We consider the classic question of state tomography: given copies of an unknown quantum state $\rho\in\mathbb{C}^{d\times d}$, output $\widehat{\rho}$ which is close to $\rho$ in some sense, e.g. trace distance or fidelity. When one is allowed to make coherent measurements entangled across all copies, $\Theta(d^2/\epsilon^2)$ copies are necessary and sufficient to get trace distance $\epsilon$. Unfortunately, the protocols achieving this rate incur large quantum memory overheads that preclude implementation on near-term devices. On the other hand, the best known protocol using incoherent (single-copy) measurements uses $O(d^3/\epsilon^2)$ copies, and multiple papers have posed it as an open question to understand whether or not this rate is tight. In this work, we fully resolve this question, by showing that any protocol using incoherent measurements, even if they are chosen adaptively, requires $\Omega(d^3/\epsilon^2)$ copies, matching the best known upper bound. We do so by a new proof technique which directly bounds the ``tilt'' of the posterior distribution after measurements, which yields a surprisingly short proof of our lower bound, and which we believe may be of independent interest. While this implies that adaptivity does not help for tomography with respect to trace distance, we show that it actually does help for tomography with respect to infidelity. We give an adaptive algorithm that outputs a state which is $\gamma$-close in infidelity to $\rho$ using only $\tilde{O}(d^3/\gamma)$ copies, which is optimal for incoherent measurements. In contrast, it is known that any nonadaptive algorithm requires $\Omega(d^3/\gamma^2)$ copies. While it is folklore that in $2$ dimensions, one can achieve a scaling of $O(1/\gamma)$, to the best of our knowledge, our algorithm is the first to achieve the optimal rate in all dimensions.

翻译：我们考虑态层析这一经典问题：给定未知量子态 $\rho\in\mathbb{C}^{d\times d}$ 的副本，输出在某种度量（如迹距离或保真度）下接近 $\rho$ 的 $\widehat{\rho}$。当允许对所有副本进行纠缠的相干测量时，$\Theta(d^2/\epsilon^2)$ 个副本足以且必须达到迹距离 $\epsilon$。然而，实现该速率的协议需要巨大的量子内存开销，阻碍了其在近期的设备上实现。另一方面，使用非相干（单副本）测量的已知最优协议需要 $O(d^3/\epsilon^2)$ 个副本，多篇论文将此速率是否紧致作为开放问题。在本工作中，我们完全解决了该问题，证明任何使用非相干测量的协议（即使自适应选择测量）都需要 $\Omega(d^3/\epsilon^2)$ 个副本，与已知最优上界匹配。我们采用一种新的证明技术，直接界定测量后后验分布的“倾斜量”，从而得到下界的极短证明，并认为该方法可能具有独立研究价值。虽然这表明适应性在迹距离层析中无帮助，但我们在保真度误差方面证明适应性确实有帮助。我们提出一种自适应算法，仅需 $\tilde{O}(d^3/\gamma)$ 个副本即可输出与 $\rho$ 在保真度上 $\gamma$-接近的态，这达到非相干测量的最优速率。相比之下，已知任何非自适应算法需 $\Omega(d^3/\gamma^2)$ 个副本。尽管在二维空间中存在 $O(1/\gamma)$ 速率的已知结果，但据我们所知，我们的算法首次在所有维度中达到最优速率。