  We define the deconstruction cost of a tripartite quantum state on systems
$ABE$ as the minimum rate of noise needed to apply to the $AE$ systems, such
that there is negligible disturbance to the marginal state on the $BE$ systems,
while the system $A$ of the resulting state is locally recoverable from the $E$
system alone. We refer to such actions as deconstruction operations and
protocols implementing them as state deconstruction protocols. State
deconstruction generalizes Landauer erasure of a single-party quantum state as
well the erasure of correlations of a two-party quantum state. We find that the
deconstruction cost of a tripartite quantum state on systems $ABE$ is equal to
its conditional quantum mutual information (CQMI) $I(A;B|E)$, thus giving the
CQMI an operational interpretation in terms of a state deconstruction protocol.
We also define a related task called conditional erasure, in which the goal is
to apply noise to systems $AE$ in order to decouple system $A$ from systems
$BE$, while causing negligible disturbance to the marginal state of systems
$BE$. We find that the optimal rate of noise for conditional erasure is also
equal to the CQMI $I(A;B|E)$. State deconstruction and conditional erasure lead
to operational interpretations of the quantum discord and squashed
entanglement, which are quantum correlation measures based on the CQMI. We find
that the quantum discord is equal to the cost of simulating einselection, the
process by which a quantum system interacts with an environment, resulting in
selective loss of information in the system. The squashed entanglement is equal
to half the minimum rate of noise needed for deconstruction/conditional erasure
if Alice has available the best possible system $E$ to help in the
deconstruction/conditional erasure task.
