Conditions for efficiently learning arbitrary continuous variable quantum channels through state learning

Characterizing quantum channels in continuous-variable (CV) (or bosonic systems) is fundamental to advancing both our understanding of diverse physical processes and the development of quantum technologies. Since CV systems are described by infinite-dimensional Hilbert spaces, a rigorous and general framework for learning arbitrary CV quantum channels has yet to be fully established, and efficient methods for such learning remain largely unexplored. In this work, we establish a framework for learning quantum channels, where learning a channel is defined as learning the characteristic function of the finite-energy Choi state associated with it. We demonstrate that such a characteristic function holds a complete description of the channel by describing its action over a complete, but non-orthogonal operator basis. We also construct an invertible relation for this characteristic function to a continuous variable analogue of the transfer matrix that can be numerically well defined for a class of channels including Gaussian channels, displacement noise, partial replacement, and bosonic dephasing channels. Building upon techniques developed in Ref. [arXiv:2501.17633] for the learning of general CV quantum states, we propose a protocol that enables efficient learning of CV quantum channels when both a 'reflected' version of the channel and joint measurements are available. In contrast, if access to either the reflected channel or joint measurements is restricted, the sample complexity increases exponentially with the number of modes in the CV system.