Majorization and additivity for multimode bosonic Gaussian channels
  Recently, the longstanding Gaussian optimizer conjecture was proven for bosonic Gaussian gauge-covariant or contravariant channels in the work of Giovannetti, Holevo and Garcia-Patron [3]. In the paper of Mari, Giovannetti and Holevo [11] this result was strengthened for one-mode channels by establishing that the output for the vacuum or coherent input majorizes the output for any other input.   In the present paper we give the multimode extension of the result of [11], including sufficient conditions under which the coherent states are the only optimizers. We also discuss direct implications of this multimode majorization result to the positive solution of the additivity problem for the Gaussian channels. In particular, we demonstrate the additivity of the output Renyi entropies of arbitrary order p>1. Finally, we present an alternative derivation of a majorization property of Glauber's coherent states by Lieb and Solovej [10], basing on the method of the work [3]. 
V. Giovannetti, A. S. Holevo, A. Mari
quant-ph math-ph math.MP