A Quantum Non-Gaussianity Criterion Based on Photon Correlations $g^{(2)}$ and $g^{(3)}$

Technical Summary: Quantum Non-Gaussianity Criterion Based on Photon Correlations $g^{(2)}$ and $g^{(3)}$

Problem Statement
In continuous variable (CV) quantum systems, Gaussian states (e.g., coherent, thermal, and squeezed states) are well-characterized but insufficient for achieving certain quantum advantages in computing, communication, and sensing. To realize these advantages, "quantum non-Gaussian" (QNG) states are required—states that cannot be expressed as statistical mixtures of Gaussian states. While non-classicality is often verified via the second-order correlation function $g^{(2)} < 1$, this metric alone is insufficient to distinguish non-Gaussian states from Gaussian ones, as Gaussian states can exhibit $g^{(2)} < 1$. Existing criteria for verifying QNG often suffer from sensitivity to optical losses and finite detection efficiencies, which are prevalent in experimental setups. There is a need for a robust, sufficient criterion that can verify QNG using standard correlation measurements while remaining resistant to attenuation.

Methodology
The authors derive a sufficient criterion for quantum non-Gaussianity based on the second-order ($g^{(2)}$) and third-order ($g^{(3)}$) photon correlation functions.

1. Theoretical Derivation:

   • The authors analyze the properties of Gaussian pure states, defined as displaced squeezed states $|\xi, \alpha\rangle = D(\alpha)S(\xi)|0\rangle$.
   • They calculate the expectation values for photon number moments $G^{(n)} = \langle (a^\dagger)^n a^n \rangle$ for these states.
   • By minimizing the relationship between $g^{(3)}$ and $g^{(2)}$ with respect to the squeezing angle $\theta$, they determine that the boundary for Gaussian pure states is reached at $\theta = 0$ (amplitude squeezing).
   • Through a Taylor expansion around the limit of zero squeezing ($r \to 0$) and small displacement, they derive a non-linear lower bound for $g^{(3)}$ as a function of $g^{(2)}$:
     $$g^{(3)} \geq \left(2 - 3\sqrt{g^{(2)}}\right)^2$$
   • This inequality is rearranged into a linear form for the region where $g^{(2)} < 4/9$:
     $$\sqrt{g^{(3)}} + 3\sqrt{g^{(2)}} \geq 2$$
   • The authors prove that this bound holds not only for Gaussian pure states but also for statistical mixtures of Gaussian states (including thermal and displaced squeezed thermal states) using Jensen's inequality and the Cauchy-Schwarz inequality.
   • They demonstrate that the criterion is strictly valid for $g^{(2)} < 4/9$. For $g^{(2)} > 4/9$, mixtures of Gaussian states can violate the bound (e.g., by mixing vacuum with a Gaussian state), meaning the criterion is a sufficient, but not necessary, condition for QNG in that region.

2. Experimental Validation:

   • The criterion is tested using a single-photon source based on an InAs quantum dot embedded in a photonic crystal waveguide.
   • The setup utilizes pulsed resonant laser excitation and superconducting nanowire single-photon detectors.
   • The experiment measures two-fold and three-fold coincidence counts to extract $g^{(2)}$ and $g^{(3)}$.
   • To ensure robustness, the authors verify that the detection efficiency is sufficiently low to match Glauber's definition of $g^{(n)}$ (avoiding artifacts from photon-number-resolving limitations) and perform control measurements with intentionally increased laser leakage to characterize background noise.

Key Results

• Theoretical Bound: The paper establishes that any state satisfying $\sqrt{g^{(3)}} + 3\sqrt{g^{(2)}} < 2$ (with $g^{(2)} < 4/9$) is unambiguously quantum non-Gaussian. This bound is derived as the lower limit for all statistical mixtures of Gaussian states.
• Experimental Verification: Using the quantum dot source, the authors measured:
  • $g^{(2)} = 0.00334(4)$
  • $g^{(3)} = 0$ (with an upper bound of $1.7 \times 10^{-4}$ at the $1\sigma$ level)
  • The resulting criterion value was $\sqrt{g^{(3)}} + 3\sqrt{g^{(2)}} = 0.174(13)$.
• Statistical Significance: The experimental result violates the Gaussian bound by more than 100 standard deviations. A p-value test under the null hypothesis that the state is Gaussian yielded $p = 4 \times 10^{-4793}$, providing extremely high confidence in the rejection of the Gaussian hypothesis.

Significance and Claims
The authors claim that this work provides a robust, sufficient criterion for verifying quantum non-Gaussianity that is fundamentally resistant to losses and finite detection efficiencies. Unlike other criteria that may require photon-number-resolving detectors or complex state tomography, this method relies solely on $g^{(2)}$ and $g^{(3)}$, which are standard measurements in quantum optics.

The paper positions this criterion as a natural extension of the non-classicality bound ($g^{(2)} < 1$), serving as a benchmark for advanced quantum light sources. The authors explicitly note limitations: the criterion is not capable of detecting QNG for Fock states with $n \geq 2$ if multi-photon contributions are significant, and it is sensitive to dark counts and mixing with coherent/thermal states which increase $g^{(2)}$ and $g^{(3)}$. They suggest that future work could extend this approach using higher-order correlations (e.g., $g^{(4)}$) to verify a broader class of non-Gaussian states. The experimental demonstration confirms that current single-photon sources can produce states well within the non-Gaussian regime, validated with high statistical significance.