The operator space entanglement entropy, or simply "operator entanglement" (OE), is an indicator of the complexity of quantum operators and of their approximability by matrix product operators (MPOs). We study the OE of the density matrix of 1D many-body models undergoing dissipative evolution. It is expected that, after an initial linear growth reminiscent of unitary quench dynamics, the OE should be suppressed by dissipative processes as the system evolves to a simple stationary state. Surprisingly, we find that this scenario breaks down for one of the most fundamental dissipative mechanisms: dephasing. Under dephasing, after the initial "rise and fall," the OE can rise again, increasing logarithmically at long times. Using a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, we demonstrate that this growth is inherent to a U(1) conservation law. We argue that in an XXZ spin model and a Bose-Hubbard model the OE grows universally as 1/4log_{2}t at long times and as 1/2log_{2}t for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.
The presence of symmetries can lead to nontrivial dynamics of operator entanglement in open quantum many-body systems, which characterizes the cost of a matrix product density operator representation of the density matrix in tensor-network methods and provides insights into the corresponding classical simulability. One example is
U(1)-symmetric open quantum systems with dephasing, in which the operator entanglement increases logarithmically at late times instead of being suppressed by the dephasing. Here we numerically study the far-from-equilibrium dynamics of operator entanglement in a dissipative quantum many-body system with the more complicated
SU(2) symmetry and dissipations beyond dephasing. We show that after the initial rise and fall, the operator entanglement increases again in a logarithmic manner at late times in the
SU(2)-symmetric case. We find that this behavior can be fully understood from the corresponding
U(1) subsymmetry by considering the symmetry-resolved operator entanglement. Especially, the probability distribution of different
U(1) sectors also follows the Gaussian distribution observed in the
U(1)-symmetric case with dephasing, with the variance growing as a power law. But unlike the latter, both the classical Shannon entropy associated with the probabilities for the half system to be in different symmetry sectors and the corresponding symmetry-resolved operator entanglement now make nontrivial contributions to the late-time logarithmic growth of operator entanglement in our
SU(2)-symmetric case. Our results show evidence that the logarithmic growth of operator entanglement at long times is a generic behavior of dissipative quantum many-body dynamics with
U(1) as the symmetry or subsymmetry and for broader dissipations beyond dephasing, although more analytical and numerical proof from future studies is still required for this conjecture. By breaking the
SU(2) symmetry of our quantum many-body dynamics to
U(1), we also show that the latter property---the logarithmic growth behavior of operator entanglement for broader dissipations beyond dephasing---is valid even for open quantum systems with only
U(1) symmetry.
Generic quantum many-body systems typically show a linear growth of the entanglement entropy after a quench from a product state. While entanglement is a property of the wave function, it is generated by the unitary time-evolution operator and is therefore reflected in its increasing complexity as quantified by the operator entanglement entropy. Using numerical simulations of a static and a periodically driven quantum spin chain, we show that there is a robust correspondence between the entanglement entropy growth of typical product states and the operator entanglement entropy of the unitary evolution operator, while special product states, e.g., σz basis states, can exhibit faster entanglement production. In the presence of a disordered magnetic field in our spin chains, we show that both the wave function and operator entanglement entropies exhibit a power-law growth with the same disorder-dependent exponent and clarify the apparent discrepancy in previous results. These systems, in the absence of conserved densities, provide further evidence for slow information spreading on the ergodic side of the many-body localization transition.
Abstract In one dimension, the area law and its implications for the approximability by matrix product states are the key to efficient numerical simulations involving quantum states. Similarly, in simulations involving quantum operators, the approximability by matrix product operators (in Hilbert–Schmidt norm) is tied to an operator area law , namely the fact that the operator space entanglement entropy (OSEE)—the natural analog of entanglement entropy for operators, investigated by Zanardi (2001 Phys. Rev . A 63 040304) and by Prosen and Pižorn (2007 Phys. Rev. A 76 032316)—is bounded. In the present paper, it is shown that the OSEE can be calculated in two-dimensional conformal field theory, in a number of situations that are relevant to questions of simulability of long-time dynamics in one spatial dimension. It is argued that: (i) thermal density matrices ρ ∝ e − β H and generalized Gibbs ensemble density matrices ρ ∝ e − H G G E with local H GGE generically obey the operator area law; (ii) after a global quench, the OSEE first grows linearly with time, then decreases back to its thermal or GGE saturation value, implying that, while the operator area law is satisfied both in the initial state and in the asymptotic stationary state at large time, it is strongly violated in the transient regime; (iii) the OSEE of the evolution operator U ( t ) = e − i H t increases linearly with t , unless the Hamiltonian is in a localized phase; (iv) local operators in Heisenberg picture, ϕ ( t ) = e i H t ϕ e − i H t , have an OSEE that grows sublinearly in time (perhaps logarithmically), however it is unclear whether this effect can be captured in a traditional CFT framework, as the free fermion case hints at an unexpected breakdown of conformal invariance.
Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.
In a many-body quantum system, local operators in the Heisenberg picture O(t)=e^{iHt}Oe^{-iHt} spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement-the entanglement entropy in operator space-is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time t in chaotic systems, we present evidence that it grows only logarithmically in generic interacting integrable systems. Although this logarithmic growth has been previously established for noninteracting fermions, there has been no progress on interacting integrable systems to date. In this Letter we provide an analytical upper bound on operator entanglement for all local operators in the "Rule 54" qubit chain, a cellular automaton model introduced in the 1990s [Bobenko et al., CMP 158, 127 (1993)CMPHAY0010-361610.1007/BF02097234], and recently advertised as the simplest representative of interacting integrable systems. Physically, the logarithmic bound originates from the fact that the dynamics of the models is mapped onto the one of stable quasiparticles that scatter elastically. The possibility of generalizing this scenario to other interacting integrable systems is briefly discussed.
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.