Abstract We show that certain singular limits of relative entropy of vacuum states in two dimensional massive free fermion theory exist and compute these limits. As an application we show that the c function computed by Casini and Huerta based on heuristic arguments arise naturally as a consequence of our results.
We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra ASDC(H,Γ). We notice, for a specific choice of f∈H, that the associated element in ASDC(H,Γ) is unitary. As a consequence, we explicitly compute the relative entropy between a quasifree state over ASDC(H,Γ) and an excitation of it with respect to the abovely mentioned unitary element. The generality of the approach, allows us to consider H as the Hilbert space of solutions of the classical Dirac equation over globally hyperbolic spacetimes, making our result, a computation of relative entropy for a Fermionic Quantum Field Theory. Our result extends those of Longo and Casini et al. for the relative entropy between a quasifree state and a coherent excitation for a free Scalar Quantum Field Theory, to the case of fermions. As a first application, we computed such a relative entropy for a Majorana field on an ultrastatic spacetime.
We prove an entropic c-theorem in two dimensions by constructing a c-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular p-nuclearity), and a finite ultraviolet bound on the resulting c-functions. We also note that, formally, the c-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.
We numerically investigate the Araki-Uhlmann relative entropy in Quantum Field Theory, focusing on a free massive scalar field in 1+1-dimensional Minkowski spacetime. Using Tomita-Takesaki modular theory, we analyze the relative entropy between a coherent state and the vacuum state, with several types of test functions localized in the right Rindler wedge. Our results confirm that relative entropy decreases with increasing mass and grows with the size of the spacetime region, aligning with theoretical expectations.
We numerically investigate the Araki-Uhlmann relative entropy in Quantum Field Theory, focusing on a free massive scalar field in 1 + 1 -dimensional Minkowski spacetime. Using Tomita-Takesaki modular theory, we analyze the relative entropy between a coherent state and the vacuum state, with several types of test functions localized in the right Rindler wedge. Our results confirm that relative entropy decreases with increasing mass and grows with the size of the spacetime region, aligning with theoretical expectations.