  We investigate entanglement breaking times of Markovian evolutions in
discrete and continuous time. In continuous time, we characterize which
Markovian evolutions are eventually entanglement breaking, that is, evolutions
for which there is a finite time after which any entanglement initially present
has been destroyed by the noisy evolution. In the discrete time framework, we
consider the entanglement breaking index, that is, the number of times a
quantum channel has to be composed with itself before it becomes entanglement
breaking. The PPT-square conjecture is that every PPT quantum channel has an
entanglement breaking index of at most 2; we prove that every faithful PPT
quantum channel has a finite entanglement breaking index, and more generally,
any faithful PPT CP map whose Hilbert-Schmidt adjoint is also faithful is
eventually entanglement breaking. We also provide a method to obtain concrete
bounds on this index for any faithful quantum channel. To obtain these
estimates, we use a notion of robustness of separability to obtain bounds on
the radius of the largest separable ball around faithful product states. We
also extend the framework of Poincar\'e inequalities for nonprimitive
semigroups to the discrete setting to quantify the convergence of quantum
semigroups in discrete time, which is of independent interest.
