Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions

We introduce two variants of the information spectrum relative entropy defined by Tomamichel and Hayashi which have the particular advantage of satisfying the data-processing inequality, i.e. monotonicity under quantum operations. This property allows us to obtain one-shot bounds for various information-processing tasks in terms of these quantities. Moreover, these relative entropies have a second order asymptotic expansion, which in turn yields tight second order asymptotics for optimal rates of these tasks in the i.i.d. setting. The tasks studied in this paper are fixed-length quantum source coding, noisy dense coding, entanglement concentration, pure-state entanglement dilution, and transmission of information through a classical-quantum channel. In the latter case, we retrieve the second order asymptotics obtained by Tomamichel and Tan. Our results also yield the known second order asymptotics of fixed-length classical source coding derived by Hayashi. The second order asymptotics of entanglement concentration and dilution provide a refinement of the inefficiency of these protocols - a quantity which, in the case of entanglement dilution, was studied by Harrow and Lo. We prove how the discrepancy between the optimal rates of these two processes in the second order implies the irreversibility of entanglement concentration established by Kumagai and Hayashi. In addition, the spectral divergence rates of the Information Spectrum Approach (ISA) can be retrieved from our relative entropies in the asymptotic limit. This enables us to directly obtain the more general results of the ISA from our one-shot bounds.