We thank J. Bollinger, A. Elben, K. Holmström, K. Jansen, M. Lukin, E. A. Martinez, S. Montangero, P. Schindler and U. J. Wiese for discussions. The numerical results presented were achieved (in part) using the HPC infrastructure LEO of the University of Innsbruck. The research at Innsbruck is supported by the ERC Synergy Grant UQUAM, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement number 741541, by the SFB FoQuS (FWF project number F4016-N23) and by QTFLAG—QuantERA. This project (or publication) has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS). We thank A. Elben and B. Vermersch for the development of the software used for evaluating the Rényi entropies measurement.

Reviewer information

Nature thanks Juergen Berges, Christof Wunderlich and the other anonymous reviewer(s) for their contribution to the peer review of this work.