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arXiv:1911.06555 · Journal of Mathematical Physics

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

Tiju Cherian John, Kalyanapuram Rangachari Parthasarathy

Published Feb 1, 2021• 12 citations
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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.

DOI
10.1063/5.0019413

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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 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.
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.
The momentum and position observables in an [Formula: see text]-mode boson Fock space [Formula: see text] have the whole real line [Formula: see text] as their spectrum. But the total number operator [Formula: see text] has a discrete spectrum [Formula: see text]. An [Formula: see text]-mode Gaussian state in [Formula: see text] is completely determined by the mean values of momentum and position observables and their covariance matrix which together constitute a family of [Formula: see text] real parameters. Starting with [Formula: see text] and its unitary conjugates by the Weyl displacement operators and operators from a representation of the symplectic group [Formula: see text] in [Formula: see text], we construct [Formula: see text] observables with spectrum [Formula: see text] but whose expectation values in a Gaussian state determine all its mean and covariance parameters. Thus measurements of discrete-valued observables enable the tomography of the underlying Gaussian state and it can be done by using five one-mode and four two-mode Gaussian symplectic gates in single and pair mode wires of [Formula: see text]. Thus the tomography protocol admits a simple description in a language similar to circuits in quantum computation theory. Such a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states permit a tomography of the channel parameters. However, in our procedure the number of counting measurements exceeds the number of channel parameters slightly. Presently, it is not clear whether a more efficient method exists for reducing this tomographic complexity. As a byproduct of our approach an elementary derivation of the probability generating function of [Formula: see text] in a Gaussian state is given. In many cases the distribution turns out to be infinitely divisible and its underlying Lévy measure can be obtained. However, we are unable to derive the exact distribution in all cases. Whether this property of infinite divisibility holds in general is left as an open problem.
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.
The requirements on the general structure of quasiprobabilities for a single boson mode are investigated. The complete Gaussian class of quasiprobabilities, which can be obtained by convolutions of the Wigner quasiprobability with the complete class of normalized Gaussian functions, is represented by a three-dimensional complex vector parameter with the property of additivity when composing convolutions and meaning that the transition between two quasiprobabilities with the vector parameters r and s is given by the convolution with a Gaussian function belonging to the vector parameter r - s . The scalar product of r with itself is related to the determinant of the second-rank symmetric tensor of the quadratic form in the exponent of the Gaussian functions. This is obtained by a mapping with the two symmetric Pauli spin matrices and the unity matrix. The Wigner quasiprobability takes on the central position within this Gaussian class with the vector parameter r = (0,0,0). The class of s-ordered quasiprobabilities is described by the vector parameters with and its diagonalization is connected with displaced Fock states . The class of quasiprobabilities corresponding to the linear interpolation between standard and antistandard ordering belongs to the vector parameter with . Its diagonalization is connected with discrete series of excitations of the eigenstates of the canonical operators Q and P. The diagonalization of the complete Gaussian class of quasiprobabilities with real vector parameter and leads to dual systems of discrete excitations of squeezed coherent states with mutually orthogonal squeezing axes and properties of orthogonality and completeness and provides a possible generalization of the displaced Fock states. The diagonalization is basically different for real and complex vector parameters r . A generalization of the coherent-state quasiprobability, which uses squeezed coherent states instead of coherent states, belongs to a vector parameter with real and imaginary and and with . New representations of special classes of quasiprobabilities are derived.
We present a novel approach to the separability problem for Gaussian quantum states of bosonic continuous variable systems. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of m versus n modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for: (i) Gaussian states of 1 versus n modes; and (ii) isotropic Gaussian states. In passing, we also retrieve (iii) the recently established equivalence between separability of a Gaussian state and and its complete Gaussian extendability. Our techniques are then applied to progress beyond the state of the art. We prove that: (iv) Gaussian states that are invariant under partial transposition are necessarily separable; (v) the PPT criterion is necessary and sufficient for separability for Gaussian states of m versus n modes that are symmetric under the exchange of any two modes belonging to one of the parties; and (vi) Gaussian states which remain PPT under passive optical operations can not be entangled by them either. This is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject. This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem.