#### Title

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

#### Article Type

Research Article

#### Publication Title

Journal of Mathematical Physics

#### Abstract

Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. 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 E2(H) of bounded operators in Γ(H). 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 ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(P)expΣr=1nλrar+Σr,s=1nαrsaras 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 H, ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C, and [αrs] is a symmetric matrix in Mn(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 E2(H)-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in Γ(Cn) by the estimation of its E2(Cn) parameters using O(n2) measurements with a finite number of outcomes.

#### DOI

10.1063/5.0019413

#### Publication Date

2-1-2021

#### Recommended Citation

John, Tiju Cherian and Parthasarathy, K. R., "A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications" (2021). *Journal Articles*. 2098.

https://digitalcommons.isical.ac.in/journal-articles/2098

## Comments

Open Access, Green