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arXiv:1305.0409 · New Journal of Physics

Detecting topological entanglement entropy in a lattice of quantum harmonic oscillators

Tommaso F. Demarie, Trond Linjordet, Nicolas C. Menicucci, Gavin K. Brennen

Published Aug 26, 2014• 20 citations
arXivHTMLPDF

Abstract

The Kitaev surface-code model is the most studied example of a topologically ordered phase and typically involves four-spin interactions on a two-dimensional surface. A universal signature of this phase is topological entanglement entropy (TEE), but due to low signal to noise, it is extremely difficult to observe in these systems, and one usually resorts to measuring anyonic statistics of excitations or non-local string operators to reveal the order. We describe a continuous-variable analog to the surface code using quantum harmonic oscillators on a two-dimensional lattice, which has the distinctive property of needing only two-body nearest-neighbor interactions for its creation. Though such a model is gapless, satisfies an area law, and the ground state can be simply prepared by measurements on a finitely squeezed and gapped two-dimensional cluster state, which does not have topological order. Asymptotically, the TEE grows linearly with the squeezing parameter, and we show that its mixed-state generalization, the topological mutual information, is robust to some forms of state preparation error and can be detected simply using single-mode quadrature measurements. Finally, we discuss scalable implementation of these methods using optical and circuit-QED technology.

DOI
10.1088/1367-2630/16/8/085011

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The Kitaev surface-code model is the most studied example of a topologically ordered phase and typically involves four-spin interactions on a two-dimensional surface. A universal signature of this phase is topological entanglement entropy (TEE), but due to low signal to noise, it is extremely difficult to observe in these systems, and one usually resorts to measuring anyonic statistics of excitations or non-local string operators to reveal the order. We describe a continuous-variable analog to the surface code using quantum harmonic oscillators on a two-dimensional lattice, which has the distinctive property of needing only two-body nearest-neighbor interactions for its creation. Though such a model is gapless, satisfies an area law, and the ground state can be simply prepared by measurements on a finitely squeezed and gapped two-dimensional cluster state, which does not have topological order. Asymptotically, the TEE grows linearly with the squeezing parameter, and we show that its mixed-state generalization, the topological mutual information, is robust to some forms of state preparation error and can be detected simply using single-mode quadrature measurements. Finally, we discuss scalable implementation of these methods using optical and circuit-QED technology.
To perform topological quantum computation we first need to experimentally realise anyons in a topological system. These systems are characterised by intriguing non-local quantum correlations that give rise to the anyonic statistics. What are the diagnostic tools we have to identify if a given system is indeed topological or not? Different phases of matter are characterised by their symmetries. This information is captured by order parameters. Usually, order parameters are defined in terms of local operators that can be measured in the laboratory. For example, the magnetisation of a spin system is given as the expectation value of a single spin with respect to the ground state. Such local properties can describe fascinating physical phenomena efficiently, such as ferromagnetism, and can pinpoint quantum phase transitions. But what about topological systems? Experimentally, we usually identify the topological character of systems, such as the fractional quantum Hall liquids, by probing the anyonic properties of their excitations (Miller et al. , 2007). However, topological order should be a characteristic of the ground state (Thouless et al. , 1982; Wen, 1995). The natural question arises: is it possible to identify a property of the ground state of a system that implies anyonic excitations? The theoretical background that made it possible to answer this question came from entropic considerations of simple topological models. Hamma et al . (2005) studied the entanglement entropy of the toric code ground state and noticed an unusual behaviour.
Topological entanglement entropy (TEE) is an efficient way to detect topological order in the ground state of gapped Hamiltonians. The seminal work of Kitaev and Preskill~\cite{preskill-kitaev-tee} and simultaneously by Levin and Wen~\cite{levin-wen-tee} proposed information quantities that can probe the TEE. In the present work, we explain why the subtraction schemes in the proposed information quantities~\cite{levin-wen-tee,preskill-kitaev-tee} work for the computation of TEE and generalize them for arbitrary number of subregions by explicitly noting the necessary conditions for an information quantity to capture TEE. Our conditions differentiate the probes defined by Kitaev-Preskill and Levin-Wen into separate classes. While there are infinitely many possible probes of TEE, we focus particularly on the cyclic quantities Q2n+1Q_{2n+1}Q2n+1​ and multi-information InI_nIn​. We also show that the holographic entropy inequalities are satisfied by the quantum entanglement entropy of the non-degenerate ground state of a topologically ordered two-dimensional medium with a mass gap.
Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the "spurious" topological entanglement entropy. We show that this spurious contribution is always non-negative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE that is invariant under constant-depth quantum circuits.
Abstract We use the topological entanglement entropy (TEE) as an efficient tool to fully characterize the Abelian phase of a Z 2 × Z 2 spin liquid emerging as the ground state of topological color code (TCC), which is a class of stabilizer states on the honeycomb lattice. We provide the fusion rules of the quasiparticle (QP) excitations of the model by introducing single- or two-body operators on physical spins for each fusion process which justify the corresponding fusion outcome. Besides this, we extract the TEE from Renyi entanglement entropy (EE) of the TCC, analytically and numerically by finite size exact diagonalization on the disk shape regions with contractible boundaries. We obtain that the EE has a local contribution, which scales linearly with the boundary length in addition to a topological term, i.e. the TEE, arising from the condensation of closed strings in the ground state. We further investigate the ground state dependence of the TEE on regions with non-contractible boundaries, i.e. by cutting the torus to half cylinders, from which we further identify multiple independent minimum entropy states (MES) of the TCC and then extract the U and S modular matrices of the system, which contain the self and mutual statistics of the anyonic QPs and fully characterize the topological phase of the TCC. Eventually, we show that, in spite of the lack of a local order parameter, TEE and other physical quantities obtained from the ground state wave function such as the entanglement spectrum (ES) and ground state fidelity are sensitive probes to study the robustness of a topological phase. We find that the topological order in the presence of a magnetic field persists until the vicinity of the transition point, where the TEE and fidelity drops to zero and the ES splits severely, signaling breakdown of the topological phase of the TCC.
Topological entanglement entropy (TEE) is a key diagnostic of topological order, allowing one to detect the presence of Abelian or non-Abelian anyons. However, there are currently no experimentally feasible protocols to measure TEE in condensed matter systems. Here, we propose a scheme to measure the TEE of chiral topological phases, carrying protected edge states, based on a nontrivial connection with the thermodynamic entropy change occurring in a quantum point contact (QPC) as it pinches off the topological liquid into two. We show how this entropy change can be extracted using Maxwell relations from charge detection of a nearby quantum dot. We demonstrate this explicitly for the Abelian Laughlin states, using an exact solution of the sine-Gordon model describing the universal crossover in the QPC. Our approach might open a new thermodynamic detection scheme of topological states also with non-Abelian statistics.
Topological entanglement entropy (TEE) provides a universal measure of long-range quantum correlations in topologically ordered systems, offering critical insights into exotic phases of matter. While theoretical frameworks, including exactly solvable lattice models, tensor networks, and field-theoretic approaches, enable precise computation of TEE in idealized settings, practical implementation faces multiple challenges. Finite-size and geometry effects, including boundary corners and lattice discretization, introduce significant corrections that can obscure the small topological contribution. Gapless edges in chiral topological phases further complicate the isolation of bulk entanglement, requiring careful separation of edge and bulk contributions. Symmetry-protected and symmetry-enriched phases necessitate additional diagnostics, such as symmetry-resolved entanglement and entanglement negativity, to fully characterize topological properties. Experimental measurement remains challenging, with current approaches limited to small systems using Rényi entropies via swap operations and interferometric methods. This work synthesizes these practical considerations, outlines mitigation strategies, and highlights ongoing directions for bridging theoretical, numerical, and experimental approaches to robustly quantify TEE. Understanding these limitations is essential for accurate characterization of topological order and for guiding the development of quantum technologies leveraging topological protection.