Efficient characterization of general Gottesman-Kitaev-Preskill qubits

Referee: [§3] §3 (operator construction): The central claim requires that each operator O_θ is positive semidefinite with kernel exactly spanned by the ideal GKP state at Bloch point θ. The manuscript must supply an explicit argument showing that O_θ ψ = 0 implies ψ is proportional to the regularized GKP wavefunction on the lattice; any additional null vectors (e.g., higher-order lattice modes or non-periodic components) would invalidate both the uniqueness statement and the factor-of-two infidelity relation inside the logical subspace.

Authors: We agree that an explicit proof of kernel uniqueness is required to rigorously support the central claims. Although the manuscript constructs each O_θ so that it annihilates the target ideal GKP state and is positive semidefinite, a complete demonstration that no other null vectors exist (such as higher-order lattice modes) was not supplied. In the revised manuscript we will add, in §3, a detailed argument in the quadrature representation: we decompose an arbitrary wavefunction into components compatible with the GKP lattice periodicity, show that the operator’s quadratic form forces all but the fundamental mode to have strictly positive expectation value, and thereby prove that O_θ ψ = 0 if and only if ψ is proportional to the regularized GKP state. This will also confirm the factor-of-two infidelity relation inside the logical subspace. revision: yes

Referee: [§4.2] §4.2 (infidelity correspondence): The equality between the expectation value and twice the logical infidelity is stated to hold only for states inside the ideal logical GKP subspace. The paper should derive the precise bound or correction term that applies when the state has support outside this subspace, because the non-Gaussianity witness interpretation otherwise rests on an unstated assumption about the support of the prepared state.

Authors: The referee correctly observes that the exact equality ⟨O_θ⟩ = 2 × logical infidelity is derived under the assumption of support strictly inside the ideal logical GKP subspace. For states with leakage outside this subspace the expectation value remains nonnegative and vanishes only for the ideal state, thereby still functioning as a non-Gaussianity witness, but the direct numerical relation to infidelity requires a correction. In the revised §4.2 we will derive the bound ⟨O_θ⟩ ≥ 2 × infidelity + c ‖P_⊥ ρ P_⊥‖, where P_⊥ projects onto the orthogonal complement of the logical subspace and c > 0 is obtained from the lowest positive eigenvalue of O_θ restricted to that complement. This supplies the requested precise correction term and clarifies the conditions under which the witness is tight. revision: yes