Non-Gaussianity and Non-Classicality Measures

• Non-Gaussianity and non-classicality are quantum resource measures defined by deviations from classical mixtures of coherent or Gaussian states.
• Phase-space methods, including the Wigner and Glauber–Sudarshan P-functions, capture state negativity and provide practical verification even under loss.
• Entropic, algebraic, and click-statistics criteria offer robust operational markers that support applications in quantum computing, communication, and metrology.

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 (Kühn et al., 2020, Lachman et al., 2016, Kühn et al., 2018).

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 (Kühn et al., 2020).
• 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 (Deepak et al., 2023). 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 (Deepak et al., 2023, Deepak et al., 2023, Malpani et al., 2021).

• $s$-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 (Kühn et al., 2020).
• 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 (Kühn et al., 2020, Kühn et al., 2018).
• 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 (Park et al., 2015, Park et al., 2016).

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 $\tau_G$ (Deepak et al., 2023, Barbieri et al., 2010, Malpani et al., 2021, Deepak et al., 2023).

• 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 (Bose, 2016).

• 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 (Allevi et al., 2012).

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 $D$ (Kühn et al., 2018).
• 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 (Lachman et al., 2016, Liu et al., 2023).
• 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” (Liu et al., 2023).
• 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 (Deepak et al., 2023, Malpani et al., 2021, Deepak et al., 2023).
• 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 (Deepak et al., 2023, Malpani et al., 2021, Deepak et al., 2023).
• 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 (Marian et al., 2019).

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 $T<0.5$ (Lachman et al., 2016, Park et al., 2015). 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 (Kühn et al., 2020, Lachman et al., 2016). QNG depth (in dB) quantifies the maximum tolerable loss for a given state before it becomes compatible with a Gaussian mixture (Liu et al., 2023).

Specific measurement prescriptions—including single-detector vacuum probabilities before and after variable attenuation (Fiurášek et al., 2021), phase-space parity measurements at a small number of optimized points (Park et al., 2016), and OPA-based intensity correlations (Kalash et al., 24 Jul 2025)—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 (Malpani et al., 2021, Deepak et al., 2023).

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 (Deepak et al., 2023, Damanik, 2012). 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 (Allevi et al., 2012).

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 (Liu et al., 2023, Kalash et al., 24 Jul 2025, Grygar et al., 2022).

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 (Lachman et al., 2016, Kühn et al., 2020, Barbieri et al., 2010).

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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.