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Lorentzian threads as 'gatelines' and holographic complexity

Juan F. Pedraza, Andrea Russo, Andrew Svesko, Zachary Weller-Davies

TL;DR

A refined measure of complexity is proposed, capturing the role of suboptimal TNs, as an ensemble average, and the bulk symplectic potential provides a "canonical" thread configuration characterizing perturbations around arbitrary CFT states.

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 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 Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of 'spacetime complexity'.

Lorentzian threads as 'gatelines' and holographic complexity

TL;DR

A refined measure of complexity is proposed, capturing the role of suboptimal TNs, as an ensemble average, and the bulk symplectic potential provides a "canonical" thread configuration characterizing perturbations around arbitrary CFT states.

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 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 Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of 'spacetime complexity'.

Paper Structure

This paper contains 1 section, 40 equations, 3 figures.

Figures (3)

  • Figure 1: Complexity is equal to the minimum number of gatelines preparing a state on maximal volume slice $\Sigma$. Optimal flow prepares optimal TN (left); suboptimal flows prepare more complex suboptimal TNs (right).
  • Figure 2: We pick a different bulk Lorentzian slice $\Sigma'$ to partition the two regions, in particular $\Sigma'$ could be the constant time surface of a Lorentz boosted observer.
  • Figure 3: Conventions for the computation of holographic complexity of a state on $\sigma_A$ (left) and subregion complexity of the bipartition $\sigma_X\cup\sigma_Y=\sigma_A$ (right).