Experimental measurement and a physical interpretation of quantum shadow enumerators
Daniel J. Miller, Kyano Levi, Lukas Postler, Alex Steiner, Lennart Bittel, Gregory A. L. White, Yifan Tang, Eric J. Kuehnke, Antonio Anna Mele, Sumeet Khatri, Lorenzo Leone, José A. Carrasco, Christian D. Marciniak, Ivan Pogorelov, Milena Guevara-Bertsch, Robert Freund, R. Blatt, Philipp Claudius Schindler, Thomas Monz, Martin Ringbauer · Physical Review Research · 2026
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.