Finding the optimal attainable precisions in quantum multiparameter metrology is a non-trivial problem. One approach to tackling this problem involves the computation of bounds which impose limits on how accurately we can estimate certain physical quantities. One such bound is the Holevo Cramér–Rao bound on the trace of the mean squared error matrix. The Holevo bound is an asymptotically achievable bound when one allows for any measurement strategy, including collective measurements on many copies of the probe. In this work, we introduce a tighter bound for estimating multiple parameters simultaneously when performing separable measurements on a finite number of copies of the probe. This makes it more relevant in terms of experimental accessibility. We show that this bound can be efficiently computed by casting it as a semidefinite programme. We illustrate our bound with several examples of collective measurements on finite copies of the probe. These results have implications for the necessary requirements to saturate the Holevo bound.

在量子多参数计量中寻找最佳可达到的精度是一个不平凡的问题。解决这个问题的一种方法涉及边界的计算，这对我们估计某些物理量的准确度施加了限制。其中一个边界是均方误差矩阵迹上的 Holevo Cramer-Rao 边界。当允许任何测量策略时，Holevo 界限是一种渐近可实现的界限，包括对探针的许多副本的集体测量。在这项工作中，我们引入了一个更严格的界限，用于在对有限数量的探针副本执行可分离测量时同时估计多个参数。这使得它在实验可访问性方面更加相关。我们表明可以通过将其转换为半定程序来有效地计算此界限。我们用几个对探针的有限副本进行集体测量的例子来说明我们的界限。这些结果对使 Holevo 边界饱和的必要要求有影响。