consider a composite system $\rho^{AB}=\rho^{A} \otimes \rho^{B}$, with tensor products of those. But one can apply a transformation:
$\chi(\xi,\xi^*) = Tr[D(\xi,\xi^*) \rho]$
where $D(\xi,\xi^*) = \exp[\xi \hat{a}^{\dagger} - \xi^* \hat{a}]$ is a displacement operator. This transformation maps the space of the $\rho^{\otimes M}$ density matrices to a space of $2M$ variable functions ($\xi$ and $\hat{a}$ are vectors when $M>1$). $\chi$ is called a characteristic function and the states for which it is Gaussian are obviously called Gaussian. The most general Gaussian state is a displaced thermal squeezed state.