Thermal states minimize the output entropy of single-mode phase-insensitive Gaussian channels with an input entropy constraint

Shannon’s entropy power inequality (EPI), e ) ≥ e + e ) for independent continuous random variables X and Y , with h(X) the differential entropy of X , has found many applications in Gaussian channel coding theorems. Equality holds iff both X and Y are Gaussian. For all the major applications of the EPI, it suffices to restrict Y to be Gaussian. The statement of this restricted EPI is that given a lower bound on h(X), where X is input to the Gaussian noise channel Z = X + Y , a Gaussian input X minimizes h(Z), the output entropy. The conjectured Entropy Photon-number Inequality (EPnI) takes on a role analogous to Shannon’s EPI in proving coding theorem converses involving quantum limits of classical communications over bosonic channels. Similar to the classical case, a restricted version of the EPnI suffices for all its applications, where one of the two states is restricted to be a thermal state—a state that has a zero-mean, circularlysymmetric, complex Gaussian distribution in phase space. The statement of this restricted EPnI is that, given a lower bound on the von Neumann entropy of the input to an n-mode lossy thermal-noise bosonic channel, an n-mode product thermal state input minimizes the output entropy. In this paper, we provide a proof that the thermal state input minimizes the output entropy of an arbitrary single-mode phase-insensitive bosonic Gaussian channel subject to an input entropy constraint. This subsumes the n = 1 case of the aforesaid restricted version of the EPnI. Our results imply that triple trade-off and broadcast capacities of quantum-limited amplifier channels are now solved. However, due to an additivity issue, several bosonic-channel coding-theorem proofs require a stronger multi-channel-use (n > 1) version of the restricted version of the EPnI we prove in this paper, a proof of which remains open.