"1. The decoupling technique is a fundamental tool in quantum information theory with applications ranging
from quantum thermodynamics to quantum many body physics to the study of black hole radiation. In this
work we introduce the notion of catalytic decoupling, that is, decoupling in the presence of an uncorrelated
ancilla system. This removes a restriction on the standard notion of decoupling, which becomes important
for structureless resources, and yields a tight characterization in terms of the max-mutual information.
Furthermore, the characterization result unifies two previously used proof techniques from one-shot
quantum Shannon theory. Catalytic decoupling can be applied to various tasks like the erasure of
correlations, and quantum state merging, and leads to a resource theory of decoupling. For erasure of
correlations the application of catalytic decoupling yields a novel one-shot generalization 2. 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 and 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."
