论文标题
量子状态估计和二进制检测的信息过度completeclete poVM
Informationally Overcomplete POVMs for Quantum State Estimation and Binary Detection
论文作者
论文摘要
在经典框架理论中众所周知，给定矢量空间的过度表示形式为未知矢量的框架系数上的加性噪声提供了鲁棒性。我们描述了如何在量子状态估计的背景下证明相同的鲁棒性。讨论的一个关键要素是将经典框架理论应用于运算符值的矢量空间或操作员空间，这些空间自然而然地在量子力学中出现。具体而言，在问题中，我们描述了框架向量由信息完整或过度（IC或IOC）POVM的元素表示，框架系数由在未知状态下进行的量子测量的结果概率表示，以及框架系数上的误差来自概率估计的有限样品样品量估计。我们表明，通过这种问题的表述，未知系统的副本数量与POVM元素数量之间的估计性能是一个权衡的。最后，我们通过模拟提供证据，表明在量子二进制状态检测的背景下存在相同的权衡 - 误差的可能性可以通过增加未知系统的副本数量或增加POVM元素的数量来降低误差的可能性。
It is well-known in classical frame theory that overcomplete representations of a given vector space provide robustness to additive noise on the frame coefficients of an unknown vector. We describe how the same robustness can be shown to exist in the context of quantum state estimation. A key element of the discussion is the application of classical frame theory to operator-valued vector spaces, or operator spaces, which arise naturally in quantum mechanics. Specifically, in the problem we describe the frame vectors are represented by the elements of an informationally complete or overcomplete (IC or IOC) POVM, the frame coefficients are represented by the outcome probabilities of a quantum measurement made on an unknown state, and the error on the frame coefficients arises from finite sample size estimations of the probabilities. We show that with this formulation of the problem, there is a tradeoff in estimation performance between the number of copies of the unknown system and the number of POVM elements. Lastly, we present evidence through simulation that the same tradeoff is present in the context of quantum binary state detection -- the probability of error can be reduced either by increasing the number of copies of the unknown system or by increasing the number of POVM elements.
