Exact distinguishability between real-valued and complex-valued Haar random quantum states
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AbstractHaar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling $t$ copies of a $d$-dimensional quantum state according to the Haar measure on the orthogonal group. In particular, we analytically compute its spectral decomposition. This allows us to compute exactly the trace distance between $t$-copies of a real Haar random state and $t$-copies of a complex Haar random state. Using this we show a lower-bound on the approximation parameter of real-valued state $t$-designs and improve the lower-bound on the number of copies required for imaginarity testing.► BibTeX data@article{Nemoz2026exact, doi = {10.22331/q-2026-05-29-2120}, url = {https://doi.org/10.22331/q-2026-05-29-2120}, title = {Exact distinguishability between real-valued and complex-valued {H}aar random quantum states}, author = {Nemoz, Tristan and All{\'{e}}aume, Romain and Brown, Peter}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2120}, month = may, year = {2026} }► References [1] Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou, ``Quantum Pseudoentanglement'' 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) 287, 2:1-2:21 (2024). https://doi.org/10.4230/LIPIcs.ITCS.2024.2 arXiv:2211.00747 [2] Andris Ambainisand Joseph Emerson ``Quantum $t$-designs: $t$-wise Independence in the Quantum World'' Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07) 129–140 (2007). https://doi.org/10.1109/CCC.2007.26 [3] Prabhanjan Ananth, Luowen Qian, and Henry Yuen, ``Cryptography from Pseudorandom Quantum States'' Advances in Cryptology – CRYPTO 2022, Part I 13507, 208–236 (2022). https://doi.org/10.1007/978-3-031-15802-5_8 arXiv:2112.10020 [4] Prabhanjan Ananth, Aditya Gulati, Luowen Qian, and Henry Yuen, ``Pseudorandom (Function-Like) Quantum State Generators: New Definitions and Applications'' TCC 2022: 20th Theory of Cryptography Conference, Part I 13747, 237–265 (2022). https://doi.org/10.1007/978-3-031-22318-1_9 arXiv:2211.01444 [5] Sheldon Axler ``HFT.m'' (2020). https://www.axler.net/HFT_Math.html [6] Sheldon Axler, Paul Bourdon, and Wade Ramey, ``Harmonic Function Theory'' Springer New York, NY (2013). https://doi.org/10.1007/978-1-4757-8137-3 https://www.axler.net/HFT.pdf [7] Bernard Beauzamy, Enrico Bombieri, Per Enflo, and Hugh Lowell Montgomery, ``Products of polynomials in many variables'' Journal of Number Theory 36, 219–245 (1990). https://doi.org/10.1016/0022-314X(90)90075-3 [8] John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba, ``Efficient Quantum Pseudorandomness from Hamiltonian Phase States'' 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025) 350, 9:1–9:18 (2025). https://doi.org/10.4230/LIPIcs.TQC.2025.9 arXiv:2410.08073 [9] Zvika Brakerskiand Omri Shmueli ``(Pseudo) Random Quantum States with Binary Phase'' TCC 2019: 17th Theory of Cryptography Conference, Part I 11891, 229–250 (2019). https://doi.org/10.1007/978-3-030-36030-6_10 arXiv:1906.10611 [10] Zvika Brakerskiand Omri Shmueli ``Scalable Pseudorandom Quantum States'' Advances in Cryptology – CRYPTO 2020, Part II 12171, 417–440 (2020). https://doi.org/10.1007/978-3-030-56880-1_15 arXiv:2004.01976 [11] Ronald Raphaël Coifmanand Guido Leopold Weiss ``Representations of compact groups and spherical harmonics'' L'Enseignement Mathématique. 2e Série 14, 121–173 (1968). https://doi.org/10.5169/seals-42346 [12] Andrew W. Cross, Lev S. Bishop, Sarah Sheldon, Paul D. Nation, and Jay M. Gambetta, ``Validating quantum computers using randomized model circuits'' Physical Review A 100 (2019). https://doi.org/10.1103/physreva.100.032328 arXiv:1811.12926 [13] Laura Cui, Thomas Schuster, Fernando Brandao, and Hsin-Yuan Huang, ``Unitary designs in nearly optimal depth'' (2025). arXiv:2507.06216 [14] Feng Daiand Yuan Xu ``Approximation Theory and Harmonic Analysis on Spheres and Balls'' Springer New York, NY chapter 1 (2013). https://doi.org/10.1007/978-1-4614-6660-4 arXiv:1304.2585 [15] Tudor Giurgica-Tironand Adam Bouland ``Pseudorandomness from Subset States'' (2023). arXiv:2312.09206 [16] Michael Hardy ``Combinatorics of Partial Derivatives'' The Electronic Journal of Combinatorics 13 (2006). https://doi.org/10.37236/1027 [17] Aram W. Harrow ``Applications of coherent classical communication and the Schur transform to quantum information theory'' thesis (2005). [18] Aram W. Harrow ``The Church of the Symmetric Subspace'' (2013). arXiv:1308.6595 [19] Tobias Haug, Kishor Bharti, and Dax Enshan Koh, ``Pseudorandom unitaries are neither real nor sparse nor noise-robust'' Quantum 9, 1759 (2025). https://doi.org/10.22331/q-2025-06-04-1759 arXiv:2306.11677 [20] Patrick Haydenand John Preskill ``Black holes as mirrors: quantum information in random subsystems'' Journal of High Energy Physics 2007, 120 (2007). https://doi.org/10.1088/1126-6708/2007/09/120 arXiv:0708.4025 [21] Alexander Hickeyand Gilad Gour ``Quantifying the imaginarity of quantum mechanics'' Journal of Physics A: Mathematical and Theoretical 51, 414009 (2018). https://doi.org/10.1088/1751-8121/aabe9c arXiv:1801.05123 [22] Hsin-Yuan Huang, Richard Kueng, and John Preskill, ``Predicting many properties of a quantum system from very few measurements'' Nature Physics 16, 1050–1057 (2020). https://doi.org/10.1038/s41567-020-0932-7 arXiv:2002.08953 [23] Fernando Granha Jeronimo, Nir Magrafta, and Pei Wu, ``Pseudorandom and Pseudoentangled States from Subset States'' (2024). arXiv:2312.15285 [24] Zhengfeng Ji, Yi-Kai Liu, and Fang Song, ``Pseudorandom Quantum States'' Advances in Cryptology – CRYPTO 2018, Part III 10993, 126–152 (2018). https://doi.org/10.1007/978-3-319-96878-0_5 https://eprint.iacr.org/2018/544.pdf [25] William Kretschmer ``Quantum Pseudorandomness and Classical Complexity'' 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021) 197, 2:1–2:20 (2021). https://doi.org/10.4230/LIPIcs.TQC.2021.2 arXiv:2103.09320 [26] William Kretschmer, Luowen Qian, Makrand Sinha, and Avishay Tal, ``Quantum Cryptography in Algorithmica'' Proceedings of the 55th Annual ACM Symposium on Theory of Computing 1589–1602 (2023). https://doi.org/10.1145/3564246.3585225 arXiv:2212.00879 [27] Chuhan Lu, Minglong Qin, Fang Song, Penghui Yao, and Mingnan Zhao, ``Quantum Pseudorandom Scramblers'' TCC 2024: 22nd Theory of Cryptography Conference, Part II 15365, 3–35 (2024). https://doi.org/10.1007/978-3-031-78017-2_1 arXiv:2309.08941 [28] Antonio Anna Mele ``Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial'' Quantum 8, 1340 (2024). https://doi.org/10.22331/q-2024-05-08-1340 arXiv:2307.08956 [29] Tony Metger, Alexander Poremba, Makrand Sinha, and Henry Yuen, ``Simple Constructions of Linear-Depth t-Designs and Pseudorandom Unitaries'' 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 485–492 (2024). https://doi.org/10.1109/FOCS61266.2024.00038 arXiv:2404.12647 [30] Francesco Mezzadri ``How to Generate Random Matrices from the Classical Compact Groups'' Notices of the American Mathematical Society 54, 592–604 (2007). https://www.ams.org/notices/200705/fea-mezzadri-web.pdf [31] Tomoyuki Morimaeand Takashi Yamakawa ``Quantum Commitments and Signatures Without One-Way Functions'' Advances in Cryptology – CRYPTO 2022, Part I 13507, 269–295 (2022). https://doi.org/10.1007/978-3-031-15802-5_10 arXiv:2112.06369 [32] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, A. Megrant, B. Chiaro, A. Dunsworth, K. Arya, R. Barends, B. Burkett, Y. Chen, Z. Chen, A. Fowler, B. Foxen, M. Giustina, R. Graff, E. Jeffrey, T. Huang, J. Kelly, P. Klimov, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, H. Neven, and J. M. Martinis, ``A blueprint for demonstrating quantum supremacy with superconducting qubits'' Science 360, 195–199 (2018). https://doi.org/10.1126/science.aao4309 arXiv:1709.06678 [33] David Pérez-García, Michael M. Wolf, Denes Petz, and Mary Beth Ruskai, ``Contractivity of positive and trace-preserving maps under $L_p$ norms'' Journal of Mathematical Physics 47, 083506 (2006). https://doi.org/10.1063/1.2218675 [34] Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, and Miguel Navascués, ``Quantum theory based on real numbers can be experimentally falsified'' Nature 600, 625–629 (2021). https://doi.org/10.1038/s41586-021-04160-4 arXiv:2101.10873 [35] Or Sattath ``Microcrypt Zoo'' (2024). https://sattath.github.io/microcrypt-zoo/ [36] Louis Schatzki ``Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished'' (2024). arXiv:2410.17213 [37] Yaoyun Shi ``Both Toffoli and controlled-NOT need little help to do universal quantum computing'' Quantum Info. Comput. 3, 84–92 (2003). https://dl.acm.org/doi/abs/10.5555/2011508.2011515 [38] Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Alexander Streltsov, ``Operational Resource Theory of Imaginarity'' Phys. Rev. Lett. 126, 090401 (2021). https://doi.org/10.1103/PhysRevLett.126.090401 arXiv:2007.14847 [39] Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Alexander Streltsov, ``Resource theory of imaginarity: Quantification and state conversion'' Physical Review A 103 (2021). https://doi.org/10.1103/physreva.103.032401 arXiv:2103.01805 [40] Shengnan Xue, Jiansheng Guo, Ping Li, Mingfei Ye, and Yongming Li, ``Quantification of resource theory of imaginarity'' Quantum Information Processing 20, 383 (2021). https://doi.org/10.1007/s11128-021-03324-5Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-29 10:16:08: Could not fetch cited-by data for 10.22331/q-2026-05-29-2120 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-29 10:16:08: Cannot retrieve data from ADS due to rate limitations.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. AbstractHaar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling $t$ copies of a $d$-dimensional quantum state according to the Haar measure on the orthogonal group. In particular, we analytically compute its spectral decomposition. This allows us to compute exactly the trace distance between $t$-copies of a real Haar random state and $t$-copies of a complex Haar random state. Using this we show a lower-bound on the approximation parameter of real-valued state $t$-designs and improve the lower-bound on the number of copies required for imaginarity testing.► BibTeX data@article{Nemoz2026exact, doi = {10.22331/q-2026-05-29-2120}, url = {https://doi.org/10.22331/q-2026-05-29-2120}, title = {Exact distinguishability between real-valued and complex-valued {H}aar random quantum states}, author = {Nemoz, Tristan and All{\'{e}}aume, Romain and Brown, Peter}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2120}, month = may, year = {2026} }► References [1] Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou, ``Quantum Pseudoentanglement'' 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) 287, 2:1-2:21 (2024). https://doi.org/10.4230/LIPIcs.ITCS.2024.2 arXiv:2211.00747 [2] Andris Ambainisand Joseph Emerson ``Quantum $t$-designs: $t$-wise Independence in the Quantum World'' Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07) 129–140 (2007). https://doi.org/10.1109/CCC.2007.26 [3] Prabhanjan Ananth, Luowen Qian, and Henry Yuen, ``Cryptography from Pseudorandom Quantum States'' Advances in Cryptology – CRYPTO 2022, Part I 13507, 208–236 (2022). https://doi.org/10.1007/978-3-031-15802-5_8 arXiv:2112.10020 [4] Prabhanjan Ananth, Aditya Gulati, Luowen Qian, and Henry Yuen, ``Pseudorandom (Function-Like) Quantum State Generators: New Definitions and Applications'' TCC 2022: 20th Theory of Cryptography Conference, Part I 13747, 237–265 (2022). https://doi.org/10.1007/978-3-031-22318-1_9 arXiv:2211.01444 [5] Sheldon Axler ``HFT.m'' (2020). https://www.axler.net/HFT_Math.html [6] Sheldon Axler, Paul Bourdon, and Wade Ramey, ``Harmonic Function Theory'' Springer New York, NY (2013). https://doi.org/10.1007/978-1-4757-8137-3 https://www.axler.net/HFT.pdf [7] Bernard Beauzamy, Enrico Bombieri, Per Enflo, and Hugh Lowell Montgomery, ``Products of polynomials in many variables'' Journal of Number Theory 36, 219–245 (1990). https://doi.org/10.1016/0022-314X(90)90075-3 [8] John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba, ``Efficient Quantum Pseudorandomness from Hamiltonian Phase States'' 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025) 350, 9:1–9:18 (2025). https://doi.org/10.4230/LIPIcs.TQC.2025.9 arXiv:2410.08073 [9] Zvika Brakerskiand Omri Shmueli ``(Pseudo) Random Quantum States with Binary Phase'' TCC 2019: 17th Theory of Cryptography Conference, Part I 11891, 229–250 (2019). https://doi.org/10.1007/978-3-030-36030-6_10 arXiv:1906.10611 [10] Zvika Brakerskiand Omri Shmueli ``Scalable Pseudorandom Quantum States'' Advances in Cryptology – CRYPTO 2020, Part II 12171, 417–440 (2020). https://doi.org/10.1007/978-3-030-56880-1_15 arXiv:2004.01976 [11] Ronald Raphaël Coifmanand Guido Leopold Weiss ``Representations of compact groups and spherical harmonics'' L'Enseignement Mathématique. 2e Série 14, 121–173 (1968). https://doi.org/10.5169/seals-42346 [12] Andrew W. Cross, Lev S. Bishop, Sarah Sheldon, Paul D. Nation, and Jay M. Gambetta, ``Validating quantum computers using randomized model circuits'' Physical Review A 100 (2019). https://doi.org/10.1103/physreva.100.032328 arXiv:1811.12926 [13] Laura Cui, Thomas Schuster, Fernando Brandao, and Hsin-Yuan Huang, ``Unitary designs in nearly optimal depth'' (2025). arXiv:2507.06216 [14] Feng Daiand Yuan Xu ``Approximation Theory and Harmonic Analysis on Spheres and Balls'' Springer New York, NY chapter 1 (2013). https://doi.org/10.1007/978-1-4614-6660-4 arXiv:1304.2585 [15] Tudor Giurgica-Tironand Adam Bouland ``Pseudorandomness from Subset States'' (2023). arXiv:2312.09206 [16] Michael Hardy ``Combinatorics of Partial Derivatives'' The Electronic Journal of Combinatorics 13 (2006). https://doi.org/10.37236/1027 [17] Aram W. Harrow ``Applications of coherent classical communication and the Schur transform to quantum information theory'' thesis (2005). [18] Aram W. Harrow ``The Church of the Symmetric Subspace'' (2013). arXiv:1308.6595 [19] Tobias Haug, Kishor Bharti, and Dax Enshan Koh, ``Pseudorandom unitaries are neither real nor sparse nor noise-robust'' Quantum 9, 1759 (2025). https://doi.org/10.22331/q-2025-06-04-1759 arXiv:2306.11677 [20] Patrick Haydenand John Preskill ``Black holes as mirrors: quantum information in random subsystems'' Journal of High Energy Physics 2007, 120 (2007). https://doi.org/10.1088/1126-6708/2007/09/120 arXiv:0708.4025 [21] Alexander Hickeyand Gilad Gour ``Quantifying the imaginarity of quantum mechanics'' Journal of Physics A: Mathematical and Theoretical 51, 414009 (2018). https://doi.org/10.1088/1751-8121/aabe9c arXiv:1801.05123 [22] Hsin-Yuan Huang, Richard Kueng, and John Preskill, ``Predicting many properties of a quantum system from very few measurements'' Nature Physics 16, 1050–1057 (2020). https://doi.org/10.1038/s41567-020-0932-7 arXiv:2002.08953 [23] Fernando Granha Jeronimo, Nir Magrafta, and Pei Wu, ``Pseudorandom and Pseudoentangled States from Subset States'' (2024). arXiv:2312.15285 [24] Zhengfeng Ji, Yi-Kai Liu, and Fang Song, ``Pseudorandom Quantum States'' Advances in Cryptology – CRYPTO 2018, Part III 10993, 126–152 (2018). https://doi.org/10.1007/978-3-319-96878-0_5 https://eprint.iacr.org/2018/544.pdf [25] William Kretschmer ``Quantum Pseudorandomness and Classical Complexity'' 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021) 197, 2:1–2:20 (2021). https://doi.org/10.4230/LIPIcs.TQC.2021.2 arXiv:2103.09320 [26] William Kretschmer, Luowen Qian, Makrand Sinha, and Avishay Tal, ``Quantum Cryptography in Algorithmica'' Proceedings of the 55th Annual ACM Symposium on Theory of Computing 1589–1602 (2023). https://doi.org/10.1145/3564246.3585225 arXiv:2212.00879 [27] Chuhan Lu, Minglong Qin, Fang Song, Penghui Yao, and Mingnan Zhao, ``Quantum Pseudorandom Scramblers'' TCC 2024: 22nd Theory of Cryptography Conference, Part II 15365, 3–35 (2024). https://doi.org/10.1007/978-3-031-78017-2_1 arXiv:2309.08941 [28] Antonio Anna Mele ``Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial'' Quantum 8, 1340 (2024). https://doi.org/10.22331/q-2024-05-08-1340 arXiv:2307.08956 [29] Tony Metger, Alexander Poremba, Makrand Sinha, and Henry Yuen, ``Simple Constructions of Linear-Depth t-Designs and Pseudorandom Unitaries'' 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 485–492 (2024). https://doi.org/10.1109/FOCS61266.2024.00038 arXiv:2404.12647 [30] Francesco Mezzadri ``How to Generate Random Matrices from the Classical Compact Groups'' Notices of the American Mathematical Society 54, 592–604 (2007). https://www.ams.org/notices/200705/fea-mezzadri-web.pdf [31] Tomoyuki Morimaeand Takashi Yamakawa ``Quantum Commitments and Signatures Without One-Way Functions'' Advances in Cryptology – CRYPTO 2022, Part I 13507, 269–295 (2022). https://doi.org/10.1007/978-3-031-15802-5_10 arXiv:2112.06369 [32] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, A. Megrant, B. Chiaro, A. Dunsworth, K. Arya, R. Barends, B. Burkett, Y. Chen, Z. Chen, A. Fowler, B. Foxen, M. Giustina, R. Graff, E. Jeffrey, T. Huang, J. Kelly, P. Klimov, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, H. Neven, and J. M. Martinis, ``A blueprint for demonstrating quantum supremacy with superconducting qubits'' Science 360, 195–199 (2018). https://doi.org/10.1126/science.aao4309 arXiv:1709.06678 [33] David Pérez-García, Michael M. Wolf, Denes Petz, and Mary Beth Ruskai, ``Contractivity of positive and trace-preserving maps under $L_p$ norms'' Journal of Mathematical Physics 47, 083506 (2006). https://doi.org/10.1063/1.2218675 [34] Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, and Miguel Navascués, ``Quantum theory based on real numbers can be experimentally falsified'' Nature 600, 625–629 (2021). https://doi.org/10.1038/s41586-021-04160-4 arXiv:2101.10873 [35] Or Sattath ``Microcrypt Zoo'' (2024). https://sattath.github.io/microcrypt-zoo/ [36] Louis Schatzki ``Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished'' (2024). arXiv:2410.17213 [37] Yaoyun Shi ``Both Toffoli and controlled-NOT need little help to do universal quantum computing'' Quantum Info. Comput. 3, 84–92 (2003). https://dl.acm.org/doi/abs/10.5555/2011508.2011515 [38] Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Alexander Streltsov, ``Operational Resource Theory of Imaginarity'' Phys. Rev. Lett. 126, 090401 (2021). https://doi.org/10.1103/PhysRevLett.126.090401 arXiv:2007.14847 [39] Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Alexander Streltsov, ``Resource theory of imaginarity: Quantification and state conversion'' Physical Review A 103 (2021). https://doi.org/10.1103/physreva.103.032401 arXiv:2103.01805 [40] Shengnan Xue, Jiansheng Guo, Ping Li, Mingfei Ye, and Yongming Li, ``Quantification of resource theory of imaginarity'' Quantum Information Processing 20, 383 (2021). https://doi.org/10.1007/s11128-021-03324-5Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-29 10:16:08: Could not fetch cited-by data for 10.22331/q-2026-05-29-2120 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-29 10:16:08: Cannot retrieve data from ADS due to rate limitations.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.