Spin qudit tomography and state reconstruction error

We consider the task of performing quantum state tomography on a $d$-level spin qudit, using only measurements of spin projection onto different quantization axes. After introducing a basis of operators closely related to the spherical harmonics, which obey the rotational symmetries of spin qudits, we map our quantum tomography task onto the classical problem of signal recovery on the sphere. We then provide algorithms with $O(rd^3)$ serial runtime, parallelizable down to $O(rd^2)$, for (i) computing a priori upper bounds on the expected error with which spin projection measurements along $r$ given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol, for which we find that using more measurement axes can yield substantial benefits that plateau after $r\approx3d$.

我们考虑对一个$d$能级自旋量子位进行量子态层析成像的任务，且仅使用在不同量子化轴上的自旋投影测量。在引入与球谐函数密切相关且遵循自旋量子位旋转对称性的算符基之后，我们将量子层析成像任务映射到球面上的经典信号恢复问题。然后，我们提供了具有$O(rd^3)$串行运行时间（可并行化为$O(rd^2)$）的算法，用于（i）计算沿$r$个给定轴的自旋投影测量能够重建一个未知量子位态的预期误差的先验上界，以及（ii）估计重建态中的后验统计误差。我们的算法激发了一种简单的随机层析成像协议，我们发现对于该协议，使用更多的测量轴可以产生显著的益处，并且在$r\approx3d$之后达到平稳。