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In continuous-variable quantum systems, a Gaussian state has a Gaussian-shaped Wigner function in phase space and is fully described by its mean values and covariance matrix. A non-Gaussian state has a different phase-space shape, so those first and second moments do not capture all its structure.
What makes a quantum state Gaussian?
In quantum optics, a system’s quadratures are observables analogous to position and momentum. Together, their possible values define a phase space. The Wigner function represents a quantum state in that space; unlike an ordinary probability distribution, it can take negative values.
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A state is Gaussian when its Wigner function has a Gaussian shape over phase space. Its first moments give the mean quadrature values, while its covariance matrix records their variances and correlations. For a Gaussian state, these quantities determine the full state description, including higher-order expectation values. In statistical terms, it resembles a multivariate normal distribution specified by its mean and covariance—but the Wigner function is a quantum representation, not always an ordinary probability distribution. See Stefano Olivares’s tutorial on Gaussian states and Mattia Walschaers’s review of non-Gaussian quantum states.
How do Gaussian and non-Gaussian states compare?
| Feature | Gaussian states | Non-Gaussian states |
|---|---|---|
| Phase-space shape | Gaussian Wigner function | Wigner function is not Gaussian |
| What describes the state | First moments and covariance matrix | Requires information beyond first and second moments |
| Examples | Vacuum, coherent, squeezed, and thermal states | Photon-number (Fock) states, cat states, and Gottesman–Kitaev–Preskill (GKP) states |
| Typical mathematical handling | Often handled through transformations of means and covariance matrices | May require higher moments, phase-space structure, or specialized measures |
| Wigner-function sign | Nonnegative | Can be negative, but need not be |
This comparison uses the continuous-variable bosonic meaning of Gaussianity. The term can have different definitions in other settings, including fermionic systems.
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Does every non-Gaussian state have a negative Wigner function?
No. Wigner negativity is a notable sign of nonclassical behavior, but it is not the definition of non-Gaussianity. In the continuous-variable setting discussed by Walschaers, pure non-Gaussian states are Wigner-negative, whereas some mixed non-Gaussian states have positive Wigner functions.
There is a further distinction: being non-Gaussian is not the same as being outside the convex hull of Gaussian states. Gaussian states do not form a convex set, so a mixture of Gaussian states can itself be non-Gaussian. “Quantum non-Gaussian” is often used for the narrower property of being outside that convex hull; Wigner negativity and stellar rank are other, distinct ways of characterizing states.
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Why are Gaussian states easier to handle?
Common quantum-optical operations such as displacement, squeezing, and mode mixing can preserve Gaussian character under the relevant conditions. Their effects can often be represented as transformations of the means and covariance matrix rather than by tracking the entire phase-space function. This compact description helps make Gaussian methods mathematically tractable and experimentally accessible.
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Non-Gaussian structure can arise through non-Gaussian operations or conditional measurements. For example, measuring part of a correlated multimode Gaussian system can produce a non-Gaussian state in the remaining modes when the necessary correlations are present.
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What is non-Gaussianity useful for?
Non-Gaussian states are studied in connection with quantum correlations, sensing, quantum information, and proposals for computational advantage. Their importance is task-dependent: non-Gaussianity does not by itself guarantee better performance for every protocol. Walschaers’s 2021 review discusses these roles alongside ways of characterizing non-Gaussian states.
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For a phase-space introduction to Gaussian states, see Olivares’s tutorial. For a broader discussion of non-Gaussian states, their characterization, and applications, see Walschaers’s 2021 PRX Quantum tutorial.
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