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In continuous-variable quantum systems, a state is Gaussian when its Wigner function has a Gaussian shape in phase space. Its mean quadrature values and covariance matrix—the first and second moments—then determine its full Gaussian description. A non-Gaussian state falls outside that family and needs additional information to describe its structure. The distinction matters because Gaussian states are often easier to calculate with, while some quantum protocols need non-Gaussian ingredients.
What makes a quantum state Gaussian?
For continuous-variable bosonic systems, such as modes of light, quantum states can be represented in phase space using a Wigner function. Phase space tracks pairs of observables called quadratures, often described as position-like and momentum-like components of a mode. The Wigner function is a quantum representation, not always an ordinary probability distribution.
A Gaussian state has a Wigner function with the shape of a multivariate normal distribution. Two sets of values specify that Gaussian shape:
- First moments: the average values of the quadratures, which locate the state in phase space.
- Covariance matrix: the quadrature variances and correlations, which describe the state’s spread and orientation.
For a Gaussian state, these first and second moments also determine all higher-order moments. This compact description is why calculations involving Gaussian states can often be handled through matrix transformations. For more detail on the phase-space formalism, see Stefano Olivares’s tutorial on Gaussian states.
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How non-Gaussian states differ
A non-Gaussian state has a Wigner function that is not Gaussian. Its mean and covariance still describe its first- and second-order features, but they do not capture its full structure. Higher-order moments or other details of the phase-space distribution may be needed.
The comparison with ordinary statistics is useful but limited: a non-Gaussian distribution may have features such as skewness, heavier tails, or multiple peaks, but a Wigner function is a quantum phase-space representation and can take negative values.
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| Feature | Gaussian states | Non-Gaussian states |
|---|---|---|
| Phase-space criterion | Wigner function is Gaussian | Wigner function is not Gaussian |
| Description | First moments and covariance matrix determine the Gaussian state | First and second moments alone are insufficient to capture the full structure |
| Examples | Vacuum, coherent, squeezed, and thermal states | Photon-number (Fock) states, cat states, and Gottesman–Kitaev–Preskill (GKP) states |
| Typical mathematical handling | Gaussian operations can often be tracked as transformations of means and covariance matrices | May require higher moments, phase-space structure, or specialized measures |
This comparison applies to continuous-variable bosonic states; “Gaussian state” can have other meanings in fermionic systems and other mathematical settings.
Does a non-Gaussian state always have a negative Wigner function?
No. Wigner negativity is a strong sign of nonclassical behavior, but it is not a complete test for non-Gaussianity. In the continuous-variable setting discussed by Mattia Walschaers, pure non-Gaussian states are Wigner-negative, while mixed non-Gaussian states can have positive Wigner functions. So a positive Wigner function does not, by itself, establish that a state is Gaussian.
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“Non-Gaussian” also differs from “quantum non-Gaussian.” The latter is a narrower description for states outside the convex hull of Gaussian states. Gaussian states do not form a convex set: a mixture of Gaussian states can itself be non-Gaussian. Wigner negativity, being outside the convex hull of Gaussian states, and stellar rank are distinct ways to characterize states, not interchangeable labels. See Walschaers’s review of non-Gaussian quantum states.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Why the distinction matters in quantum optics
Gaussian states are often easier to manipulate
Standard quantum-optical methods can generate and manipulate Gaussian states. Displacement, squeezing, and mode mixing are examples of operations that, under the relevant conditions, transform the means and covariance matrix while preserving Gaussian character. This makes Gaussian states experimentally accessible and mathematically tractable.
Non-Gaussian elements extend what can be prepared or characterized
Non-Gaussian states can arise through non-Gaussian operations or conditional measurements. In a multimode Gaussian state, measuring some modes can create a non-Gaussian state in the remaining modes when suitable correlations are present. Non-Gaussian states are studied in connection with quantum correlations, sensing, and quantum information, but non-Gaussianity alone does not guarantee an advantage for every task.
For an overview of how non-Gaussian states arise and where they appear in quantum information, read the PRX Quantum tutorial by Mattia Walschaers. A broader introduction to light’s phase-space description is available in Oxford Academic’s “Quantum States of Light” chapter.
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