The study of relative entropy between states in quantum field theory has recently attracted much attention in connection with the quantum null energy condition; usually one considers the vacuum and a coherent excitation, and the relative entropy with respect to a wedge algebra. In generalization of this, we study the relative entropy for a subalgebra of a generic CCR algebra between a general (possibly mixed) quasifree state and a coherent excitation of it. We give a formula for this entropy in terms of single-particle modular data. We also investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces, and study lower estimates for the second derivative of the relative entropy along this family, which replace the usual notion of convexity of the entropy. Our main input is a regularity condition for the family of subspaces (“differential modular position”) which generalizes the notion of half-sided modular inclusions. Examples include thermal states for the conformal U(1)-current.
Thomas Faulkner (University of Illinois)
Some new results on reflected entropy
It has been known for over 50 years that quantum fields cannot respect positivity of energy density in all states. It is over 40 years since Ford gave a physical argument to indicate the existence of quantum energy inequalities (QEIs) that restrict the duration and magnitude of energy conditions violation, and 30 years since Ford gave the first proof of such an inequality in 1991. In this talk I will survey what is known about QEIs, focussing particularly on rigorous results. I will also describe more recent work on the probability distribution of individual measurements of averaged energy densities and, if time allows, some applications.
Marius Junge (University of Illinois)
Poissonization, boundary and relative entropy
Inspired by a suggestion of A. Connes, operator algebra theory has developed a rich theory of boundaries. The aim of this talk is to present tools from quantum probability, operator-valued Poisson processes, which allow to combine boundaries, toy models for quantum field theory and relative entropy. For simplicity and transparency this talk will focus on the semifinite version of this imperfect quantum field theory, and indicate the challenges in developing the set up for true CFT's.
I present recent results on improvements of the data processing inequality for the relative entropy. Such inequalities have important consequences for how well a state can be recovered after it has been passed to a noisy channel.
We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. These spacetimes possess in general no timelike Killing vector field, but admit a Kodama vector field which provides a replacement. Spherically symmetric spacelike cross-sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time-parameter of the Kodama flow, where the inverse temperature is related to the surface gravity of the horizon cross-section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunneling probability in the scaling limit between in- and outgoing Fourier modes with respect to the the Kodama time shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling-limit-theory have relative entropies behaving proportional to the cross-sectional horizon area. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking.
This is joint work with F. Kurpicz and N. Pinamonti (arXiv:2102.11547).
We will present some rigorous results about Relative entropy in QFT, motivated in part by recent physicists work which however depends on heuristic arguments such as introducing cut off and using path integrals. In the particular case of CFT, we will discuss interesting relations between relative entropy, central charge and global dimension of conformal net.