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连续时间测量下量子比特态的最优估计

Optimal estimation of qubit states with continuous time measurements

Madalin Guta, Bas Janssens, Jonas Kahn

备注：
32 pages, 3 figures, to appear in Commun. Math. Phys
发表信息：
Commun. Math. Phys. 277, 127-160 (2008)
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摘要：

我们提出了一种自适应的两步策略，用于估计混合量子比特状态。我们表明，该策略是最佳的，在一个局部极小极大意义上的跟踪范数距离以及其他局部二次品质因数。局部极小极大最优性意味着给定$n$相同的量子比特，不存在可以在任意状态的大小为$n^{-1/2}$的邻域上比所提出的估计器表现得更好的估计器。特别是，它是渐近贝叶斯最优的一大类先验分布。我们提出了一个物理实现的最佳估计策略的基础上，在一个领域的连续时间测量耦合的量子位。结果的关键成分是量子比特的局部渐近正态性（或LAN）概念。这意味着，对于大的$n$，由$n$相同制备的量子比特描述的统计模型局部等价于仅具有量子谐振子的经典高斯分布和高斯态的模型。术语“局部”指的是固定状态$ρ_{0}$周围的收缩邻域。一个基本的结果是，邻域半径可以任意选择接近$n^{-1/4}$。这允许我们使用两个步骤的过程，通过该过程，我们首先将状态定位在半径$n^{-1/2+ε}$的较小邻域内，然后使用LAN来执行最优估计。

英文摘要：

We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given $n$ identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large $n$, the statistical model described by $n$ identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term `local' refers to a shrinking neighborhood around a fixed state $ρ_{0}$. An essential result is that the neighborhood radius can be chosen arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius $n^{-1/2+ε}$, and then use LAN to perform optimal estimation.

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