Local solutions of maximum likelihood estimation in quantum state tomography
Local solutions of maximum likelihood estimation in quantum state tomography
复制标题
量子态层析成像中最大似然估计的局部解
DOI：
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复制
发表时间：
2011
影响因子：
1
通讯作者：
P. S. Ribeiro
中科院分区：
文献类型：
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作者：
D. Gonçalves;M. A. Gomes;C. Lavor;O. J. Farias;P. S. Ribeiro
最大似然估计是量子态层析成像中最常用的方法之一，其目的是根据测量结果重建物理系统的密度矩阵。处理正性和单位迹约束的一种策略是参数化要重构的矩阵，以确保它是物理的。在这种情况下，就参数而言的负对数似然函数可能有几个局部最小值。在该领域的各种论文中，该过程中的错误来源与大多数局部最小值不是全局的可能性相关，因此优化方法可能陷入错误的最小值，从而导致错误的密度矩阵。在这里，我们表明，对于凸负对数似然函数，无约束参数化问题的所有局部最小值都是全局的，因此任何最小值都会导致密度矩阵的最大似然估计。我们还讨论了一些实际的错误来源。
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.