During time evolution of many-body systems entanglement grows rapidly, limiting exact simulations to small-scale systems or small timescales. Quantum information tends, however, to flow towards larger scales without returning to local scales, such that its detailed large-scale structure does not directly affect local observables. This allows for the removal of large-scale quantum information in a way that preserves all local observables and gives access to large-scale and large-time quantum dynamics. To this end, we use the recently introduced to organize quantum information into different scales, allowing us to define and that we employ to systematically discard long-range quantum correlations in a controlled way. Our approach relies on decomposing the system into subsystems up to a maximum scale and time evolving the subsystem density matrices by solving the subsystem von Neumann equations in parallel. Importantly, the information flow needs to be preserved during the discarding of large-scale information. To achieve this without the need to make assumptions about the microscopic details of the information current, we introduce a second scale at which information is discarded, while using the state at the maximum scale to accurately obtain the information flow. The resulting algorithm, which we call local-information time evolution, is highly versatile and suitable for investigating many-body quantum dynamics in both closed and open quantum systems with diverse hydrodynamic behaviors. We present results for the energy transport in the mixed-field Ising model and the magnetization transport in the X X spin chain with onsite dephasing where we accurately determine the power-law exponent and the diffusion coefficients. Furthermore, the information lattice framework employed here promises to offer insightful results about the spatial and temporal behavior of entanglement in many-body systems.
During time evolution of many-body systems entanglement grows rapidly, limiting exact simulations to small-scale systems or small timescales. Quantum information tends, however, to flow towards larger scales without returning to local scales, such that its detailed large-scale structure does not directly affect local observables. This allows for the removal of large-scale quantum information in a way that preserves all local observables and gives access to large-scale and large-time quantum dynamics. To this end, we use the recently introduced to organize quantum information into different scales, allowing us to define and that we employ to systematically discard long-range quantum correlations in a controlled way. Our approach relies on decomposing the system into subsystems up to a maximum scale and time evolving the subsystem density matrices by solving the subsystem von Neumann equations in parallel. Importantly, the information flow needs to be preserved during the discarding of large-scale information. To achieve this without the need to make assumptions about the microscopic details of the information current, we introduce a second scale at which information is discarded, while using the state at the maximum scale to accurately obtain the information flow. The resulting algorithm, which we call local-information time evolution, is highly versatile and suitable for investigating many-body quantum dynamics in both closed and open quantum systems with diverse hydrodynamic behaviors. We present results for the energy transport in the mixed-field Ising model and the magnetization transport in the X X spin chain with onsite dephasing where we accurately determine the power-law exponent and the diffusion coefficients. Furthermore, the information lattice framework employed here promises to offer insightful results about the spatial and temporal behavior of entanglement in many-body systems.
Quantum many-body dynamics generically result in increasing entanglement that eventually leads to thermalization of local observables. This makes the exact description of the dynamics complex despite the apparent simplicity of (high-temperature) thermal states. For accurate but approximate simulations one needs a way to keep track of essential (quantum) information while discarding inessential one. To this end, we first introduce the concept of the information lattice, which supplements the physical spatial lattice with an additional dimension and where a local Hamiltonian gives rise to well-defined locally conserved von Neumann information current. This provides a convenient and insightful way of capturing the flow, through time and space, of information during quantum time-evolution, and gives a distinct signature of when local degrees of freedom decouple from long-range entanglement. As an example, we describe such decoupling of local degrees of freedom for the mixed-field transverse Ising model. Building on this, we secondly construct algorithms to time-evolve sets of local density matrices without any reference to a global state. With the notion of information currents, we motivate algorithms based on the intuition that information for statistical reasons flows from small to large scales. Using this guiding principle, we construct an algorithm that, at worst, shows two-digit convergence in time-evolutions up to very late times for diffusion process governed by the mixed-field transverse Ising Hamiltonian. While we focus on dynamics in 1D with nearest-neighbor Hamiltonians, the algorithms do not essentially rely on these assumptions and can in principle be generalized to higher dimensions and more complicated Hamiltonians.
Unitary time evolution of initially simple quantum many-body states rapidly generates entanglement and complex correlations, which limits direct numerical simulations. The late-time dynamics of physical observables, however, typically exhibits an effective simplicity in the form of hydrodynamics or kinetic theory. This leads to the question whether microscopic equations of motion can remain accurate and tractable up to long time scales by discarding irrelevant information in a controlled manner. Here, we introduce compressed minimum-purity time evolution (CoMPuTE) as an approach to keep track of a consistent set of reduced local density matrices, closing the hierarchical equations of motion using a minimum-purity principle. In benchmark applications we demonstrate (i) accurate description of energy diffusion in the one-dimensional mixed-field Ising model, (ii) the applicability to genuinely out-of-equilibrium Floquet dynamics starting from a pure state, and (iii) the limitations of the local reduced density matrix approximation when describing transport in the XXZ chain at Δ=1 that is governed by increasingly non-local integrals of motion. The CoMPuTE method enhances computational efficiency in comparison to the closely related local-information time evolution algorithm, opening a possible route towards an extension to systems in higher spatial dimensions.
Understanding the entanglement dynamics in quantum many-body systems under steady-state transport conditions is an actively pursued challenging topic. Hydrodynamic equations, akin to transport equations for charge or heat, would be of great interest but face severe challenges because of the inherent nonlocality of entanglement and the difficulty of identifying conservation laws. We show that progress is facilitated by using information as key quantity related to - but distinct from - entanglement. Employing the recently developed "information lattice" framework, we formulate general continuity equations governing the flow of spatially and scale-resolved information currents in nonequilibrium open quantum systems. To illustrate our theory, we consider noninteracting fermion chains coupled to dissipative reservoirs, using Lindblad master equations. By relating the information lattice to a noise lattice constructed from particle-number fluctuations, we show that information is experimentally accessible via noise measurements. Similarly, local information currents can be obtained by measuring particle currents, onsite occupations, and covariances of particle numbers and/or particle currents. Using the fermionic negativity to quantify bipartite entanglement, we also study transport-induced entanglement and its relation to information currents. For a clean particle-hole symmetric chain, we find that information currents are shielded from entering the information lattice. Impurities or particle-hole asymmetry break this effect, causing information current flow and entanglement between end segments of the chain. Our work opens the door to systematic investigations of information transport and entanglement generation in driven open quantum systems far from equilibrium.
Developing a local description of quantum information is a central problem in quantum many-body physics: the goal is to identify how correlations are organized across positions and scales without choosing specific observables. We introduce a higher-dimensional information lattice, an inclusion-exclusion construction that decomposes the von Neumann information of reduced density matrices into contributions labeled by subsystem position and scale. Above one dimension, overlapping subsystems can form loops, allowing information associated with one subsystem to be inferred nonlocally from the density matrices of others. The sign of the inclusion-exclusion information resolves such overlap redundancies in position and scale: positive values mark positions and scales with net information gain, while negative values occur where redundant contributions dominate new ones. We implement the construction in two dimensions and apply it to many-body ground states with distinct entanglement structures. For localized states, such as disordered-localized states or gapped ground states of local Hamiltonians, the local information decays exponentially along different directions, defining direction-dependent information length scales. In critical states with a Fermi surface, decay becomes algebraic, with dominant directions tied to the velocity of gapless modes. In the toric code, topological order manifests as a negative information value at the largest lattice scale, interpreted as global information redundancy between the smallest loop-enclosing overlapping subsystems. We also isolate the universal information carried by edge states and locate the signatures of non-Abelian fusion channels in position and scale. Our results establish the higher-dimensional information lattice as a framework for resolving universal correlation structures through the spatial and scale distribution of quantum information.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
Using a Krylov-subspace time evolution algorithm, we simulate the real-time dynamics of translation invariant non-integrable finite spin rings to quite long times with high accuracy. We systematically study the finite-size deviation between the resulting equilibrium state and the thermal state, and we highlight the importance of the energy variance on the deviations. We find that the deviations are well described by the eigenstate thermalization hypothesis, and that the von Neumann entropy correction scaling is the square of the local operator scaling. We reveal also an area law contribution to the relaxed von Neumann entropy, which we connect to the mutual information between the considered subsystem and its immediate environment. We also find that local observables relax towards equilibrium exponentially with a relaxation time scale that grows linearly with system length and is somewhat independent of the local operator, but depends strongly on the energy of the initial state, with the fastest relaxation times found towards one end of the overall energy spectrum. To contrast this behaviour we also study domain wall initial states, which exhibit clear diffusive behaviour, with a Thouless time scaling quadratically with the systems size, leading to a rather precise estimate for the diffusion constant for states in the vicinity of the middle of the energy spectrum.