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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.
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Quantum states with geometric duals are known to satisfy a stricter set of entropy inequalities than those obeyed by general quantum systems. The set of allowed entropies derived using the Ryu-Takayanagi (RT) formula defines the Holographic Entropy Cone (HEC). These inequalities are no longer satisfied once general quantum corrections are included by employing the Quantum Extremal Surface (QES) prescription. Nevertheless, the structure of the QES formula allows for a controlled study of how quantum contributions from bulk entropies interplay with HEC inequalities. In this paper, we initiate an exploration of this problem by relating bulk entropy constraints to boundary entropy inequalities. In particular, we show that requiring the bulk entropies to satisfy the HEC implies that the boundary entropies also satisfy the HEC. Further, we also show that requiring the bulk entropies to obey monogamy of mutual information (MMI) implies the boundary entropies also obey MMI.
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
In holographic duality, if a boundary state has a geometric description that realizes the Ryu-Takayanagi proposal then its entanglement entropies must obey certain inequalities that together define the so-called holographic entropy cone. A large family of such inequalities have been proven under the assumption that the bulk geometry is static, using a method involving contraction maps. By using kinematic space techniques, we show that in two boundary (three bulk) dimensions, all entropy inequalities that can be proven in the static case by contraction maps must also hold in holographic states with time dependence.
Since the work of Ryu and Takayanagi, deep connections between quantum entanglement and spacetime geometry have been revealed. The negative eigenvalues of the partial transpose of a bipartite density operator is a useful diagnostic of entanglement. In this paper, we discuss the properties of the associated entanglement negativity and its Renyi generalizations in holographic duality. We first review the definition of the Renyi negativities, which contain the familiar logarithmic negativity as a special case. We then study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics. Finally, we study entanglement negativity in holographic theories with a gravity dual, where we find that Renyi negativities are often dominated by bulk solutions that break the replica symmetry. From these replica symmetry breaking solutions, we derive general expressions for Renyi negativities and their special limits including the logarithmic negativity. In fixed-area states, these general expressions simplify dramatically and agree precisely with our results in the random tensor network model. This provides a concrete setting for further studying the implications of replica symmetry breaking in holography.
We examine the mechanical matrix model that can be derived from the SU(2) Yang-Mills light-cone field theory by restricting the gauge fields to depend on the light-cone time alone. We use Diracs generalized Hamiltonian approach. In contrast to its well-known instant-time counterpart the light-cone version of SU(2) Yang-Mills mechanics has in addition to the constraints, generating the SU(2) gauge transformations, the new first and second class constraints also. On account of all of these constraints a complete reduction in number of the degrees of freedom is performed. It is argued that the classical evolution of the unconstrained degrees of freedom is equivalent to a free one-dimensional particle dynamics. Considering the complex solutions to the second class constraints we show at this time that the unconstrained Hamiltonian system represents the well-known model of conformal mechanics with a ``strength of the inverse square interaction determined by the value of the gauge field spin.
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