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Experimental measurement and a physical interpretation of quantum shadow enumerators

Daniel Miller1,2,*, Kyano Levi1, Lukas Postler3, Alex Steiner3, Lennart Bittel1, Gregory A. L. White1, Yifan Tang1, Eric J. Kuehnke1, Antonio A. Mele1 et al.

Sumeet Khatri1,4,5, Lorenzo Leone1, Jose Carrasco1, Christian D. Marciniak3, Ivan Pogorelov3, Milena Guevara-Bertsch3, Robert Freund3, Rainer Blatt3,6, Philipp Schindler3, Thomas Monz3,7, Martin Ringbauer3, and Jens Eisert1

  • *Contact author: d.miller@fu-berlin.de
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Phys. Rev. Research 8, 023318 – Published 18 June, 2026

DOI: https://doi.org/10.1103/8h3b-brg1

Abstract

Throughout its history, the theory of quantum error correction has heavily benefited from translating classical concepts into the quantum setting. In particular, classical notions of weight enumerators, which relate to the performance of an error-correcting code, and MacWilliams’ identity, which links enumerators of a code to the ones of its dual, have been generalized to the quantum case. In this work, we establish a relationship between the theoretical machinery of quantum weight enumerators and a seemingly unrelated physics experiment: we prove that Rains’ quantum shadow enumerators—a powerful mathematical tool—arise as probabilities of observing fixed numbers of triplets in a two-copy Bell sampling experiment. This insight allows us to develop here a rigorous framework for the direct measurement of quantum weight enumerators, thus enabling experimental and theoretical studies of the entanglement structure of any quantum error-correcting code or quantum state under investigation. On top of that, we derive concrete sample complexity bounds and physically motivated robustness guarantees against unavoidable experimental imperfections. Finally, we demonstrate the feasibility of experimentally learning weight enumerators on a trapped-ion quantum computer. Our experimental findings are in good agreement with theoretical predictions and illuminate how entanglement theory and quantum error correction cross-fertilize each other once two-copy Bell sampling experiments are combined with the theoretical machinery of quantum weight enumerators.

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Phys. Rev. Research 8, 023318– Published 18 June, 2026
  • Received 8 October 2024
  • Revised 25 November 2025
  • Accepted 6 March 2026
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