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Verification of the area law of mutual information in a quantum field simulator

Nature Physics volume 19, pages 1022–1026 (2023)Cite this article

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Abstract

The theoretical understanding of scaling laws of entropies and mutual information has led to substantial advances in the study of correlated states of matter, quantum field theory and gravity. Experimentally measuring von Neumann entropy in quantum many-body systems is challenging, as it requires complete knowledge of the density matrix, which normally requires the implementation of full state reconstruction techniques. Here we measure the von Neumann entropy of spatially extended subsystems in an ultracold atom simulator of one-dimensional quantum field theories. We experimentally verify one of the fundamental properties of equilibrium states of gapped quantum many-body systems—the area law of quantum mutual information. We also study the dependence of mutual information on temperature and on the separation between the subsystems. Our work represents a step towards employing ultracold atom simulators to probe entanglement in quantum field theories.

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Fig. 1: Schematic of the experimental protocol.
Fig. 2: Area law of MI and volume law of vN entropy.
Fig. 3: Shared information content between two spatially separated subsystems.
Fig. 4: Temperature dependence of MI and vN entropy.

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Data availability

The experimental raw data containing all the information required to extract and calculate the results presented in Figs. 2–4 and Extended Data Fig. 1 are available in ref. 41. Sample code for calculating absorption images from raw data is also available there. All other data are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We thank M. Gluza, J. Eisert, I. Cirac, I. Mazets and S. Erne for helpful discussions. This work is supported by the DFG/FWF Research Unit FOR 2724 ‘Thermal machines in the quantum world’, the FQXi programme on ‘Information as fuel’, and DFG/FWF CRC 1225, ‘Isolated quantum systems and universality in extreme conditions (ISOQUANT)’. The Flatiron Institute is a division of the Simons Foundation. E.D. acknowledges support from ARO grant no. W911NF-20-1-0163. The work of I.K. was supported by the Max-Planck-Harvard Research Center for Quantum Optics (MPHQ). F.C., F.M. and J. Sabino acknowledge support from the Austrian Science Fund (FWF) in the framework of the Doctoral School on Complex Quantum Systems (CoQuS). J. Sabino acknowledges support by the Fundação para a Ciência e Tecnologia, Portugal (PD/BD/128641/2017). T.S. acknowledges support from the Max Kade Foundation through a postdoctoral fellowship. B.R. acknowledges support by the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement no. 888707. D.S. is partially supported by AFOSR (grant no. FA9550-21-1-0236) and NSF (grant no. OAC-2118310). S.S. acknowledges support by the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement no. 101030988 and by the Slovenian Research Agency (ARRS) under a QTE grant (no. N1-0109).

Author information

Authors and Affiliations

  1. Vienna Center for Quantum Science and Technology (VCQ), Atominstitut, TU Wien, Vienna, Austria

    Mohammadamin Tajik, Bernhard Rauer, Thomas Schweigler, Federica Cataldini, João Sabino, Frederik Møller, Philipp Schüttelkopf, Si-Cong Ji & Jörg Schmiedmayer

  2. Max-Planck Institute of Quantum Optics, Garching, Germany

    Ivan Kukuljan

  3. Munich Center for Quantum Science and Technology (MCQST), Munich, Germany

    Ivan Kukuljan

  4. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany

    Spyros Sotiriadis

  5. Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, Paris, France

    Bernhard Rauer

  6. Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    João Sabino

  7. Department of Physics, New York University, New York, NY, USA

    Dries Sels

  8. Center for Computational Quantum Physics, Flatiron Institute, New York, NY, USA

    Dries Sels

  9. Institute for Theoretical Physics, ETH Zurich, Zurich, Switzerland

    Eugene Demler

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Contributions

M.T. and B.R. performed the experiment with contributions by T.S., F.C., S.-C.J., J. Sabino, F.M. and P.S. M.T. analysed the experimental data. I.K., S.S. and M.T. provided the theoretical methodology and calculations, with helpful suggestions from D.S. J. Schmiedmayer and E.D. provided scientific guidance on experimental and theoretical questions. J. Schmiedmayer conceived the experiment. All authors contributed to the interpretation of the data and to the writing of the manuscript.

Corresponding authors

Correspondence to Mohammadamin Tajik or Jörg Schmiedmayer.

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Extended data

Extended Data Fig. 1 Additional results for area law of MI and volume law of vN entropy.

a, Experimental results for I(A: AC), SA, and \({S}_{{A}^{{{{\rm{C}}}}}}\), calculated based on N = 7 modes (left) and N = 8 modes (right) (see the caption of Fig. 2 for detailed explanation). The extracted parameters including their 95% confidence intervals in parenthesis are given above. b, Close-up of the measured MI.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–5, discussion of the experimental results and Table 1.

Source data

Source Data Fig. 2

Numerical values of the plotted quantities.

Source Data Fig. 3

Numerical values of the plotted quantities.

Source Data Fig. 4

Numerical values of the plotted quantities.

Source Data Extended Data Fig. 1

Numerical values of the plotted quantities.

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Tajik, M., Kukuljan, I., Sotiriadis, S. et al. Verification of the area law of mutual information in a quantum field simulator. Nat. Phys. 19, 1022–1026 (2023). https://doi.org/10.1038/s41567-023-02027-1

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