01182nas a2200169   4500000000100000000000100001000000100002008004100003100001800044700002200062700001700084700002200101245005600123300001100179490000700190520081500197       2013                            d1 aWojciech Roga1 aZbigniew Puchała1 aŁ. Rudnicki1 aKarol Życzkowski00aEntropic trade-off relations for quantum operations  a0323080 v873 aSpectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel Φ. We prove that for any map acting on an N-dimensional quantum system the sum of both entropies is not smaller than lnN. For any bistochastic map this lower bound reads 2lnN. We investigate also the corresponding Rényi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.