Abstract
We obtain new constraints for the modular energy of general states by using the monotonicity property of relative entropy. In some cases, modular energy can be related to the energy density of states and these constraints lead to interesting relations between energy and entropy. In particular, we derive new quantum energy inequalities that improve some previous bounds for the energy density of states in a conformal field theory. Additionally, the inequalities derived in this manner also lead us to conclude that the entropy of the state further restricts the possible amount of negative energy allowed by the theory.
Highlights
Equation (1.3) has proven to be of wide use in a variety of topics
Modular energy can be related to the energy density of states and these constraints lead to interesting relations between energy and entropy
For instance, if we consider a Completely Positive Trace-Preserving (CPTP) map Φ that preserves the mean value of the energy and that keeps invariant the Gibbs thermal state ρT, it is straightforward to show that the entanglement entropy of the system does not decrease with the evolution under
Summary
Modular hamiltonians are in general non-local objects and the evolution they generate does not correspond to a local geometric flow. This result follows from analicity properties originating in Lorentz invariance and positivity of energy In this case, the modular hamiltonian is given by an integral of the energy density operator, weighted by the coordinate x1 in which the region V extends (this is the operator that generates the boost transformations in the plane (x0, x1)). For two-dimensional CFTs, there are some other cases in which the modular hamiltonian of the vacuum is local and can be written again as an integral of the energy-momentum tensor times a local weight. A remarkable result arises for a two-dimensional CFT in a thermal state at inverse temperature β reduced to the half spatial line V In this case, the modular Hamiltonian is a local object that can be expressed as an integral of the energy density [21]. We will use these results (equation (2.13)) to show how the modular energy inequalities (2.8) and (2.9) can be used to generate QEIs and energy-entropy bounds
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