Lower bound on locality for the symmetric filtered operator used in catalytic tomography

Derive a locality lower bound for the filtered operator \(\hat{A}_f\) defined by time-weighted Heisenberg evolution with a real-valued filter (such as a Gaussian or compactly supported bump function), as used in the catalytic ground-state tomography protocol, analogous to the bound obtained via correlation decay for asymmetric filters; ascertain how the minimal locality radius required to implement \(\hat{A}_f\) scales with the spectral gap or ground-state correlation length.

Background

The authors prove a locality lower bound by relating quasi-local filtering to decay of correlations, but this argument relies on an asymmetric filter that zeroes out upward energy transitions. Their actual tomography construction employs a symmetric filter $\hat{A}_f$ (e.g., Gaussian or bump functions centered at zero frequency) and they are unable to provide the analogous locality lower bound for that operator.

Establishing such a lower bound for the symmetric $\hat{A}_f$ would clarify optimal locality requirements of the proposed catalytic tomography protocol and whether the required locality must scale with inverse gap or correlation length across models.