一个简单的量子态极小极大估计
A simple minimax estimator for quantum states
摘要：
量子层析成像需要重复测量物理系统的许多副本，所有副本都是由未知状态的源准备的。在测量非常多副本的限制下，用于将收集的数据转换为状态估计的常用最大似然（ML）方法工作得非常好。然而，对于较小的数据集，它往往遭受的问题，秩不足的估计状态。对于与量子信息处理相关的许多系统来说，制备非常大量的相同量子态副本仍然是一个技术挑战，这促使我们寻找即使在没有太多数据的情况下也能表现良好的估计策略。本文回顾了极大极小态估计的概念，并利用极大极小的思想构造了一个简单的量子态估计器。我们证明，对于单个量子比特的层析成像的情况下，我们的估计量显着优于ML估计量的状态测量的副本的数量少。我们的估计总是满秩的，而且，对测量的拷贝数有一个自然的依赖性，这在ML估计中是缺失的。
英文摘要：
Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. In this article, we review the concept of minimax state estimation, and use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.
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