Convex functions of quantum states play a key role in quantum physics, with examples ranging from Bell inequalities to von Neumann entropy. However, in experimental scenarios, direct measurements of these functions are often impractical. We address this issue by introducing two methods for determining rigorous confidence bounds for convex functions based on informationally incomplete measurements. Our approach outperforms existing protocols by providing tighter bounds for a fixed confidence level and number of measurements. We evaluate the performance of our methods using both numerical and experimental data. Our findings demonstrate the efficacy of our approach, paving the way for improved quantum state certification in real-world applications.

中文翻译：

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部分信息下量子态函数的证明


量子态的凸函数在量子物理学中发挥着关键作用，其例子包括从贝尔不等式到冯诺依曼熵。然而，在实验场景中，直接测量这些函数通常是不切实际的。我们通过引入两种基于信息不完整的测量来确定凸函数的严格置信界限的方法来解决这个问题。我们的方法通过为固定置信水平和测量数量提供更严格的界限，优于现有协议。我们使用数值和实验数据评估我们方法的性能。我们的研究结果证明了我们方法的有效性，为现实应用中改进量子态认证铺平了道路。