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Klauder-Bargmann Integral Representation of Gaussian Symmetries and Generating Functions of Gaussian States

Tiju Cherian John, Kalyanapuram Rangachari Parthasarathy

Published Nov 15, 2019• 0 citations
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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 $\rho = Z_{1}^{\dagger}Z_{1}$, where $Z_{1}$ is an element of $\mathcal{E}_2(\mathcal{H})$ and has the form $Z_{1} = \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}$ on the dense linear manifold generated by all exponential vectors, $\Lambda$ being a positive operator in $\mathcal{H}$, $a_{r}, 1\leq r \leq n$ 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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Let Γ(H)\Gamma(\mathcal{H})Γ(H) be the boson Fock space over a finite dimensional Hilbert space H\mathcal{H}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 E2(H)\mathcal{E}_2(\mathcal{H})E2​(H) of bounded operators in Γ(H)\Gamma(\mathcal{H})Γ(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 ρ\rhoρ admits a factorization ρ=Z1†Z1\rho = Z_{1}^{\dagger}Z_{1}ρ=Z1†​Z1​, where Z1Z_{1}Z1​ is an element of E2(H)\mathcal{E}_2(\mathcal{H})E2​(H) and has the form Z1=cΓ(Λ)exp⁡∑r=1nλrar+∑r,s=1nαrsarasZ_{1} = \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}Z1​=c​Γ(Λ​)exp∑r=1n​λr​ar​+∑r,s=1n​αrs​ar​as​ on the dense linear manifold generated by all exponential vectors, Λ\LambdaΛ being a positive operator in H\mathcal{H}H, ar,1≤r≤na_{r}, 1\leq r \leq nar​,1≤r≤n are the annihilation operators corresponding to the nnn different modes in Γ(H)\Gamma(\mathcal{H})Γ(H), λr∈C\lambda_r\in \mathbb{C}λr​∈C and [αrs][\alpha_{rs}][αrs​] is a symmetric matrix in Mn(C)M_n(\mathbb{C})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)\mathcal{E}_2(\mathcal{H})E2​(H)-parameters;
(iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure nnn-mode gaussian states which are completely entangled;
(iv) tomography of an unknown gaussian state in Γ(Cn)\Gamma(\mathbb{C}^n)Γ(Cn) by the estimation of its E2(Cn)\mathcal{E}_2(\mathbb{C}^n)E2​(Cn)-parameters using O(n2)O(n^2)O(n2) measurements with a finite number of outcomes.
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.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.
Given a set of ordered quantum states, described by density operators {ρj}j=1n, the Bargmann invariant of {ρj}j=1n is defined as tr(ρ1ρ2, …, ρn). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr(ρ1ρ2, …, ρn) for any m-mode bosonic Gaussian states {ρj}j=1n in terms of the means and covariance matrices of {ρj}j=1n. We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.
It is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group. Although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis. Next a coherent state ‘‘wave function’’ is defined in analogy with the Schrödinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions. The function space in this realization is Bargmann’s Hilbert space of analytic functions. It is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of Bargmann. The transpose of this matrix element yields the kernel for the inversion of the transform. In this construction the Bargmann transform appears as the unitary transformation connecting the Schrödinger and coherent state wave functions. This considerably simplifies Bargmann’s original arguments and yields the integral transform as well as its inversion formula in a simple way.
This paper defines coherent manifolds and discusses their properties and their application in quantum mechanics. Every coherent manifold with a large group of symmetries gives rise to a Hilbert space, the completed quantum space of ZZZ, which contains a distinguished family of coherent states labeled by the points of the manifold. The second quantization map in quantum field theory is generalized to quantization operators on arbitrary coherent manifolds. It is shown how the Schrödinger equation on any such completed quantum space can be solved in terms of computations only involving the coherent product. In particular, this applies to a description of Bosonic Fock spaces as completed quantum spaces of a class of coherent manifolds called Klauder spaces.