Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication

The optimal rate of reliable communication over a quantum channel can be enhanced by pre-shared entanglement. Whereas the enhancement may be unbounded in infinite-dimensional settings even when the input power is constrained, a long-standing conjecture asserts that the ratio between the entanglement-assisted and unassisted classical capacities is bounded in finite-dimensional settings [Bennett et al., IEEE Trans. Inf. Theory 48, 2637 (2002)]. In this work, we prove this conjecture by showing that their ratio is upper bounded by $o(d^2)$, where $d$ is the input dimension of the channel. An application to quantum communication with noisy encoders and decoders is given.