  Quantum teleportation is one of the crucial protocols in quantum information
processing. It is important to accomplish an efficient teleportation under
practical conditions, aiming at a higher fidelity desirably using fewer
resources. The continuous-variable (CV) version of quantum teleportation was
first proposed using a Gaussian state as a quantum resource, while other
attempts were also made to improve performance by applying non-Gaussian
operations. We investigate the CV teleportation to find its ultimate fidelity
under energy constraint identifying an optimal quantum state. For this purpose,
we present a formalism to evaluate teleportation fidelity as an expectation
value of an operator. Using this formalism, we prove that the optimal state
must be a form of photon-number entangled states. We further show that Gaussian
states are near-optimal while non-Gaussian states make a slight improvement and
therefore are rigorously optimal, particularly in the low-energy regime.
