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

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)exp∑r=1nλrar+∑r,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.

Computation of sandwiched relative alpha-entropy of two n-mode gaussian states

“…is the contraction diagonal matrix and Γ(K) ∈ E 2 (H) is the corresponding positive contraction operator [4]. Thus ρ(s)…”

“…All the theorems and proofs that are readily available in previous Refs. [1][2][3][4][5] are only stated.…”

“…where c = 0 is a scalar; λ, µ ∈ C n ; A, B and Λ are complex n × n matrices, with A, B being symmetric. We list the properties [4] of Z belonging to the operator semigroup E 2 :…”

“…Any unitary operator U ∈ E 2 (H) is a gaussian symmetry i.e., U ρ U † is a gaussian state whenever ρ is also a gaussian state (see Proposition V.10.1 of Ref. [4]). Every gaussian symmetry operation belongs to E 2 (H).…”