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Comparing bipartite entropy growth in open-system matrix product simulation methods

Guillermo PreisserDavid WellnitzThomas BotzungJohannes Schachenmayer
Mar 2023
The dynamics of one-dimensional quantum many body systems is oftennumerically simulated with matrix product states (MPS). The computationalcomplexity of MPS methods is known to be related to the growth of entropies ofreduced density matrices for bipartitions of the chain. While for closedsystems the entropy relevant for the complexity is uniquely defined by theentanglement entropy, for open systems it depends on the choice of therepresentation. Here, we systematically compare the growth of differententropies relevant to the complexity of matrix product representations inopen-system simulations. We simulate an XXZ spin-1/2 chain in the presence ofspontaneous emission/absorption and dephasing. We compare simulations using arepresentation of the full density matrix as a matrix product density operator(MPDO) with a quantum trajectory unravelling, where each trajectory is itselfrepresented by an MPS (QT+MPS). We show that the bipartite entropy in the MPDOdescription (operator entanglement, OE) generally scales more favorable withtime than the entropy in QT+MPS (trajectory entanglement, TE): i) Forspontaneous emission/absorption the OE vanishes while the TE grows and reachesa constant value for large dissipative rates and sufficiently long times; ii)for dephasing the OE exhibits only logarithmic growth while the TE growspolynomially. Although QT+MPS requires a smaller local state space, the morefavorable entropy growth can thus make MPDO simulations fundamentally moreefficient than QT+MPS. Furthermore, MPDO simulations allow for easierexploitation of higher order Trotter decompositions and translationalinvariance, allowing for larger time steps and system sizes.