The modular Hamiltonian in asymptotically flat spacetime conformal to Minkowski

We consider a four-dimensional globally hyperbolic spacetime (M, g) conformal to Minkowski spacetime, together with a massless, conformally coupled scalar field. Using a bulk-to-boundary correspondence, one can establish the existence of an injective *-homomorphism M between W (M), the Weyl algebra of observables on M and a counterpart which is defined intrinsically on future null infinity ^+², a component of the conformal boundary of (M, g). Using invariance under the asymptotic symmetry group of ^+, we can individuate thereon a distinguished two-point correlation function whose pull-back to M via M identifies a quasi-free Hadamard state for the bulk algebra of observables. In this setting, if we consider V^+ₓ, a future light cone stemming from x M as well as W (V^+ₓ) =W (M) |ₕ^+䂲, its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in Kₓ, a positive half strip on ^+. To each such cone, we associate a standard subspace of the boundary one-particle Hilbert space, which coincides with the one associated naturally to Kₓ. We extend such correspondence replacing Kₓ and V^+ₓ with deformed counterparts, denoted by SC and VC. In addition, since the one particle Hilbert space at the boundary decomposes as a direct integral on the sphere of U (1) -currents defined on the real line, we prove that also the generator of the modular group associated to the standard subspace of VC decomposes as a suitable direct integral. This result allows us to study the relative entropy between coherent states of the algebras associated to the deformed cones VC establishing the quantum null energy condition.