Fujiwara Akio, Yamagata Koichi
Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan.
Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan.
Entropy (Basel). 2018 Aug 16;20(8):609. doi: 10.3390/e20080609.
Suppose that a -dimensional Hilbert space H ≃ C d admits a full set of mutually unbiased bases | 1 ( a ) 〉 , ⋯ , | d ( a ) 〉 , where a = 1 , ⋯ , d + 1 . A randomized quantum state tomography is a scheme for estimating an unknown quantum state on H through iterative applications of measurements M ( a ) = | 1 ( a ) 〉 〈 1 ( a ) | , ⋯ , | d ( a ) 〉 〈 d ( a ) | for a = 1 , ⋯ , d + 1 , where the numbers of applications of these measurements are random variables. We show that the space of the resulting probability distributions enjoys a mutually orthogonal dualistic foliation structure, which provides us with a simple geometrical insight into the maximum likelihood method for the quantum state tomography.
假设一个(d)维希尔伯特空间(H\cong\mathbb{C}^d)允许一组完备的相互无偏基(\vert1^{(a)}\rangle,\cdots,\vert d^{(a)}\rangle)，其中(a = 1,\cdots,d + 1)。随机量子态断层扫描是一种通过对(a = 1,\cdots,d + 1)迭代应用测量(M^{(a)}=\vert1^{(a)}\rangle\langle1^{(a)}\vert,\cdots,\vert d^{(a)}\rangle\langle d^{(a)}\vert)来估计(H)上未知量子态的方案，其中这些测量的应用次数是随机变量。我们表明，所得概率分布的空间具有相互正交的二元叶状结构，这为我们提供了对量子态断层扫描的最大似然方法的简单几何见解。
