The Holographic Entropy Cone (1505.07839v1)

Abstract: We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

• The paper defines the holographic entropy cone by systematically analyzing entropy inequalities in AdS/CFT frameworks.
• It employs combinatorial graph models to classify finite polyhedral structures and identify novel cyclic inequalities for five or more regions.
• The study provides insights into multipartite entanglement, influencing quantum gravity theories and potential applications in quantum error correction.

An Analysis of "The Holographic Entropy Cone"

This paper presents a systematic examination of the entropy inequalities related to the Ryu-Takayanagi formula, a cornerstone in the paper of entanglement entropy in holographic dual conformal field theories (CFTs). By investigating the entropy vectors formed by partitioning boundary regions within a bulk holographic geometry, the authors define what they term the "holographic entropy cone" for different numbers of regions.

Key Findings

The paper classifies the known entropy inequalities in holographic scenarios and identifies new ones, especially when dealing with five or more boundary regions, beyond the familiar strong subadditivity and monogamy inequalities recognized for fewer regions. In particular, the work reveals an infinite family of cyclic inequalities for a larger number of regions, governed by relationships between conditional and total entropies that were undetermined in non-holographic quantum systems.

The authors establish the holographic entropy cone as a polyhedral convex cone and emphasize its computational characterization using graph models. These models offer a combinatorial perspective by associating graph cuts with holographic entropy calculations, revealing that the entropy cone has a finite number of facets defined by linear inequalities.

Numerical and Theoretical Implications

The numerical results emphasize that even for five regions, the set of holographic inequalities expands significantly. This enriched set of inequalities insists that geometric wormhole states cannot be simply understood in terms of bipartite entangled structures like EPR pairs alone, aligning with the broader ER=EPR conjecture but requiring enhanced structures for multipartite understanding.

Theoretically, the paper provides stronger groundwork in recognizing that holographic states follow distinct entropy patterns from generic quantum states. The new cyclic inequalities and their implications for multipartite entanglement suggest novel ways through which holographic constraints could influence our understanding of entanglement in quantum gravity scenarios.

Prospective Developments

The research points to numerous avenues for extending this work. Notably, investigating whether these geometric constraints hold under perturbative corrections or within covariant holographic frameworks could deepen understanding. Additionally, the connections to stabilizer states in quantum information theory hint at potential insights into quantum error correction codes derived from holographic principles.

Beyond the current results, the approach using combinatorial graph models offers a promising technique not just for theoretical insights but also for computational advancements in studying entanglement structures. These can further inform potential physical scenarios in higher dimensions and inform future conjectures on quantum geometry's role in holography and beyond.

In sum, "The Holographic Entropy Cone" delineates a substantial advance in understanding entropy inequalities in holography. The work has profound implications for how we approach quantum states with holographic interpretations, especially entanglement structures fundamental in quantum gravity theories like the AdS/CFT correspondence. These findings establish a precise framework for further exploration in high energy theory, quantum gravity, and quantum information science.