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Timelike entanglement entropy
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We define a new complex-valued measure of information called the timelike entanglement entropy (EE) which in the boundary theory can be viewed as a Wick rotation that changes a spacelike boundary subregion to a timelike one. An explicit definition of the timelike EE in 2d field theories is provided followed by numerical computations which agree with the analytic continuation of the replica method for CFTs. We argue that timelike EE should be correctly interpreted as another measure previously considered, the pseudo entropy, which is the von Neumann entropy of a reduced transition matrix. Our results strongly imply that the imaginary part of the pseudo entropy describes an emergent time which generalizes the notion of an emergent space from quantum entanglement. For holographic systems we define the timelike EE as the total complex valued area of a particular stationary combination of both space and timelike extremal surfaces which are homologous to the boundary region. For the examples considered we find explicit matching of our optimization procedure and the careful implementation of the Wick rotation in the boundary CFT. We also make progress on higher dimensional generalizations and relations to holographic pseudo entropy in de Sitter space.
Forward citations
Cited by 31 Pith papers
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de Sitter holography from a Lorentzian torus
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For timelike boundary regions, the entanglement first law ΔS = Δ⟨H⟩ is equivalent, by the paper's proof, to the linearized Einstein equations around AdS.
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Timelike Holographic Complexity
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De Sitter Complexity Grows Linearly in the Static Patch
Timelike extremal volume in the de Sitter static patch gives a holographic complexity that grows linearly with time and is proportional to horizon entropy times temperature.
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Entanglement measures for causally connected subregions and holography
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Temporal Entanglement from Holographic Entanglement Entropy
Holographic timelike entanglement entropy is defined by analytically continuing all candidate extremal surfaces through the light cone and selecting the one with smallest real part of the area.
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Holographic timelike entanglement in AdS$_{3}$ Vaidya
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Constraints from Entanglement Wedge Nesting for Holography at a Finite Cutoff
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Brane Cosmology from AdS/BCFT
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Holographic Timelike Entanglement Entropy in Non-relativistic Theories
The complex timelike entanglement entropy is computed in non-relativistic holographic theories, with a logarithmic real part and a constant imaginary part proposed as Fermi-surface signatures.
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Renormalized pseudoentropy in dS/CFT
Renormalized holographic pseudoentropy in dS/CFT is constructed from conformal-gravity actions in four and six dimensions, yielding finite sphere values and Mezei-like shape dependence for small deformations.
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Timelike entanglement and central charge for quantum BTZ black holes
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Thermal Pseudo-Entropy
Thermal pseudo-entropy is the analytic continuation S(β+it) of thermal entropy, equals the pseudo-entropy of a Thermofield Double transition matrix, and its averaged real part tracks the spectral form factor.
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Correlation functions of von Neumann entropy
Two-point correlators of modular Hamiltonians obey entropy-like properties and equal the stress-tensor conformal block for spherical regions in CFT, including imaginary-time separations.
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Emergent Holographic Spacetime from Quantum Information
Takayanagi's essay outlines a research program in which holographic spacetime, including the time direction, may emerge from entanglement, complexity, and complex-valued pseudo-entropy, without presenting a new derivation.
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