Quasi-local soliton-gas mappings, ancestry algorithms, and sharp logarithmic operator-entanglement growth in interacting integrable quantum systems beyond the Rule 54 chain

The operator entanglement (operator-space entanglement entropy) of Heisenberg-picture local operators controls the efficiency of matrix-product-operator simulations of quantum many-body dynamics and has been proposed as a diagnostic distinguishing integrable from chaotic evolution. Alba, Dubail and Medenjak [Phys. Rev. Lett. 122, 250603 (2019)] proved that in the integrable Rule 54 cellular automaton the operator entanglement of every local operator grows at most logarithmically in time, by combining (a) a quasi-local mapping $M$ that conjugates the evolution operator into soliton-gas dynamics with (b) a solitonic algorithm deciding whether a quasiparticle descends from the operator's initial location. The open problem posed there is to extend this solvability to genuinely interacting, dispersive integrable models such as the spin-1/2 XXZ Heisenberg chain or the Lieb-Liniger gas: construct (exactly, or approximately with quantified locality) an analogous quasi-local mapping; devise an efficient ancestry algorithm for general soliton gases; and determine whether the logarithmic upper bound is tight and whether logarithmic growth for times shorter than the system size is universal for interacting integrable lattice models. Later work — numerical saturation of a $\frac{1}{2}\ln t$ bound in the XXZ chain, logarithmic temporal entanglement in interacting integrable circuits, a proof that Rule 54's late-time saturation stays logarithmic in system size, and heuristics that generic integrable models instead saturate at volume law — has refined but not resolved any strand: no quasi-local mapping, no general ancestry algorithm, and no matching lower bounds exist. The problem remains open.

Background

In a many-body quantum system, a local operator $O$ in the Heisenberg picture, $O(t)=e^{iHt}Oe^{-iHt}$, spreads over an expanding region of the lattice as time increases. The operator entanglement (also called operator-space entanglement entropy, OSE) quantifies the complexity of this spreading: one maps $O$ to a vector $|O\rangle$ in a doubled Hilbert space $\mathcal{H}\otimes\tilde{\mathcal{H}}$ by turning bras into kets, and computes the von Neumann or Rényi entanglement entropy of the normalized state $|O\rangle$ across a spatial bipartition. In the Rényi $\alpha\to 0$ limit this entropy is the logarithm of the minimal bond dimension of a matrix-product-operator (MPO) representation of $O(t)$ — an operator that is a tensor network with a bond dimension governing its compressibility — so slow operator-entanglement growth is the key to simulating long-time quantum dynamics with tensor networks. For systems that map onto free fermions, at most logarithmic OSE growth was found numerically by Prosen and Žnidarič, with analytic conformal-field-theory derivations developed subsequently; in contrast, chaotic systems are expected to show linear growth, as captured by the membrane picture of operator entanglement (a geometric description in which the entanglement grows like the area of an effective tension-loaded membrane anchored on the bipartition). Whether interacting integrable systems behave like the free or the chaotic case was, before 2019, accessible only through short-time numerics. The Rule 54 chain is a one-dimensional reversible cellular automaton on qubits, introduced by Bobenko, Bordemann, Gunn and Pinkall as an exactly integrable cellular automaton, and later developed — via a time-dependent matrix product ansatz by Klobas, Medenjak, Prosen and Vanicat and related works — into the simplest representative of interacting integrable quantum dynamics. Its configurations decompose exactly into two species of stable solitons (left and right movers) that scatter elastically with a nonzero time delay, making the model genuinely interacting in the quasiparticle sense while remaining exactly solvable. Alba, Dubail and Medenjak proved that the operator entanglement of every local operator in the Rule 54 chain grows at most logarithmically in time — the first rigorous result on operator entanglement in an interacting integrable system. The proof has two ingredients: (a) a quasi-local mapping $M$ — a mapping that is local up to correction tails decaying with distance — conjugating the evolution operator $U$ into the evolution of a soliton gas, $MUM^{\dagger}$, where a soliton gas is a classical gas of dispersing solitons with two-body scattering phase shifts — the structure underlying generalized hydrodynamics, the hydrodynamic theory of integrable systems based on their infinitely many conserved quasiparticle occupation numbers; and (b) a solitonic algorithm that decides, for any configuration at time $t$, whether a given soliton descends from the origin at time $t=0$ (its ancestry), efficiently enough that the operator-space decomposition of $O(t)$ contains only logarithmically many independent terms. The paper then poses the generalization to interacting integrable models with dispersive quasiparticles — quasiparticles whose velocities depend on their momenta, unlike the fixed-velocity solitons of Rule 54 — such as the spin-1/2 XXZ Heisenberg chain (a chain of qubits with anisotropic nearest-neighbour exchange interactions) or the Lieb-Liniger continuum gas (the exactly solved one-dimensional Bose gas with contact interactions), and the design of an ancestry algorithm for general soliton gases such as the one constructed by Doyon, Yoshimura and Caux; the questions of tightness (matching lower bounds) and of universality of logarithmic growth follow from the paper's having proven only an upper bound while presenting evidence for logarithmic growth in generic interacting integrable systems. Subsequent work both strengthened and refined this picture: numerics show a $\frac{1}{2}\ln t$ bound to be saturated for simple diagonal operators in Rule 54 and in the XXZ chain irrespective of anisotropy; temporal entanglement — the entanglement of the Feynman-Vernon influence matrix, a tensor encoding the effect of the remainder of the system on a local subsystem in time, which is the space-time dual of operator entanglement — was found to grow logarithmically in interacting integrable circuits by Giudice et al.; and Jacoby and Gopalakrishnan proved that the late-time ($t\gg L$) saturation of the operator entanglement stays at most logarithmic in system size $L$ for Rule 54, but argued heuristically and numerically that generic interacting integrable models such as the Heisenberg chain saturate at volume law (entanglement extensive in the number of lattice sites), because Rule 54 possesses only two quasiparticle species and hence only two possible phase-shift values. What is missing is precisely the source's program: a mechanism-level construction for a dispersive interacting integrable model, and the sharp asymptotic statements it should deliver.

Problem Statement

The problem, posed in the source paper's discussion, is to extend the exact operator-entanglement solvability established for the Rule 54 chain to interacting integrable quantum systems with dispersive quasiparticles. It comprises three jointly linked components serving one overarching objective — a general, mechanism-level theory of operator-entanglement growth in interacting integrable dynamics. First (quasi-local mapping): construct, for at least one genuinely interacting integrable model beyond the cellular-automaton class — meaning a regime with a nonzero two-body scattering phase shift, so that free-fermion-equivalent points such as the XXZ chain at zero anisotropy and the Lieb-Liniger gas at zero or infinite coupling do not qualify; the source's paradigm examples are the spin-1/2 XXZ Heisenberg chain and the Lieb-Liniger gas — a quasi-local mapping $M$ that conjugates the evolution into soliton-gas dynamics, exactly as in Rule 54, or an approximate version whose locality is quantified well enough to transfer the operator-space decomposition and yield controlled operator-entanglement bounds for local operators; or prove that no such sufficiently local mapping exists — either outcome discharges this first component, which by itself does not resolve the task. Second (ancestry algorithm): given a soliton gas describing a general integrable model (for instance the construction of Doyon, Yoshimura and Caux), devise an efficient algorithm, analogous to the Rule 54 solitonic algorithm, that decides the ancestry of quasiparticles — whether a marked quasiparticle observed at position $x$ at time $t$ originated from a given location at $t=0$; or establish that no such algorithm exists for general soliton gases — either outcome discharges this second component, which by itself does not resolve the task. Third (sharp asymptotics and universality): determine whether the at-most-logarithmic operator-entanglement growth is asymptotically tight — proving matching lower bounds or exact $\sim c\log t$ behavior in the target models — and whether logarithmic growth for times $t\ll L$ is a universal feature of all interacting integrable quantum lattice models or only of a subclass, in view of the evidence that late-time ($t\gg L$) saturation is volume-law for generic interacting integrable systems. The first two components are the two ingredients the source identifies as jointly sufficient for a general theoretical explanation of the logarithmic growth, and the third is the sharp asymptotic statement such a theory should deliver; a resolution may proceed by techniques other than the specific construction of $M$ and of the ancestry algorithm the source proposes, but it must discharge all three components — the two mechanism components as well as establishing rigorous two-sided control of operator-entanglement growth for at least one genuinely interacting dispersive integrable model and settling the universality question by a model-independent proof or by a controlled counterexample.

References

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