Gaussian Phase-Space Representation

Updated 3 December 2025

Gaussian phase-space representations are mathematical frameworks that model quantum states using Gaussian distributions characterized by displacement and covariance, applicable to both bosonic and fermionic systems.
They facilitate the mapping of quantum operations and measurements—such as displacements, squeezing, and homodyne detection—via symplectic transformations that preserve Gaussianity.
The formalism supports efficient stochastic simulations through Fokker–Planck equations and diffusion gauges, enabling practical computation of observables in strongly correlated many-body systems.

A Gaussian phase-space representation is a formalism in which quantum states, operators, and dynamics are described using phase-space distributions whose main structure is Gaussian—characterized by their first and second moments (displacement and covariance). This methodology plays a foundational role in quantum optics, continuous-variable quantum information, quantum statistical mechanics, and the simulation of strongly correlated many-body systems. In the bosonic context, phase-space Gaussians correspond to coherent, squeezed, thermal, and general multimode Gaussian states; for fermions, the formalism extends to Gaussian operator bases and covariance-matrix-valued phase-space distributions. The Gaussian representation offers analytic tractability, transparent visualization of quantum features, efficient computation of entropic and informational quantities, and direct links to measurement scenarios such as quantum tomography and homodyne detection.

1. Mathematical Structure of Gaussian Phase-Space Distributions

Gaussian phase-space representations are built from distributions on phase-space vectors. For bosonic systems, phase-space coordinates are $R = (x_1, p_1, ..., x_n, p_n)^T$ with canonical commutators $[R_k, R_l]=i\Omega_{kl}$ where $\Omega$ is the symplectic form. A generic $n$ -mode quantum state $\varrho$ admits a Wigner function $W(\xi)$ , and when this is Gaussian: $W(\xi) = \frac{1}{(2\pi)^n \sqrt{\det \sigma}} \exp\left[ -\frac{1}{2} (\xi-d)^T \sigma^{-1} (\xi-d) \right]$ where $d$ is the displacement vector and $\sigma$ the $2n \times 2n$ covariance matrix. The covariance must satisfy the quantum uncertainty constraint: $\sigma + \frac{i}{2} \Omega \geq 0$ Physical meaning is encoded entirely in $d$ and $\sigma$ ; all quantum operations, measurements, and evolutions correspond to transformations of these objects (Olivares, 2011, Brask, 2021).

Fermionic systems are represented via Gaussian operator bases such as: $\widehat\Lambda(n) = \frac{1}{\det[I+n]} :\exp[-\hat a^\dagger (I+n)^{-1} \hat a]:$ for bosons, or analogous forms for fermions, where $n$ is a matrix of correlation functions. States are represented as probability distributions over the phase-space of Green’s functions, supporting stochastic simulation and explicit calculations of purity, entanglement, and observable averages (Rosales-Zárate et al., 2011, Ogren et al., 2010, Rousse et al., 30 Nov 2025).

2. Gaussian Smoothing, s-Ordered Distributions, and Measurement

Gaussian phase-space representations generalize the Wigner function by convolution with Gaussian kernels. Given a quantum state with Wigner function $W(q,p)$ , the Gaussian-smoothed Wigner function is: $W_G(q,p) = \int d\xi\,d\eta\,\frac{1}{2\pi\sigma_q\sigma_p}\exp\left\{-\frac{(q-\xi)^2}{2\sigma_q^2} -\frac{(p-\eta)^2}{2\sigma_p^2}\right\} W(\xi,\eta)$ This defines a family of $s$ -parameterized distributions—Cahill-Glauber distributions—with $s = -2\sigma_q \sigma_p / \hbar$ . In this hierarchy: $s=0$ is Wigner, $s=-1$ is Husimi Q (positive-definite), and $s=+1$ is Glauber-Sudarshan P (1311.0666). Through this smoothing, quantum negativity and fine phase-space structures are suppressed, but analytic positivity and noise resilience are gained. Phase-space statistics extracted from imperfect detectors are precisely modeled by adjusted Gaussian widths, enabling direct quantum tomography even in lossy measurement scenarios such as eight-port homodyne detection (1311.0666). Expectation values of operators $\langle \hat O \rangle$ are recovered from convolutions of Weyl symbols: $\langle \hat O \rangle = \int dq\,dp\,W_G(q,p) A_G(q,p)$ with $A_G$ the Gaussian-convolved operator symbol.

3. Symplectic Operations, Quantum Dynamics, and Frames

Any operation generated by quadratic Hamiltonians—displacements, squeezing, phase shifts, beam-splitters—corresponds to affine transformations of $d,\sigma$ : $d \to S d + \delta, \qquad \sigma \to S \sigma S^T$ with $S \in Sp(2n, \mathbb{R})$ the symplectic matrix. This directly maps to phase-space Wigner functions, maintaining Gaussianity (Brask, 2021, Olivares, 2011). For operator sampling and quantum error correction (e.g., GKP codes), Gaussian unitaries likewise act by symplectic transformations on Wigner functions (Mensen et al., 2020). The phase-space representation underpins the design of Gabor and Weyl-Heisenberg frames, where Gaussian symbols in phase-space admit expansions into shifted elementary Gaussians with exponential convergence and explicit error bounds (Faulhuber et al., 2017).

Quantum dynamics—either Hamiltonian evolution or Lindblad-type dissipative processes—admit closed-form equations for $d,\sigma$ ; for example, under a lossy channel: $d(t) = e^{-\Gamma t/2}d(0), \quad \sigma(t) = e^{-\Gamma t}\sigma(0) + \left(1-e^{-\Gamma t}\right)\sigma_{\infty}$ for damping rate $\Gamma$ and asymptotic noise floor $\sigma_{\infty}$ (Olivares, 2011).

4. Generalized Gaussian Operator Representations and Entropic Quantities

Generalized Gaussian phase-space representations use overcomplete operator bases parameterized by stochastic Green’s functions $n$ for bosons/fermions. The density operator $\hat\rho$ is encoded as: $\hat\rho = \int P(n)\, \widehat\Lambda(n) dn$ Averages of quantum observables reduce to statistical averages over $P(n)$ (Rosales-Zárate et al., 2011). This representation supports efficient Monte Carlo computation of purity, linear (Rényi) entropy, fidelity, and entanglement:

Linear entropy: $S_2 = -\ln\Tr[\rho^2]$ computed by sampling pairwise overlaps of Gaussian kernels.
Purity: $(2\pi)^n\int [W(\xi)]^2\,d^{2n}\xi = 1/\sqrt{\det \sigma}$ for a Gaussian state.

In comparison to standard Wigner and Husimi methods, Gaussian kernel overlaps are non-singular and support positive-definite sampling even for nonclassical states (Rosales-Zárate et al., 2011). Fermionic generalizations employ covariance-matrix-valued phase-spaces, including Majorana representations on symmetry class D, facilitating direct Fokker–Planck and SDE-based simulation of topologically nontrivial systems (Joseph et al., 2017).

5. Gaussian Cat States, Interference, and Nonlinear Extensions

Superpositions of Gaussian states—"generalized Gaussian cat states"—exhibit Wigner functions with two “Gaussian hills” and an interference term featuring phase-space oscillations determined by a quadratic form. The general structure for a superposition $|\Psi\rangle = a |V_1,u\rangle + b |V_2,v\rangle$ yields: $W_\Psi(r) = |a|^2 W_1(r) + |b|^2 W_2(r) + 2|ab| \mathcal{I}(r)$ where $W_j(r)$ are Gaussians centered at $u$ and $v$ , and $\mathcal{I}(r)$ is governed by: $\mathcal{I}(r) = \frac{1}{(\pi\hbar)^n} \Re\{e^{i\Phi(r)} D(r)\}$ with $D(r)$ a positive Gaussian envelope and $\Phi(r)$ a phase comprising symplectic products and quadratic forms. In one degree of freedom, interference fringes are hyperbolic and remain robust under Lindblad (thermal) evolution, preserving their phase-space structure (Nicacio et al., 2010).

Mixed-state cats, constructed by conditional Gaussian operations or via Kerr-type nonlinear dynamics, yield interference terms whose phase-space profiles—the Gaussian envelope and oscillatory fringe structure—are analytically tractable and exhibit characteristic hyperbolic or elliptical patterns depending on system parameters (Nicacio et al., 2010).

6. Stochastic Simulation, Diffusion Gauges, and Many-Body Systems

The Gaussian phase-space approach supports exact (up to boundary terms) sampling of quantum dynamics via stochastic differential equations (SDEs) derived from Fokker–Planck equations associated with operator correspondences. For the Fermi–Hubbard model: $\frac{\partial P}{\partial t} = \left[-\sum \frac{\partial}{\partial n} A(n) + \frac{1}{2} \sum \frac{\partial^2}{\partial n^2} D(n) \right] P$ with drift and diffusion matrices $A$ , $D$ constructed from the Hamiltonian. Equivalent SDEs are obtained by factorizing $D = BB^T$ , with the non-uniqueness exploited as “diffusion gauge” freedom (Rousse et al., 30 Nov 2025, Ogren et al., 2010). Computational performance is optimized by selecting numerically compact gauges (e.g., SVD-based), thus suppressing the development of “fat tails” in the sampled trajectories and considerably extending practical simulation times for large and strongly correlated systems (Rousse et al., 30 Nov 2025). Observables, such as density and correlation functions, correspond to moments of the phase-space variables, and ensemble averages reproduce quantum expectation values exactly while the sampled distribution remains bounded.

7. Quantum Camouflage, Stationarity, and Nonlinear Hamiltonian Extensions

In specific systems—e.g., one-dimensional Hamiltonians with $H^W(q,p)$ such that $\partial^2 H^W/\partial q \partial p = 0$ —the quantum flow of Wigner functions for Gaussian ensembles coincides with the classical Liouville flow. The quantum continuity equation: $\partial_{t} W(q,p;t) + \partial_q J_q + \partial_p J_p = 0$ has currents $J_q$ , $J_p$ that, under appropriate Gaussian ansatz and parameter matching, become identical to their classical counterparts, a phenomenon termed “quantum camouflage” (Bernardini et al., 24 Sep 2024). In these situations, the higher-order quantum corrections encoded in Gaussian fluctuation profiles can be tuned to exactly mimic classical stationarity conditions, emphasizing the analytic power and flexibility of the Gaussian ensemble approach even when confronting non-standard, nonlinear Hamiltonians.