The entropy photon-number inequality (Guha-Erkmen-Shapiro)

Theorem 1.1: for finite $n\ge1$, $\eta\in[0,1]$ and $n$-mode states $\rho_A,\rho_B$ of finite mean photon number, the output of passive mixing satisfies $g^{-1}(S(\rho_C)/n)\ge\eta g^{-1}(S(\rho_A)/n)+(1-\eta)g^{-1}(S(\rho_B)/n)$, with equality for product thermal inputs. Corollaries: the exact minimum output entropy at fixed input entropy for tensor powers of the thermal attenuator, and the capacity region of the degraded pure-loss bosonic broadcast channel. Not covered: infinite-energy finite-entropy inputs, amplifiers and additive-noise channels, and uniqueness of optimizers.

Theorem 1.1: for finite $n\ge1$, $\eta\in[0,1]$ and $n$-mode states $\rho_A,\rho_B$ of finite mean photon number, the output of passive mixing satisfies $g^{-1}(S(\rho_C)/n)\ge\eta g^{-1}(S(\rho_A)/n)+(1-\eta)g^{-1}(S(\rho_B)/n)$, with equality for product thermal inputs. Corollaries: the exact minimum output entropy at fixed input entropy for tensor powers of the thermal attenuator, and the capacity region of the degraded pure-loss bosonic broadcast channel. Not covered: infinite-energy finite-entropy inputs, amplifiers and additive-noise channels, and uniqueness of optimizers.