Abstract
We develop new techniques for studying the modular and the relative modular flows of general excited states. We show that the class of states obtained by acting on the vacuum (or any cyclic and separating state) with invertible operators from the algebra of a region is dense in the Hilbert space. This enables us to express the modular and the relative modular operators, as well as the relative entropies of generic excited states in terms of the vacuum modular operator and the operator that creates the state. In particular, the modular and the relative modular flows of any state can be expanded in terms of the modular flow of operators in vacuum. We illustrate the formalism with simple examples including states close to the vacuum, and coherent and squeezed states in generalized free field theory.
Highlights
Entanglement and quantum information have played increasingly important roles in our understanding of quantum field theory (QFT), equilibrium and non-equilibrium dynamics of strongly correlated condensed matter systems, and quantum gravity
We develop new techniques for studying the modular and the relative modular flows of general excited states
Results from operator algebras and techniques developed in algebraic approach to quantum field theory provide powerful tools for organizing and obtaining quantum information properties of QFT and quantum statistical systems
Summary
Entanglement and quantum information have played increasingly important roles in our understanding of quantum field theory (QFT), equilibrium and non-equilibrium dynamics of strongly correlated condensed matter systems, and quantum gravity. Deep and rich mathematical structures have been uncovered about quantum field theory using this approach These structures provide powerful tools to study the quantum information properties of states in QFT and quantum statistical systems. A key observation is that generic excited states can be obtained by acting on the vacuum (or any cyclic and separating state) with invertible operators from the algebra of a region. We show that such states are dense in the Hilbert space This observation enables us to express, in a simple way, the modular and the relative modular operator, as well as relative entropies, of generic excited states in terms of the modular operator of the vacuum.
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