The estimation of multiple parameters in quantum metrology is important for a vast array of applications in quantum information processing. However, the unattainability of fundamental precision bounds for incompatible observables greatly diminishes the applicability of estimation theory in many practical implementations. The Holevo Cramér-Rao bound (HCRB) provides the most fundamental, simultaneously attainable bound for multiparameter estimation problems. A general closed form for the HCRB is not known given that it requires a complex optimization over multiple variables. In this work, we develop an analytic approach to solving the HCRB for two parameters. Our analysis reveals the role of the HCRB and its interplay with alternative bounds in estimation theory. For more parameters, we generate a lower bound to the HCRB. Our work greatly reduces the complexity of determining the HCRB to solving a set of linear equations that even numerically permits a quadratic speedup over previous state-of-the-art approaches. We apply our results to compare the performance of different probe states in magnetic field sensing and characterize the performance of state tomography on the code space of noisy bosonic error-correcting codes. The sensitivity of state tomography on noisy binomial code states can be improved by tuning two coding parameters that relate to the number of correctable phase and amplitude damping errors. Our work provides fundamental insights and makes significant progress toward the estimation of multiple incompatible observables.

中文翻译：

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不相容参数同时估计的严格界

量子计量学中多个参数的估计对于量子信息处理中的大量应用非常重要。但是，对于不兼容的可观测量，基本精度范围的无法达到极大地降低了估计理论在许多实际实现中的适用性。HolevoCramér-Rao界限（HCRB）为多参数估计问题提供了最基本的，同时可达到的界限。鉴于HCRB需要对多个变量进行复杂的优化，因此不知道其通用封闭形式。在这项工作中，我们开发了一种解析方法来求解HCRB的两个参数。我们的分析揭示了HCRB的作用及其与估计理论中其他界限的相互作用。对于更多参数，我们生成了HCRB的下限。我们的工作极大地降低了确定HCRB的难度，从而可以解决一组线性方程，该线性方程甚至在数值上都比以前的最新方法有了二次加速。我们应用我们的结果来比较磁场探测中不同探头状态的性能，并在嘈杂的玻色子纠错码的代码空间上表征状态层析成像的性能。可以通过调整与可校正相位和幅度阻尼误差的数量有关的两个编码参数来提高状态层析成像对有噪二项编码状态的敏感性。我们的工作提供了基本的见识，并在估计多个不兼容的可观测量方面取得了重大进展。