Witnessing and quantifying stellar rank with the Wigner function

Abstract

Quantifying non-Gaussianity is essential for assessing the computational power of bosonic quantum systems. We introduce a framework to witness and quantify non-Gaussianity from displaced parity measurements, and we directly apply it to experimental data obtained with superconducting cavities. Our method estimates fidelities with target pure states from the displaced parity measurements, enabling certification of non-Gaussianity via the stellar rank. We further introduce refined witnesses computed by measuring the value of the Wigner function at a few points, providing efficient certification tools outperforming existing ones. Applying these techniques to Fock, binomial, cat, and GKP experimental data, we rigorously certify non-Gaussian resources and establish a stellar rank of $8$ for a GKP state, setting a new benchmark across all quantum platforms. Our methods enable a rigorous quantification of non-Gaussian quantum resources directly applicable to superconducting devices.

The paper studies how to quantify bosonic non-Gaussianity through stellar-rank lower bounds using displaced-parity measurements, without full tomography. It develops fidelity-based, few-point, and phase-averaged witnesses adapted to platforms with direct access to the Wigner function. The fidelity method combines importance sampling from the absolute target-state Wigner function with previously defined stellar-fidelity thresholds. For one mode, the stated sample bound is $N\leq 8\|W_\psi\|_1^2\epsilon^{-2}\log(1/\delta)$ for additive error $\epsilon$ and failure probability $\delta$. Retrospective simulations using reconstructed superconducting-cavity density matrices yield stellar-rank lower bounds of $4$ for a binomial state, $1$ for an odd cat, and $8$ for a GKP state with $\Delta=0.35$. Few-point witnesses are weighted sums of displaced parity, with rank-dependent thresholds obtained by optimizing truncated-operator eigenvalues over Gaussian transformations. Five settings detect rank at least $4$ for the experimental binomial state and $4$ for the GKP state; four settings detect rank at least $2$ for the cat. For equal-and-opposite two-point weights, thresholds are independent of point locations, and the optimal expectation equals $\pi/4$ times the Wigner-function range. Phase averaging exploits rotational symmetry and reduces threshold optimization from four real parameters to three. The applications use reconstructed states rather than new direct measurement records. Crucially, numerical thresholds involve nonconvex optimization without global-optimality certificates, so the reported rank bounds are numerical evidence rather than unconditional rigorous certifications. Further limitations include target-dependent witness design, sensitivity to loss and displacement errors, and single-mode demonstrations; multimode generalization remains open.