Interval signature spectra for structured Pauli error families

Determine whether the interval behavior of signature spectra persists for more structured Pauli error families, such as those defined by a weight cutoff.

Background

The paper studies the signature spectrum, defined as the Euclidean-norm image of a joint higher-rank numerical range for a family of Hermitian Pauli observables. Earlier results cited in the paper observed closed-interval signature spectra for unrestricted Pauli-detection problems, but the paper constructs a three-qubit example in which additional noncommuting Pauli constraints reduce the attainable monitored coefficient range to the barycenter and four vertices of a tetrahedron. Its signature spectrum is therefore the disconnected set {0, √3}.

The authors explicitly leave unresolved whether interval signature spectra nevertheless occur for more structured and coding-theoretically natural error families, including families specified by a Pauli-weight cutoff. Resolving this would determine whether the interval phenomenon survives under standard distance-based detection constraints despite failing for arbitrary prescribed finite Pauli families.