  Entanglement serves as the resource to empower quantum computing. Recent
progress has highlighted its positive impact on learning quantum dynamics,
wherein the integration of entanglement into quantum operations or measurements
of quantum machine learning (QML) models leads to substantial reductions in
training data size, surpassing a specified prediction error threshold. However,
an analytical understanding of how the entanglement degree in data affects
model performance remains elusive. In this study, we address this knowledge gap
by establishing a quantum no-free-lunch (NFL) theorem for learning quantum
dynamics using entangled data. Contrary to previous findings, we prove that the
impact of entangled data on prediction error exhibits a dual effect, depending
on the number of permitted measurements. With a sufficient number of
measurements, increasing the entanglement of training data consistently reduces
the prediction error or decreases the required size of the training data to
achieve the same prediction error. Conversely, when few measurements are
allowed, employing highly entangled data could lead to an increased prediction
error. The achieved results provide critical guidance for designing advanced
QML protocols, especially for those tailored for execution on early-stage
quantum computers with limited access to quantum resources.
