  Insights from quantum information theory show that correlation measures based
on quantum entropy are fundamental tools that reveal the entanglement structure
of multipartite states. In that spirit, Groisman, Popescu, and Winter [Physical
Review A 72, 032317 (2005)] showed that the quantum mutual information $I(A;B)$
quantifies the minimal rate of noise needed to erase the correlations in a
bipartite state of quantum systems $AB$. Here, we investigate correlations in
tripartite systems $ABE$. In particular, we are interested in the minimal rate
of noise needed to apply to the systems $AE$ in order to erase the correlations
between $A$ and $B$ given the information in system $E$, in such a way that
there is only negligible disturbance on the marginal $BE$. We present two such
models of conditional decoupling, called deconstruction and conditional erasure
cost of tripartite states $ABE$. Our main result is that both are equal to the
conditional quantum mutual information $I(A;B|E)$ -- establishing it as an
operational measure for tripartite quantum correlations.
