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# Entanglement Hamiltonian tomography in quantum simulation

## Abstract

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

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank P. Calabrese, M. Dalmonte, G. Giudici, L.K. Joshi, B. Kraus, R. Kueng, C. Roos, L. Sieberer, J. Yu and W. Zhu for discussions, and members of the Innsbruck trapped-ion group for generously sharing the experimental data of ref. 4. Work at Innsbruck is supported by the European Union programme Horizon 2020 under grants 817482 (PASQuanS) and 731473 (FWF QuantERA via QTFLAG I03769), the US Air Force Office of Scientific Research (AFOSR) via IOE grant FA9550-19-1-7044 LASCEM and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, P.Z.). B.V. acknowledges funding from the Austrian Science Foundation (FWF, P 32597 N), and the French National Research Agency (ANR-20-CE47-0005 JCJC QRand). The computational results presented here have been achieved (in part) using the LEO HPC infrastructure of the University of Innsbruck. Numerical calculations were performed (in part) using the ITensor library60.

## Author information

Authors

### Contributions

The research topic was developed by C.K., R.v.B., A.E. and B.V., following suggestions by P.Z. C.K., R.v.B. and A.E. developed the theoretical protocols. C.K., R.v.B., A.E. and P.Z. wrote the manuscript. All authors contributed to the discussion of the results and the manuscript.

### Corresponding authors

Correspondence to Christian Kokail or Rick van Bijnen or Andreas Elben or Benoît Vermersch or Peter Zoller.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

Supplementary information

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Kokail, C., van Bijnen, R., Elben, A. et al. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936–942 (2021). https://doi.org/10.1038/s41567-021-01260-w

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