发表机构
Inria, ENS de Lyon, UCBL, LIP; Department of Mathematics, University of York（法国国家信息与自动化研究所、里昂高等师范学院、里昂第一大学、信息科学实验室; 约克大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入 $k$ 重无偏测量（$k$-UMs）将 $k$ 重无偏基推广到任意秩投影测量，证明其与测量不相容性的精确对应，并构造出高秩三元组及噪声阈值，数值表明四结果 3-UMs 具有最大不相容性。
AI 中文摘要
互无偏基刻画了两个量子测量之间的完美互补性。已有研究提出了超越成对无偏性的扩展，但本质上没有已知的非平凡高阶构造。我们引入了 $k$ 重无偏测量（$k$-UMs），将 [arXiv:1706.04446] 中的 $k$ 重无偏基概念从秩一基测量扩展到任意秩的投影测量，并表明这种高秩设置支持更丰富的理论。我们发展了代数与谱 $k$-UMs 的概念，并证明它们对于秩一测量和三元组测量（3-UMs）是一致的。我们建立了高阶秩一构造和三结果 3-UMs 的强否定结果，但利用 Hadamard 矩阵和 Clifford 代数获得了无穷多个高秩三元组。然后，我们赋予这些结构在测量不相容性方面的精确操作解释。对于任意结果数的 3-UMs，我们精确确定了它们的广义不相容鲁棒性，并在相容性阈值处为其噪声版本构造了显式的联合测量。最后，我们实现了最近在 [New J. Phys. 28, 064509 (2026)] 中引入的平方和层级对称性约简，并给出数值证据表明，由 $k$-UM 分析产生的噪声阈值可能刻画该层级的渐近行为。特别地，我们以非常高的精度数值表明，我们构造的四结果 3-UMs 是四结果测量中最不相容的三元组之一。
英文摘要
Mutually unbiased bases capture perfect complementarity between two quantum measurements. Extensions beyond pairwise unbiasedness have been proposed, but essentially no non-trivial higher-order constructions are known. We introduce $k$-fold unbiased measurements ($k$-UMs), extending the $k$-fold unbiased bases notion of [arXiv:1706.04446] from rank-one basis measurements to arbitrary-rank projective measurements, and show that this higher-rank setting supports a much richer theory. We develop the notions of algebraic and spectral $k$-UMs and prove that they coincide for rank-one measurements and triples of measurements (3-UMs). We establish strong no-go results for higher-order rank-one constructions and three-outcome 3-UMs, but obtain infinitely many higher-rank triples using Hadamard matrices and Clifford algebras. We then give these structures an exact operational interpretation in terms of measurement incompatibility. For 3-UMs with any number of outcomes, we determine their generalised incompatibility robustness exactly and construct an explicit joint measurement for their noisy versions at the compatibility threshold. Finally, we implement the symmetry reduction of the sum-of-squares hierarchy recently introduced in [New J. Phys. 28, 064509 (2026)] and give numerical evidence that the noise thresholds arising from the $k$-UM analysis may characterise the asymptotic behaviour of this hierarchy. In particular, with very high precision, we numerically show that our constructed four-outcome 3-UMs are among the most incompatible triples of four-outcome measurements.
Comments10+14 pages, 3 tables. Comments are welcome!
