Quantum state tomography is a fundamental problem in quantum computing. Given $n$ copies of an unknown $N$-qubit state $\rho \in \mathbb{C}^{d \times d},d=2^N$, the goal is to learn the state up to an accuracy $\epsilon$ in trace distance, with at least probability 0.99. We are interested in the copy complexity, the minimum number of copies of $\rho$ needed to fulfill the task. Pauli measurements have attracted significant attention due to their ease of implementation in limited settings. The best-known upper bound is $O(\frac{N \cdot 12^N}{\epsilon^2})$, and no non-trivial lower bound is known besides the general single-copy lower bound $\Omega(\frac{8^n}{\epsilon^2})$, achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of $O(\frac{10^N}{\epsilon^2})$. To complement it with a lower bound of $\Omega(\frac{9.118^N}{\epsilon^2})$, which holds under adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage over prior methods is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements. Moreover, we connect the copy-complexity lower bound to the eigenvalues of the measurement information channel, which governs the measurement's capacity to distinguish states. To demonstrate the generality of the new framework, we obtain tight-bounds for adaptive quantum tomography with $k$-outcome measurements, where we recover existing results and establish new ones.

翻译：量子态层析是量子计算中的一个基本问题。给定未知 $N$ 量子比特态 $\rho \in \mathbb{C}^{d \times d},d=2^N$ 的 $n$ 个副本，目标是以至少 0.99 的概率在迹距离意义下以精度 $\epsilon$ 学习该态。我们关注副本复杂度，即完成任务所需 $\rho$ 的最小副本数。泡利测量因其在受限场景中易于实现而受到广泛关注。已知最佳上界为 $O(\frac{N \cdot 12^N}{\epsilon^2})$，而除了一般单拷贝下界 $\Omega(\frac{8^n}{\epsilon^2})$（由难以实现的结构化 POVM 如 MUB、SIC-POVM 和均匀 POVM 达成）外，尚无非平凡下界。我们针对这一长期存在的问题取得了重要进展。首先证明了更强的上界 $O(\frac{10^N}{\epsilon^2})$，并辅以在自适应条件下成立的 $\Omega(\frac{9.118^N}{\epsilon^2})$ 下界。据我们所知，这首次展示了泡利测量与结构化 POVM 之间的已知分离性。新下界源于我们提出的具有测量约束的自适应量子态层析新框架。相较于先前方法的主要优势在于，我们可以利用测量相关的困难实例来证明泡利测量的紧致下界。此外，我们将副本复杂度下界与测量信息通道的特征值联系起来，该通道决定了测量区分量子态的能力。为证明新框架的普适性，我们获得了 $k$ 结果自适应量子层析的紧致界，其中既复现了现有结果，也建立了新的结论。