Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this work, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that $\Theta(2^n\varepsilon^{-2})$ rounds of measurements are required to estimate each eigenvalue of an $n$-qubit Pauli channel to $\varepsilon$ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs $\Theta(\varepsilon^{-2})$ copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.

翻译：量子纠缠是学习自然性质的关键资源，但精确刻画其优势颇具挑战。本研究将无纠缠学习算法定义为仅使用主系统与辅助系统之间可分离的状态、测量和操作的算法。有趣的是，我们证明这类算法等价于在主系统上交替进行量子电路、中间测量与经典前馈操作的方案。在此框架下，我们证明了无纠缠条件下泡利信道学习的紧下界，填补了已知最优上下界之间的空白。特别地，研究表明：当无纠缠学习时，为以高概率将n量子比特泡利信道的每个特征值估计至ε误差范围内，需要Θ(2ⁿε⁻²)轮测量；而具有纠缠的学习算法仅需Θ(ε⁻²)个泡利信道副本。这一紧下界为实验验证泡利噪声表征中的纠缠增强优势奠定了理论基础。