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# Quantum simulation and spectroscopy of entanglement Hamiltonians

## Abstract

The properties of a strongly correlated many-body quantum system, from the presence of topological order to the onset of quantum criticality, leave a footprint in its entanglement spectrum. The entanglement spectrum is composed by the eigenvalues of the density matrix representing a subsystem of the whole original system, but its direct measurement has remained elusive due to the lack of direct experimental probes. Here we show that the entanglement spectrum of the ground state of a broad class of Hamiltonians becomes directly accessible via the quantum simulation and spectroscopy of a suitably constructed entanglement Hamiltonian, building on the Bisognano–Wichmann theorem of axiomatic quantum field theory. This theorem gives an explicit physical construction of the entanglement Hamiltonian, identified as the Hamiltonian of the many-body system of interest with spatially varying couplings. On this basis, we propose a scalable recipe for the measurement of a system’s entanglement spectrum via spectroscopy of the corresponding Bisognano–Wichmann Hamiltonian realized in synthetic quantum systems, including atoms in optical lattices and trapped ions. We illustrate and benchmark this scenario on a variety of models, spanning phenomena as diverse as conformal field theories, topological order and quantum phase transitions.

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## Acknowledgements

We thank V. Alba, P. Calabrese, L. Chomaz, R. Fazio, C. Roos, E. Tonni and R. van Bijnen for useful discussions. M.D. thanks M. Falconi for useful discussions and clarifications. Work in Innsbruck was supported in part by the ERC Synergy Grant UQUAM, SIQS and the SFB FoQuS (FWF project no. F4016-N23). Work in Trieste was supported in part by the ERC Starting grant AGEnTh.

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All authors devised the project, discussed the results and wrote the manuscript.

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Correspondence to M. Dalmonte.

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The authors declare no competing interests.

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## Supplementary information

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Supplementary notes, Supplementary Figures 1–6, Supplementary References 1–17

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### Cite this article

Dalmonte, M., Vermersch, B. & Zoller, P. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nature Phys 14, 827–831 (2018). https://doi.org/10.1038/s41567-018-0151-7

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