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Subdimensional entanglement entropy: from geometric-topological response to entanglement-induced mixed-state landscape

Meng-Yuan Li, Peng Ye

TL;DR

Subdimensional entanglement entropy (SEE) provides a unified framework to disentangle geometric and topological contributions by evaluating entropy on subdimensional entanglement subsystems (SESs) embedded in the bulk. SEE, via $S_A = \alpha|A| + \zeta(A)$, reveals geometric (gSEE) and topological (tSEE) responses across cluster states, $\mathbb{Z}_q$ topological orders, and fracton orders, and its mixed-state interpretation exposes strong and weak symmetries on SESs that are organized into transparent composite symmetries (TCS) with t-patch operators. A key insight is that SESs with nonzero SEE holographically encode a $(D+1)$-dimensional topological order, enabling a holographic perspective that links stabilizer structure, mixed-state symmetries, and topological holography. The results establish a robust, circuit-insensitive (under finite-depth circuits) framework for diagnosing and classifying complex quantum phases, including SSPTs and fracton orders, and point to broad future applications in numerical studies and higher-dimensional code states.

Abstract

A growing body of examples reveals that topology and geometry are deeply intertwined in shaping universal properties of quantum matter, as seen in fracton orders with size-dependent topological ground-state degeneracy and in cluster states with spurious topological entanglement entropy. We introduce the \textit{subdimensional entanglement entropy} (SEE), defined on \textit{subdimensional entanglement subsystems} (SESs) embedded in the bulk, as a direct probe of this intertwinement. By virtually tuning SES dimension, geometry and topology, the subleading components of SEE show sharply distinct geometric and topological responses in cluster states, $\mathbb{Z}_q$ topological orders, and fracton orders. Further viewing the SES reduced density matrix as a mixed state on the SES manifold, we establish an exact correspondence between stabilizers and mixed-state symmetries, distinguishing \textit{strong} from \textit{weak} classes. Nontrivial SEE enforces weak symmetries to act as \textit{transparent patch operators} of strong ones, together forming \textit{transparent composite symmetries} that are robust against finite depth quantum circuits. Due to the presence of transparent patch operators, each $D$-dimensional SES holographically encodes a $(D+1)$-dimensional topological order. Thus, SEE simultaneously accesses various forefront directions, providing a unified and versatile nexus linking phenomena previously regarded as distinct.

Subdimensional entanglement entropy: from geometric-topological response to entanglement-induced mixed-state landscape

TL;DR

Subdimensional entanglement entropy (SEE) provides a unified framework to disentangle geometric and topological contributions by evaluating entropy on subdimensional entanglement subsystems (SESs) embedded in the bulk. SEE, via , reveals geometric (gSEE) and topological (tSEE) responses across cluster states, topological orders, and fracton orders, and its mixed-state interpretation exposes strong and weak symmetries on SESs that are organized into transparent composite symmetries (TCS) with t-patch operators. A key insight is that SESs with nonzero SEE holographically encode a -dimensional topological order, enabling a holographic perspective that links stabilizer structure, mixed-state symmetries, and topological holography. The results establish a robust, circuit-insensitive (under finite-depth circuits) framework for diagnosing and classifying complex quantum phases, including SSPTs and fracton orders, and point to broad future applications in numerical studies and higher-dimensional code states.

Abstract

A growing body of examples reveals that topology and geometry are deeply intertwined in shaping universal properties of quantum matter, as seen in fracton orders with size-dependent topological ground-state degeneracy and in cluster states with spurious topological entanglement entropy. We introduce the \textit{subdimensional entanglement entropy} (SEE), defined on \textit{subdimensional entanglement subsystems} (SESs) embedded in the bulk, as a direct probe of this intertwinement. By virtually tuning SES dimension, geometry and topology, the subleading components of SEE show sharply distinct geometric and topological responses in cluster states, topological orders, and fracton orders. Further viewing the SES reduced density matrix as a mixed state on the SES manifold, we establish an exact correspondence between stabilizers and mixed-state symmetries, distinguishing \textit{strong} from \textit{weak} classes. Nontrivial SEE enforces weak symmetries to act as \textit{transparent patch operators} of strong ones, together forming \textit{transparent composite symmetries} that are robust against finite depth quantum circuits. Due to the presence of transparent patch operators, each -dimensional SES holographically encodes a -dimensional topological order. Thus, SEE simultaneously accesses various forefront directions, providing a unified and versatile nexus linking phenomena previously regarded as distinct.
Paper Structure (20 sections, 2 theorems, 50 equations, 11 figures, 1 table)

This paper contains 20 sections, 2 theorems, 50 equations, 11 figures, 1 table.

Table of Contents

  1. Review of von Neumann entanglement entropy in stabilizer states
  2. Computation details of SEE
  3. Example 1: subsystem cat state
  4. Example 2: 2D cluster state
  5. Example 3: $[[15,1,3]]$ 3D color code
  6. Example 4: 2D $\mathbb{Z}_q$ topological orders
  7. Example 5: 3D toric code model
  8. Example 6: Decoupled intersecting 2D toric code stack (ITCS) model
  9. Example 7: X-cube model
  10. Proof of strong and weak symmetries of SESs in Pauli stabilizer states(Theorem 1)
  11. Review of t-patch operators
  12. Basic examples of t-patch operators
  13. The first example: 1D spin-$\frac{1}{2}$ chain with global $\mathbb{Z}_{2}$ symmetry
  14. The second example: 2D spin-$\frac{1}{2}$ model on square lattice with global $\mathbb{Z}_{2}$ symmetry
  15. The third example: 2D spin-$\frac{1}{2}$ model on square lattice with $\mathbb{Z}_{2}$ $1$-form symmetry
  16. ...and 5 more sections

Key Result

Theorem 1

Given a Pauli stabilizer state $|\psi\rangle$ and an entanglement subsystem $A$ composed of a set of spins, any stabilizer fully supported on $A$ is a strong symmetry of $\rho_A$, while any stabilizer partially supported on $A$ is a weak symmetry of $\rho_A$ (where the symmetry transformation is the

Figures (11)

  • Figure 1: Representative Hamiltonians and SESs of the (a,d) 2D plaquette Ising model, (b,e) 2D cluster-state model, and (c,f) 2D $\mathbb{Z}_q$ toric code. Panels (a–c) show representative Hamiltonian terms; the remaining terms follow by lattice translations under PBC. Panels (d–f) illustrate SESs with nonzero (orange) and zero (purple) subleading terms $\zeta(A)$.
  • Figure 2: Representative stabilizers and SESs of (a,b) 3D toric code and (c,d) X-cube model. Panels (a,c) show representative local stabilizers, where qubits reside on plaquettes and links, respectively. In (a) and (c), Pauli-$X$ and Pauli-$Z$ stabilizers are colored red and blue and denoted by $A$ and $B$ terms (see SM supp for details). Panels (b,d) illustrate SESs with nonzero (orange) and zero (purple) $\zeta(A)$. Panel (d) highlights a planar contractible loop SES whose $\zeta(A)$ simultaneously exhibits tSEE and gSEE contributions.
  • Figure 3: TCS of a flat, noncontractible 2D closed membrane SES $A$ in the X-cube model. Panels (a) and (b) show a portion of $A$. The operators generating the strong and weak symmetries are depicted in (a) and (b), respectively, together constituting the TCS of this SES.
  • Figure S1: Hamiltonian and linear subsystem symmetries of 2D plaquette Ising model. In (a), we present a typical Hamiltonian term $B_i$. In (b), we present a generator of linear subsystem symmetries, that is the product of Pauli $X$ operators along a straight line. In cat state, such symmetry generators are also used as nonlocal stabilizers to specify the cat state as the unique ground state.
  • Figure S2: SES in 2D plaquette Ising model. (a) and (b) respectively show a straight line SES along direction $x$ and a straight diagonal line SES. In (c), we present an SES that is regards as a closed loop. In (d), we present a discrete rectangle fully embedded in the loop in (c) that contributes a subleading entanglement entropy term. In (e), we present a line at angle $\theta$ to $x$-axis. In (f), we present a special case where $L=6$ and $tan(\theta)=3/2$, such that $A$ reduces to an array of sites.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem
  • proof