A Common Parametrization for Finite Mode Gaussian States, their Symmetries and associated Contractions with some Applications

Abstract

Let $\Gamma(\mathcal{H})$ be the boson Fock space over a finite dimensional Hilbert space $\mathcal{H}$. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup $\mathcal{E}_2(\mathcal{H})$ of bounded operators in $\Gamma(\mathcal{H})$. This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every gaussian state $\rho$ admits a factorization $\rho = Z_{1}^{\dagger}Z_{1}$, where $Z_{1}$ is an element of $\mathcal{E}_2(\mathcal{H})$ and has the form $Z_{1} = \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}$ on the dense linear manifold generated by all exponential vectors, $\Lambda$ being a positive operator in $\mathcal{H}$, $a_{r}, 1\leq r \leq n$ are the annihilation operators corresponding to the $n$ different modes in $\Gamma(\mathcal{H})$, $\lambda_r\in \mathbb{C}$ and $[\alpha_{rs}]$ is a symmetric matrix in $M_n(\mathbb{C})$; (ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its $\mathcal{E}_2(\mathcal{H})$-parameters; (iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure $n$-mode gaussian states which are completely entangled; (iv) tomography of an unknown gaussian state in $\Gamma(\mathbb{C}^n)$ by the estimation of its $\mathcal{E}_2(\mathbb{C}^n)$-parameters using $O(n^2)$ measurements with a finite number of outcomes.