Modular-operator structure and locality in Tomita–Takesaki-based relational construction

Characterize the modular operator for general states in the Tomita–Takesaki-based relational construction and determine whether conjugation by the modular operator maps local operators to local operators beyond special cases, thereby clarifying the locality properties of the resulting relational observables.

Background

As an alternative to the projector-based scheme, the authors discuss a Tomita–Takesaki approach that ensures commutation with a coarse algebra including the Hamiltonian on a code subspace. This method involves conjugation with the modular operator to define improved operators in a putative commutant.

They highlight a key unresolved issue: the structure of the modular operator is not known for general states, and it is unclear whether the induced map preserves locality except in special cases. This uncertainty limits the ability to establish well-behaved local algebras suitable for fine-grained entropies.

References

We do not understand the structure of the modular operator for general states and it is not clear if the map so obtained maps local operators to local operators except in special states where the modular operator has a nice form.

— Seeing Page Curves and Islands with Blinders On  (2602.06543 - Geng et al., 6 Feb 2026) in Section 6.4, “Tomita–Takesaki construction”

However, these previous works leave open which structural property determines whether projection preserves locality and what forms of non-locality emerge when it does not.

— Structured Non-Locality and Emergent Locality in Cavity-QED Many-Body Dynamics  (2608.25908 - Oehlgrien et al., 26 Aug 2026) in Introduction