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Lorentzian threads as gatelines and holographic complexity

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 نشر من قبل Juan Pedraza
 تاريخ النشر 2021
  مجال البحث فيزياء
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The continuous min flow-max cut principle is used to reformulate the complexity=volume conjecture using Lorentzian flows -- divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a CFT state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a canonical thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einsteins equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of spacetime complexity.



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Holographic entanglement entropy was recently recast in terms of Riemannian flows or bit threads. We consider the Lorentzian analog to reformulate the complexity=volume conjecture using Lorentzian flows -- timelike vector fields whose minimum flux th rough a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows, holographic complexity is shown to obey a number of properties. Particularly, the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final target states. We provide multiple explicit geometric realizations of Lorentzian flows in AdS backgrounds, including their time-dependence and behavior near the singularity in a black hole interior. Conceptually, discretized flows are interpreted as Lorentzian threads or gatelines. Upon selecting a reference state, complexity thence counts the minimum number of gatelines needed to prepare a target state described by a tensor network discretizing the maximal volume slice, matching its quantum information theoretic definition. We point out that suboptimal tensor networks are important to fully characterize the state, leading us to propose a refined notion of complexity as an ensemble average. The bulk symplectic potential provides a specific canonical thread configuration characterizing perturbations around arbitrary CFT states. Consistency of this solution requires the bulk satisfy the linearized Einsteins equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating for a principle of spacetime complexity. Lastly, we argue Lorentzian threads provide a notion of emergent time. This article is an expanded and detailed version of [arXiv:2105.12735], including several new results.
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