1. Definition

The non-Gaussianity (nonG) of a continuous-variable (CV) quantum state $ \rho $ is defined as the quantum relative entropy distance between $ \rho $ and a reference Gaussian state $ \rho_G $ that has the same first moments and the same covariance matrix as $ \rho $:

\[\delta[\rho] = S(\rho \| \rho_G) = S(\rho_G) - S(\rho)\]

where:

  • $ S(\rho) = -\mathrm{Tr}(\rho \log \rho) $ is the von Neumann entropy,
  • $ \rho_G $ is uniquely fixed by matching first moments and covariance matrix of $ \rho $.

For a single-mode Gaussian state,

\[S(\rho_G) = h\!\left(\sqrt{\det \sigma[\rho]}\right)\]

with

\[h(t) = \left(t+\tfrac12\right)\ln\!\left(t+\tfrac12\right) + \left(t-\tfrac12\right)\ln\!\left(t-\tfrac12\right).\]

Hence,

\[\boxed{\;\delta[\rho] = h\!\left(\sqrt{\det \sigma[\rho]}\right) - S(\rho)\;}\]

The covariance matrix of a continuous-variable quantum state $ \rho $ is defined in terms of the quadrature operators collected in the vector

\[\hat R = (\hat x,\hat p)^T .\]

Its elements are

\[\sigma_{ij} = \frac12\langle \hat R_i \hat R_j + \hat R_j \hat R_i \rangle - \langle \hat R_i \rangle \langle \hat R_j \rangle ,\]

where expectation values are taken with respect to $ \rho $:

\[\langle \hat O \rangle = \mathrm{Tr}(\rho\,\hat O).\]

For a single-mode state, this gives

\[\sigma = \begin{pmatrix} \langle x^2\rangle-\langle x\rangle^2 & \frac12\langle xp+px\rangle-\langle x\rangle\langle p\rangle \\[6pt] \frac12\langle xp+px\rangle-\langle x\rangle\langle p\rangle & \langle p^2\rangle-\langle p\rangle^2 \end{pmatrix}.\]

If the first moments vanish, $ \langle x\rangle=\langle p\rangle=0 $, this simplifies to

\[\sigma = \begin{pmatrix} \langle x^2\rangle & \frac12\langle xp+px\rangle \\ \frac12\langle xp+px\rangle & \langle p^2\rangle \end{pmatrix}.\]

The covariance matrix completely characterizes any Gaussian state and fixes the reference Gaussian state $ \rho_G $ used in non-Gaussianity measures.

Example: Non-Gaussian state defined by

The non-Gaussian pure state is defined by the wavefunction

\[\phi_0(x;\lambda)=\frac{e^{-x^2/2}}{\pi^{1/4}}\left[1-\frac{\lambda}{\sqrt{2}}\operatorname{Erf}(x)\right]\]

The corresponding density operator of the system is

\[\rho = |\phi_0\rangle\langle \phi_0| ,\]

which fully characterizes the quantum state.

Since $|\phi_0\rangle$ is a pure state, its von Neumann entropy vanishes:

\[S(\rho)=0 .\]

The reference Gaussian state $ \rho_G $ is defined as the unique Gaussian state having the same first moments and the same covariance matrix as $ \rho $.
It represents the closest Gaussian approximation to $ \rho $ used to quantify non-Gaussianity.

Using parity properties of $e^{-x^2}$ and $\operatorname{Erf}(x)$:

  • Mean position: \(\langle x\rangle = \int x|\phi_0(x;\lambda)|^2 dx = 0\)

  • Mean momentum: \(\langle p\rangle = 0\)

Thus, the first-moment vector is \(E[\rho]=(0,0)^T\)

Why the reference Gaussian state shares the same first moments:
In the relative-entropy definition of non-Gaussianity, the reference Gaussian state $ \rho_G $ is chosen to match the first moments of $ \rho $ so that non-Gaussianity is not contaminated by trivial phase-space displacements. Since first moments can always be changed by a unitary displacement without altering the Gaussian or non-Gaussian nature of a state, fixing them ensures that $ \delta[\rho] $ quantifies only genuine shape deviations from Gaussianity.

If the first moments were nonzero:
For $E[\rho]\neq 0$, the reference Gaussian state $ \rho_G $ would have the same nonzero first-moment vector. Equivalently, one could apply a displacement operator to shift both $ \rho $ and $ \rho_G $ to zero mean without changing $ \delta[\rho] $, because quantum relative entropy is invariant under unitary displacements.

Define quadratures (with $\hbar=1$): \(\hat x = x, \qquad \hat p=-i\frac{d}{dx}\)

The second moments are

\[\langle x^2\rangle = \frac12 + \frac{\lambda^2}{2\pi}\] \[\langle p^2\rangle = \frac12 + \frac{\lambda^2}{2\pi}\]

Cross term: \(\langle xp+px\rangle = 0\)

Why the reference Gaussian state shares the same covariance matrix:
The covariance matrix fixes all second-order moments. For a given set of first and second moments, there exists a unique Gaussian state. Choosing $ \rho_G $ with the same covariance matrix makes it the closest Gaussian state in the sense of relative entropy.

The covariance matrix is therefore

\[\sigma[\rho] = \begin{pmatrix} \langle x^2\rangle & 0 \\ 0 & \langle p^2\rangle \end{pmatrix} = \begin{pmatrix} \frac12+\frac{\lambda^2}{2\pi} & 0 \\ 0 & \frac12+\frac{\lambda^2}{2\pi} \end{pmatrix}\] \[\det \sigma = \left(\frac12+\frac{\lambda^2}{2\pi}\right)^2\] \[\sqrt{\det\sigma} = \frac12+\frac{\lambda^2}{2\pi}\]

The reference Gaussian state $ \rho_G $ has the same first moments and the same covariance matrix as $ \rho $. Its entropy is

\[S(\rho_G) = h\!\left(\frac12+\frac{\lambda^2}{2\pi}\right)\]

with \(h(t)=\left(t+\frac12\right)\ln\!\left(t+\frac12\right) +\left(t-\frac12\right)\ln\!\left(t-\frac12\right)\)

Since $S(\rho)=0$,

\[\boxed{ \delta[\rho] = h\!\left(\frac12+\frac{\lambda^2}{2\pi}\right) }\]

The function $h(t)$ is the von Neumann entropy of a single-mode Gaussian state written in terms of its symplectic eigenvalue $t$.

It is defined as \(h(t)=\left(t+\frac12\right)\ln\!\left(t+\frac12\right) +\left(t-\frac12\right)\ln\!\left(t-\frac12\right)\)

In this example, \(t=\sqrt{\det\sigma} = \frac12+\frac{\lambda^2}{2\pi}\)

Hence, the explicit value of $h(t)$ is \(h\!\left(\frac12+\frac{\lambda^2}{2\pi}\right) = \left(1+\frac{\lambda^2}{2\pi}\right)\ln\!\left(1+\frac{\lambda^2}{2\pi}\right) + \frac{\lambda^2}{2\pi}\ln\!\left(\frac{\lambda^2}{2\pi}\right)\)

For $\lambda=0$, \(h\!\left(\frac12\right)=0\)

For $\lambda\neq0$, $h(t)>0$ and increases monotonically with $\lambda^2$.

Interpretation

  • For $ \lambda=0 $:
    $\delta[\rho]=0$ → Gaussian vacuum state.
  • For $ \lambda\neq0 $:
    $\delta[\rho]>0$ → genuine non-Gaussianity induced by the error-function deformation.
  • The non-Gaussianity increases monotonically with $ \lambda^2 $.

This explicitly shows how the relative-entropy non-Gaussianity is computed step-by-step for the state shown in the image.

2. Origin / History

  • Introduced in quantum information theory as an application of quantum relative entropy (Umegaki, 1962).
  • First systematically applied to quantify non-Gaussianity by
    Genoni, Paris, and Banaszek (2007–2008).
  • Motivated by the central role of Gaussian states as free states in CV quantum optics and quantum information.

3. Why Important & Comparative Analysis

Importance

  • Vanishes iff the state is Gaussian.
  • Operationally meaningful: measures statistical distinguishability from the closest Gaussian state.
  • Directly linked to resource theories of non-Gaussianity.
  • Relevant for tasks like:
    • entanglement distillation,
    • quantum error correction,
    • universal CV quantum computation.

Comparison with Other Measures

Measure Basis Advantages Limitations
Relative-entropy nonG (this) Information-theoretic Unique Gaussian reference, monotone, additive Requires entropy evaluation
Wigner negativity Phase-space Easy visualization Zero for many non-Gaussian states
Hilbert–Schmidt distance Geometric Simple form Not monotone under CPTP maps
Kurtosis / higher moments Statistical Easy experimentally Not invariant, not operational

Key distinction: this measure is operational, monotonic, and reference-free (Gaussian reference fixed uniquely).