论文标题
在非功率尺寸的希尔伯特空间中的Qudits的压缩传感层析成像
Compressed Sensing Tomography for qudits in Hilbert spaces of non-power-of-two dimensions
论文作者
论文摘要
先前由D. Gross等人改编了量子状态断层扫描（QST）的低级基质恢复的技术。 [物理。莱特牧师。 105，150401（2010）]，他们考虑$ N $ spin- $ 1/2 $的系统。对于尺寸的密度矩阵$ d = 2^n $和与$ r \ ll 2^n $的等级$ r $，显示出$ o（dr \ log（d）^2）$的随机选择的Pauli测量值足以通过运行特定的CONVEX优化algorithm来充分重建密度矩阵。结果利用了Pauli操作员基础的低操作员表，这使其与低级矩阵“不连贯”。对于尺寸$ d $不是两个功率的量子系统，Pauli测量不可用，并且可以考虑使用SU（$ d $）测量。在这里，我们指出，由于其高运营商的规范，SU（$ d $）运营商在成功恢复所有排名所需的测量设置数量上并不能节省大量资金。我们提出了一种替代策略，其中仅使用$ \ textrm {poly}（\ log（d）^2）$门的量子信息将量子信息交换到Power-Two系统的子空间中，随后通过执行$ O（DR \ log（dr \ log（d）^2）$ Pauli umesurements执行QST。我们表明，尽管维度有所提高，但此方法比使用SU（$ d $）测量的方法更有效。
The techniques of low-rank matrix recovery were adapted for Quantum State Tomography (QST) previously by D. Gross et al. [Phys. Rev. Lett. 105, 150401 (2010)], where they consider the tomography of $n$ spin-$1/2$ systems. For the density matrix of dimension $d = 2^n$ and rank $r$ with $r \ll 2^n$, it was shown that randomly chosen Pauli measurements of the order $O(dr \log(d)^2)$ are enough to fully reconstruct the density matrix by running a specific convex optimization algorithm. The result utilized the low operator-norm of the Pauli operator basis, which makes it `incoherent' to low-rank matrices. For quantum systems of dimension $d$ not a power of two, Pauli measurements are not available, and one may consider using SU($d$) measurements. Here, we point out that the SU($d$) operators, owing to their high operator norm, do not provide a significant savings in the number of measurement settings required for successful recovery of all rank-$r$ states. We propose an alternative strategy, in which the quantum information is swapped into the subspace of a power-two system using only $\textrm{poly}(\log(d)^2)$ gates at most, with QST being implemented subsequently by performing $O(dr \log(d)^2)$ Pauli measurements. We show that, despite the increased dimensionality, this method is more efficient than the one using SU($d$) measurements.
