Entanglement spectrum of geometric states

Abstract

The reduced density matrix of a given subsystem, denoted by ${\rho }_A$, contains the information on subregion duality in a holographic theory. We may extract the information by using the spectrum (eigenvalue) of the matrix, called entanglement spectrum in this paper. We evaluate the density of eigenstates, one-point and two-point correlation functions in the microcanonical ensemble state ${\rho }_{A,m}$ associated with an eigenvalue $\lambda$ for some examples, including a single interval and two intervals in vacuum state of 2D CFTs. We find there exists a microcanonical ensemble state with ${\lambda }_0$ which can be seen as an approximate state of ${\rho }_A$. The parameter ${\lambda }_0$ is obtained in the two examples. For a general geometric state, the approximate microcanonical ensemble state also exists. The parameter ${\lambda }_0$ is associated with the entanglement entropy of $A$ and Rényi entropy in the limit $n\to \infty$. As an application of the above conclusion we reform the equality case of the Araki-Lieb inequality of the entanglement entropies of two intervals in vacuum state of 2D CFTs as conditions of Holevo information. We show the constraints on the eigenstates. Finally, we point out some unsolved problems and their significance on understanding the geometric states.

arXiv commentreferences added

References in corpus (51)

• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Building up spacetime with quantum entanglement
• Entanglement Renormalization and Holography
• Entanglement Renyi entropies in holographic theories
• Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality
• Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy
• The Gravity Dual of a Density Matrix
• Causality & holographic entanglement entropy
• Entanglement spectrum in one-dimensional systems
• Entanglement entropy of two disjoint intervals in conformal field theory
• Relative Entropy and Holography
• Form factors of branch-point twist fields in quantum integrable models and entanglement entropy
• The Gravity Dual of Renyi Entropy
• Holographic Entanglement of Purification
• Entanglement entropy of two disjoint intervals in conformal field theory II
• Holographic Mutual Information is Monogamous
• Entanglement hamiltonians in two-dimensional conformal field theory
• Holographic Calculations of Renyi Entropy
• Entanglement of purification: from spin chains to holography
• Comments on quantum gravity and entanglement
• Flat entanglement spectra in fixed-area states of quantum gravity
• Holographic Renyi Entropy from Quantum Error Correction
• A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots
• On short interval expansion of Rényi entropy
• Relative Entropies in Conformal Field Theory
• Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT
• Modular Hamiltonian of Excited States in Conformal Field Theory
• The boundary dual of the bulk symplectic form
• Holographic Rényi entropy for CFT with $W$ symmetry
• From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories
• Probing thermality beyond the diagonal
• Holographic Phases of Renyi Entropies
• Twist operators in higher dimensions
• Relative entropy of excited states in two dimensional conformal field theories
• On excited states in real-time AdS/CFT
• A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled
• Subsystem Trace Distance in Quantum Field Theory
• Relative entropy of excited states in conformal field theories of arbitrary dimensions
• One-loop universality of holographic codes
• On Distinguishability of Black Hole Microstates
• Subsystem trace distance in low-lying states of $(1+1)$-dimensional conformal field theories
• Holographic Renyi Entropy at High Energy Density
• Holographic excited states in AdS Black Holes
• Modular Hamiltonian for (holographic) excited states
• Interacting fields in real-time AdS/CFT
• Distinguishing Black Hole Microstates using Holevo Information
• From sources to initial data and back again: on bulk singularities in Euclidean AdS/CFT
• Holographic CFT states for localized perturbations to AdS black holes
• Non-geometric States in a Holographic Conformal Field Theory
• Rényi entropy at large energy density in 2D CFT

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