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.
While local unitary transformations are used for identifying quantum states that are in the same topological class, nonlocal unitary transformations are also important for studying the transition between different topological classes. In particular, it is an important task to find suitable nonlocal transformations that systematically sweep different topological classes. Here, regarding the role of dimension in the topological classes, we introduce partially local unitary transformations, namely Greenberger-Horne-Zeilinger (GHZ) disentanglers, which reduce the dimension of the initial topological model by a layer-by-layer disentangling mechanism. We justify the importance of such a mechanism for characterizing topological phases by considering two important examples. To this end, we apply GHZ disentanglers to two-dimensional (2D) topological quantum codes and show that they are converted to many copies of Kitaev's ladders. It implies that the above disentanglers cause a transition from an intrinsic topological phase to a symmetry-protected topological phase. Then, we show that while Kitaev's ladders are building blocks of both color code and toric code, there are different patterns of entangling ladders in 2D color code and toric code. It shows that different topological features of these topological codes are reflected in different patterns of entangling ladders. In this regard, we propose that the layer-by-layer disentangling mechanism can be used as a systematic method for classification of topological orders based on finding different patterns of long-range entanglement in topological lattice models.
Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.
As a quantum-informative window into quantum many-body physics, the concept and application of the entanglement renormalization group (ERG) have been playing a vital role in the study of novel quantum phases of matter, especially long-range entangled (LRE) states in topologically ordered systems. For instance, by recursively applying local unitaries as well as adding and removing qubits that form product states, the 2D toric code ground states, i.e., the fixed point of
Z2 topological order, are efficiently coarse-grained with respect to the system size. As a further improvement, the addition and removal of 2D toric codes in the ground states of the 3D X-cube model is shown to be indispensable and, remarkably, leads to well-defined fixed points of a large class of fracton orders that are non-liquid-like. Here, we present a substantially unified ERG framework in which general degrees of freedom are allowed to be recursively added or removed. Specifically, we establish an exotic hierarchy of ERG and LRE states in Pauli stabilizer codes, where the 2D toric code and 3D X-cube models are naturally included. In the hierarchy, LRE states like 3D X-cube and 3D toric code ground states can be added or removed in ERG processes of more complex LRE states. In this way, a large group of Pauli stabilizer codes are categorized into a series of state towers; within each tower, in addition to local unitaries including cnot (controlled-NOT) gates, lower LRE states of level-
n are added or removed in the level-
n ERG process of an upper LRE state of level
(n+1), connecting LRE states of different levels and unveiling complex relations among LRE states. As future directions, we expect this hierarchy can be applied to more general LRE states, leading to a unified ERG scenario of LRE states and exact tensor-network representations in the form of more generalized branching multiscale entanglement renormalization ansatz.
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.
We study the entanglement structure of topological orders subject to decoherence on the bipartition boundary. Focusing on the toric codes in
d space dimensions for
d=2,3,4, we explore whether the boundary decoherence may be able to induce a disentangling transition, characterized by the destruction of mixed-state long-range entanglement across the bipartition, measured by topological entanglement negativity. A key insight of our approach is the connection between the negativity spectrum of the decohered mixed states and emergent symmetry-protected topological orders under certain symmetry-preserving perturbation localized on the bipartition boundary. This insight allows us to analytically derive the exact results of entanglement negativity and characterize the universality class of the disentangling transition without using a replica trick.