  Every multipartite entangled quantum state becomes fully separable after an
entanglement breaking quantum channel acted locally on each of its subsystems.
Whether there are other quantum channels with this property has been an open
problem with important implications for entanglement theory (e.g., for the
distillation problem and the PPT squared conjecture). We cast this problem in
the general setting of proper convex cones in finite-dimensional vector spaces.
The entanglement annihilating maps transform the $k$-fold maximal tensor
product of a cone $C_1$ into the $k$-fold minimal tensor product of a cone
$C_2$, and the pair $(C_1,C_2)$ is called resilient if all entanglement
annihilating maps are entanglement breaking. Our main result is that
$(C_1,C_2)$ is resilient if either $C_1$ or $C_2$ is a Lorentz cone. Our proof
exploits the symmetries of the Lorentz cones and applies two constructions
resembling protocols for entanglement distillation: As a warm-up, we use the
multiplication tensors of real composition algebras to construct a finite
family of generalized distillation protocols for Lorentz cones, containing the
distillation protocol for entangled qubit states by Bennett et al. as a special
case. Then, we construct an infinite family of protocols using solutions to the
Hurwitz matrix equations. After proving these results, we focus on maps between
cones of positive semidefinite matrices, where we derive necessary conditions
for entanglement annihilation similar to the reduction criterion in
entanglement distillation. Finally, we apply results from the theory of Banach
space tensor norms to show that the Lorentz cones are the only cones with a
symmetric base for which a certain stronger version of the resilience property
is satisfied.
