Entanglement Hamiltonian tomography in quantum simulation

Entanglement is the crucial ingredient of quantum many-body physics, and characterizing and quantifying entanglement in the closed-system dynamics of quantum simulators remains a 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 quasilocal few-body terms. This ansatz is fitted to, and can be independently verified from, a small number of randomized measurements. By analysing data from trapped-ion quantum simulators for quench dynamics of a one-dimensional long-range Ising model, we demonstrate the ability of the protocol to measure the time evolution of the entanglement spectrum, in agreement with theoretical expectations. Furthermore, we develop the protocol as a testbed for predictions of entanglement structure in quantum field theories, which we illustrate for conformal field theory in quench dynamics, as well as the Bisognano–Wichmann theorem for ground states. In theoretical simulations, we demonstrate favourable scaling of sampling efficiency with subsystem size. Although the post-processing might ultimately be exponential, our protocol addresses the bottleneck of exponential sampling complexity in the investigation of entanglement structure in quantum simulation, and brings subsystems of tens of spins into reach for present experiments Entanglement is central to theories of quantum many-body systems but is very resource intensive to measure. A protocol based on a quasilocal parametrization of physical states allows entanglement structures to be studied using very few measurements.