  Suppose that Alice and Bob are located in distant laboratories, which are
connected by an ideal quantum channel. Suppose further that they share many
copies of a quantum state $\rho_{ABE}$, such that Alice possesses the $A$
systems and Bob the $BE$ systems. In our model, there is an identifiable part
of Bob's laboratory that is insecure: a third party named Eve has infiltrated
Bob's laboratory and gained control of the $E$ systems. Alice, knowing this,
would like use their shared state and the ideal quantum channel to communicate
a message in such a way that Bob, who has access to the whole of his laboratory
($BE$ systems), can decode it, while Eve, who has access only to a sector of
Bob's laboratory ($E$ systems) and the ideal quantum channel connecting Alice
to Bob, cannot learn anything about Alice's transmitted message. We call this
task the conditional one-time pad, and in this paper, we prove that the optimal
rate of secret communication for this task is equal to the conditional quantum
mutual information $I(A;B|E)$ of their shared state. We thus give the
conditional quantum mutual information an operational meaning that is different
from those given in prior works, via state redistribution, conditional erasure,
or state deconstruction. We also generalize the model and method in several
ways, one of which demonstrates that the negative tripartite interaction
information $-I_{3}(A;B;E) = I(A;BE)-I(A;B)-I(A;E)$ of a tripartite state
$\rho_{ABE}$ is an achievable rate for a secret-sharing task, i.e., the case in
which Alice's message should be secure from someone possessing only the $AB$ or
$AE$ systems but should be decodable by someone possessing all systems $A$,
$B$, and $E$.
