Entanglement Hamiltonian Tomography in Quantum Simulation

TL;DR

This paper introduces an efficient protocol for reconstructing entanglement spectra in quantum simulators using a parametrization based on entanglement Hamiltonians, validated through numerical simulations and experimental data.

Contribution

It proposes a novel, experimentally feasible method for entanglement Hamiltonian tomography applicable to quantum simulators and beyond, inspired by conformal field theory and the Bisognano-Wichmann theorem.

Findings

Validated protocol on a 1D long-range Ising model

Demonstrated measurement of entanglement spectrum evolution in ion traps

Efficiently reconstructs reduced density matrices with limited measurements

Abstract

Entanglement is the crucial ingredient of quantum many-body physics, and characterizing and quantifying entanglement in closed system dynamics of quantum simulators is an outstanding challenge in today's era of intermediate scale quantum devices. Here we discuss an efficient tomographic protocol for reconstructing reduced density matrices and entanglement spectra for spin systems. The key step is a parametrization of the reduced density matrix in terms of an entanglement Hamiltonian involving only quasi local few-body terms. This ansatz is fitted to, and can be independently verified from, a small number of randomised measurements. The ansatz is suggested by Conformal Field Theory in quench dynamics, and via the Bisognano-Wichmann theorem for ground states. Not only does the protocol provide a testbed for these theories in quantum simulators, it is also applicable outside these regimes. We show the validity and efficiency of the protocol for a long-range Ising model in 1D using numerical simulations. Furthermore, by analyzing data from $10$ and $20$ ion quantum simulators [Brydges \textit{et al.}, Science, 2019], we demonstrate measurement of the evolution of the entanglement spectrum in quench dynamics.