Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases
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AbstractEfficient verification of multipartite quantum states is crucial to many applications in quantum information processing. By virtue of Schmidt decomposition and mutually unbiased bases, here we propose a universal protocol to verify arbitrary multipartite pure quantum states using adaptive local projective measurements. Moreover, we establish a universal upper bound on the sample complexity that is independent of the local dimensions. Numerical calculations further indicate that Haar-random pure states can be verified with a constant sample cost, irrespective of the qudit number and local dimensions, even in the adversarial scenario in which the source cannot be trusted. As alternatives, we provide several simpler variants that can achieve similar high efficiencies without using Schmidt decomposition. The simplest variant consists of only two distinct tests.Featured image: Universal and efficient protocol for verifying multipartite pure quantum states based on Schmidt decomposition and mutually unbiased bases (MUB). Each of the first $n-1$ parties performs a projective measurement in succession in either the Schmidt basis or a MUB, while the last party performs a projective measurement onto the conditional reduced state of the target state, determined by the measurement choices and outcomes of the other parties.Popular summaryWe present a universal and efficient method to verify multipartite pure quantum states, an essential task for quantum technologies. Leveraging Schmidt decomposition and mutually unbiased bases, our adaptive local measurement protocol applies to any pure quantum state, with sample complexity independent of local dimensions. Remarkably, even in the adversarial scenario with untrusted sources, Haar-random pure states can be verified using only a constant number of samples. We also provide simpler variants that achieve comparable performance with far fewer tests and no Schmidt decomposition. Our work makes efficient and reliable quantum state verification more accessible, advancing quantum characterization and verification, with potential applications to many tasks in quantum information processing.► BibTeX data@article{Li2026universalefficient, doi = {10.22331/q-2026-03-04-2011}, url = {https://doi.org/10.22331/q-2026-03-04-2011}, title = {Universal and {E}fficient {Q}uantum {S}tate {V}erification via {S}chmidt {D}ecomposition and {M}utually {U}nbiased {B}ases}, author = {Li, Yunting and Zhu, Huangjun}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2011}, month = mar, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). https://doi.org/10.1103/RevModPhys.81.865 [2] Otfried Gühne and Géza Tóth. ``Entanglement detection''. Phys. 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Could not fetch ADS cited-by data during last attempt 2026-03-04 10:07:33: 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. AbstractEfficient verification of multipartite quantum states is crucial to many applications in quantum information processing. By virtue of Schmidt decomposition and mutually unbiased bases, here we propose a universal protocol to verify arbitrary multipartite pure quantum states using adaptive local projective measurements. Moreover, we establish a universal upper bound on the sample complexity that is independent of the local dimensions. Numerical calculations further indicate that Haar-random pure states can be verified with a constant sample cost, irrespective of the qudit number and local dimensions, even in the adversarial scenario in which the source cannot be trusted. As alternatives, we provide several simpler variants that can achieve similar high efficiencies without using Schmidt decomposition. The simplest variant consists of only two distinct tests.Featured image: Universal and efficient protocol for verifying multipartite pure quantum states based on Schmidt decomposition and mutually unbiased bases (MUB). Each of the first $n-1$ parties performs a projective measurement in succession in either the Schmidt basis or a MUB, while the last party performs a projective measurement onto the conditional reduced state of the target state, determined by the measurement choices and outcomes of the other parties.Popular summaryWe present a universal and efficient method to verify multipartite pure quantum states, an essential task for quantum technologies. Leveraging Schmidt decomposition and mutually unbiased bases, our adaptive local measurement protocol applies to any pure quantum state, with sample complexity independent of local dimensions. Remarkably, even in the adversarial scenario with untrusted sources, Haar-random pure states can be verified using only a constant number of samples. We also provide simpler variants that achieve comparable performance with far fewer tests and no Schmidt decomposition. Our work makes efficient and reliable quantum state verification more accessible, advancing quantum characterization and verification, with potential applications to many tasks in quantum information processing.► BibTeX data@article{Li2026universalefficient, doi = {10.22331/q-2026-03-04-2011}, url = {https://doi.org/10.22331/q-2026-03-04-2011}, title = {Universal and {E}fficient {Q}uantum {S}tate {V}erification via {S}chmidt {D}ecomposition and {M}utually {U}nbiased {B}ases}, author = {Li, Yunting and Zhu, Huangjun}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2011}, month = mar, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). https://doi.org/10.1103/RevModPhys.81.865 [2] Otfried Gühne and Géza Tóth. ``Entanglement detection''. Phys. Rep. 474, 1–75 (2009). https://doi.org/10.1016/j.physrep.2009.02.004 [3] Michael Walter, David Gross, and Jens Eisert. ``Multipartite entanglement''. Chapter 14, pages 293–330. John Wiley & Sons. (2016). https://doi.org/10.1002/9783527805785.ch14 [4] Jose Carrasco, Andreas Elben, Christian Kokail, Barbara Kraus, and Peter Zoller. ``Theoretical and experimental perspectives of quantum verification''. PRX Quantum 2, 010102 (2021). https://doi.org/10.1103/PRXQuantum.2.010102 [5] Joshua Morris, Valeria Saggio, Aleksandra Gočanin, and Borivoje Dakić. ``Quantum verification and estimation with few copies''. Adv. Quantum Technol. 5, 2100118 (2022). https://doi.org/10.1002/qute.202100118 [6] Xiao-Dong Yu, Jiangwei Shang, and Otfried Gühne. ``Statistical methods for quantum state verification and fidelity estimation''. Adv. Quantum Technol. 5, 2100126 (2022). https://doi.org/10.1002/qute.202100126 [7] Masahito Hayashi, Keiji Matsumoto, and Yoshiyuki Tsuda. ``A study of LOCC-detection of a maximally entangled state using hypothesis testing''. J. Phys. A: Math. Gen. 39, 14427 (2006). https://doi.org/10.1088/0305-4470/39/46/013 [8] Leandro Aolita, Christian Gogolin, Martin Kliesch, and Jens Eisert. ``Reliable quantum certification of photonic state preparations''. Nat. Commun. 6, 8498 (2015). https://doi.org/10.1038/ncomms9498 [9] Dominik Hangleiter, Martin Kliesch, Martin Schwarz, and Jens Eisert. ``Direct certification of a class of quantum simulations''. Quantum Sci. Technol. 2, 015004 (2017). https://doi.org/10.1088/2058-9565/2/1/015004 [10] Yuki Takeuchi and Tomoyuki Morimae. ``Verification of many-qubit states''. Phys. Rev. X 8, 021060 (2018). https://doi.org/10.1103/PhysRevX.8.021060 [11] Sam Pallister, Noah Linden, and Ashley Montanaro. ``Optimal verification of entangled states with local measurements''. Phys. Rev. Lett. 120, 170502 (2018). https://doi.org/10.1103/PhysRevLett.120.170502 [12] Huangjun Zhu and Masahito Hayashi. ``Efficient verification of pure quantum states in the adversarial scenario''. Phys. Rev. Lett. 123, 260504 (2019). https://doi.org/10.1103/PhysRevLett.123.260504 [13] Huangjun Zhu and Masahito Hayashi. ``General framework for verifying pure quantum states in the adversarial scenario''. Phys. Rev. A 100, 062335 (2019). https://doi.org/10.1103/PhysRevA.100.062335 [14] Ya-Dong Wu, Ge Bai, Giulio Chiribella, and Nana Liu. ``Efficient verification of continuous-variable quantum states and devices without assuming identical and independent operations''. Phys. Rev. Lett. 126, 240503 (2021). https://doi.org/10.1103/PhysRevLett.126.240503 [15] Ye-Chao Liu, Jiangwei Shang, Rui Han, and Xiangdong Zhang. ``Universally optimal verification of entangled states with nondemolition measurements''. Phys. Rev. Lett. 126, 090504 (2021). https://doi.org/10.1103/PhysRevLett.126.090504 [16] Aleksandra Gočanin, Ivan Šupić, and Borivoje Dakić. ``Sample-efficient device-independent quantum state verification and certification''. PRX Quantum 3, 010317 (2022). https://doi.org/10.1103/PRXQuantum.3.010317 [17] Masahito Hayashi. ``Group theoretical study of LOCC-detection of maximally entangled states using hypothesis testing''. New J. Phys. 11, 043028 (2009). https://doi.org/10.1088/1367-2630/11/4/043028 [18] Huangjun Zhu and Masahito Hayashi. ``Optimal verification and fidelity estimation of maximally entangled states''. Phys. Rev. A 99, 052346 (2019). https://doi.org/10.1103/PhysRevA.99.052346 [19] Kun Wang and Masahito Hayashi. ``Optimal verification of two-qubit pure states''. Phys. Rev. A 100, 032315 (2019). https://doi.org/10.1103/PhysRevA.100.032315 [20] Zihao Li, Yun-Guang Han, and Huangjun Zhu. ``Efficient verification of bipartite pure states''. Phys. Rev. A 100, 032316 (2019). https://doi.org/10.1103/PhysRevA.100.032316 [21] Xiao-Dong Yu, Jiangwei Shang, and Otfried Gühne. ``Optimal verification of general bipartite pure states''. npj Quantum Inf. 5, 112 (2019). https://doi.org/10.1038/s41534-019-0226-z [22] Masahito Hayashi and Tomoyuki Morimae. ``Verifiable measurement-only blind quantum computing with stabilizer testing''. Phys. Rev. Lett. 115, 220502 (2015). https://doi.org/10.1103/PhysRevLett.115.220502 [23] Keisuke Fujii and Masahito Hayashi. ``Verifiable fault tolerance in measurement-based quantum computation''. Phys. Rev. A 96, 030301(R) (2017). https://doi.org/10.1103/PhysRevA.96.030301 [24] Masahito Hayashi and Michal Hajdušek. ``Self-guaranteed measurement-based quantum computation''. Phys. Rev. A 97, 052308 (2018). https://doi.org/10.1103/PhysRevA.97.052308 [25] Damian Markham and Alexandra Krause. ``A simple protocol for certifying graph states and applications in quantum networks''. Cryptogr. 4, 3 (2020). https://doi.org/10.3390/cryptography4010003 [26] Zihao Li, Yun-Guang Han, and Huangjun Zhu. ``Optimal verification of Greenberger-Horne-Zeilinger states''. Phys. Rev. Appl. 13, 054002 (2020). https://doi.org/10.1103/PhysRevApplied.13.054002 [27] Huangjun Zhu and Masahito Hayashi. ``Efficient verification of hypergraph states''. Phys. Rev. Appl. 12, 054047 (2019). https://doi.org/10.1103/PhysRevApplied.12.054047 [28] Ye-Chao Liu, Xiao-Dong Yu, Jiangwei Shang, Huangjun Zhu, and Xiangdong Zhang. ``Efficient verification of Dicke states''. Phys. Rev. Appl. 12, 044020 (2019). https://doi.org/10.1103/PhysRevApplied.12.044020 [29] Zihao Li, Yun-Guang Han, Hao-Feng Sun, Jiangwei Shang, and Huangjun Zhu. ``Verification of phased Dicke states''. Phys. Rev. A 103, 022601 (2021). https://doi.org/10.1103/PhysRevA.103.022601 [30] Tianyi Chen, Yunting Li, and Huangjun Zhu. ``Efficient verification of Affleck-Kennedy-Lieb-Tasaki states''. Phys. Rev. A 107, 022616 (2023). https://doi.org/10.1103/PhysRevA.107.022616 [31] Huangjun Zhu, Yunting Li, and Tianyi Chen. ``Efficient verification of ground states of frustration-free Hamiltonians''. Quantum 8, 1221 (2024). https://doi.org/10.22331/q-2024-01-10-1221 [32] Wen-Hao Zhang, Chao Zhang, Zhe Chen, Xing-Xiang Peng, Xiao-Ye Xu, Peng Yin, Shang Yu, Xiang-Jun Ye, Yong-Jian Han, Jin-Shi Xu, Geng Chen, Chuan-Feng Li, and Guang-Can Guo. ``Experimental optimal verification of entangled states using local measurements''. Phys. Rev. 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