  The quantum Renyi relative entropies play a prominent role in quantum
information theory, finding applications in characterizing error exponents and
strong converse exponents for quantum hypothesis testing and quantum
communication theory. On a different thread, quantum Gaussian states have been
intensely investigated theoretically, motivated by the fact that they are more
readily accessible in the laboratory than are other, more exotic quantum
states. In this paper, we derive formulas for the quantum Renyi relative
entropies of quantum Gaussian states. We consider both the traditional (Petz)
Renyi relative entropy as well as the more recent sandwiched Renyi relative
entropy, finding formulas that are expressed solely in terms of the mean
vectors and covariance matrices of the underlying quantum Gaussian states. Our
development handles the hitherto elusive case for the Petz--Renyi relative
entropy when the Renyi parameter is larger than one. Finally, we also derive a
formula for the max-relative entropy of two quantum Gaussian states, and we
discuss some applications of the formulas derived here.
