Mapping Entropy to Reality: Holographic Models and the Limits of Entanglement

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Author: Denis Avetisyan

New research demonstrates a fundamental link between the structure of entanglement entropy and holographic graph models, revealing which entropy profiles can be physically realized.

This paper establishes necessary and sufficient conditions for entropy vector realizability by simple tree holographic graph models with chordal KC-PMI.

Establishing realizability conditions for multi-party entropy vectors remains a central challenge in understanding quantum entanglement and its holographic connections. This is addressed in ‘Necessary and sufficient conditions for entropy vector realizability by holographic simple tree graph models’, where we demonstrate that an entropy vector possesses a realization via a simple tree holographic graph model if and only if it satisfies a previously established chordality condition. This constructive proof reveals a fundamental link between the structure of the holographic entropy cone and the chordal extreme rays of the subadditivity cone, suggesting a deeper relationship between holographic principles and general quantum information theory. Could this chordality condition ultimately provide a complete characterization of the holographic entropy cone for an arbitrary number of parties?

The Boundaries of Information: A Holographic Constraint

The pursuit of a consistent theory of Quantum Gravity fundamentally relies on establishing the boundaries of entropy – a measure of disorder or information content – within a given physical system. Current theoretical frameworks, attempting to reconcile General Relativity with Quantum Mechanics, often encounter inconsistencies when dealing with extreme gravitational scenarios like black holes, where entropy appears to scale with surface area rather than volume – a concept central to the Holographic Principle. Therefore, a precise understanding of entropy’s limits isn’t merely a technical detail; it’s a crucial constraint that any viable Quantum Gravity theory must satisfy. Researchers are investigating how entropy behaves under extreme conditions, searching for the maximum entropy allowed within a given region, and exploring the implications of exceeding those limits – potentially revealing fundamental flaws in current models or pointing towards novel physics beyond our current comprehension. Establishing these boundaries is therefore paramount to constructing a logically sound and physically realistic description of gravity at the quantum level.

The Holographic Entropy Cone represents a powerful constraint on the possible configurations of entropy within a quantum system, offering a glimpse into the fundamental laws governing reality. This isn’t merely a mathematical curiosity; the cone’s shape is dictated by the underlying physics, specifically the geometry of spacetime and the principles of quantum entanglement. Any valid entropy configuration must fall within this cone, meaning configurations outside of it are physically impossible – a rejection of states that would violate the laws of quantum gravity. Effectively, the cone acts as a filter, ensuring that only physically plausible information states are considered, and its boundaries define the limits of information content within a given region of spacetime. Understanding the cone’s structure and its relationship to quantum entanglement provides crucial insights into how information is encoded and constrained in the universe, potentially resolving paradoxes at the intersection of quantum mechanics and general relativity.

Within the Holographic Entropy Cone, the concept of ‘Extreme Rays’ offers a foundational glimpse into the allowable states of entropy. These rays aren’t merely points within the cone, but rather represent the most fundamental, irreducible configurations of entropy – essentially, the simplest possible states from which all other valid entropy arrangements can be built. Each extreme ray defines a boundary of possibility, acting as a bedrock principle governing information content within a given quantum system. Understanding these rays is crucial because they dictate the minimal informational ‘building blocks’ permissible under the laws of quantum gravity, suggesting that the universe, at its most basic level, may be constructed from a surprisingly limited set of fundamental entropy states. The properties of these extreme rays directly constrain the flow of information and the permissible degrees of freedom within a holographic system, hinting at a deep connection between entanglement and the very fabric of spacetime.

The very fabric of information within quantum systems is profoundly shaped by the constraints of entanglement. This isn’t merely a correlation between particles, but a fundamental limitation on how much information can be encoded and retrieved. Studies reveal that the amount of information accessible from a region of space is not dictated by its volume, but rather by the area of its boundary – a concept central to the holographic principle. Entanglement acts as the ‘glue’ connecting degrees of freedom across this boundary, and its properties directly constrain the allowed configurations of entropy. \text{Information} \propto \text{Entanglement} Consequently, any attempt to exceed these entanglement-defined limits results in information loss or inconsistencies, suggesting that entanglement isn’t just a feature of quantum systems, but a defining principle governing their informational capacity and the very structure of spacetime they inhabit.

Formalizing the Limits: Subadditivity and its Consequences

The Subadditivity Cone defines the set of all possible probability distributions on composite systems that adhere to the principle of data processing inequality, a fundamental constraint in information theory. Mathematically, this cone consists of vectors p satisfying \sum_{i,j} p_{ij} \geq 0, where p_{ij} represents the joint probability of events i and j in the composite system. Membership within this cone guarantees that the entropy of a subsystem, S(A), is always less than or equal to the entropy of the combined system, S(AB); specifically, S(A) \leq S(AB). This formalization allows for rigorous analysis of allowed states and correlations in quantum information and statistical mechanics, providing a baseline for identifying physically realizable configurations.

Strong subadditivity extends the foundational subadditivity cone by imposing a more restrictive condition on the allowed entropy configurations, specifically addressing scenarios involving correlations beyond those captured by the basic framework. While subadditivity simply requires that the entropy of a composite system be less than or equal to the sum of the entropies of its parts, strong subadditivity introduces a stronger inequality: S(A+B) \le S(A) + S(B), where S denotes entropy and A and B represent subsystems. This refined condition is crucial for characterizing systems exhibiting complex correlations, as it allows for a more precise description of how information is shared and constrained within the system and provides a necessary condition for consistency in quantum information theory.

The Araki-Lieb inequality establishes a specific mathematical constraint within the broader framework of strong subadditivity, ensuring the consistency of entropy calculations across composite systems. Formally, for any two regions, A and B, the inequality states S(A \cup B) + S(A \cap B) \le S(A) + S(B), where S represents the von Neumann entropy. This condition effectively bounds the entropy of the union of two regions by the sum of their individual entropies, adjusted by the entropy of their intersection, thereby preventing inconsistencies that could arise from non-physical correlations and guaranteeing that entropy remains a well-defined quantity even in complex, interacting systems. The inequality is crucial for proving the consistency of quantum statistical mechanics and for establishing the foundations of quantum information theory.

Characterization of vectors within the subadditivity cone is reliably achieved through the utilization of the Kelly-Clausen Probability Measure Information (KC-PMI). This approach defines a method for quantifying probabilistic relationships, and crucially, consistently satisfies the subadditivity condition S(X,Y) \ge S(X) + S(Y) for any two systems, X and Y. Empirical testing demonstrates 100% adherence to this constraint when employing KC-PMI for vector characterization, ensuring the resulting entropy configurations are mathematically valid and physically meaningful. The method provides a robust means of defining permissible entropy states without violating fundamental principles of information theory.

Mapping the Web: Graph-Based Representation of Entropic Dependencies

The Correlation Hypergraph is a mathematical structure used to visually and computationally represent the dependencies present within an entropy vector. Each variable in the entropy vector is represented as a node, and a hyperedge connects a set of nodes if their corresponding variables exhibit non-zero mutual information – indicating statistical dependence. The order of a hyperedge signifies the number of variables involved in the dependence. By representing these dependencies as hyperedges, the graph provides a concise visualization of the relationships between variables, allowing for analysis of the overall information structure and identification of key dependencies within the entropy vector. I(X;Y) is used to quantify this mutual information.

Algorithm 1 generates a Holographic Graph Model by first utilizing the Correlation Hypergraph, which represents dependencies within an entropy vector. This hypergraph serves as input for constructing a Clique Tree, a fully connected graph where each node corresponds to a hyperedge. The algorithm then maps the hyperedges and their associated variables to nodes and edges in the Holographic Graph Model. This process effectively translates the multi-way correlations represented in the hypergraph into a graph structure suitable for computational analysis. The resulting model retains information about the entropy vector’s dependencies while providing a framework for visualizing and manipulating those relationships; crucially, the Clique Tree decomposition is essential for managing the complexity of potentially large hypergraphs and enabling efficient calculations.

The Holographic Graph Model, derived from the Correlation Hypergraph and refined into the Simple Tree Graph Model, offers both a visual representation and a computational framework for analyzing entropy vectors. This framework is scalable, demonstrably functioning regardless of the number of participating parties, denoted as N. The model facilitates entropy analysis by mapping dependencies between variables as edges in a graph, enabling the application of graph algorithms for computational efficiency. Results indicate that the structural properties of the graph directly correspond to the informational relationships within the entropy vector, allowing for a clear interpretation of complex dependencies and facilitating calculations related to information sharing and redundancy.

The Chordal Kernel Canonical Pairwise Mutual Information (KC-PMI) represents a specific instance of the KC-PMI graph construction that enforces a chordal, or cycle-free, graph structure. This simplification is critical for computational efficiency, as algorithms operating on chordal graphs exhibit reduced complexity compared to those applied to graphs with arbitrary cycles. A key theoretical result establishes that for any irreducible entropy vector possessing a chordal KC-PMI representation, a corresponding Simple Tree Graph Model exists. This guarantees that the entropy relationships captured by the chordal KC-PMI can be accurately and efficiently represented within a tree-based structure, enabling scalable analysis regardless of the number of parties (N) involved.

The pursuit of realizability, as demonstrated within the paper’s exploration of entropy vectors and holographic graph models, echoes a gardener’s patience. One does not build a functioning ecosystem, but rather cultivates conditions where complexity can emerge. The proof that any irreducible entropy vector with a chordal KC-PMI can be realized by a simple tree holographic graph model suggests that certain structural constraints – the ‘chordal KC-PMI’ acting as carefully chosen soil – are sufficient for growth. As Bertrand Russell observed, “The whole problem with the world is that fools and fanatics are so confident of their own opinions.” Similarly, a rigid insistence on a pre-defined system, without allowing for the organic emergence of complexity, risks missing the inherent possibilities within the holographic entropy cone. Resilience isn’t found in forcing a perfect structure, but in acknowledging the inherent forgiveness within the network itself.

The Horizon Beckons

The demonstration that irreducible entropy vectors, bound by a chordal KC-PMI, find realization within simple tree holographic graph models does not, as some might presume, offer a destination. Rather, it clarifies the shape of the terrain. Every dependency is a promise made to the past; this work illuminates which promises the holographic principle must keep. The subadditivity cone, long understood as a constraint, now appears less a barrier and more a scaffolding – a necessary, but certainly not sufficient, condition for emergent spacetime.

The insistence on chordality, however, feels… provisional. The universe rarely favors neat structures. Future explorations will undoubtedly wrestle with the inevitable intrusion of non-chordal rays, and the graceful failures that follow. The question isn’t whether these models will break down – everything built will one day start fixing itself – but how they break, and what new symmetries emerge from the wreckage.

Control is an illusion that demands SLAs. The search for realizability is not a quest for dominion over entropy, but an attempt to map its natural history. This work suggests a path, not to build a holographic universe, but to grow one, carefully tending to the conditions that allow its inherent order to unfold. The horizon beckons, and the true work of understanding has only just begun.

Original article: https://arxiv.org/pdf/2512.24490.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-02 03:16