Quadrature Witness: Nonclassicality Diagnostic

• Quadrature witnesses are experimentally accessible observables that certify nonclassical and non-Gaussian properties by measuring canonical quadratures like position and momentum.
• They include techniques such as the quadrature coherence scale, single-quadrature protocols, and nonlinear squeezing, each offering scalable and efficient diagnostics.
• Applications span quantum state engineering and gravitational-wave detection, with protocols designed to mitigate noise and handle loss effectively.

A quadrature witness is any experimentally accessible observable or protocol that certifies nonclassical properties or reconstructs key features of a quantum state by exploiting properties of canonical quadratures, typically position $\hat x$ and momentum $\hat p$, in bosonic systems such as quantum optics. Quadrature witnesses underlie a unifying class of nonclassicality and non-Gaussianity diagnostics, ranging from the quadrature coherence scale to single-quadrature or nonlinear-squeezing observables. They provide scalable, often tomography-free, tools for certifying quantum resources, quantifying state quality, and mitigating noise, with practical applications in quantum state engineering and gravitational-wave detection.

1. Mathematical Definition and Types of Quadrature Witness

Quadrature witnesses are defined through expectation values or derived functionals of combinations of position and/or momentum operators; their vanishing or violation signals non-Gaussianity, nonclassicality, or other phenomena. The most analytically developed is the Quadrature Coherence Scale (QCS), given for an $n$-mode bosonic state $\hat\rho$ with quadratures $$\hat{\mathbf r}=(\hat x_1, \hat p_1, \ldots, \hat x_n, \hat p_n)$$: $$\mathcal{C}^2(\hat\rho) = \frac{1}{2n\,\mathcal{P}(\hat\rho)} \sum_{j=1}^{2n} \operatorname{Tr}[\hat\rho, \hat r_j][\hat r_j, \hat\rho]$$ where purity is $$\mathcal{P}(\hat\rho) = \operatorname{Tr}[\hat\rho^2]$$. In phase space, for single mode: $$\mathcal{C}^2(\hat\rho) = \frac{1}{4} \frac{\int |\nabla_{\alpha} W(\alpha)|^2\, d^2\alpha} {\int |W(\alpha)|^2\, d^2\alpha}$$ for Wigner function $W(\alpha)$; $|\nabla_\alpha W(\alpha)|^2$ quantifies phase-space “roughness.”

For single-quadrature measurement protocols, the witness takes the form: $$\operatorname{Tr}[\hat\rho\,\hat W_{\theta,x,\eta}] = \int_{x-\eta/2}^{x+\eta/2} p_{\hat\rho, \theta}(q) dq$$ where $p_{\hat\rho, \theta}(q)$ is the homodyne marginals, and $\hat W_{\theta,x,\eta}$ is a projector onto an interval of quadrature $$\hat q_\theta = \cos\theta\,\hat x + \sin\theta\,\hat p$$.

Nonlinear-squeezing quadrature witnesses are given by operators such as: $$\hat{W}_m(u,\phi,c) = [(\hat x^2-u^2)^2]^m + c[\sin(u\hat p + \phi/2)]^{2m}$$ with their expectation values serving as witnesses and quantifiers of non-Gaussianity or fidelity in superposition-of-quadrature-eigenstate (SQE) generation.

2. Nonclassicality and Non-Gaussianity Certification

Quadrature witnesses serve as necessary and in some cases sufficient criteria for detecting nonclassicality. The QCS provides a strong operational bound: for states admitting a positive Glauber-Sudarshan $P$-function (i.e., “classical” mixtures of coherent states), $\mathcal{C}(\hat\rho)\leq 1$, so $\mathcal{C}(\hat\rho)>1$ certifies nonclassicality. However, the converse is not true: $\mathcal{C}(\hat\rho)\leq 1$ does not guarantee classicality (Griffet et al., 2022, Hertz et al., 6 Feb 2024).

Single-quadrature protocols can also serve as non-Gaussianity witnesses. A homodyne distribution $p_{\hat\rho,\theta}(q)$ that vanishes at some $(\theta,x)$ signals quantum non-Gaussianity; the minimal probability achievable by any Gaussian state of a given energy $E$ serves as a threshold $w^E_{\theta,x,\eta}$. Observing a measured probability window $\bar w<w^E_{\theta,x,\eta}$ certifies quantum non-Gaussianity (Wassner et al., 30 Jul 2025). The underlying criterion exploits Hudson’s theorem: any pure state whose quadrature wavefunctions vanish at real points must be non-Gaussian.

For SQE states, nonlinear-squeezing witnesses $\langle\hat{W}_m\rangle$ drop below all Gaussian thresholds if and only if the state is non-Gaussian, and the violation quantifies the degree of “nonlinear squeezing” (Kuchař et al., 20 Jun 2025).

3. Interferometric and Single-Quadrature Measurement Protocols

a. Interferometric QCS Measurement

QCS can be directly measured without full state tomography using a two-copy protocol (Griffet et al., 2022). The procedure is:

• Prepare two identical copies of $\hat\rho$.
• Interfere them on a balanced beam splitter.
• Discard one output mode; perform photon-number-resolving detection on the other.

The measured photon statistics $p_n$ allow evaluation of both purity and the QCS numerator via: $$\mathcal{P}(\hat\rho) = \sum_{n=0}^{\infty} (-1)^n p_n$$

$$\mathcal{N}(\hat\rho) = \sum_{n=0}^{\infty} (-1)^n (1 + 2n) p_n$$

$$\mathcal{C}^2(\hat\rho) = \frac{\mathcal{N}(\hat\rho)}{\mathcal{P}(\hat\rho)}$$

This protocol is operationally efficient and extensible to multimode systems by stacking beam splitters.

b. Single-Quadrature Witness Protocols

For homodyne-based witnesses (Wassner et al., 30 Jul 2025):

• Choose quadrature angle $\theta$, point $x$ (with $p_{\hat\rho,\theta}(x)\approx 0$), small bin-width $\eta$, and bound energy $E$.
• Numerically compute $$w^E_{\theta,x,\eta} = \inf_{\sigma \in \text{Gaussian}, \langle \hat n\rangle \leq E} \operatorname{Tr}[\sigma \hat W_{\theta,x,\eta}]$$.
• Perform $M$ homodyne measurements to estimate $\bar w$.
• Violation, $\bar w<w^E_{\theta,x,\eta}$, certifies non-Gaussianity.

Table: Summary of Measurement Protocols

Witness Type	Observable	Protocol
QCS (interfero)	$(1+2n)\cdot (-1)^n p_n$	Two-copy beam splitter + PNR det.
Single-quad	$p_{\hat\rho,\theta}(x)$	Homodyne, threshold on window bin
Nonlinear sqz.	$\langle \hat W_m \rangle$	Two homodyne runs (x, p)

4. Operational Metrics, Applications, and Scaling

QCS is operationally significant:

• For $n$-mode pure and Gaussian states, closed formulas link QCS directly to quadrature variances and covariance matrices (Hertz et al., 6 Feb 2024).
• For mixed or non-Gaussian states, sums of Gaussian Wigner components suffice for efficient computation.

QCS bounds the trace distance $D(\rho,\mathcal{E}_{\mathrm{cl}})$ to the nearest classical state: $$\mathcal{C}(\hat\rho)-1 \leq D(\rho,\mathcal{E}_{\mathrm{cl}})\leq \mathcal{C}(\hat\rho)$$ Large QCS values thus indicate both strong nonclassicality and macroscopic phase-space coherence.

Practical applications:

• Certification of macroscopicity in Schrödinger cat and GKP grid states via explicit QCS calculation.
• Rapid assessment of nonclassicality degradation under loss; for any single-mode pure state, QCS drops to unity precisely at 50% photon loss, matching the threshold for Wigner function positivity (Hertz et al., 6 Feb 2024). For $\eta \leq 0.5$, QCS witnesses no nonclassicality.
• Figure of merit in GKP “breeding” protocols; QCS tracks the buildup of grid structure and thus the fidelity of encoded quantum information (Hertz et al., 6 Feb 2024).

In gravitational-wave detectors, the orthogonal (“witness”) quadrature—measured in parallel with the GW signal quadrature—serves as a reference channel for reconstructing and subtracting classical non-stationary disturbances (e.g., back-scattered stray light), with precise noise-suppression and compatibility with frequency-dependent squeezing (Böttner et al., 5 Nov 2025).

5. Experimental Realization and Limitations

Quadrature witness protocols exhibit broad experimental accessibility and efficiency:

• Two-copy QCS interferometry: Only a balanced beam splitter and photon-number-resolving detectors are required, bypassing full state tomography and high-dimensional fits; recently implemented in superconducting detector platforms (Griffet et al., 2022).
• Single-quadrature and nonlinear-squeezing witnesses: Routine homodyne detection, windowing, and classical post-processing suffice; optimal window parameters maximize signal-to-noise for target states (Wassner et al., 30 Jul 2025, Kuchař et al., 20 Jun 2025).

Key limitations and considerations:

• Interferometric approaches require interferometric stability, mode matching, and high-quantum-efficiency PNR detectors; losses and dark counts degrade witness sharpness and require calibration (Griffet et al., 2022).
• For single-quadrature protocols, sample complexity obeys rigorous Hoeffding-type bounds; even modest increases in the number of quadrature angles dramatically improve robustness against loss (Wassner et al., 30 Jul 2025).
• Loss or noise rapidly erodes witnessed nonclassicality; e.g., QCS threshold behavior at $\eta=0.5$ for all pure states is both a universal limitation and a sharp transition for phase-space negativity (Hertz et al., 6 Feb 2024).
• For nonlinear-squeezing witnesses, finite homodyne sample size, detector inefficiencies, and windowing must be statistically accounted for (Kuchař et al., 20 Jun 2025).

6. Connections and Generalizations

Quadrature witnesses unify apparently disparate diagnostics:

• The QCS unifies phase-space “roughness,” quadrature commutator variance, and quantum purity under a single functional.
• Single-quadrature and nonlinear-squeezing witnesses generalize the intuition of negativity or zeros in measurement distributions to practical, threshold-based (non-)Gaussianity tests.
• For multimode systems, QCS generalizes directly via sums over all canonical quadratures, and the associated measurement protocols scale linearly with mode number via stacked beam splitters.

In gravitational-wave detection, the “quadrature-witness readout” constructs a classical-noise mitigation protocol by directly measuring the cross-coupled disturbance on the orthogonal quadrature channel, enabling effective subtraction in data analysis without requiring quantum memory or adaptive control (Böttner et al., 5 Nov 2025). This extends quadrature witness concepts beyond quantum information to precision measurement and metrology.

A plausible implication is that further generalizations—e.g., to higher-order cumulants or multimode nonlinear correlators—could produce even stronger quantum resource witnesses or more robust protocols for noise subtraction in large-scale quantum-limited detectors.

7. Outlook and Current Directions

Quadrature witness techniques provide powerful tools for both foundational characterization and operational tasks:

• Continued refinement of measurement protocols, especially in the context of incomplete or noisy detectors, is likely to drive further advances in practical quantum information certification.
• The conjecture that QCS cannot exceed unity after 50% loss for any pure state sets a fundamental limit for bosonic channel engineering and error correction (Hertz et al., 6 Feb 2024).
• Cross-fertilization between quantum state certification and precision metrology (e.g., in gravitational-wave observatories) highlights the versatility and impact of quadrature witness approaches for both experimental and theoretical frontiers.

These developments suggest quadrature witnesses will remain a central component in scalable characterization and deployment of quantum technologies based on continuous-variable platforms.