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Operational interpretation of the Stabilizer Entropy

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Operational interpretation of the Stabilizer Entropy

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AbstractMagic-state resource theory is a fundamental framework with far-reaching applications in quantum error correction and the classical simulation of quantum systems. Recent advances have significantly deepened our understanding of magic as a resource across diverse domains, including many-body physics, nuclear and particle physics, and quantum chemistry. Central to this progress is the stabilizer Rényi entropy, a computable and experimentally accessible magic monotone. Despite its widespread adoption, a rigorous operational interpretation of the stabilizer entropy has remained an open problem. In this work, we provide such an interpretation in the context of quantum property testing. By showing that the stabilizer entropy is the most robust measurable magic monotone, we demonstrate that the Clifford orbit of a quantum state becomes exponentially indistinguishable from Haar-random states, at a rate governed by the stabilizer entropy $M_{\alpha}(\psi)$ and the number of available copies. This implies that the Clifford orbit forms an approximate state $k$-design, with an approximation error $\exp(-\Theta (M_{\alpha}(\psi)))$ for $\alpha\ge2$. Conversely, we establish that the optimal probability of distinguishing a given quantum state from the set of stabilizer states is also governed by its stabilizer entropy. These results reveal that the stabilizer entropy quantitatively characterizes the transition from stabilizer states to universal quantum states, thereby offering a comprehensive operational perspective of the stabilizer entropy as a quantum resource.► BibTeX data@article{Bittel2026operational, doi = {10.22331/q-2026-04-15-2069}, url = {https://doi.org/10.22331/q-2026-04-15-2069}, title = {Operational interpretation of the {S}tabilizer {E}ntropy}, author = {Bittel, Lennart and Leone, Lorenzo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2069}, month = apr, year = {2026} }► References [1] Scott Aaronson. Shadow tomography of quantum states, 2018. URL https://arxiv.org/abs/1711.01053. arXiv:1711.01053 [2] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70: 052328–052328, November 2004. 10.1103/PhysRevA.70.052328. https://doi.org/10.1103/PhysRevA.70.052328 [3] Noga Alon. Combinatorial nullstellensatz. Combinatorics, Probability and Computing, 8 (1-2): 7–29, 1999. [4] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80: 517–576, May 2008. 10.1103/RevModPhys.80.517. URL. https://doi.org/10.1103/RevModPhys.80.517 [5] Rafael Aoude, Hannah Banks, Chris D. White, and Martin J. White. Probing new physics in the top sector using quantum information, 2025. URL https://arxiv.org/abs/2505.12522. arXiv:2505.12522 [6] Srinivasan Arunachalam and Arkopal Dutt. Polynomial-time tolerant testing stabilizer states, 2024. URL https://arxiv.org/abs/2408.06289. arXiv:2408.06289 [7] Srinivasan Arunachalam, Sergey Bravyi, and Arkopal Dutt. A note on polynomial-time tolerant testing stabilizer states, 2024. URL https://arxiv.org/abs/2410.22220. arXiv:2410.22220 [8] Zongbo Bao, Philippe van Dordrecht, and Jonas Helsen. Tolerant testing of stabilizer states with a polynomial gap via a generalized uncertainty relation, 2024. URL https://arxiv.org/abs/2410.21811. arXiv:2410.21811 [9] Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov. Lower bounds on the non-clifford resources for quantum computations. Quantum Science and Technology, 5 (3): 035009, June 2020. ISSN 2058-9565. 10.1088/2058-9565/ab8963. URL http://dx.doi.org/10.1088/2058-9565/ab8963. https://doi.org/10.1088/2058-9565/ab8963 [10] Lennart Bittel and al. The theory of the real clifford commutant. In preparation, 2025. [11] Lennart Bittel, Jens Eisert, Lorenzo Leone, Antonio A. Mele, and Salvatore F. E. Oliviero. A complete theory of the clifford commutant, 2025. URL https://arxiv.org/abs/2504.12263. arXiv:2504.12263 [12] Zvika Brakerski, Nir Magrafta, and Tomer Solomon. State-based classical shadows, 2025. URL https://arxiv.org/abs/2507.10362. arXiv:2507.10362 [13] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Physical Review A, 86: 052329–052329, November 2012. 10.1103/PhysRevA.86.052329. https://doi.org/10.1103/PhysRevA.86.052329 [14] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71: 022316–022316, February 2005. 10.1103/PhysRevA.71.022316. https://doi.org/10.1103/PhysRevA.71.022316 [15] Raphael Brieger, Markus Heinrich, Ingo Roth, and Martin Kliesch. Stability of classical shadows under gate-dependent noise.

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Could not fetch ADS cited-by data during last attempt 2026-04-15 11:41:01: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractMagic-state resource theory is a fundamental framework with far-reaching applications in quantum error correction and the classical simulation of quantum systems. Recent advances have significantly deepened our understanding of magic as a resource across diverse domains, including many-body physics, nuclear and particle physics, and quantum chemistry. Central to this progress is the stabilizer Rényi entropy, a computable and experimentally accessible magic monotone. Despite its widespread adoption, a rigorous operational interpretation of the stabilizer entropy has remained an open problem. In this work, we provide such an interpretation in the context of quantum property testing. By showing that the stabilizer entropy is the most robust measurable magic monotone, we demonstrate that the Clifford orbit of a quantum state becomes exponentially indistinguishable from Haar-random states, at a rate governed by the stabilizer entropy $M_{\alpha}(\psi)$ and the number of available copies. This implies that the Clifford orbit forms an approximate state $k$-design, with an approximation error $\exp(-\Theta (M_{\alpha}(\psi)))$ for $\alpha\ge2$. Conversely, we establish that the optimal probability of distinguishing a given quantum state from the set of stabilizer states is also governed by its stabilizer entropy. These results reveal that the stabilizer entropy quantitatively characterizes the transition from stabilizer states to universal quantum states, thereby offering a comprehensive operational perspective of the stabilizer entropy as a quantum resource.► BibTeX data@article{Bittel2026operational, doi = {10.22331/q-2026-04-15-2069}, url = {https://doi.org/10.22331/q-2026-04-15-2069}, title = {Operational interpretation of the {S}tabilizer {E}ntropy}, author = {Bittel, Lennart and Leone, Lorenzo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2069}, month = apr, year = {2026} }► References [1] Scott Aaronson. Shadow tomography of quantum states, 2018. URL https://arxiv.org/abs/1711.01053. arXiv:1711.01053 [2] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70: 052328–052328, November 2004. 10.1103/PhysRevA.70.052328. https://doi.org/10.1103/PhysRevA.70.052328 [3] Noga Alon. Combinatorial nullstellensatz. Combinatorics, Probability and Computing, 8 (1-2): 7–29, 1999. [4] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80: 517–576, May 2008. 10.1103/RevModPhys.80.517. URL. https://doi.org/10.1103/RevModPhys.80.517 [5] Rafael Aoude, Hannah Banks, Chris D. White, and Martin J. White. Probing new physics in the top sector using quantum information, 2025. URL https://arxiv.org/abs/2505.12522. arXiv:2505.12522 [6] Srinivasan Arunachalam and Arkopal Dutt. Polynomial-time tolerant testing stabilizer states, 2024. URL https://arxiv.org/abs/2408.06289. arXiv:2408.06289 [7] Srinivasan Arunachalam, Sergey Bravyi, and Arkopal Dutt. A note on polynomial-time tolerant testing stabilizer states, 2024. URL https://arxiv.org/abs/2410.22220. arXiv:2410.22220 [8] Zongbo Bao, Philippe van Dordrecht, and Jonas Helsen. Tolerant testing of stabilizer states with a polynomial gap via a generalized uncertainty relation, 2024. URL https://arxiv.org/abs/2410.21811. arXiv:2410.21811 [9] Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov. Lower bounds on the non-clifford resources for quantum computations. Quantum Science and Technology, 5 (3): 035009, June 2020. ISSN 2058-9565. 10.1088/2058-9565/ab8963. URL http://dx.doi.org/10.1088/2058-9565/ab8963. https://doi.org/10.1088/2058-9565/ab8963 [10] Lennart Bittel and al. The theory of the real clifford commutant. In preparation, 2025. [11] Lennart Bittel, Jens Eisert, Lorenzo Leone, Antonio A. Mele, and Salvatore F. E. Oliviero. A complete theory of the clifford commutant, 2025. URL https://arxiv.org/abs/2504.12263. arXiv:2504.12263 [12] Zvika Brakerski, Nir Magrafta, and Tomer Solomon. State-based classical shadows, 2025. URL https://arxiv.org/abs/2507.10362. arXiv:2507.10362 [13] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Physical Review A, 86: 052329–052329, November 2012. 10.1103/PhysRevA.86.052329. https://doi.org/10.1103/PhysRevA.86.052329 [14] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71: 022316–022316, February 2005. 10.1103/PhysRevA.71.022316. https://doi.org/10.1103/PhysRevA.71.022316 [15] Raphael Brieger, Markus Heinrich, Ingo Roth, and Martin Kliesch. Stability of classical shadows under gate-dependent noise.

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Operational interpretation of the Stabilizer Entropy

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