Abstract Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Rényi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Rényi family is the sandwiched Rényi divergence and we obtain a simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Rényi divergence is non-negative which is a generalization of the quantum null energy condition to the Rényi family.
We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length
R in a theory of central charge
c, and states of mean energy at most
E, the supremum of the modular energy is
Mmax=2πRE+6c−288REπc2+O((RE)−2), and neither constant can be improved. The infimum is logarithmic,
Mmin≃−12c[ln(24πRE/c)−1], so the range is linear above and logarithmic below, the two meeting as
RE→0; it is strictly wider above, by more than
52πRE at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient
πR/2 is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant
c/6 is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being
nc/6 for
n disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound,
ΔSI≤2πRE+6c, recovering the original form of that inequality from the modular form that by itself imposes no constraint.
In the last decades, it has been understood that a wide variety of phenomena in quantum field theory (QFT) can be characterised using quantum information measures, such as the entanglement entropy of a state and the relative entropy between quantum states in the same Hilbert space. In this thesis, we use these and other tools from quantum information theory to study several interesting problems in quantum field theory. The topics analysed range from the study of the Aharonov-Bohm effect in QFT using entanglement entropy, to the consistence of the Ryu-Takayanagi formula (proposed in the context of the AdS/CFT duality) using properties of relative entropy. We show that relative entropy can also be used to obtain new interesting quantum energy inequalities, that constrain the spatial distribution of negative energy density.
It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional conformal field theory that such negative energy densities are present in any nontrivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.
Energy conditions play an important role in constraining the dynamics of quantum field theories as well as gravitational theories. For example, in semi-classical gravity, the achronal averaged null energy condition (AANEC) can be used to prove that it is always slower to traverse through a wormhole than to travel around via its exterior. Such conditions prevent causality violations that would lead to paradoxes. Recent advances have been made in proving previously conjectured energy conditions directly in quantum field theory (QFT) as well as in uncovering new ones. This thesis will be an exploration of various energy inequalities in conformal field theories (CFTs) as well as semi-classical quantum gravity. At the core of this work lies the recently proved quantum null energy condition (QNEC). The QNEC bounds the null energy flowing past a point by a certain second shape derivative of entanglement entropy. We will demonstrate that the QNEC represents a deep connection between causality, energy and entanglement in quantum field theories. We explore this connection first in the context of holographic CFTs. Calculations in holographic theories will lead us to conjecture that the so-called diagonal QNEC is saturated in all interacting QFTs. We will then provide further, independent evidence that this conjecture holds for all CFTs with a twist gap by explicitly calculating shape derivatives of entanglement entropy using defect CFT techniques.