  It is widely accepted that the dynamic of entanglement in presence of a
generic circuit can be predicted by the knowledge of the statistical properties
of the entanglement spectrum. We tested this assumption by applying a
Metropolis-like entanglement cooling algorithm generated by different sets of
local gates, on states sharing the same statistic. We employ the ground states
of a unique model, namely the one-dimensional Ising chain with a transverse
field, but belonging to different macroscopic phases such as the paramagnetic,
the magnetically ordered, and the topological frustrated ones. Quite
surprisingly, we observe that the entanglement dynamics are strongly dependent
not just on the different sets of gates but also on the phase, indicating that
different phases can possess different types of entanglement (which we
characterize as purely local, GHZ-like, and W-state-like) with different degree
of resilience against the cooling process. Our work highlights the fact that
the knowledge of the entanglement spectrum alone is not sufficient to determine
its dynamics, thereby demonstrating its incompleteness as a characterization
tool. Moreover, it shows a subtle interplay between locality and non-local
constraints.
