This paper presents a practical way to estimate a central quantum quantity called relative entropy. In quantum field theory the relative entropy between two states is given by the expectation value of a difficult operator, the relative modular Hamiltonian. The authors show how to bound that operator from above and below using the modular Hamiltonian of a different, better understood reference state. This gives bounds on the relative entropy even when the relative modular Hamiltonian cannot be computed exactly.

The method uses locality properties of algebraic quantum field theory. The authors consider three nested regions: a smaller region V1, the region of interest V2, and a larger region V3. Under suitable assumptions about the states, they prove an upper bound for the relative modular Hamiltonian on V2 in terms of the modular Hamiltonian of the reference state on the larger region V3. They likewise prove a lower bound in terms of the reference modular Hamiltonian on the smaller region V1. In plain terms, operators on a bigger or smaller region give control over the hard operator on the middle region.

The applicability of these bounds is linked to a version of Sorkin’s paradox, which concerns so‑called “impossible measurements” and apparent superluminal signalling. If the reference state equals one of the pair, then the upper bound holds exactly when there exists a unitary operation that maps the reference to the other state on V3 and does not allow signalling from the spacelike complement of V3 into V2. A similar statement holds for the lower bound with V1 and the complement of V2. The paper notes an open question: it is not yet proved that the existence of such a non‑signalling unitary is always a sufficient condition for the bounds to apply.

To test the strength of the estimates the authors study coherent states in canonical commutation relation (CCR) or Weyl algebras, and then focus on free scalar fields in the Minkowski vacuum. They show the bounds apply even when the relative modular Hamiltonian cannot be written down. For sufficiently regular excitations they can shrink the gap between upper and lower bounds arbitrarily by a squeezing trick and recover an exact formula for the relative entropy. They also obtain the analogous exact result for massless fields in double cone regions. These examples suggest the bounds do not lose much information in realistic cases.