  We show that quantum entanglement can provide an exponential advantage in
learning properties of a bosonic continuous-variable (CV) system. The task we
consider is estimating a probabilistic mixture of displacement operators acting
on $n$ bosonic modes, called a random displacement channel. We prove that if
the $n$ modes are not entangled with an ancillary quantum memory, then the
channel must be sampled a number of times exponential in $n$ in order to
estimate its characteristic function to reasonable precision; this lower bound
on sample complexity applies even if the channel inputs and measurements
performed on channel outputs are chosen adaptively. On the other hand, we
present a simple entanglement-assisted scheme that only requires a number of
samples independent of $n$, given a sufficient amount of squeezing. This
establishes an exponential separation in sample complexity. We then analyze the
effect of photon loss and show that the entanglement-assisted scheme is still
significantly more efficient than any lossless entanglement-free scheme under
mild experimental conditions. Our work illuminates the role of entanglement in
learning continuous-variable systems and points toward experimentally feasible
demonstrations of provable entanglement-enabled advantage using CV quantum
platforms.
