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 [40] 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 quanti-ties. 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. set-ting. 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 [41]. Our results also yield the known second order asymptotics of fixed-length classical source coding derived by Hayashi [16]. 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 [14]. 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 [24]. In addition, the spectral divergence rates of the Information Spectrum Approach (ISA) can be retrieved from our relative en-tropies in the asymptotic limit. This enables us to directly obtain the more general results of the ISA from our one-shot bounds.