---
title: Non-Gaussianity and Non-Classicality Measures
url: https://www.emergentmind.com/topics/non-gaussianity-and-non-classicality-measures
type: topic
---

# Non-Gaussianity and Non-Classicality Measures

Non-Gaussianity and non-classicality are central, yet distinct, resource concepts in quantum optics and continuous-variable quantum information. Non-classicality distinguishes quantum states that cannot be described as mixtures of coherent states, typically via the negativity or singularity of their Glauber–Sudarshan $P$-function. Non-Gaussianity refines this notion further, identifying states not expressible as convex mixtures of Gaussian states (i.e., squeezed, displaced, and thermal states). A multiplicity of operationally relevant measures—based on phase-space, entropic, and algebraic criteria—have been developed to quantify, detect, and compare these resources. These measures differ in their resource-theoretic implications, experimental accessibility, robustness to loss, and sensitivity to detector imperfections.

## 1. Conceptual Distinction: Classicality, Non-Classicality, and Non-Gaussianity

Classical states in quantum optics are those that admit a regular and positive $P$-function; any such state admits a physical interpretation as a statistical mixture of coherent states. Non-classical states, in contrast, exhibit $P$-functions that are negative or more singular than a delta function. However, the set of all mixtures of coherent states (classical) is a strict subset of mixtures of Gaussian states (which include squeezed and thermal states), and non-Gaussianity is only present if a state lies outside this Gaussian convex hull [2010.02173, 1611.07737, 1803.03133].

This hierarchy is summarized as:
\[
\text{Coherent mixtures} \subset \text{Gaussian mixtures} \subset \text{all states}
\]
With important implications: non-Gaussianity is always a form of non-classicality, but not all non-classical states are non-Gaussian (e.g., pure squeezed vacuum states are non-classical but Gaussian).

## 2. Phase-Space Measures: Wigner Function, Quasiprobabilities, and Logarithmic Negativity

A foundational suite of non-classicality measures utilize phase-space quasiprobability distributions such as the Wigner, $P$, and $Q$ functions:

- **Glauber–Sudarshan $P$-function**: Its negativity or singularity is the definitive identifier of non-classicality, but it is not directly accessible except for simple states [2010.02173].
- **Wigner function $W(\gamma,\gamma^*)$**: Negativity in the Wigner function is a sufficient (but not necessary) test for non-classicality. For pure states, Hudson’s theorem stipulates positivity implies the state is Gaussian [2304.06046]. The *Wigner logarithmic negativity*,
  \[
  W_{\rm LN}(\rho) = \log_2\Bigl[\int d^2\gamma\, |W_\rho(\gamma)|\Bigr],
  \]
  serves as an additive measure quantifying the “volume” of negative values of $W(\gamma)$, thus capturing both non-classicality and non-Gaussianity [2304.06046, 2306.04490, 2109.12145].

- **$s$-parametrized quasiprobabilities**: The convolution of the $P$-function with a Gaussian kernel yields distributions indexed by $s\in[-1,1]$ (with $s=0$ the Wigner function, $s=-1$ the $Q$-function). Sufficiently negative $s$ ensure positivity for all states with overall detection efficiency $\eta \leq \frac{1}{2}$, limiting their utility as non-classicality witnesses under loss [2010.02173].

- **Non-Gaussian filtered quasiprobabilities**: Convolution with suitable non-Gaussian kernels yields regularized phase-space functions whose negativities directly and unambiguously witness non-classicality—and, crucially, remain robust even under high loss or low detector efficiency [2010.02173, 1803.03133].

- **Bell-type phase-space rectangles**: Tests such as the functional
  \[
  \mathcal{J}[\rho] = \frac{\pi}{2}\left\{W(x_0,y_0)+W(x_1,y_0)+W(x_0,y_1)-W(x_1,y_1)\right\},
  \]
  derived by Banaszek–Wódkiewicz, provide a nonclassicality witness with algebraic bounds ($|\mathcal{J}| \leq 2$ for mixtures of coherent states). Extensions set stricter bounds for mixtures of Gaussian states ($\mathcal{J} \leq 8/3^{9/8} \approx 2.32$), so exceeding these certifies genuine quantum non-Gaussianity [1505.01470, 1601.00279].

## 3. Entropic and Reference-State Measures

Several entropic quantifiers compare a quantum state to a reference state, either classical or Gaussian:

- **Relative entropy of non-Gaussianity ($\delta_{\rm RE}$)**:
  \[
  \delta_{\rm RE}[\rho] = S(\rho\|\tau_G) = \mathrm{Tr}[\rho\log\rho - \rho\log\tau_G],
  \]
  where $\tau_G$ is the (unique) Gaussian state with the same first and second moments as $\rho$. For pure states, this reduces to the von Neumann entropy of $\tau_G$ [2304.06046, 1012.0466, 2109.12145, 2306.04490].

- **Wehrl entropy excess**: Defined via the Husimi $Q$-function, the Wehrl entropy $H_w(\rho)$, and a “classical” reference maximized under fixed von Neumann entropy,
  \[
  \Delta_{\rm NC}(\rho) = H_w(\rho) - H_w^{\max, cl}[S(\rho)],
  \]
  which is analytic for pure and Gaussian states, and quantifies how much more “phase-space spread” the state possesses compared to any classical state of the same mixedness [1701.00064].

- **Hilbert–Schmidt distance-based and entropy-based non-Gaussianity measures ($\delta_A$, $\delta_B$)**: For diagonal states, $\delta_A$ uses quadrature overlap and purity; $\delta_B$ is the difference in von Neumann entropy between the state and its Gaussian reference. Both are experimentally accessible for Fock-diagonal states [1210.2747].

The table summarizes key reference-state measures:

| Measure                      | Reference State             | Analytic for           |
|------------------------------|----------------------------|------------------------|
| $\delta_{\rm RE}$            | Covariance-matched Gaussian| All states (simple for pure) |
| $\Delta_{\rm NC}$            | Classical state (max Wehrl)| Pure, Gaussian, all with S(ρ)=0 or known S(ρ) |
| $\delta_A$                   | Covariance-matched Gaussian| Fock-diagonal, others numerically |
| $\delta_B$                   | Covariance-matched Gaussian| Fock-diagonal         |

## 4. Operational and Resource-Theoretic Measures

Several criteria and quantifiers are grounded in resource theory and operational interpretations:

- **Superposition number (degree of non-classicality $D$)**: Minimal number of coherent states needed in a pure-state superposition; for mixed states, the minimal $r$ such that $\rho$ is in the convex hull of pure superpositions with $r$ components. Regularized nonclassicality quasiprobabilities (via non-Gaussian filtering) yield tight lower bounds for $D$ [1803.03133].

- **Click-statistics criteria for quantum non-Gaussianity**: Dividing an incoming field into $n+1$ channels and detecting multi-detector click events yields linear inequalities between probabilities that are saturated by all Gaussian mixtures but violated by non-Gaussian states. These are robust to loss and require only “on/off” (non-resolving) detection [1611.07737, 2307.04531].

- **Photon-counting witnesses**: For heralded or multiphoton states, inequalities such as $P_{2+} < \frac{2}{3} P_1^3$ (where $P_{2+}$ is the two-or-more-photon probability) serve as explicit single-mode QNG criteria; generalizations apply to multimode entangled photon pairs and “coincidences depth” [2307.04531].

- **Beam-splitter entanglement potential (linear entropy $L_E$)**: Mixing a state with vacuum on a 50:50 beam splitter and computing the linear entropy of the output reduced state yields an accessible measure of non-classicality, with $L_E=0$ for all classical inputs [2304.06046, 2109.12145, 2306.04490].

- **Skew information–based measures**: Quantum coherence relative to $a$ and $a^\dagger$ (Wigner–Yanase skew information) quantifies non-classicality since coherent states minimize this metric ($N(\rho) = 1/2$), while higher values signal Poissonian or super-Poissonian statistics [2304.06046, 2109.12145, 2306.04490].

- **Geometric quantifiers**: For pure states, the minimum distance (Hilbert–Schmidt, Bures, etc.) to the (unique) set of coherent states may be cast as $D_g(|\psi\rangle) = \sqrt{1 - \pi\,\max_\beta Q(\beta)}$, depending solely on the peak value of the $Q$-function [1909.09433].

## 5. Experimental Robustness, Loss, and Detector Efficiency

Many measures become insensitive to non-classicality under loss or limited detection efficiency. For Wigner-function-based criteria, negativity is erased for transmission $T<0.5$ [1611.07737, 1505.01470]. In contrast, non-Gaussian filtered quasiprobabilities and click-statistics criteria remain applicable under arbitrarily strong losses, as linear optical losses correspond to classical Gaussian operations and cannot generate or hide true quantum non-Gaussianity [2010.02173, 1611.07737]. QNG depth (in dB) quantifies the maximum tolerable loss for a given state before it becomes compatible with a Gaussian mixture [2307.04531].

Specific measurement prescriptions—including single-detector vacuum probabilities before and after variable attenuation [2107.09380], phase-space parity measurements at a small number of optimized points [1601.00279], and OPA-based intensity correlations [2507.18296]—provide experimentally viable, loss-robust, and mode-insensitive means to certify non-classicality and non-Gaussianity.

## 6. Comparative Evaluation of Measures and Contextual Trends

Multiple measures are often consistent in their qualitative ordering of states but—especially for mixed or complex structured states—can disagree or introduce non-monotonicities. For example, in photon-added or photon-subtracted displaced Fock states, measures such as $L_E$, skew information, and Wigner logarithmic negativity grow with photon-adding (or subtracting) number at small displacement but can cross and even decrease at large displacements. The role of the underlying Fock index $n$ becomes dominant at large displacement, while photon addition/subtraction dominates at low displacement [2109.12145, 2306.04490].

Relative-entropy non-Gaussianity typically mirrors the behavior of Wigner negativity-based measures for pure states but can differ for highly mixed or phase-randomized states [2304.06046, 1201.2747]. Some quantifiers, such as $\delta_B$ (relative-entropy-based non-Gaussianity), are more sensitive to small deviations but may overstate resource content in near-classical or nearly Gaussian regimes [1210.2747].

## 7. Relation to Quantum Information Tasks and Practical Implementation

The selection of non-Gaussian and non-classical states is critical for a wide range of quantum technologies, including quantum computation (linear-optical QC, boson sampling), communication (QKD with non-Gaussian resources), metrology, and foundational tests (e.g., Bell inequality violations). Robust, operationally accessible measures—especially those that tolerate moderate to high loss or rely on straightforward click-counting or phase-space sampling—are essential for practical resource certification [2307.04531, 2507.18296, 2208.12521].

The construction of QNG witnesses based on operational parameters, photon-number statistics, and realistic measurement scenarios ensures that both theoretical resource quantification and experimental benchmarking proceed on common, physically meaningful footing [1611.07737, 2010.02173, 1012.0466].

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In summary, the landscape of non-Gaussianity and non-classicality measures encompasses phase-space, entropic, algebraic, and operationally defined quantifiers, each tailored to different resource, implementation, and robustness requirements. The consistent theme is that quantum non-Gaussianity marks an essential quantum resource beyond mere non-classicality, often tightly coupled to experimental certifiability and direct relevance for quantum technological protocols.

Source: https://www.emergentmind.com/topics/non-gaussianity-and-non-classicality-measures