标题： 一种关于泡利层析的随机矩阵理论 显示英文标题

标题： A Random Matrix Theory of Pauli Tomography

摘要： 量子态层析（QST），即从对所述态的多个副本进行重复测量中重建未知量子态 $\hat\rho$ 的过程，在量子计算和模拟的背景下是一个基础性的重要任务。 因此，对 QST 重建中的误差 $\Delta\hat\rho = \hat\rho-\hat\rho^\prime$ 和重建 $\hat\rho^\prime$ 的详细表征对于量子理论和实验具有明显的重要性。 在本工作中，我们开发了信息完备基下状态层析的完全随机矩阵理论（RMT）处理方法；并通过这样做，揭示了 QST 误差 $\Delta\hat\rho$ 与高斯酉系综（GUE）之间的深刻联系。 通过利用这一联系，我们证明了 $\Delta\hat\rho$ 的谱的广泛类别的函数可以通过将适当 GUE 的样本替换为 $\Delta\hat\rho$ 的实现来评估。 这一强大且灵活的结果使得可以通过迹距离 $\|\Delta\hat\rho\|_\mathrm{Tr}$ 来量化误差的均值和方差的简单解析处理成为可能（我们对常见的层析协议进行了数值验证），允许我们推导出 QST 样本复杂度的界限，并随后证明该界限在最常用的重新物理化过程中不会发生变化。 这些结果共同展示了我们方法的灵活性、强度和广泛应用性；并为未来更广泛的 RMT 处理 QST 的研究奠定了基础。 显示英文摘要

摘要： Quantum state tomography (QST), the process of reconstructing some unknown quantum state $\hat\rho$ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error $\Delta\hat\rho = \hat\rho-\hat\rho^\prime$ in a QST reconstruction $\hat\rho^\prime$ is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors $\Delta\hat\rho$ and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of $\Delta\hat\rho$ can be evaluated by substituting samples of an appropriate GUE for realizations of $\Delta\hat\rho$. This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance $\|\Delta\hat\rho\|_\mathrm{Tr}$ (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future.