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arXiv:2411.05077

Towards a Topological Proof of the Strong Subadditivity

Chih-Yu Lo, Po-Yao Chang

Published Nov 7, 2024• 0 citations
arXivHTMLPDF

Abstract

Topological entanglement entropy (TEE) represents an intrinsic contribution to the entanglement entropy (EE) in topologically ordered systems. In quantum information theory, strong subadditivity (SSA) is a fundamental property of EE, reflecting the non-negativity of conditional mutual information. TEE was originally believed to be a universal correction to the area law of EE, suggesting that its SSA would directly follow from the SSA of EE. However, due to spurious contributions, the correction term is not universal; consequently, the value predicted by topological quantum field theory (TQFT) provides only a lower bound. In this work, we present a topological analysis showing that the SSA for TEE is equivalent to a specific inequality within the TQFT framework. We further verify that this inequality holds for all known unitary modular tensor categories (UMTCs) up to rank 11, supporting the conjecture that SSA holds universally in the TQFT framework. Conversely, assuming the validity of the SSA condition, the inequality can be interpreted as a consistency criterion for candidate UMTCs.

DOI
10.48550/arxiv.2411.05077

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Topological entanglement entropy (TEE) represents an intrinsic contribution to the entanglement entropy (EE) in topologically ordered systems. In quantum information theory, strong subadditivity (SSA) is a fundamental property of EE, reflecting the non-negativity of conditional mutual information. TEE was originally believed to be a universal correction to the area law of EE, suggesting that its SSA would directly follow from the SSA of EE. However, due to spurious contributions, the correction term is not universal; consequently, the value predicted by topological quantum field theory (TQFT) provides only a lower bound. In this work, we present a topological analysis showing that the SSA for TEE is equivalent to a specific inequality within the TQFT framework. We further verify that this inequality holds for all known unitary modular tensor categories (UMTCs) up to rank 11, supporting the conjecture that SSA holds universally in the TQFT framework. Conversely, assuming the validity of the SSA condition, the inequality can be interpreted as a consistency criterion for candidate UMTCs.
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.
We classify canonical Rényi strong subadditivity on an unpunctured torus for finite unitary modular categories. For all pure logical states and admissible region pairs, validity at 0<alpha<1 is equivalent to Abelianity; at alpha=1 it is universal. Each finite non-Abelian group G has a pure untwisted D(G) fixed-point ground state with an order-two violation under ordinary edge partial traces. A specified local-filter D(S_3) family changes reduced spectra; its fixed-cut deficit tends to the negative fixed-point value over a small size-independent interval in a controlled joint limit.
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 derive the property of strong superadditivity of mutual information arising from the Markov property of the vacuum state in a conformal field theory and strong subadditivity of entanglement entropy. We show this inequality encodes unitarity bounds for different types of fields. These unitarity bounds are precisely the ones that saturate for free fields. This has a natural explanation in terms of the possibility of localizing algebras on null surfaces. A particular continuity property of mutual information characterizes free fields from the entropic point of view. We derive a general formula for the leading long distance term of the mutual information for regions of arbitrary shape which involves the modular flow of these regions. We obtain the general form of this leading term for two spheres with arbitrary orientations in spacetime, and for primary fields of any tensor representation. For free fields we further obtain the explicit form of the leading term for arbitrary regions with boundaries on null cones.
Abstract We derive the property of strong superadditivity of mutual information arising from the Markov property of the vacuum state in a conformal field theory and strong subadditivity of entanglement entropy. We show this inequality encodes unitarity bounds for different types of fields. These unitarity bounds are precisely the ones that saturate for free fields. This has a natural explanation in terms of the possibility of localizing algebras on null surfaces. A particular continuity property of mutual information characterizes free fields from the entropic point of view. We derive a general formula for the leading long distance term of the mutual information for regions of arbitrary shape which involves the modular flow of these regions. We obtain the general form of this leading term for two spheres with arbitrary orientations in spacetime, and for primary fields of any tensor representation. For free fields we further obtain the explicit form of the leading term for arbitrary regions with boundaries on null cones.
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.