Emergence of Kinematic Space from Quantum Modular Geometric Tensor

TL;DR

This paper introduces a generalized Quantum Geometric Tensor replacing Hamiltonians with modular Hamiltonians, revealing a quantum kinematic space and modular Berry curvature, with implications for holography and quantum information.

Contribution

It presents the Quantum Modular Geometric Tensor, linking quantum entanglement, kinematic space, and holography, extending previous geometric frameworks to quantum gravity contexts.

Findings

Derived a quantum kinematic space from modular Hamiltonians.

Established a holographic entanglement formula for two intervals.

Connected Wilson line correlators to mutual information.

Abstract

We generalize the Quantum Geometric Tensor by replacing a Hamiltonian with a modular Hamiltonian. The symmetric part of the Quantum Geometric Tensor provides a Fubini-Study metric, and its anti-symmetric sector gives a Berry curvature. Now the generalization or Quantum Modular Geometric Tensor gives a Kinematic Space and a modular Berry curvature. Here we demonstrate the emergence by focusing on a spherical entangling surface. We also use the result of the identity Virasoro block to relate the connected correlator of two Wilson lines to the two-point function of a modular Hamiltonian. This result realizes a novel holographic entanglement formula for two intervals of a general separation. This formula does not only hold for a classical gravity sector but also Quantum Gravity. The formula also provides a new Quantum Information interpretation to the connected correlators of Wilson lines as the mutual information. Our study provides an opportunity to explore Quantum Kinematic Space through Quantum Modular Geometric Tensor and hence go beyond symmetry.