Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases

Yunting Li; Huangjun Zhu

doi:10.22331/q-2026-03-04-2011

Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases

State Key Laboratory of Surface Physics, Department of Physics, and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China
Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

Published:	2026-03-04, volume 10, page 2011
Editor:	Jiangwei Shang
Eprint:	arXiv:2506.19809v3
Doi:	https://doi.org/10.22331/q-2026-03-04-2011
Citation:	Quantum 10, 2011 (2026).

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Abstract

Efficient 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 summary

We 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}
}

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[1] Liang 亮 Mao 毛, Yifei 逸飞 Wang 王, Yingfei 颖飞 Gu 顾, and Chengshu 成疏 Li 李, "Certifying Quantum States with Uniform Measurements", Chinese Physics Letters 43 7, 070601 (2026).

[2] Xiaodi Li, "A new general quantum state verification protocol by the classical shadow method", Quantum Information Processing 24 9, 276 (2025).

[3] Sami Abdul Sater, Maxime Garnier, Thierry Martinez, Harold Ollivier, and Ulysse Chabaud, "Efficient certification of intractable quantum states with few Pauli measurements", arXiv:2511.07300, (2025).

[4] Liang Mao, Yifei Wang, Yingfei Gu, and Chengshu Li, "Certifying Quantum States with Uniform Measurements", arXiv:2508.09259, (2025).

[5] Andrea Coladangelo, Jerry Li, Joseph Slote, and Ellen Wu, "The Power of Two Bases: Robust and copy-optimal certification of nearly all quantum states with few-qubit measurements", arXiv:2602.11616, (2026).

[6] Xiao-Dong Zhang, Bin-Bin Cai, and Song Lin, "Verifcation of general multi-qudit pure states", arXiv:2511.17901, (2025).

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