Entanglement is the crucial ingredient of quantum many-body physics, and characterizing and quantifying entanglement in quantum dynamics is an outstanding challenge in today's era of intermediate scale quantum devices. Quantum simulators allow us to observe in quench dynamics the increasing complexity of the many-body wavefunction in evolution towards thermodynamic equilibrium, including regimes inaccessible to classical computation. Here we discuss quantum protocols for spin systems, allowing an efficient tomographic reconstruction of the reduced density matrix of a subsystem including the entanglement spectrum [1]. The key step is the parametrization of the reduced density matrix in terms of an entanglement Hamiltonian, which is quasi-local and involves only few-body terms. This is then fitted to observations from a small number of single-spin random rotations and measurements. The ansatz is suggested by Conformal Field Theory for quench dynamics, and the Bisognano-Wichmann theorem for many-body 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 [2], we demonstrate measurement of the evolution of the entanglement spectrum in quench dynamics.
[1] C. Kokail, R. van Bijnen, A. Elben, B. Vermersch, and P. Zoller, arXiv:2009.09000 (2020)
[2] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B.P. Lanyon, P. Zoller, R. Blatt, and C.F. Roos, Science 364, 6437, p. 260 (2019)
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