  The dynamics of one-dimensional quantum many-body systems is often
numerically simulated with matrix-product states (MPSs). The computational
complexity of MPS methods is known to be related to the growth of entropies of
reduced density matrices for bipartitions of the chain. While for closed
systems the entropy relevant for the complexity is uniquely defined by the
entanglement entropy, for open systems it depends on the choice of the
representation. Here, we systematically compare the growth of different
entropies relevant to the complexity of matrix-product representations in
open-system simulations. We simulate an XXZ spin-1/2 chain in the presence of
spontaneous emission and absorption, and dephasing. We compare simulations
using a representation of the full density matrix as a matrix-product density
operator (MPDO) with a quantum trajectory unraveling, where each trajectory is
itself represented by an MPS (QT+MPS). We show that the bipartite entropy in
the MPDO description (operator entanglement, OE) generally scales more
favorably with time than the entropy in QT+MPS (trajectory entanglement, TE):
i) For spontaneous emission and absorption the OE vanishes while the TE grows
and reaches a constant value for large dissipative rates and sufficiently long
times; ii) for dephasing the OE exhibits only logarithmic growth while the TE
grows polynomially. Although QT+MPS requires a smaller local state space, the
more favorable entropy growth can thus make MPDO simulations fundamentally more
efficient than QT+MPS. Furthermore, MPDO simulations allow for easier
exploitation of higher-order Trotter decompositions and translational
invariance, allowing for larger time steps and system sizes.
