Topological entanglement entropy meets holographic entropy inequalities

This contribution elucidates the mechanism behind subtraction schemes for topological entanglement entropy, derives necessary conditions for arbitrary subregion probes to detect topological order, and demonstrates that holographic entropy inequalities hold for the ground states of gapped two-dimensional topologically ordered systems.

The Gist

The Big Picture: Finding the "Hidden Shape" of the Universe

Imagine a piece of fabric. If you look at the surface, you see patterns, colors, and textures. But what if the fabric underneath has a hidden shape – like a knot or a hole – that you cannot recognize just by looking at the surface? In physics, certain materials (so-called "topological phases") possess these hidden shapes. They are special because their properties do not change even if you stretch or squeeze the material, as long as you do not tear it.

Physicists want to find a way to "see" these hidden shapes without tearing the fabric apart. One way to do this is to measure the entanglement entropy. Think of entanglement as a measure of how strongly two pieces of fabric are "connected" or "tangled" with each other.

Normally, this measurement depends on the size of the piece being examined (how much surface area it has). However, hidden within this measurement is a tiny, constant "correction." This correction is called Topological Entanglement Entropy (TEE). It is like a secret code that reveals the hidden shape of the fabric to you, regardless of how large or small the piece is.

The Problem: How to Isolate the Secret Code

The paper begins by examining two famous methods (developed by Kitaev/Preskill and by Levin/Wen) that attempt to isolate this secret code. They use a "subtraction scheme."

The Analogy: Imagine you are trying to hear a whisper (the TEE) in a noisy room. The noise is the "surface" of the fabric.

• Method A says: "Take three pieces of fabric, measure the noise in each of them, and subtract them in a specific way so that the noise cancels out and only the whisper remains."
• Method B says: "Take a different arrangement of three pieces and subtract them in another way to isolate the whisper."

The authors ask: Are there other ways to perform this subtraction? Can we use more than three pieces? And what rules must these subtraction methods follow so that they actually work?

The Solution: Borrowing Ideas from "Holograms"

The authors decided to adopt ideas from a field called Holography. In physics, a hologram is a 2D surface that contains all the information about a 3D object. There are strict mathematical rules (so-called holographic entropy inequalities) that govern how information is exchanged in these holographic systems.

The paper establishes a surprising connection: The rules that govern holograms also apply to these topological materials.

Here is what they discovered:

1. The "Superbalanced" Rule: They found that the subtraction method must be "superbalanced" to successfully isolate the secret code (TEE).

   • Analogy: Imagine a scale. If you place weights on the left side, you must place exactly the same total weight on the right side to keep it balanced. "Superbalanced" means, however, that it is not only balanced for the entire scale, but also for every single small group of weights you select.
   • If a subtraction method is "superbalanced," it automatically cancels out all the noise (surface) and leaves you with the whisper (the topological code).

2. New Measurement Methods: Based on this rule, the authors showed that many different combinations of fabric pieces (not just three) can be used to find the TEE. As long as the mathematics is "superbalanced," it works. They proved this using a mathematical tool called Topological Quantum Field Theory (TQFT), which acts like a rulebook for the behavior of these special fabrics.

3. The "Holographic" Connection: They proved that for these special materials, the "holographic rules" (which were thought to apply only to black holes and gravity) are actually obeyed. This means that the way information is entangled in these materials is highly ordered and follows the same strict laws as the holographic universe.

The Two Types of "Detectors"

The paper classifies the tools used to find this hidden shape into two categories:

• Fixed-topological Probes: These are the "superbalanced" tools. They work regardless of how you arrange the fabric pieces, as long as the overall shape (the topology) remains the same. They are robust and reliable.
• Fixed-geometric Probes: These are tools that only work if you arrange the fabric in a very specific, rigid shape. If you change the shape slightly, they no longer work. The authors show that the famous "Levin-Wen" method falls into this category – it is somewhat more fragile.

The Conclusion

In simple words, this paper says:

• We have a new, generalized method to find the hidden "shape" of special materials.
• The key is to use subtraction methods that are "superbalanced" (perfectly balanced in every possible way).
• These materials follow the same strict mathematical rules as holograms, which is a big surprise and a powerful new tool for physicists.
• By using these rules, we can create many new "detectors" to discover topological order, which is a crucial step toward better quantum computers in the future (although the paper focuses on the mathematics and not on building the computers themselves).

The authors have essentially developed a universal "filter" that can remove the noise of size and shape to reveal the pure, hidden topological nature of the material.

Technical Summary

Technical Summary: Topological Entanglement Entropy Meets Holographic Entropy Inequalities

Problem Statement
Topological Entanglement Entropy (TEE), denoted as $\gamma$, is a universal constant in the area law of von Neumann (VN) entanglement entropy for gapped ground states ($S = \alpha L - \gamma + \dots$). It serves as a diagnostic for topological order and encodes non-local features of the ground state. Historically, TEE has been defined via specific subtraction schemes proposed by Kitaev and Preskill (KP) and by Levin and Wen (LW). These schemes utilize specific tripartite information quantities (related to the monogamy of mutual information or strong subadditivity) to eliminate boundary contributions. However, there is no consensus on the precise definition of TEE, and it remains an open question whether other information quantities can serve as equivalent definitions or what general properties such quantities must possess. Furthermore, the relationship between these topological probes and the broader class of holographic entropy inequalities (HEIs) derived from the AdS/CFT correspondence requires clarification in the context of topological phases.

Methodology
The authors employ a topological quantum field theory (TQFT) framework to analyze entanglement entropy on a two-dimensional, disk-like geometry. Key methodological steps include:

1. TQFT Calculation: Instead of relying solely on the area law approach, the authors perform explicit TQFT calculations. They utilize the technique of "stitching" a time-reversed (TR) copy of the system with the original and introducing punctures at the interfaces of subregions. This maps the VN entropy of subregions to the entropy of a sphere hosting $n$ punctures containing anyons.
2. Generalization of Subtraction Schemes: The study generalizes subtraction schemes to an arbitrary number of subregions ($n$) and arbitrary information quantities. They analyze cyclic information quantities ($Q_{2n+1}$) and multi-information ($I_n$).
3. Analysis of the Holographic Entropy Cone (HEC): The authors investigate whether the facet inequalities of the holographic entropy cone (HEC), which constrain the entanglement entropy of holographic states, are satisfied by the ground states of two-dimensional topologically ordered systems. They distinguish between "facet" inequalities (tight constraints) and non-facet inequalities.
4. Balance Criteria: The paper introduces and utilizes the concepts of "balanced" and "superbalanced" (2-balanced) information quantities. An information quantity is superbalanced if every $k$-cluster formation of subregions (for $k=1, 2$) occurs with equal positive and negative coefficients.

Main Contributions and Results

1. Generalized Framework for TEE: The authors propose a generalized framework for defining TEE based on information quantities satisfying specific conditions. They demonstrate that any 2-balanced (superbalanced) information quantity $Q$ with a negative sum of coefficients constitutes a valid probe for TEE on a two-dimensional disk geometry.
2. Equivalence of Definitions: The work classifies existing TEE definitions into two classes:
   • Topology-fixed Probes: Quantities such as the cyclic inequalities $Q_{2n+1}$ and multi-information $I_n$ (for $n \ge 3$) prove to be topological, independent of the specific geometric arrangement of subregions, provided the topology (disk) remains unchanged. These quantities yield $Q = -c \log D$, where $D$ is the total quantum dimension and $c$ is the sum of the coefficients.
   • Geometry-fixed Probes: Quantities such as the LW definition (conditional mutual information) prove to be topological only for specific geometric configurations (e.g., specific connectivity of regions) and may fail for other geometries (e.g., disconnected regions).
3. Validity of Holographic Inequalities: The paper proves that the quantum entanglement entropy of the unique (non-degenerate) ground state of a two-dimensional topologically ordered medium with a mass gap satisfies all facet HEIs of the holographic entropy cone.
   • The authors show that for $n \ge 3$, all facet HEIs on the two-dimensional disk are topological and non-negative.
   • Specifically, it is shown that the multi-information $I_n$ in this context is sign-definite for $n \ge 3$ ($I_n = -\log D$), contrasting with its sign-indefinite nature in general holographic systems.
4. Superbalancedness as a Condition: The authors put forward a proposition (supported by heuristic proofs and anyonic derivations) that every 2-balanced information quantity is topological with a sign-indefinite sign. Combined with a negative sum of coefficients, such quantities become sign-definite probes for TEE.

Significance and Claims
The paper claims to provide a unified understanding of TEE by directly linking it to the structure of holographic entropy inequalities.

• Unification: It demonstrates that the subtraction schemes used by Kitaev-Preskill and Levin-Wen are specific instances of a broader class of superbalanced information quantities that naturally eliminate boundary terms in TQFT.
• New Probes: The work suggests that any superbalanced inequality with a negative coefficient sum can be used to extract TEE ($\gamma = \log D$) from a gapped system, offering a systematic way to generate new TEE probes beyond traditional tripartite definitions.
• Constraints on Ground States: The results imply that the space of entanglement entropy vectors for unique ground states of gapped two-dimensional Hamiltonians is contained within the holographic entropy cone. This points to a deep structural connection between topological order and holographic states, although the latter represent a distinct subset of quantum mechanical states.
• Robustness: The authors note that their results hold for non-degenerate ground states on a disk. They acknowledge that "spurious" contributions (occurring in symmetry-protected topological phases) or degenerate ground states (on higher-genus manifolds) could alter these inequalities; this remains a topic reserved for future discussion.

The paper concludes that the holographic entropy cone offers a robust set of constraints and probes for topological order, with multi-information $I_n$ and cyclic inequalities serving as primary examples of quantities that successfully capture the topological entanglement entropy $\gamma = \log D$.