Meng-Yuan Li of Sun Yat-sen University and Peng Ye have introduced a new measure, subdimensional entanglement entropy (SEE), to investigate how the geometry and topology of quantum matter jointly influence its universal properties. The researchers define SEE as an entanglement-based probe applied to embedded within a material, allowing them to distinguish geometric and topological contributions to a system’s behavior.

Their work demonstrates that the subleading term of SEE exhibits in phases like cluster states, Z_q topological orders and fracton orders, revealing a connection between bulk entanglement and mixed-state symmetry on these subdimensional manifolds. These results establish SEE as a tool to explore both geometric-topological responses and a potential route to holography on subdimensional manifolds.

Subdimensional Entanglement Entropy: Probing Geometry and Topology

This approach establishes a direct link between how a material’s shape and connectivity influence its fundamental quantum properties, moving beyond traditional classifications based solely on topology. A key finding centers on the relationship between bulk stabilizers and the symmetries present within these SESs, categorizing them as either “strong” or “weak” classes. This classification unveils a potential mechanism for “strong-to-weak spontaneous symmetry breaking” occurring within the subsystems themselves, a phenomenon previously difficult to observe directly.

Researchers found that a correspondence exists between these symmetries, with weak symmetries acting as for their stronger counterparts. This observation led to the proposal of a robust form of symmetry preserved by specific quantum circuits that maintain SEE values. The study suggests a surprising holographic connection; a D-dimensional SES may, in effect, encode a (D + 1)-dimensional topological order.

By treating the reduced density matrix of an SES as a mixed state existing on the SES manifold, the team established a bridge between pure-state entanglement in the bulk material and mixed-state physics in lower dimensions. “Surprisingly, this perspective allows us to study mixed-state symmetries on SESs while varying the dimension, geometry, and topology of the underlying submanifold,” the authors state, highlighting the potential for exploring complex quantum phenomena through this novel entanglement-based framework. This approach offers a new avenue for understanding how geometry and topology jointly shape universal properties of quantum matter.

SEE Calculation in Stabilizer States: Cluster, Z_q, and Fracton Orders

Researchers demonstrated these variations by altering the dimension, geometry and topology of subdimensional entanglement subsystems (SESs) to probe the underlying physics. The analysis focused on stabilizer states, providing a common framework for examining these diverse phases and yielding insights into their properties. The volume of a subdimensional SES is equivalent to its surface measure, which is an important geometric detail for the calculations. Beyond the specific phases examined, the research suggests avenues for extending SEE calculations to more complex scenarios.

Future work will explore SEE in systems beyond stabilizer states, including fracton phases and cluster states subjected to external fields, using numerical methods like tensor-network wavefunctions and quantum Monte Carlo simulations. Analytical exploration within each phase’s theoretical framework will require further investigation, and free-fermion studies on fractal and hyperbolic lattices offer additional opportunities to examine geometry-dependent entanglement scaling.

A key aspect of this work lies in the ability of SEE to categorize symmetries within these subsystems, separating them into “strong” and “weak” classes based on their relationship to bulk stabilizers and mixed-state symmetries.

Defining Subdimensional Entanglement Subsystems (SESs) and Their Volume

The newly defined subdimensional entanglement entropy (SEE) functions as a probe into the interplay between geometry and topology in quantum materials, distinguishing itself from conventional entanglement diagnostics by focusing on subdimensional entanglement subsystems (SESs). These SESs, unlike typical bulk or skeletal subsets, possess well-defined intrinsic dimensions and codimensions, allowing for controlled variation of their geometric and topological attributes to directly examine responses. This approach allows for a systematic analysis of subdimensional manifolds across diverse quantum phases, revealing structures previously hidden from standard entanglement measurements.

By varying the dimension, geometry and topology of the SES, the work shows that the subleading term of SEE exhibits sharply distinct responses in a series of stabilizer states representing fixed points of different phases, including cluster states, $Z_{q}$ topological orders and fracton orders. A set of weak symmetries acts as transparent patch operators of the corresponding strong symmetries.

This motivates the notion of transparent composite symmetry, which remains robust under bipartition-factorized finite-depth circuits that preserve SEE, and implies that a D-dimensional SES may holographically encode 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.

Reduced Density Matrix: Mixed-State Physics on SES Manifolds

Subdimensional entanglement entropy (SEE) offers a novel approach to understanding quantum many-body states by focusing on information theory rather than conventional geometric or topological responses. Unlike methods that require deforming entire spacetime manifolds, SEE analyzes entanglement data associated with subdimensional entanglement subsystems (SESs) of varying dimensions, geometries and topologies, providing a more structured diagnostic tool.

This framework moves beyond simply refining entanglement diagnostics, establishing a direct link between pure-state entanglement and mixed-state physics, including symmetry structures and topological holography on SESs. The researchers found that SEE provides a unifying framework connecting stabilizer codes, topological holography and mixed-state symmetries.

Strong and Weak Symmetries Originating from Bulk Stabilizers

Stabilizer states reveal a direct link between their internal structure and the symmetries observed in the mixed states of subdimensional entanglement subsystems (SESs), according to new work establishing a correspondence between bulk stabilizers and mixed-state symmetries. Researchers found that stabilizers fully contained within an SES generate strong symmetries, while those partially contained generate weak symmetries, a distinction formalized in Theorem 1. The analysis extends to open quantum systems, where symmetries are traditionally classified as strong or weak based on their effect on mixed states.

A strong symmetry transforms a state without altering it beyond a phase factor, while a weak symmetry leaves the state invariant without such a restriction. These composite symmetries remain stable even under specific quantum circuit operations, bipartition-factorized finite-depth circuits that preserve subdimensional entanglement entropy (SEE).

Explicit calculations across a series of SESs with nontrivial SEE confirm these findings, demonstrating that the strong and weak symmetries form these transparent composite symmetries. The paper states, detailing the derivation of mixed-state symmetries from stabilizers and proving Theorem 1. The team’s work establishes a rigorous connection between stabilizers and mixed-state symmetries, offering a new framework for understanding the interplay between geometry, topology and quantum entanglement.

Spontaneous Symmetry Breaking Within Subdimensional Entanglement Systems

Calculations across various stabilizer states, including those exhibiting spontaneous subsystem symmetry breaking, topological order, SSPT order and fracton order, demonstrate that SEE resolves these differences, offering a refined probe of universal quantum properties. Researchers found that these symmetry classes are not static; within subdimensional entanglement subsystems (SESs) possessing nontrivial SEE, weak symmetries act as of their corresponding strong symmetries. The team’s work establishes a general relation between SESs and mixed-state symmetries in stabilizer states, and uses it to uncover strong-to-weak spontaneous symmetry breaking (SW-SSB) in SESs with nontrivial SEE.

“By systematically analyzing subdimensional manifolds in a range of quantum phases, we show that SEE resolves structures that are invisible to conventional entanglement diagnostics,” the paper states. The findings illuminate the rich structure of SEE and associated SESs from a distinct perspective, potentially opening new avenues for understanding and manipulating complex quantum systems.

Transparent Composite Symmetries: Robustness via SEE Preservation

Subdimensional entanglement subsystems (SESs) exhibiting nontrivial subdimensional entanglement entropy (SEE), weak symmetries function as transparent patch operators of corresponding strong symmetries, establishing a novel form of symmetry interaction. This correspondence is preserved even when subjected to bipartition-factorized finite-depth circuits that maintain SEE, a key indicator of robustness. The resulting combination of strong and weak symmetries defines what researchers term transparent composite symmetries (TCSs). The formation of these transparent composite symmetries has significant implications for understanding how information is encoded in quantum systems.

The paper demonstrates that a mixed state residing on a D-dimensional SES can holographically encode a topological order in one higher dimension, a (D+1)-dimensional space. This suggests a deeper connection between the geometry of lower-dimensional systems and the topological properties of their higher-dimensional counterparts, potentially offering new avenues for designing robust quantum materials.

Holographic Encoding: D-Dimensional SES to (D+1)-Dimensional Order

A newly identified relationship allows researchers to connect the symmetries within lower-dimensional systems to the topological order of one dimension higher. This correspondence remains stable even when subjected to specific circuits designed to preserve SEE, suggesting a robust mechanism for holographic encoding. The algebra generated by the mixed-state symmetries of these SESs aligns with a framework for t-patch operators, local and extended symmetric operators, previously developed to study topological holography. This interplay defines what they term transparent composite symmetry, a combined symmetry that encodes topological order.

👉 More information
🗞 Subdimensional entanglement entropy: From geometric-topological response to entanglement-induced mixed-state holography
✍️ Meng-Yuan Li and Peng Ye
🧠 DOI: http://link.aps.org/doi/10.1103/kds6-fvp4

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