Entanglement cost hierarchies in quantum fragmented mixed states

Strong symmetries enforce non-trivial quantum entanglement patterns on the stationary states of symmetric open quantum dynamics. Specifically, non-commuting conserved quantities lead to long-range quantum entanglement even for infinite temperature mixed states within fixed symmetry sectors. Leveraging the commutant algebra framework, we show that various bipartite entanglement measures for mixed states -- including exact and asymptotically-exact entanglement costs and squashed entanglement, which are generally intractable for a generic many-body mixed state -- can be computed for this class of states. In particular, we focus on strongly symmetric maximally mixed states arising from the Temperley-Lieb model, which features quantum Hilbert space fragmentation with exponentially large (in system size) non-Abelian commutants. We find that while both the logarithmic negativity and the `exact' entanglement cost for equal-size bipartitions scale with the volume of the system, the entanglement of formation, squashed entanglement, entanglement cost, and distillable entanglement exhibit subextensive scaling. We relate this separation in entanglement measures to a parametric difference between the entanglement cost of exact and asymptotically-exact state preparations, and infer this to be a consequence of a particular pattern of quantum Hilbert space fragmentation.