Abstract The nature of the famed long-range entanglement of quantum many-body systems is the paradoxical coexistence of short-range correlation and nonlocal information which cannot be encoded through constant-depth local quantum circuits. This coexistence is only proved in topologically ordered states and their extensions in 2D or higher dimensions, and is in charge of their fundamental importance in fault-tolerant quantum computation. Here, we broaden the knowledge of long-range entangled states beyond the manifold-topology picture by proving the coexistence in a new concrete example in addition to those well-known ones like the toric-code ground state. The new long-range entangled state reveals an intrinsically different paradigm, it describes quantum matter on the newly experimentally realized fractal lattice geometry in 1.58D where the topological framework is negated, but a self-similarity structure shapes how the nonlocal information is encoded. The new paradigm portrays an exotic quantum order and uncovers the potential of fractional-dimension quantum states in studies relevant to quantum computation.
Abstract The nature of the famed long-range entanglement of quantum many-body systems is the paradoxical coexistence of short-range correlation and nonlocal information which cannot be encoded through constant-depth local quantum circuits. This coexistence is only proved in topologically ordered states and their extensions in 2D or higher dimensions, and is in charge of their fundamental importance in fault-tolerant quantum computation. Here, we broaden the knowledge of long-range entangled states beyond the manifold-topology picture by proving the coexistence in a new concrete example in addition to those well-known ones like the toric-code ground state. The new long-range entangled state reveals an intrinsically different paradigm, it describes quantum matter on the newly experimentally realized fractal lattice geometry in 1.58D where the topological framework is negated, but a self-similarity structure shapes how the nonlocal information is encoded. The new paradigm portrays an exotic quantum order and uncovers the potential of fractional-dimension quantum states in studies relevant to quantum computation.
The essence of the famed long-range entanglement as revealed in topologically ordered state is the paradoxical coexistence of short-range correlation and nonlocal information that cannot be removed through constant-depth local quantum circuits. Its realization in different quantum states is a focus research topic in both quantum computation and quantum matter. However, the proved realizations are subject to the paradigm of topological order (including its extensions), i.e. via a quantum code structure with macroscopic code distance. Here, we broaden the knowledge of long-range entangled states by rigorously proving the coexistence in a new concrete state. The state describes qudits on the newly experimentally discovered fractal lattice geometry (1.58D) on which the quantum code structure has been shown not to exist, i.e., there is no topological order. Our result might reveal a new paradigm for the realization of the long-range entanglement in many-body quantum states, and might stimulate new studies connecting quantum information and quantum matter.
A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z 2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G × H symmetry gives rise to anomalous LRE upon measuring G -charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way. Published by the American Physical Society 2024
We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.
Abstract Toric Code, a cornerstone of the topological order, admits numerous realizations on different planar graphs. We prove that all these realizations share a single, universal entanglement structure in a sense that any Toric Code state can be generated by applying a fixed set of non-local unitaries between similar quasi-one-dimensional quantum ladders. In particular, we consider Toric Code states corresponding to various planar graphs and apply non-local dientanglers to qubits corresponding to non-contractible cycles that satisfy a topological constraint. We demonstrate that, independent of the geometry of the underlying graph, disentanglers convert Toric Code states into a tensor product of Kitaev's Ladder states. This exact and graph-independent construction reveals that the long-range entanglement characterizing the Toric Code's topological phase has a precise decomposable structure in a sense that it arises entirely from a specific pattern of entanglement between lower-dimensional subsystems. This structural insight opens new avenues for classification and simulation of topological orders.
Topological states of matter, such as quantum Hall states, are characterized by the absence of local order parameters. In the continuum, their topological order has been reinterpreted as the condensation of nonlocal composite bosons, but this condensation and the resulting long-range correlations have so far eluded direct confirmation beyond the simplest trial states. Here, we apply tensor-network methods to lattice Hamiltonians hosting quantum Hall states, revealing this exotic order directly in interacting ground states and demonstrating both the long-range correlations and the condensation of composite bosons. These correlations define a nonlocal order parameter that identifies phase transitions between trivial and topological phases, for both bosonic and fermionic models. The corresponding operators are accessible through site-resolved, local measurements on an extensive number of particles, making them directly probable in quantum simulators. Our work opens new avenues for microscopic studies of topological quantum matter beyond the reach of traditional solid-state approaches.
We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.