Quantum Non-Gaussian States

• Quantum non-Gaussian states are quantum states that cannot be represented as convex mixtures of Gaussian states, making them essential for advanced quantum computation and metrology.
• They are engineered through nonlinear processes such as photon subtraction and deterministic operations in optical, mechanical, and multimode platforms to achieve high-fidelity nonclassicality.
• Their quantification relies on measures like relative entropy, Wigner negativity, and Fock-population thresholds, which certify non-Gaussian resources crucial for fault-tolerant quantum technologies.

Quantum non-Gaussian states are quantum states of bosonic modes (typically photonic, phononic, or collective motional degrees of freedom) that cannot be expressed as convex mixtures of Gaussian states—i.e., states obtained by acting with linear optics, squeezers, and displacements on the vacuum (or thermal) state. Their significance lies in their indispensability for quantum computational advantage, high-fidelity quantum error correction, quantum metrology beyond standard quantum limits, and the realization of quantum networks with continuous-variable resources that cannot be classically simulated. This article provides a comprehensive overview of quantum non-Gaussian states, encompassing their mathematical characterization, state engineering protocols, resource-theoretic properties, experimental realizations, and applications.

1. Definition and Characterization of Quantum Non-Gaussian States

Quantum non-Gaussian states (QNG states) are defined by their inability to be represented as any mixture (convex combination) of Gaussian states. In the phase-space formalism, Gaussian states are completely characterized by their first and second quadrature moments and have Wigner functions of the form

$$W_\text{G}({\bf x}) = \frac{1}{\pi \sqrt{\det \sigma}} \exp\left[-({\bf x}-{\bf r})^\mathrm{T} \sigma^{-1} ({\bf x}-{\bf r})\right]$$

where $\sigma$ is the covariance matrix and ${\bf r}$ the displacement vector.

A state $\rho$ is quantum non-Gaussian if

$$\rho \neq \int d\lambda\,p(\lambda)\,|\psi_\mathrm{G}(\lambda)\rangle\langle\psi_\mathrm{G}(\lambda)|$$

where $|\psi_\mathrm{G}(\lambda)\rangle$ runs over all pure Gaussian states and $p(\lambda)$ is a probability measure (Lachman et al., 2022). Any pure state with negative Wigner function is necessarily QNG, but mixtures with positive Wigner functions may also be QNG if they cannot be decomposed as above.

Several operationally meaningful QNG witnesses and quantifiers have been proposed:

• Faithful relative entropy measure: The "convex-roof" relative entropy of QNG is defined by (Park et al., 2018):

$$Q(\rho) = \min_{\{p_i,\rho_i\}} \sum_i p_i\,S(\rho_i \| (\rho_i)_\mathrm{G})$$

where $S(\rho\|\sigma)$ is the quantum relative entropy, the minimization is over all convex decompositions, and $(\rho_i)_\mathrm{G}$ is the Gaussian state with the same first and second moments as $\rho_i$. This measure vanishes if and only if $\rho$ is a Gaussian mixture.

• Photon correlation criterion: For a single-mode field, the violation of the bound

$\sqrt{g^{(3)} + 3\sqrt{g^{(2)}}} \geq 2$

where $g^{(k)}$ are normalized photon correlation functions, is a sufficient, attenuation-invariant witness of QNG (Hotter et al., 11 Nov 2025).

• Fock-state population bounds: The maximal $n$-Fock population achievable by any Gaussian state sets a threshold, e.g., for single phonons, $P_1^\mathrm{G} \approx 0.478$. Observing $P_n > P_n^\mathrm{G}$ certifies genuine $n$-phonon QNG (Bemani et al., 20 Nov 2025, Rakhubovsky et al., 14 Jun 2024, Podhora et al., 2021).
• Phase-space functional criteria: Inequalities involving $s$-parametrized quasiprobabilities, such as Husimi Q-functions at the origin, can witness QNG (Hughes et al., 2014).

2. Engineering and Control Protocols: Optical and Mechanical Platforms

Quantum non-Gaussian states are produced via fundamentally nonlinear (nonquadratic) processes. The two principal paradigms are probabilistic heralding via measurement and deterministic unitary transformations.

Optical State Engineering:

• Photon Subtraction and Addition: Heralded subtraction (via detection of a photon in an ancillary mode after a tap on a squeezed vacuum) produces non-Gaussian states with negative Wigner functions and is instrumental in generating Schrödinger cat, Gottesman-Kitaev-Preskill (GKP), and photon-added/subtracted states (Endo et al., 2023, Takeda et al., 2012, Barbieri et al., 2010). Multi-photon subtraction enables higher-fidelity non-Gaussian resource states for universal quantum computation (Crescimanna et al., 20 Feb 2025).
• Adaptive Gaussian Boson Sampling architectures: Adaptive feedforward schemes based on Gaussian interferometers with photon-number-resolving detection increase heralding probability and robustness, enabling the preparation of high-fidelity non-Gaussian resource states (cat and GKP states) beyond the reach of passive single-shot methods. These schemes exploit conditional updating of the circuit parameters based on measurement outcomes (Crescimanna et al., 20 Feb 2025).
• Postselected von Neumann Measurement: Coupling a two-level system to a Gaussian pointer followed by postselection can generate a broad family of single- and two-mode non-Gaussian states, including squeezed and entangled cat states, GKP-like states, and Bell-state analogues, tunable via the weak value and interaction strength (Yao et al., 29 Sep 2025).

Mechanical and Multimode Platforms:

• Pulsed Optomechanics: In levitated optomechanical systems, pulsed interactions enable the preparation of mechanical Fock-like states via photon-count heralding, with explicit certification of QNG depth using Fock-probability witnesses (Bemani et al., 20 Nov 2025).
• Superfluid Helium Vibrations: Protocols for photon-count heralded generation of phonon-added states in superfluid helium achieve robust QNG with respect to mechanical heating, enabling QNG depth estimation and enhanced sensing (Rakhubovsky et al., 14 Jun 2024).
• Multimode Photon Subtraction: Programmable subtraction of photons in chosen supermodes of an optical frequency comb reveals non-Gaussianity that can be distributed and concentrated among cluster states for scalable quantum computation (Ra et al., 2019).
• Deterministic Quantum Switches: Schemes utilizing indefinite causal order of Gaussian unitaries, controlled by a qubit, yield deterministic generation of QNG single-mode states without nonlinear photonics (Koudia et al., 2021).
• Beam-splitter-Induced Non-Gaussianity: Applying a balanced beam splitter to two-mode squeezed states generates pure non-Gaussian vortex states deterministically, with enhanced entanglement and negative Wigner volume (Banerji et al., 2021).

3. Structural and Hierarchical Properties

Hierarchies of QNG:

• The "stellar hierarchy" organizes single-mode pure states by the count of zeros of the Husimi Q-function; the minimal number of photon additions required to reach a state equals its stellar rank (Chabaud et al., 2019). This defines an operational hierarchy, with each rank corresponding to resource cost in state engineering.
• For Fock states, the faithful QNG hierarchy uses tailored thresholds for population $P_n$ versus the best achievable by any Gaussianized $(n-1)$-dimensional subspace, establishing a strict hierarchy of robustness against Gaussian convexification. This is key for resource certification in quantum computation and sensing (Podhora et al., 2021).

Multimode Structure:

• Hypergraph calculus for non-Gaussian states formalizes multimode correlations as higher-order hyperedges, with transformation, generation, and measurement rules mapped to Gaussian and non-Gaussian operations (Vandré et al., 11 Sep 2024). This structure provides insight into resource engineering, such as embedding cubic-phase gates or fusing cluster-state fragments efficiently.

4. Quantification, Witnesses, and Loss Robustness

Quantitative Measures:

• Relative-entropy QNG: $Q(\rho)$ is convex, monotonic under Gaussian channels and conditional Gaussian operations, and faithful (Park et al., 2018).
• Wigner negativity ($N_W$): Measures phase-space non-classicality and is strictly necessary for pure QNG states but not sufficient for all mixtures.
• Husimi Q-function witnesses and higher $s$-parameter quasiprobabilities: Remain sensitive to QNG beyond the Wigner-negativity regime, particularly useful in high-loss channels (Hughes et al., 2014).

Loss Tolerance:

• Attenuation-insensitive criteria based on $g^{(2)}, g^{(3)}$ (Hotter et al., 11 Nov 2025) or vacuum probabilities before and after loss (Fiurášek et al., 2021) enable QNG certification under significant losses, a critical requirement for quantum technologies involving transmission channels and imperfect detectors.
• Fock-population-based witnesses have numerically tabulated thresholds for arbitrary mode number and can be applied to output states of highly multimode photonic or phononic systems (Bemani et al., 20 Nov 2025, Rakhubovsky et al., 14 Jun 2024).

5. Experimental Realizations

Quantum non-Gaussian states have been realized and certified in a variety of physical platforms:

• Optical field: Non-Gaussianity is generated by photon addition/subtraction from squeezed states or coherent states, with entropy-based or Husimi-based measures confirming QNG in both pulsed and continuous-wave regimes (Endo et al., 2023, Barbieri et al., 2010, Takeda et al., 2012).
• Multimode photonics: Active shaping and selection of subtraction modes in optical frequency combs enable the controlled distribution and transfer of QNG among entangled nodes (Ra et al., 2019).
• Single quantum emitters: Attenuation-resilient QNG witnessed by $g^{(2)}, g^{(3)}$ have been demonstrated for quantum dot sources, providing benchmarks for scalable quantum networks (Hotter et al., 11 Nov 2025).
• Levitated optomechanics & superfluid helium: Pulsed schemes with single-photon and multiphoton detection yield mechanical Fock-like states with high QNG depth—these have direct implications for force sensing and macroscopic quantum coherence tests (Bemani et al., 20 Nov 2025, Rakhubovsky et al., 14 Jun 2024).
• Trapped ions: Hierarchical QNG criteria up to $n=10$ phonons have been implemented with single trapped ions, with QNG survival tracked under engineered heating and applied to quantum sensing protocols (Podhora et al., 2021).

6. Applications in Quantum Technologies

Quantum non-Gaussian states are essential resources in multiple domains:

• Fault-tolerant quantum computation: Preparation of resource states such as GKP and cat states is both necessary and enabled by high-fidelity, high-heralding-probability QNG protocols (Crescimanna et al., 20 Feb 2025).
• Quantum sensing and metrology: Fock-like and phonon-added states exhibit enhanced Fisher information for displacement and force estimation, outperforming any Gaussian probe (Bemani et al., 20 Nov 2025, Podhora et al., 2021, Rakhubovsky et al., 14 Jun 2024).
• Entanglement distillation and error correction: Non-Gaussianity enables entanglement concentration and supports bosonic codes for error correction (cat, GKP, binomial codes), which are provably impossible to realize with Gaussian states alone (Lachman et al., 2022).
• Quantum communication: Verified non-Gaussian single-photon sources are necessary for secure quantum key distribution and entanglement swapping across repeater nodes (Hotter et al., 11 Nov 2025, Bemani et al., 20 Nov 2025).

The development of robust, scalable QNG state generation, detection, and manipulation continues to be a central challenge and opportunity in quantum information science. The integration of advanced protocols—adaptive, deterministic, or highly multimode—and loss-robust certification methodologies underpins progress toward practical fault-tolerant quantum technologies (Crescimanna et al., 20 Feb 2025, Hotter et al., 11 Nov 2025, Rakhubovsky et al., 14 Jun 2024, Podhora et al., 2021).