A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications

Journal of Mathematical Physics, Volume 62, Issue 2, February 2021. Let [math] be the boson Fock space over a finite dimensional Hilbert space [math]. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup [math] of bounded operators in [math]. This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a 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 ρ admits a factorization [math], where Z1 is an element of [math] and has the form [math] on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in [math], ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in [math], [math], and [αrs] is a symmetric matrix in [math]; (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 [math]-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in [math] by the estimation of its [math] parameters using O(n2) measurements with a finite number of outcomes.