Conditional quantum one-time pad

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 $ρ_{ABE}$, 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 $ρ_{ABE}$ 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 $E$.