  Macroscopic behavior such as the lack of interference patterns has been
attributed to "decoherence", a word with several possible definitions such as
(1) the loss of off-diagonal density matrix elements, (2) the flow of
information to the environment, (3) the loss of complementary information, and
(4) the loss of the ability to create entanglement in a measurement. In this
article, we attempt to unify these distinct definitions by providing general
quantitative connections between them, valid for all finite-dimensional quantum
systems or quantum processes. The most important application of our results is
to the understanding of quantum discord, a measure of the non-classicality of
the correlations between quantum systems. We show that some popular measures of
discord measure the information missing from the purifying system and hence
quantify security, which can be stated operationally in terms of distillable
secure bits. The results also give some strategies for constructing discord
measures.
