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 $\rho =Z_1^{†}{Z}_1$, where $Z_1$ is an element of ${\mathcal{E}}_2(\mathcal{H})$ and has the form $Z_1=\sqrt{c}\Gamma (\sqrt{\Lambda })\exp {\sum }_{r=1}^n{\lambda }_r{a}_r+{\sum }_{r,s=1}^n{\alpha }_{rs}{a}_r{a}_s$ on the dense linear manifold generated by all exponential vectors, $\Lambda$ being a positive operator in $\mathcal{H}$, $a_r,1\le r\le n$ 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.

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