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arxiv: 2510.15766 · v3 · submitted 2025-10-17 · ❄️ cond-mat.str-el · cond-mat.stat-mech· hep-th· quant-ph

Subdimensional Entanglement Entropy: From Geometric-Topological Response to Mixed-State Holography

Meng-Yuan Li , Peng Ye This is my paper

Pith reviewed 2026-05-18 06:08 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechhep-thquant-ph
keywords subdimensional entanglement entropymixed-state symmetrytransparent composite symmetryholographyfracton ordertopological ordergeometric-topological responsestrong-weak symmetry breaking
0
0 comments X p. Extension

The pith

Each D-dimensional subdimensional entanglement subsystem holographically encodes a (D+1)-dimensional topological order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces subdimensional entanglement entropy measured on subdimensional entanglement subsystems as a probe of how geometry and topology shape quantum matter. Varying the dimension, shape, and topology of these subsystems produces distinct subleading entropy responses across cluster states, Z_q topological orders, and fracton orders. Treating the reduced density matrix on each subsystem as a mixed state supported on that manifold maps bulk stabilizers onto mixed-state symmetries and divides them into strong and weak classes. For subsystems with nontrivial entropy, weak symmetries act as transparent patch operators of the corresponding strong symmetries, which motivates a transparent composite symmetry that survives finite-depth circuits preserving the entropy and implies the holographic encoding of higher-dimensional topological order.

Core claim

For subdimensional entanglement subsystems with nontrivial subdimensional entanglement entropy, weak symmetries function as transparent patch operators for the corresponding strong symmetries. This gives rise to a transparent composite symmetry that is preserved by finite-depth quantum circuits maintaining the entropy. Consequently, each D-dimensional SES holographically encodes a (D+1)-dimensional topological order.

What carries the argument

The correspondence between bulk stabilizers and mixed-state symmetries on SESs, which classifies them into strong and weak classes, with weak symmetries acting as transparent patch operators for strong ones.

If this is right

  • The subleading term of SEE responds differently across cluster states, Z_q orders, and fracton orders when SES dimension and geometry are varied.
  • Strong-to-weak spontaneous symmetry breaking is identified inside the mixed states defined on the SES.
  • Transparent composite symmetry remains robust under finite-depth quantum circuits that preserve SEE.
  • Each D-dimensional SES holographically encodes a (D+1)-dimensional topological order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same correspondence may supply a practical route to detect higher-dimensional topology by measuring entanglement on accessible lower-dimensional cuts in experiments.
  • The construction suggests that mixed-state symmetry classifications could be extended to other submanifold geometries beyond those explicitly treated.
  • Numerical simulations of larger stabilizer codes could directly test whether the transparent composite symmetry survives under local perturbations that preserve SEE.

Load-bearing premise

The reduced density matrix of an SES can be treated as a many-body mixed state supported on the SES manifold.

What would settle it

A calculation in a concrete model such as the toric code or a fracton Hamiltonian where a finite-depth circuit that preserves SEE nevertheless breaks the proposed transparent composite symmetry.

Figures

Figures reproduced from arXiv: 2510.15766 by Meng-Yuan Li, Peng Ye.

Figure 1
Figure 1. Figure 1: FIG. 1. Representative Hamiltonians and SESs of the (a,d) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Representative stabilizers and SESs of (a,b) 3D toric [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. TCS of a flat, noncontractible 2D closed membrane [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We introduce the subdimensional entanglement entropy (SEE), defined on subdimensional entanglement subsystems (SESs) embedded in the bulk, as an entanglement-based probe of how geometry and topology jointly shape universal properties of quantum matter. By varying the dimension, geometry, and topology of the SES, we show that the subleading term of SEE exhibits sharply distinct responses in different phases, including cluster states, $\mathbb{Z}_q$ topological orders, and fracton orders. Treating the reduced density matrix of an SES as a many-body mixed state supported on the SES manifold, we further establish a general correspondence between bulk stabilizers and mixed-state symmetries on SESs, separating them into strong and weak classes, and use it to identify strong-to-weak spontaneous symmetry breaking within SESs. Finally, for SESs with nontrivial SEE, we show that weak symmetries act as transparent patch operators of the corresponding strong symmetries. This motivates the notion of transparent composite symmetry, which remains robust under finite-depth quantum circuits that preserve SEE, and implies that each $D$-dimensional SES holographically encodes a $(D+1)$-dimensional topological order. These results establish SEE not only as a sharp probe of geometric-topological response, but also as a route from bulk pure-state entanglement to mixed-state symmetry and holography on subdimensional manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces subdimensional entanglement entropy (SEE) defined on subdimensional entanglement subsystems (SESs) embedded in the bulk as an entanglement-based probe of geometric-topological responses in quantum matter. By varying the dimension, geometry, and topology of SESs, it shows that the subleading term of SEE exhibits distinct responses in phases including cluster states, Z_q topological orders, and fracton orders. Treating the SES reduced density matrix as a many-body mixed state on the SES manifold, the work establishes a correspondence between bulk stabilizers and mixed-state symmetries, classifying them into strong and weak classes and identifying strong-to-weak spontaneous symmetry breaking. For SESs with nontrivial SEE, weak symmetries act as transparent patch operators for the corresponding strong symmetries; this motivates the notion of transparent composite symmetry, which remains robust under finite-depth quantum circuits preserving SEE and implies that each D-dimensional SES holographically encodes a (D+1)-dimensional topological order.

Significance. If the results hold, the manuscript provides a new entanglement diagnostic linking geometry, topology, and mixed-state symmetries in subdimensional structures, with potential relevance to fracton phases and holographic dualities in condensed-matter settings. Explicit demonstrations of SEE responses across known phases and the introduction of transparent composite symmetry are constructive contributions. The proposed route from bulk pure-state entanglement to mixed-state holography via symmetry robustness could open avenues for subdimensional probes, provided the encoding is made rigorous.

major comments (2)
  1. [Abstract and section on mixed-state holography / transparent composite symmetry] The step from the robustness of transparent composite symmetry under SEE-preserving finite-depth quantum circuits to the claim that each D-dimensional SES holographically encodes a (D+1)-dimensional topological order (as stated in the abstract and the final section) lacks an explicit encoding map or verification that full topological data such as anyon content, braiding phases, or ground-state degeneracy appear in SES observables. Invariance of the composite symmetry alone does not automatically establish the correspondence to higher-dimensional topological order.
  2. [Section establishing bulk-stabilizer to mixed-state symmetry correspondence] The foundational assumption that the reduced density matrix of an SES can be treated as a many-body mixed state supported on the SES manifold, which enables the bulk-stabilizer correspondence and the separation into strong and weak symmetries, requires additional justification regarding the precise support and operator algebra on the subdimensional manifold; this underpins the subsequent definitions of transparent patch operators and strong-to-weak breaking.
minor comments (2)
  1. [Introduction / early results section] Add an explicit example or figure illustrating the embedding of an SES in a simple lattice model (e.g., the cluster state) to clarify the geometric construction before presenting the SEE responses.
  2. [Section on geometric-topological responses] Specify the precise form of the subleading term of SEE (e.g., constant, logarithmic, or area-law correction) for each phase examined, and include a table comparing the responses across cluster, Z_q, and fracton orders.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and have made revisions to improve clarity and justification where the concerns are valid.

read point-by-point responses

  1. Referee: [Abstract and section on mixed-state holography / transparent composite symmetry] The step from the robustness of transparent composite symmetry under SEE-preserving finite-depth quantum circuits to the claim that each D-dimensional SES holographically encodes a (D+1)-dimensional topological order (as stated in the abstract and the final section) lacks an explicit encoding map or verification that full topological data such as anyon content, braiding phases, or ground-state degeneracy appear in SES observables. Invariance of the composite symmetry alone does not automatically establish the correspondence to higher-dimensional topological order.

    Authors: We agree that invariance of the transparent composite symmetry under SEE-preserving circuits does not by itself constitute a complete holographic encoding with all topological data. The manuscript motivates the claim through the correspondence between bulk stabilizers and the strong/weak symmetry structure on the SES, where the transparent patch operators capture essential features analogous to topological order protection. In the revised manuscript we have modified the abstract and final section to state the implication more precisely as a symmetry-based correspondence rather than a full encoding of anyon content or braiding phases. We have added a clarifying paragraph noting that explicit verification of ground-state degeneracy or braiding statistics on SES observables would require further operator-algebra analysis and is left for future work. This addresses the concern without overstating the current results. revision: partial

  2. Referee: [Section establishing bulk-stabilizer to mixed-state symmetry correspondence] The foundational assumption that the reduced density matrix of an SES can be treated as a many-body mixed state supported on the SES manifold, which enables the bulk-stabilizer correspondence and the separation into strong and weak symmetries, requires additional justification regarding the precise support and operator algebra on the subdimensional manifold; this underpins the subsequent definitions of transparent patch operators and strong-to-weak breaking.

    Authors: We accept that the treatment of the SES reduced density matrix as a mixed state on the subdimensional manifold requires explicit justification. In the revised manuscript we have inserted a new explanatory paragraph immediately preceding the symmetry correspondence section. This paragraph defines the support as the geometric submanifold of the SES and specifies that the operator algebra is obtained by restricting the bulk stabilizer generators to this submanifold, thereby inducing strong symmetries (non-localizable on proper subsets) and weak symmetries (patch operators). This added justification directly supports the subsequent definitions of transparent patch operators and strong-to-weak spontaneous symmetry breaking. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation generates independent distinctions from definitions

full rationale

The paper defines SEE on SESs of varying dimension/geometry/topology and computes its subleading term to exhibit distinct responses across cluster states, Z_q TOs, and fracton orders. It then treats the SES reduced density matrix as a mixed state on the SES manifold to establish a bulk-stabilizer to mixed-state symmetry correspondence, partitioning symmetries into strong and weak classes and identifying strong-to-weak SSB. For nontrivial SEE it shows weak symmetries act as transparent patch operators of strong symmetries, motivating a transparent composite symmetry that is invariant under SEE-preserving FDQCs; the holographic-encoding claim is presented as a consequence of this invariance rather than a redefinition of the input symmetries or a fitted parameter. No load-bearing step reduces by construction to prior inputs, self-citations, or ansatzes; the constructions produce new symmetry classifications and response distinctions that are not tautological with the initial definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on newly introduced definitions and a correspondence between bulk stabilizers and mixed-state symmetries whose validity is asserted but not derived in the abstract.

axioms (1)
  • standard math Standard quantum-information definitions of reduced density matrix and entanglement entropy apply to subdimensional subsystems.
    Invoked to define SEE on SESs.
invented entities (2)
  • Subdimensional entanglement entropy (SEE) no independent evidence
    purpose: Entanglement-based probe of geometric-topological response
    Newly defined quantity whose subleading term is claimed to distinguish phases.
  • Transparent composite symmetry no independent evidence
    purpose: Symmetry that remains robust under SEE-preserving circuits and encodes holographic information
    Introduced to link weak symmetries to strong ones and to the holographic claim.

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