  If a quantum system A, which is initially correlated to another system, E,
undergoes an evolution separated from E, then the correlation to E generally
decreases. Here, we study the conditions under which the correlation disappears
(almost) completely, resulting in a decoupling of A from E. We give a criterion
for decoupling in terms of two smooth entropies, one quantifying the amount of
initial correlation between A and E, and the other characterizing the mapping
that describes the evolution of A. The criterion applies to arbitrary such
mappings in the general one-shot setting. Furthermore, the criterion is tight
for mappings that satisfy certain natural conditions. Decoupling has a number
of applications both in physics and information theory, e.g., as a building
block for quantum information processing protocols. As an example, we give a
one-shot state merging protocol and show that it is essentially optimal in
terms of its entanglement consumption/production.
