We consider the transition between short-range entangled (SRE) and long-range entangled states of infinite quantum spin chains which is induced by local measurements. Specifically, we assume that the initial state is in a non-trivial symmetry-protected topological phase of a compact local symmetry group $\mathcal{G}$. We show that the on-site measurements of an appropriate local charge on intervals of increasing lengths transform the initial SRE state into a family of states with increasingly long-range correlations. In particular, the post-measurement states cannot be uniformly short-range entangled. In the case where the initial state is obtained from a product state using a quantum cellular automaton, we construct the infinite-volume post-measurement state and exhibit almost local observables that are maximally correlated.
Comments
v1 --> v2. Major generalization. The result applies to an arbitrary compact group. Additional minor changes
We consider the transition between short-range entangled (SRE) and long-range entangled states of infinite quantum spin chains which is induced by local measurements. Specifically, we assume that the initial state is in a non-trivial symmetry-protected topological phase of a compact local symmetry group G. We show that the on-site measurements of an appropriate local charge on intervals of increasing lengths transform the initial SRE state into a family of states with increasingly long-range correlations. In particular, the post-measurement states cannot be uniformly short-range entangled. In the case where the initial state is obtained from a product state using a quantum cellular automaton, we construct the infinite-volume post-measurement state and exhibit almost local observables that are maximally correlated.
Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.
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
Topological properties are associated with nonlocal entanglement and global properties. On the other hand, in some topological systems, it has been shown that quasilocal operators detect topological transitions. We show in this work that in the case of noninteracting long-range symmetry-protected topological models, energy bonds signal topological transitions. Interestingly, we also find that they display some signatures at the Berezinskii-Kosterlitz-Thouless transition that occurs in a spin chain with first and second neighbor interactions, if we consider the first excited state instead of the ground state of the system. Moreover, we show that the ground-state topological transition in the spin Kitaev model in a honeycomb lattice, which displays topological long-range order, is also detected by the energy bond correlator. Despite the model being interacting, in the ground state it reduces to a noninteracting model. Even for the spin-liquid phase of the two-dimensional Heisenberg model with first and second neighbor interactions, where the system has true long-range entanglement and topological order, the local bonds do signal the topological phase transition.
We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.
The formulation of quantum measurements compatible with relativistic causality is obstructed by the tension between the locality of observable algebras and the dynamical state transformations induced by measurement processes. We develop a local measurement theory for quantum cellular automata, lattices of finite-dimensional quantum systems with a causal reversible dynamics evolving in discrete time steps. Measurement processes are described by localized completely positive maps and organized into operational measurement protocols, finite non-adaptive sequences of local instruments distributed across the causal cone induced by the automaton local dynamics. Each protocol induces an effective measurement on the initial time slice, whose support is constrained by the causal structure generated by the automaton. Non-local measurement effects therefore arise through causal propagation and composition of local operations, rather than from primitive non-local interventions or instantaneous collapse prescriptions. We establish the corresponding spatial support bounds and characterize the resulting measurement architecture. As an explicit example, we construct a two-step protocol for a one-dimensional nearest-neighbour qubit automaton which, for suitable automaton parameters, realizes a Bell-basis projective measurement on two neighbouring sites after coarse-graining of the local outcomes. Away from this point, the same protocol generates a continuous family of deformed Bell measurements whose entanglement and stabilizer Rényi non-classicality can be quantified explicitly. The framework thus provides a causally consistent operator-algebraic mechanism for synthesizing spatially extended quantum measurements from strictly local operations.
Recent years have witnessed a surge of interest in performing measurements within topological phases of matter, e.g., symmetry-protected topological (SPT) phases and topological orders. Notably, measurements of certain SPT states have been known to be related to Kramers-Wannier duality and Jordan-Wigner transformations, giving rise to long-range entangled states and invertible phases, such as the Kitaev chain. Moreover, measurements of topologically ordered states correspond to charge condensations. In this work, we present a field-theoretic framework for describing measurements within topological field theories. We employ various lattice models as examples to illustrate the outcomes of measuring local symmetry operators within topological phases, demonstrating their agreement with the predictions from field-theoretic descriptions. We demonstrate that these measurements can lead to SPT, spontaneous symmetry-breaking, and topologically ordered phases. Specifically, when there is emergent symmetry after measurement, the remaining symmetry and emergent symmetry will have a mixed anomaly, which leads to long-ranged entanglement.