Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity

Abstract

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 through 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 Einstein’s 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 [1], including several new results.

Bath deformations, islands and holographic complexity

“…An outstanding question will be to understand the corresponding jump in subregion complexity and its dependence on the perturbation parameters if one adds a second brane in the model [16] and considering a gravitating bath. Further, recently a formulation of the subregion volume has been given in terms of bit threads [107,108]. The exciting open question is to understand whether bit thread formulation can provide a more vivid picture of the discontinuity of the subregion complexity in the doubly holographic braneworld model, perhaps along the line of [109,110].…”

Holographic complexity and de Sitter space

“…Previous attempts to study quantum computational complexity in dS space appeared in[48][49][50] 3. This conjecture was reformulated using Lorentzian flows in[60,61] similarly to the earlier reformulation of the Ryu-Takayanagi prescription[62] using 'bit threads'[63].…”

Holographic Complexity in dS$_{d+1}$

“…In the de Sitter setting, this leads to two distinct proposals: the monolayer [1], and bilayer [41] approaches for entanglement entropysee [42] for a discussion of the differences between the two approaches. However, we note that there is an analogous 'gate line' description of complexity=volume [94][95][96]. Hence it would be helpful to examine this description of holographic complexity in the context of de Sitter space and explore if analogous subtleties arise as were found for the holographic entanglement entropy.…”