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arXiv:1306.5711 · Physical Review A

Entanglement negativity and topological order

Yirun Arthur Lee, Guifré Vidal

Published Oct 16, 2013• 130 citations
arXivHTMLPDF

Abstract

We use the entanglement negativity, a measure of entanglement for mixed states, to probe the structure of entanglement in the ground state of a topologically ordered system. Through analytical calculations of the negativity in the ground state(s) of the toric code model, we explicitly show that the pure-state entanglement of a region $A$ and its complement $B$ is the sum of two types of contributions: boundary entanglement and long-range entanglement. Boundary entanglement is seen to be insensitive to tracing out the degrees of freedom in the interior of regions $A$ and $B$, and therefore it entangles only degrees of freedom in $A$ and $B$ that are close to their common boundary. We recover the well-known result that boundary entanglement is proportional to the size of each boundary separating $A$ and $B$ and it includes an additive, universal correction. The second, long-range, contribution to pure-state entanglement appears only when $A$ and $B$ are noncontractible regions (e.g., on a torus) and it is seen to be destroyed when tracing out a noncontractible region in the interior of $A$ or $B$. In the toric code, only the long-range contribution to the entanglement depends on the specific ground state under consideration.

DOI
10.1103/physreva.88.042318

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We use the entanglement negativity, a measure of entanglement for mixed states, to probe the structure of entanglement in the ground state of a topologically ordered system. Through analytical calculations of the negativity in the ground state(s) of the toric code model, we explicitly show that the pure-state entanglement of a region AAA and its complement BBB is the sum of two types of contributions: boundary entanglement and long-range entanglement. Boundary entanglement is seen to be insensitive to tracing out the degrees of freedom in the interior of regions AAA and BBB, and therefore it entangles only degrees of freedom in AAA and BBB that are close to their common boundary. We recover the well-known result that boundary entanglement is proportional to the size of each boundary separating AAA and BBB and it includes an additive, universal correction. The second, long-range, contribution to pure-state entanglement appears only when AAA and BBB are noncontractible regions (e.g., on a torus) and it is seen to be destroyed when tracing out a noncontractible region in the interior of AAA or BBB. In the toric code, only the long-range contribution to the entanglement depends on the specific ground state under consideration.
We study the entanglement structure of topological orders subject to decoherence on the bipartition boundary. Focusing on the toric codes in ddd space dimensions for d=2,3,4d=2,3,4d=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.
In this paper we study the behavior of the entanglement measure dubbed negativity in the context of the toric code model. Using a replica method introduced recently by Calabrese, Cardy, and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the nonlocal correlations present in a topologically ordered state reflect in the behavior of the negativity of the system. We find that the negativity has a leading area-law contribution if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically nontrivial in both directions on a torus. We further confirm by explicit calculation that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered eight-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.
Here we show 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, for the topological universality class of the toric code model. As happens for the entanglement entropy, we find that for large block sizes the geometric entanglement is, up to possible subleading corrections, the sum of two contributions: a non-universal bulk contribution obeying a boundary law times the number of blocks, and a universal contribution quantifying the underlying pattern of long-range entanglement of a topologically-ordered state.
We study how the color code under decoherence gives rise to an intrinsic mixed-state topological order (imTO), which has no counterpart in pure ground states of local gapped Hamiltonians. For decoherence induced by XX-type operators on red links of the honeycomb lattice, we show that the resulting mixed state inherits half of the topological properties of the color code, including anyon content, logical operators, and topological entanglement structure. Using a gauging procedure for mixed stabilizer states, we identify the emergent phase as closely related to a single toric code. We characterize this phase by topological entanglement negativity (TEN) and perform efficient stabilizer-formalism simulations. While the pure color code has TEN=2ln⁡2{\rm TEN} = 2 \ln 2TEN=2ln2, the maximally decohered state has TEN=ln⁡2{\rm TEN} = \ln 2TEN=ln2, indicating emergence of a single toric code. By tuning the decoherence strength, we find a smooth crossover in TEN accompanied by a pronounced, nearly system-size-independent peak in its variance. We further show that the negativity exhibits characteristic scaling only for subsystem partitions commensurate with the triangular lattice of the emergent toric code. Our results demonstrate that negativity-based quantities provide powerful probes of mixed-state topological order generated by decoherence.
We propose a diagnostic for finite temperature topological order using "topological entanglement negativity," the long-range component of a mixed-state entanglement measure. As a demonstration, we study the toric code model in d spatial dimensions for d=2,3,4, and find that when topological order survives thermal fluctuations, it possesses a nonzero topological entanglement negativity, whose value is equal to the topological entanglement entropy at zero temperature. Furthermore, we show that the Gibbs state of 2D and 3D toric code at any nonzero temperature, and that of 4D toric code above a certain critical temperature, can be expressed as a convex combination of short-range entangled pure states, consistent with the absence of topological order.
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.