  A longstanding open problem in quantum information theory is to find the
classical capacity of an optical communication link, modeled as a Gaussian
bosonic channel. It has been conjectured that this capacity is achieved by a
random coding of coherent states using an isotropic Gaussian distribution in
phase space. We show that proving a Gaussian minimum entropy conjecture for a
quantum-limited amplifier is actually sufficient to confirm this capacity
conjecture, and we provide a strong argument towards this proof by exploiting a
connection between quantum entanglement and majorization theory.
