Non-Gaussianity in cat codes: global incompatibility and local geometric alignment with the magic resource

Technical Summary: Non-Gaussianity in Cat Codes – Global Incompatibility and Local Geometric Alignment with the Magic Resource

Problem Statement
Non-Gaussianity (NG) is a fundamental resource for achieving genuine quantum advantage in continuous-variable (CV) systems, analogous to the "magic" resource (non-stabilizerness) in discrete-variable (DV) systems. Recent work established an exact quantitative correspondence between NG and magic resources within the Gottesman–Kitaev–Preskill (GKP) encoding framework, where the Wigner logarithmic negativity (WLN) of the physical state aligns perfectly with the magic measure of the logical state. However, it remains unclear whether this correspondence persists beyond the specific lattice structure of GKP codes. This paper investigates whether a similar relationship exists for cat codes, a prominent bosonic encoding platform utilizing superpositions of coherent states. Specifically, the authors ask if the global geometric structures of the WLN and the magic measure are compatible in cat codes, or if their relationship is limited to local geometric alignments.

Methodology
The authors employ a resource-theoretic and geometric analysis of $d$-peaked cat states in the large-separation regime.

1. Asymptotic WLN Analysis: They define a generalized $d$-peaked cat state as a superposition of $d$ coherent states. By taking the non-degenerate large-separation limit ($|\alpha_i - \alpha_j| \to \infty$), they derive a closed-form expression for the asymptotic WLN. This limit allows the decomposition of the Wigner function into non-overlapping Gaussian peaks and interference terms, simplifying the calculation of the absolute-value integral required for WLN.
2. Magic Measure Characterization: They analyze the magic resource using a measure $M$ defined as the $\ell_1$ norm of a quasiprobability distribution associated with a phase-operator basis. They demonstrate that the state space for this measure is partitioned into "sign sectors" based on the signs of the quasiprobability components. Within each sector, the magic measure exhibits a local phase-rotation invariance determined by the eigenbasis of a signature operator.
3. Geometric Comparison: The authors compare the level-set structures (contours of constant resource value) of the asymptotic WLN and the magic measure. They first test for global compatibility in standard even/odd cat codes and then construct a specific "Asymptotic Cat Code" (ACC) basis transformation to test for local alignment.
4. Physical Benchmarking: To assess the physical realizability of these geometric alignments, they numerically compare the geometric error (deviation from ideal circular level sets) of finite-energy physical cat codes against approximate finite-energy GKP codes.

Key Contributions and Results

• Asymptotic WLN Geometry: The authors derive that in the large-separation limit, the WLN of a $d$-peaked cat state depends solely on the amplitudes of the coherent components ($|c_i|$) and is invariant under relative phase rotations. Consequently, the asymptotic WLN level sets form rigid structures characterized by fixed amplitude constraints and $U(1)^{d-1}$ phase-rotation symmetries.
• Global Incompatibility: The paper establishes that the global level-set structures of the WLN and the magic measure are fundamentally incompatible in standard cat encodings.
  • The asymptotic WLN geometry is globally organized by the coherent-state basis and is invariant under all relative phases.
  • The magic measure geometry is locally organized by sign sectors, where the level sets are centered around sector-dependent eigenaxes.
  • In the qubit case, the WLN level sets are circles around the logical $x$-axis, while the magic measure level sets are circles around axes defined by the sign sector (e.g., $r_x = -r_y = r_z$). This proves that no global, value-preserving correspondence exists between the two resources in standard cat codes, unlike in GKP codes.
• Local SU(d) Alignment: Despite global incompatibility, the authors identify a constructive local relationship. Both the asymptotic WLN and the magic measure share a local $U(1)^{d-1}$ phase-rotation invariance within specific regions of the state space.
  • By defining a distinguished Asymptotic Cat Code (ACC) where the logical basis is aligned with the eigenbasis of the magic measure's signature operator in a chosen sign sector, the authors achieve a local geometric alignment.
  • In this constructed ACC, the WLN level sets locally align with the magic measure level sets. Specifically, the WLN minimizer maps to the magic measure maximizer, and the circular contour structures coincide within the target sign sector.
• Physical Robustness: Numerical benchmarks comparing finite-energy cat codes and approximate GKP codes reveal that the physical cat construction achieves comparable geometric tolerance (low deviation from ideal circular level sets) at moderate coherent amplitudes ($|\alpha|$). In contrast, the approximate GKP family requires significantly larger scale-separation parameters to achieve similar geometric fidelity.

Significance
The paper concludes that the exact magic–NG correspondence observed in GKP codes is not a universal feature of all bosonic codes but is encoding-dependent. While cat codes do not inherit the global value-preserving equivalence found in GKP encodings, they possess a rich local geometric structure that allows for a constructive alignment between the magic resource and non-Gaussianity. This finding suggests that resource correspondences in quantum computing can be understood through a geometry-based approach that transcends the specific lattice constraints of GKP codes. The work provides a distinct, physically accessible route (via cat codes) to probing the local geometric connections between magic and non-Gaussian resources, offering a new perspective on resource theories in continuous-variable quantum information.