The method of classical shadows proposed by Huang, Kueng, and Preskill heralds remarkable opportunities for quantum estimation with limited measurements. Yet its relationship to established quantum tomographic approaches, particularly those based on likelihood models, remains unclear. In this article, we investigate classical shadows through the lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME is found to attain significantly lower error on average, but classical shadows prove remarkably more accurate in specific situations—such as high-fidelity ground truth states—which are improbable in a fully uniform Hilbert space. We then introduce an observable-oriented pseudo-likelihood that successfully emulates the dimension-independence and state-specific optimality of classical shadows, but within a Bayesian framework that ensures only physical states. Our research reveals how classical shadows effect important departures from conventional thinking in quantum state estimation, as well as the utility of Bayesian methods for uncovering and formalizing statistical assumptions.

Huang、Kueng 和 Preskill 提出的经典阴影方法预示着利用有限测量进行量子估计的绝佳机会。然而，它与已建立的量子断层扫描方法的关系，尤其是那些基于似然模型的方法，仍不清楚。在本文中，我们通过贝叶斯均值估计 (BME) 的镜头研究经典阴影。在对数值数据的直接测试中，发现 BME 平均获得显着更低的误差，但经典阴影在特定情况下被证明更加准确——例如高保真地面实况状态——这在完全均匀的希尔伯特空间中是不可能的。然后，我们引入了一个面向可观察的伪似然，它成功地模拟了经典阴影的维度独立性和特定于状态的最优性，但在贝叶斯框架内，只确保物理状态。我们的研究揭示了经典阴影如何影响与量子状态估计中传统思维的重要偏离，以及贝叶斯方法在揭示和形式化统计假设方面的效用。