  We investigate the phenomenon of Hilbert space fragmentation (HSF) in open
quantum systems and find that it can stabilize highly entangled steady states.
For concreteness, we consider the Temperley-Lieb model, which exhibits quantum
HSF in an entangled basis, and investigate the Lindblad dynamics under two
different couplings. First, we couple the system to a dephasing bath that
reduces quantum fragmentation to a classical one with the resulting stationary
state being separable. We observe that despite vanishing quantum correlations,
classical correlations develop due to fluctuations of the remaining conserved
quantities, which we show can be captured by a classical stochastic circuit
evolution. Second, we use a coupling that preserves the quantum fragmentation
structure. We derive a general expression for the steady state, which has a
strong coherent memory of the initial state due to the extensive number of
non-commuting conserved quantities. We show that it is highly entangled as
quantified by the logarithmic negativity.
