Tight bounds for Pauli channel learning with and without entanglement*

Entanglement is a useful quantum resource for learning, but a precise characterization of its advantage can be challenging. In this work, we define learning algorithms without entanglement to be those that only utilize separable states, measurements, and operations between the main system of interest and an ancillary system. These algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. We prove a tight lower bound for learning Pauli channels without entanglement that closes a cubic gap between the best-known upper and lower bound. In particular, we show that Θ(n^2/ε^2) rounds of measurements are required to estimate each eigenvalue of an n-qubit Pauli channel to ε error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs Θ(1/ε^2) copies of the Pauli channel. The tight lower bounds enable an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.