Holographic Codes and Bulk Renormalization-group Flows Enable Construction of Minimal-size ‘IR’ Codes with Seed States

The quest to understand the relationship between gravity and quantum information receives a significant boost from new work exploring how information is encoded and transformed in holographic systems. Xi Dong, Donald Marolf from the University of California, Santa Barbara, and Pratik Rath from the University of California, Berkeley, and colleagues investigate how holographic error correcting codes behave when subjected to a process mimicking the way physical systems simplify at different scales. Their research demonstrates a direct link between the mathematical tools used to describe these codes and the established ideas of renormalization group flow, a cornerstone of modern physics. This connection provides a novel perspective on the arguments proposed by Susskind and Uglum, and importantly, the team explicitly constructs simplified ‘infrared’ codes from more complex ‘ultraviolet’ codes, guaranteeing the preservation of key quantum information, offering a deeper understanding of how gravity emerges from quantum mechanics.

Holographic Codes and Information Recovery Frameworks

This work explores the construction and analysis of holographic codes within the framework of AdS/CFT, a duality between gravity and quantum field theory. The central goal is to develop a robust framework for understanding how information is encoded and recovered in these systems, particularly when standard holographic tools become limited due to quantum effects. The authors are essentially creating a quantum error correction scheme for gravity, demonstrating that information falling into a black hole, or encoded in a gravitational system, can be reliably recovered despite quantum fluctuations. They achieve this by constructing codes that map information between the bulk (gravity) and the boundary (field theory) and back again.

Holographic codes provide a way to encode information in a gravitational system (the bulk) and decode it on the boundary. The Ryu-Takayanagi formula connects entanglement entropy in the field theory to the area of surfaces in the bulk, while entanglement entropy quantifies the amount of information encoded in a region of space. This research addresses scenarios where the reduced density matrix, describing a subsystem, has very small eigenvalues, which can cause problems for standard holographic techniques. The team’s code construction filters out these problematic eigenvalues, enabling a more accurate description of information recovery.

The research also explores the trade-offs between code size, the number of degrees of freedom needed to encode information, and the ability to correct errors. They demonstrate that reducing code size is possible, even if it introduces some errors in the reconstruction process. The work confirms that the constructed code is consistent with the expected behavior of the boundary field theory, allowing for the recovery of information about the boundary. Furthermore, the team investigates the relationship between modular flows, describing changes in the modular Hamiltonian, and the JLMS formula, which connects modular flows in the bulk and boundary theories.

They demonstrate how their code construction can be used to derive this formula. Key results include a novel approach to building holographic codes directly in the bulk gravitational theory, a method for handling small eigenvalues of the reduced density matrix, an analysis of the trade-offs between code size and error correction, confirmation of consistency with the boundary field theory, and a derivation of the JLMS formula using the code construction. In essence, this research provides a way to protect information from being lost, even in the presence of quantum fluctuations and imperfections, by building a code entirely within the gravitational system and confirming its consistency with the boundary theory.

Holographic Codes Transform Under Renormalization Flows

This research presents a breakthrough in understanding how holographic error-correcting codes behave under transformations analogous to renormalization group flows, crucial for exploring the deep connection between gravity and quantum information. Scientists have explicitly constructed a method to move between ‘UV’ and ‘IR’ codes, effectively changing the scale at which information is encoded in a holographic system, while preserving key properties of the original code. The core achievement lies in demonstrating how to truncate a large ‘UV’ Hilbert space, representing many degrees of freedom, into a smaller ‘IR’ Hilbert space, maintaining the ability to recover information and ensuring a flat entanglement spectrum, a characteristic of well-behaved holographic codes. The team began by establishing a framework for constructing ‘IR’ codes from a given set of ‘seed’ states present in the ‘UV’ code, guaranteeing that these states remain within the reduced ‘IR’ Hilbert space.

This construction ensures exact two-sided recovery, meaning information can be reliably retrieved from both sides of the holographic boundary, and maintains the flat entanglement spectrum, vital for consistency with the AdS/CFT correspondence. Crucially, the method doesn’t simply assume the existence of related codes, but actively builds them, starting from a defined ‘UV’ state and constructing the corresponding ‘IR’ representation. The research demonstrates that this transformation preserves the essential properties of the holographic code, even as the underlying degrees of freedom are reduced. This is achieved by carefully selecting states within the ‘IR’ code based on eigenvalue windows of the boundary modular Hamiltonian, effectively ‘chopping’ the seed state into pieces that fit within the reduced Hilbert space. The scientists have shown that this process maintains the ability to accurately map information between the bulk and boundary of the holographic system, even as the scale of the description changes. This work provides a powerful new tool for investigating the emergence of spacetime and the quantum properties of black holes, paving the way for future studies of modular flow and addressing challenges related to small eigenvalues in the density matrix of bulk subregions.

Holographic Codes Evolve Under Renormalization Flow

This research establishes a method for understanding how holographic error correcting codes, a key concept linking gravity and quantum information, change under renormalization group flow, a process describing how physical systems evolve at different scales. The team demonstrates that these codes can be systematically ‘coarse-grained’, effectively reducing the complexity of the system while preserving crucial quantum information. Importantly, they achieve this by explicitly constructing new, simplified codes from initial ‘seed’ states, ensuring these states remain identifiable within the reduced system. The work extends previous investigations by considering scenarios where the underlying quantum structures possess non-trivial centers, allowing for a more nuanced treatment of gravitational effects.

The researchers also maintain a flat entanglement spectrum throughout this ‘coarse-graining’ process, a characteristic vital for holographic codes. This systematic approach moves beyond simply assuming the existence of related codes, instead providing a concrete method for generating them. The authors acknowledge that their current construction focuses on specific scenarios and does not fully address all possible complexities of renormalization group flow in holographic systems. Future work will explore the application of this method to study modular flow, a concept related to the symmetries of quantum systems, and to address limitations encountered in previous related calculations.