Abstract
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 multistep 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 conformal field theory (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 Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of "spacetime complexity."
Highlights
The sharpest realization of this is captured by the Ryu-Takayanagi (RT) formula [1,2], relating the area of minimal codimension-2 surfaces mðAÞ in a (d þ 1)dimensional anti–de Sitter (AdS) spacetime to the entanglement entropy (EE) SðAÞ of a conformal field theory (CFT) state restricted to a (d − 1)-dimensional boundary subregion A homologous to m
The RT prescription was reformulated in terms of flows or holographic “bit threads” [8], where mðAÞ is replaced by the maximum flux of a divergenceless normbounded Riemannian vector field v through A, Z SðAÞ 1⁄4 max v; v∈F A
Does (1) have technical advantages, it offers conceptual insight: a thread emanating from A is interpreted as a channel carrying a singlebit encoding the microstate of A, where the maximum number of threads gives SðAÞ, which may be distilled as Bell pairs
Summary
The CV conjecture says the complexity C of a CFT state defined on a Cauchy slice σA delimiting a boundary region A, so that ∂A 1⁄4 σA, is dual to the volume of an extremal codimension-1 bulk hypersurface Σ homologous A Lorentzian manifold M is equal to the volume V of the maximal bulk codimension-1 Cauchy slice Σ ∼ A: Z
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