Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.27053

Resolving spurious topological entanglement entropy in stabilizer codes

Peilun Han, Zijian Liang, Yifei Wang, Bowen Yang, Yingfei Gu, Yu-An Chen

quant-phcond-mat.str-elmath.QASubmitted Apr 29, 2026• Announced May 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Topological entanglement entropy (TEE) is a key diagnostic of long-range entanglement in two-dimensional gapped phases of matter, but it can suffer from spurious contributions that overestimate the total quantum dimension of the underlying topological order. In this work, we identify the microscopic origin of spurious TEE and introduce a concave partition for computing the Levin-Wen TEE of translation-invariant stabilizer codes of prime-dimensional qudits. We rigorously prove that this prescription is free of spurious contributions. As a complementary probe, we study bivariate bicycle codes on a bipartite cylinder and show that the entanglement entropy depends sensitively on the cylinder circumference, revealing topological frustration of the underlying anyons.

Comments
7+46 pages, 15 figures, 1 table

Related papers

Universal Lower Bound on Topological Entanglement Entropy

Isaac H. Kim, Michael Levin, Ting-Chun Lin +2 more

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.

arXiv:2302.00689•Physical Review Letters•Oct 19, 2023•23 citations

Spurious long-range entanglement and replica correlation length

Liujun Zou, Jeongwan Haah

Topological entanglement entropy has been regarded as a smoking-gun signature of topological order in two dimensions, capturing the total quantum dimension of the topological particle content. An extrapolation method on cylinders has been used frequently to measure the topological entanglement entropy. Here, we show that a class of short-range entangled 2D states, when put on an infinite cylinder of circumference $L$, exhibits the entanglement R\'enyi entropy of any integer index $\ensuremath{\alpha}\ensuremath{\ge}2$ that obeys ${S}_{\ensuremath{\alpha}}=aL\ensuremath{-}\ensuremath{\gamma}$, where $a,\ensuremath{\gamma}>0$. Under the extrapolation method, the subleading term $\ensuremath{\gamma}$ would be identified as the topological entanglement entropy, which is spurious. A nonzero $\ensuremath{\gamma}$ is always present if the 2D state reduces to a certain symmetry-protected topological 1D state, upon disentangling spins that are far from the entanglement cut. The internal symmetry that stabilizes $\ensuremath{\gamma}>0$ is not necessarily a symmetry of the 2D state, but should be present after the disentangling reduction. If the symmetry is absent, $\ensuremath{\gamma}$ decays exponentially in $L$ with a characteristic length, termed as a replica correlation length, which can be arbitrarily large compared to the two-point correlation length of the 2D state. We propose a simple numerical procedure to measure the replica correlation length through replica correlation functions. We also calculate the replica correlation functions for representative wave functions of Abelian discrete gauge theories and the double semion theory in 2D, to show that they decay abruptly to zero. This supports a conjecture that the replica correlation length being small implies that the subleading term from the extrapolation method determines the total quantum dimension.

arXiv:1604.06101•Physical review. B./Physical review. B•Aug 25, 2016•74 citations

Topological entanglement entropy meets holographic entropy inequalities

Joydeep Naskar, Sai Satyam Samal

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 $Q_{2n+1}$ and multi-information $I_n$. 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.

arXiv:2412.05484•Physical review. D/Physical review. D.•Jun 25, 2026•0 citations

Accuracy of Topological Entanglement Entropy on Finite Cylinders

Hong‐Chen Jiang, Rajiv R. P. Singh, Leon Balents

Topological phases are unique states of matter which support nonlocal excitations which behave as particles with fractional statistics. A universal characterization of gapped topological phases is provided by the topological entanglement entropy (TEE). We study the finite size corrections to the TEE by focusing on systems with a Z2 topological ordered state using density-matrix renormalization group and perturbative series expansions. We find that extrapolations of the TEE based on the Renyi entropies with a Renyi index of n≥2 suffer from much larger finite size corrections than do extrapolations based on the von Neumann entropy. In particular, when the circumference of the cylinder is about ten times the correlation length, the TEE obtained using von Neumann entropy has an error of order 10(-3), while for Renyi entropies it can even exceed 40%. We discuss the relevance of these findings to previous and future searches for topological ordered phases, including quantum spin liquids.

arXiv:1304.0780•Physical Review Letters•Sep 5, 2013•18 citations

Entanglement Entropy as a Probe for Topological Order in Quantum Systems

Dr. Rajendra Kumar, Dr. Aram Singh

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.

IOSR Journal of Applied Physics•Apr 1, 2025•0 citations

Entanglement entropy of gapped phases and topological order in three dimensions

Tarun Grover, Ari M. Turner, Ashvin Vishwanath

We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an ``entropy density'' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size-independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the ``zeroth Betti number''). In $D>3$ dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in $D\ensuremath{\ge}3$.

arXiv:1108.4038•Physical Review B•Nov 23, 2011•136 citations
Topological entanglement entropy (TEE) is a key diagnostic of long-range entanglement in two-dimensional gapped phases of matter, but it can suffer from spurious contributions that overestimate the total quantum dimension of the underlying topological order. In this work, we identify the microscopic origin of spurious TEE and introduce a concave partition for computing the Levin-Wen TEE of translation-invariant stabilizer codes of prime-dimensional qudits. We rigorously prove that this prescription is free of spurious contributions. As a complementary probe, we study bivariate bicycle codes on a bipartite cylinder and show that the entanglement entropy depends sensitively on the cylinder circumference, revealing topological frustration of the underlying anyons.
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.
Topological entanglement entropy has been regarded as a smoking-gun signature of topological order in two dimensions, capturing the total quantum dimension of the topological particle content. An extrapolation method on cylinders has been used frequently to measure the topological entanglement entropy. Here, we show that a class of short-range entangled 2D states, when put on an infinite cylinder of circumference LLL, exhibits the entanglement R\'enyi entropy of any integer index $\ensuremath{\alpha}\ensuremath{\ge}2$ that obeys ${S}_{\ensuremath{\alpha}}=aL\ensuremath{-}\ensuremath{\gamma}$, where $a,\ensuremath{\gamma}>0$. Under the extrapolation method, the subleading term $\ensuremath{\gamma}$ would be identified as the topological entanglement entropy, which is spurious. A nonzero $\ensuremath{\gamma}$ is always present if the 2D state reduces to a certain symmetry-protected topological 1D state, upon disentangling spins that are far from the entanglement cut. The internal symmetry that stabilizes $\ensuremath{\gamma}>0$ is not necessarily a symmetry of the 2D state, but should be present after the disentangling reduction. If the symmetry is absent, $\ensuremath{\gamma}$ decays exponentially in LLL with a characteristic length, termed as a replica correlation length, which can be arbitrarily large compared to the two-point correlation length of the 2D state. We propose a simple numerical procedure to measure the replica correlation length through replica correlation functions. We also calculate the replica correlation functions for representative wave functions of Abelian discrete gauge theories and the double semion theory in 2D, to show that they decay abruptly to zero. This supports a conjecture that the replica correlation length being small implies that the subleading term from the extrapolation method determines the total quantum dimension.
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.
Topological phases are unique states of matter which support nonlocal excitations which behave as particles with fractional statistics. A universal characterization of gapped topological phases is provided by the topological entanglement entropy (TEE). We study the finite size corrections to the TEE by focusing on systems with a Z2 topological ordered state using density-matrix renormalization group and perturbative series expansions. We find that extrapolations of the TEE based on the Renyi entropies with a Renyi index of n≥2 suffer from much larger finite size corrections than do extrapolations based on the von Neumann entropy. In particular, when the circumference of the cylinder is about ten times the correlation length, the TEE obtained using von Neumann entropy has an error of order 10(-3), while for Renyi entropies it can even exceed 40%. We discuss the relevance of these findings to previous and future searches for topological ordered phases, including quantum spin liquids.
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.
We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an ``entropy density'' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size-independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the ``zeroth Betti number''). In D>3D>3D>3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in $D\ensuremath{\ge}3$.