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.