Modular Flow of Excited States

TL;DR

This work develops a general framework to study modular and relative modular flows for excited states in algebraic quantum field theory. By proving that states generated from the vacuum by invertible region-algebra operators form a dense set, the authors express the modular operator, relative modular operator, and relative entropy of generic excitations in terms of the vacuum data and the state-creating operator, leveraging a unitary cocycle to relate excited-state flows to vacuum flows. They introduce a perturbative expansion for relative modular Hamiltonians and entropies, and apply the method to generalized free fields, obtaining explicit modular flows for coherent and squeezed states and showing how non-unitary excitations can be treated within the same formalism. The approach provides a versatile toolkit for analyzing entanglement structure and modular dynamics in excited states, with potential implications for holography, bulk reconstruction, and conformal perturbation theory.

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.

Figures (6)

• Figure 1: (a) A tensor with $m$ lines attached to the bottom and $n$ lines to the top represents a linear operator from $\mathcal{H}^{\otimes m}\to \mathcal{H}^{\otimes n}$ (b) A ket vector represents a quantum state $|v\rangle\in\mathcal{H}^{\otimes n}$ in a particular basis. (c) The dual bra $\langle v|$ (d) An example of an inner product between states with different number of legs: $\langle w|v\rangle\in\mathcal{H}$.
• Figure 2: Tensor diagram representation of (a) density matrix $\sigma$ (b) The unnormalized maximally entangled state built from $\sigma$ (c) The purification of $\sigma$ in a two-copy Hilbert space. (d) The expectation value of an operator in a state: $\sigma(A)=\langle E_\Omega|(\sigma^{1/2}\otimes \mathcal{I})A(\sigma^{1/2}\otimes \mathcal{I})|E_\Omega\rangle=tr(\sigma A)$.
• Figure 3: The definition of the anti-linear operator $T_\sigma$.
• Figure 4: The definition of the anti-linear operator $J_\Omega$.
• Figure 5: Tensor diagram identities.