How well can multiple incompatible observables be implemented by a single measurement? This is a fundamental problem in quantum mechanics with wide implications for the performance optimization of numerous tasks in quantum information science. While existing studies have been mostly focusing on the approximation of two observables with a single measurement, in practice multiple observables are often encountered, for which the errors of the approximations are little understood. Here we provide a framework to study the implementation of an arbitrary finite number of observables with a single measurement. Our methodology yields novel analytical bounds on the errors of these implementations, significantly advancing our understanding of this fundamental problem. Additionally, we introduce a more stringent bound utilizing semi-definite programming that, in the context of two observables, generates an analytical bound tighter than previously known bounds. The derived bounds have direct applications in assessing the trade-off between the precision of estimating multiple parameters in quantum metrology, an area with crucial theoretical and practical implications. To validate the validity of our findings, we conducted experimental verification using a superconducting quantum processor. This experimental validation not only confirms the theoretical results but also effectively bridges the gap between the derived bounds and empirical data obtained from real-world experiments. Our work paves the way for optimizing various tasks in quantum information science that involve multiple noncommutative observables.

通过一次测量可以有多好地实现多个不兼容的可观测值？这是量子力学中的一个基本问题，对量子信息科学中众多任务的性能优化具有广泛的影响。虽然现有的研究主要集中在一次测量的两个可观测量的近似上，但在实践中经常遇到多个可观测量，对此近似误差知之甚少。在这里，我们提供了一个框架来研究通过单个测量实现任意有限数量的可观测值。我们的方法对这些实现的错误产生了新颖的分析界限，显着增进了我们对这一基本问题的理解。此外，我们利用半定规划引入了更严格的界限，在两个可观察量的上下文中，生成比先前已知的界限更严格的分析界限。导出的界限可直接应用于评估量子计量中估计多个参数的精度之间的权衡，这是一个具有重要理论和实践意义的领域。为了验证我们研究结果的有效性，我们使用超导量子处理器进行了实验验证。该实验验证不仅证实了理论结果，而且有效地弥合了推导的界限与现实世界实验获得的经验数据之间的差距。我们的工作为优化量子信息科学中涉及多个非交换可观测量的各种任务铺平了道路。