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.
Let K be a complex separable Hilbert space of finite or infinite dimensions. We prove that the Petz-Rényi relative entropy of any two quantum states ρ and σ on K, denoted by D α (ρ||σ) reduces to the Rényi relative entropy (divergence) of two classical probability measures P and Q obtained from ρ and σ. This leads to a number of new results in the infinite dimensions, and new proofs for some known results in the finite dimensional setting. Our method provides a general framework for proving a quantum counterpart of any result about the classical Rényi divergence involving only two probability distributions. Furthermore, we construct an example to show that the information theoretic definition of the von Neumann relative entropy is different from Araki's definition of relative entropy. This disproves a recent attempt in the literature to prove such a result. All the results proved here are valid in both finite and infinite dimensions and hence can be applied to continuous variable systems as well.
Quantum Gaussian states on Bosonic Fock spaces are quantum versions of Gaussian distributions. In this paper we explore infinite mode quantum Gaussian states. We extend many of the results of Parthasarathy in [Par10] and [Par13] to the infinite mode case, which includes various characterizations, convexity and symmetry properties.
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