Northeastern & Brookhaven Labs Derive Maximum Distance for Holographic Codes

Researchers at Northeastern University and Brookhaven National Laboratory have derived the maximum distance within novel combinatorial holographic quantum secret sharing (CHQSS) schemes, characterizing how logical information of a bulk subregion can be encoded in the boundary and protected from erasures of boundary subregions. This work leverages the connection between gravity, as described by Anti-de Sitter space (AdS3), and quantum information theory via Conformal Field Theory (CFT2), utilizing the AdS3/CFT2 correspondence as its framework. The study identifies multiple distinct phase transition points in multipartite entanglement wedges within these schemes, revealing a complex relationship between information encoding and quantum entanglement. These findings demonstrate the ability to construct CHQSS schemes, including both perfect threshold and non-threshold varieties, and define key parameters for characterizing their performance.

Combinatorial Holographic Quantum Secret Sharing Introduction

The very structure of spacetime may hold the key to unbreakable data security, according to new research examining a novel approach to quantum secret sharing. This framework, linking Anti-de Sitter space (AdS3) to Conformal Field Theory (CFT2), allows researchers to study how information is encoded and protected in a way previously unexplored. The work, available on July 17, 2026, introduces key metrics for evaluating CHQSS schemes, notably deriving the maximum distance, a quantifiable limit on how widely information can be distributed while still guaranteeing reliable reconstruction. This is not simply an observed limit, but a mathematically derived value, suggesting a precise understanding of the system’s capabilities. The paper reports that “We present the phase transitions of multipartite entanglement wedges in a symmetric setup and observe multiple distinct phase transition points.” This approach differs from traditional quantum secret sharing by framing the problem within a holographic context, reminiscent of bulk reconstruction.

The team explains that an access structure, defining which combinations of shares can reconstruct a secret, can be naturally defined by entanglement wedge reconstruction, as studied in Gottesman (2000). The study highlights the differences between pure-state and mixed-state holographic encoding, particularly concerning redundancy and how reconstruction works in different holographic phases. The authors state that “The main goal of this paper is to characterize the quality of holographic encoding in terms of quantum secret sharing and to understand the difference between pure-state and mixed-state holographic encoding,” suggesting a path toward more robust and secure quantum communication protocols.

These transitions are not isolated shifts, but rather fundamental changes in how information is encoded and accessed. Crucially, the team’s analysis extends beyond simple reconstruction to consider redundancy, the ability of the system to maintain information integrity even when portions of the boundary are inaccessible.

While quantum error correction focuses on protecting encoded information from disturbances, holographic quantum secret sharing (HQSS) examines how readily bulk logical information is encoded onto the boundary of a holographic system, and how easily it can be reconstructed. The study introduces key metrics to characterize these schemes, including a “distance” representing how far information can be “spread” and thresholds for reconstruction and secrecy. Notably, the researchers derive the maximum distance, rather than simply observing a limit. The work further clarifies the difference between pure and mixed-state holographic encoding. Specifically, the team found differences between pure and mixed states in holographic encoding, and that redundancy in holographic encoding, the ability to reconstruct information even after partial erasure, varies significantly depending on the holographic phase and connectivity of entanglement wedges. For instance, bulk logical information can be reconstructed from certain combinations of boundary subregions, but not from individual ones, describing how reconstruction works in different holographic phases.

The work reveals fundamental differences in how effectively information can be secured depending on whether the system operates in a “pure-state” or “mixed-state” configuration. Unlike traditional methods, this approach leverages the AdS3/CFT2 correspondence, linking gravity to quantum information to frame the problem of information encoding. The distinction between pure and mixed states hinges on access to the boundary of the holographic system. A pure-state scheme allows access to the entire boundary, offering numerous reconstruction possibilities for bulk logical information, protected unless the entanglement wedge of an erased boundary subregion contains the target region. Conversely, mixed-state QSS restricts access to a portion of the boundary, making reconstruction dependent on the shape and configuration of entanglement wedges. The team observed “multiple distinct phase transition points” within these mixed-state schemes, indicating a complex interplay between entanglement and information encoding.

A crucial element of their investigation is the derivation of the maximum distance within these CHQSS schemes. The researchers found that this distance, along with associated “thresholds,” are dependent on the specific holographic phase and the chosen bulk subregion being examined. They found that in certain configurations, bulk logical information remains reconstructible even after the erasure of boundary subregions, while in others, access to multiple boundary regions is required for any recovery.