TY - JOUR
T1 - Second-Order Asymptotics for Source Coding, Dense Coding, and Pure-State Entanglement Conversions
AU - Datta, Nilanjana
AU - Leditzky, Felix
N1 - Publisher Copyright:
© 1963-2012 IEEE.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.
PY - 2015/1/1
Y1 - 2015/1/1
N2 - 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 independent and identically distributed 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.
AB - 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 independent and identically distributed 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.
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U2 - 10.1109/TIT.2014.2366994
DO - 10.1109/TIT.2014.2366994
M3 - Article
AN - SCOPUS:84937559565
SN - 0018-9448
VL - 61
SP - 582
EP - 608
JO - IRE Professional Group on Information Theory
JF - IRE Professional Group on Information Theory
IS - 1
M1 - 6945831
ER -