  Given a bipartite system, correlations between its subsystems can be
understood as information that each one carries about the other. In order to
give a model-independent description of secure information disposal, we propose
the paradigm of private quantum decoupling, corresponding to locally reducing
correlations in a given bipartite quantum state without transferring them to
the environment. In this framework, the concept of private local randomness
naturally arises as a resource, and total correlations get divided into
eliminable and ineliminable ones. We prove upper and lower bounds on the amount
of ineliminable correlations present in an arbitrary bipartite state, and show
that, in tripartite pure states, ineliminable correlations satisfy a monogamy
constraint, making apparent their quantum nature. A relation with entanglement
theory is provided by showing that ineliminable correlations constitute an
entanglement parameter. In the limit of infinitely many copies of the initial
state provided, we compute the regularized ineliminable correlations to be
measured by the coherent information, which is thus equipped with a new
operational interpretation. In particular, our results imply that two
subsystems can be privately decoupled if their joint state is separable.
