Decomposition of time-covariant operations on quantum systems with continuous and/or discrete energy spectrum

Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over CP-maps Gσ where σ is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each Gσ is an energy shift followed by a dephasing channel. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. The dephasing is given by the Hadamard product of the density operator with a positive operator. As an example, I calculate this decomposition explicitly for the gaussian channel on a single mode. For a special type of channels, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (Gσ). ∗e-mail: [email protected]