arXiv:2602.08611 · Journal of Physics A Mathematical and Theoretical

Representation theory of inhomogeneous Gaussian unitaries

Jingqi Sun, Joshua Combes, Lucas Hackl

quant-phSubmitted Feb 9, 2026• Announced Feb 10, 2026• 0 citations

Abstract

Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. Their structures correspond to symplectic (bosons) and orthogonal (fermions) groups, but physical realizations give rise to their respective double covers, introducing phase and sign ambiguities. The homogeneous (quadratic-only) case has been resolved through a parameterization constructed in a recent work [arXiv:2409.11628]. We extend the previous framework to inhomogeneous Gaussian unitaries parameterized by $(M,z,Ψ)$. The Baker-Campbel-Hausdorff formula allows us then to factor any Gaussian unitary into a squeezing and a displacement transformation, from which we derive the group multiplication law.

Comments: 14+10 pages, 3 figures, 4 tables

Related papers

Representation theory of Gaussian unitary transformations for bosonic and fermionic systems

Tommaso Guaita, Lucas Hackl, Thomas Quella

arXiv:2409.11628Annals of PhysicsOct 17, 20263 citations

From Hamiltonians to complex symplectic transformations

Gianfranco Cariolaro, Gianfranco L. Pierobon

arXiv:1704.02008Apr 6, 20170 citations

Quadratic Hamiltonians

Didier Robert, Monique Combescure

Theoretical and mathematical physicsJan 1, 20211 citation

Gaussian states and operations -- a quick reference

Jonatan Bohr Brask

arXiv:2102.05748Feb 10, 20218 citations

Uncovering Non-Gaussianity through Multi-Copy Symmetries

Hao Dai, Yue Zhang

quant-pharXiv:2608.03755Aug 5, 20260 citations

Exact gaussian states under quadratic non-hermitian deformations

Kapil Shrivastava

Abstract We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map \[ T_\phi=\exp\{2\phi(\partial_x)\}, \qquad \phi(z)=c_1+c_2z+c_3z^2 . \] The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization \[ H_\phi=T_\phi^{-1}HT_\phi, \] the physical inner product is generated by $M_\phi=T_\phi^\dagger T_\phi$, time evolution is governed by the Schr"odinger equation with $H_\phi$, and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite--Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by $\kappa=e^{i\theta}\tanh r$, $\Delta_\phi=1-\kappa+8c_3\kappa$, and $\Gamma_\phi=1+\kappa+8c_3\kappa$, together with their Wigner profiles. In these dimensionless coefficients, $c_3$ is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.

Journal of Physics A Mathematical and TheoreticalOct 28, 20260 citations

arXiv:2602.08611 · Journal of Physics A Mathematical and Theoretical

Representation theory of inhomogeneous Gaussian unitaries

Jingqi Sun, Joshua Combes, Lucas Hackl

quant-phSubmitted Feb 9, 2026• Announced Feb 10, 2026• 0 citations

arXiv HTML PDF

Abstract

(M,z,Ψ)

. The Baker-Campbel-Hausdorff formula allows us then to factor any Gaussian unitary into a squeezing and a displacement transformation, from which we derive the group multiplication law.

Comments: 14+10 pages, 3 figures, 4 tables

Related papers

Representation theory of Gaussian unitary transformations for bosonic and fermionic systems

Tommaso Guaita, Lucas Hackl, Thomas Quella

Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $\langle 0|\exp{(-i\hat{H})}|0\rangle$ , where $|0\rangle$ is a Gaussian state and $\hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $\langle 0|\exp{(-i\hat{H})}|0\rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $\langle 0|\exp{(-i\hat{H})}|0\rangle$, where $|0\rangle$ is a Gaussian state and $\hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $\langle 0|\exp{(-i\hat{H})}|0\rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.

arXiv:2409.11628Annals of PhysicsOct 17, 20263 citations

From Hamiltonians to complex symplectic transformations

Gianfranco Cariolaro, Gianfranco L. Pierobon

Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such as Bogoliubov and real symplectic transformations. The paper develops an alternative representation, called complex symplectic transformation, which is more compact and is comprehensive of both Bogoliubov and real symplectic transformations. Moreover, it has other advantages. One of the main results of the theory, not available in the literature, is that the final displacement is not simply given by the linear part of the Hamiltonian, but depends also on the quadratic part. In particular, it is shown that by combining squeezing and rotation, it is possible to achieve a final displacement with an arbitrary amount.Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such as Bogoliubov and real symplectic transformations. The paper develops an alternative representation, called complex symplectic transformation, which is more compact and is comprehensive of both Bogoliubov and real symplectic transformations. Moreover, it has other advantages. One of the main results of the theory, not available in the literature, is that the final displacement is not simply given by the linear part of the Hamiltonian, but depends also on the quadratic part. In particular, it is shown that by combining squeezing and rotation, it is possible to achieve a final displacement with an arbitrary amount.

arXiv:1704.02008Apr 6, 20170 citations

Quadratic Hamiltonians

Didier Robert, Monique Combescure

The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in $L^2(\mathbb R^n)$ corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in $L^2(\mathbb R^n)$ corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.

Theoretical and mathematical physicsJan 1, 20211 citation

Gaussian states and operations -- a quick reference

Jonatan Bohr Brask

Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.

arXiv:2102.05748Feb 10, 20218 citations

Uncovering Non-Gaussianity through Multi-Copy Symmetries

Hao Dai, Yue Zhang

Gaussian states are fundamental in continuous-variable quantum information, yet characterizing non-Gaussianity remains challenging due to the non-convexity of the Gaussian set. Existing witnesses typically rely on Wigner negativity or other information-theoretic quantities. In this work, we develop a group-theoretic, multi-copy approach to detect non-Gaussianity in bosonic systems. We study passive linear optical transformations that mix copies of a quantum state and analyze their commutation with identical Gaussian unitaries applied to each copy. Orthogonal copy-mixing transformations commute with the symplectic part of the Gaussian action, while the displacement part restricts the symmetry to the stabilizer of the collective mode. This structure yields a family of witnesses satisfied by all single-mode Gaussian states. Fixing the thermal reference parameter via the purity, violation of these identities certifies non-Gaussianity. We illustrate the method with several single-mode examples and present an experimental protocol based on passive interferometry and photon-number-resolved detection, showing that the relevant multi-copy expectation values can be estimated from bounded phase observables. Finally, we extend the construction to multi-mode systems and discuss how the same symmetry framework may lead to quantitative measures of non-Gaussianity.Gaussian states are fundamental in continuous-variable quantum information, yet characterizing non-Gaussianity remains challenging due to the non-convexity of the Gaussian set. Existing witnesses typically rely on Wigner negativity or other information-theoretic quantities. In this work, we develop a group-theoretic, multi-copy approach to detect non-Gaussianity in bosonic systems. We study passive linear optical transformations that mix copies of a quantum state and analyze their commutation with identical Gaussian unitaries applied to each copy. Orthogonal copy-mixing transformations commute with the symplectic part of the Gaussian action, while the displacement part restricts the symmetry to the stabilizer of the collective mode. This structure yields a family of witnesses satisfied by all single-mode Gaussian states. Fixing the thermal reference parameter via the purity, violation of these identities certifies non-Gaussianity. We illustrate the method with several single-mode examples and present an experimental protocol based on passive interferometry and photon-number-resolved detection, showing that the relevant multi-copy expectation values can be estimated from bounded phase observables. Finally, we extend the construction to multi-mode systems and discuss how the same symmetry framework may lead to quantitative measures of non-Gaussianity.

quant-pharXiv:2608.03755Aug 5, 20260 citations

Exact gaussian states under quadratic non-hermitian deformations

Kapil Shrivastava

Abstract We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map $T_\phi=\exp\{2\phi(\partial_x)\}, \qquad \phi(z)=c_1+c_2z+c_3z^2 .$ The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization $H_\phi=T_\phi^{-1}HT_\phi,$ the physical inner product is generated by $M_\phi=T_\phi^\dagger T_\phi$ , time evolution is governed by the Schr"odinger equation with $H_\phi$ , and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite--Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by $\kappa=e^{i\theta}\tanh r$ , $\Delta_\phi=1-\kappa+8c_3\kappa$ , and $\Gamma_\phi=1+\kappa+8c_3\kappa$ , together with their Wigner profiles. In these dimensionless coefficients, $c_3$ is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.Abstract We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map \[ T_\phi=\exp\{2\phi(\partial_x)\}, \qquad \phi(z)=c_1+c_2z+c_3z^2 . \] The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization \[ H_\phi=T_\phi^{-1}HT_\phi, \] the physical inner product is generated by $M_\phi=T_\phi^\dagger T_\phi$, time evolution is governed by the Schr"odinger equation with $H_\phi$, and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite--Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by $\kappa=e^{i\theta}\tanh r$, $\Delta_\phi=1-\kappa+8c_3\kappa$, and $\Gamma_\phi=1+\kappa+8c_3\kappa$, together with their Wigner profiles. In these dimensionless coefficients, $c_3$ is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.

Journal of Physics A Mathematical and TheoreticalOct 28, 20260 citations

(M,z,Ψ)

. The Baker-Campbel-Hausdorff formula allows us then to factor any Gaussian unitary into a squeezing and a displacement transformation, from which we derive the group multiplication law.

\langle 0|\exp{(-i\hat{H})}|0\rangle

, where

|0\rangle

is a Gaussian state and

\hat{H}

an arbitrary quadratic Hamiltonian. We provide closed formulas for

\langle 0|\exp{(-i\hat{H})}|0\rangle

and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.

L^2(\mathbb R^n)

corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.

Abstract We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map

T_\phi=\exp\{2\phi(\partial_x)\}, \qquad \phi(z)=c_1+c_2z+c_3z^2 .

The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization

H_\phi=T_\phi^{-1}HT_\phi,

the physical inner product is generated by

M_\phi=T_\phi^\dagger T_\phi

, time evolution is governed by the Schr"odinger equation with

H_\phi

, and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite--Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by

\kappa=e^{i\theta}\tanh r

\Delta_\phi=1-\kappa+8c_3\kappa

, and

\Gamma_\phi=1+\kappa+8c_3\kappa

, together with their Wigner profiles. In these dimensionless coefficients,

c_3

is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.