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.
Highlights
Introduction and summary1.1 The big pictureSpacetime physics and quantum information are fundamentally intertwined
This statement is most precisely formulated in the context of the AdSd+1/CFTd correspondence, where S(A) is the entanglement entropy of a CFT state confined to a boundary region A, and m(A) is the minimal codimension-2 bulk surface that is homologous to A
In this article we provided a Lorentzian flow based reformulation of the complexity=volume proposal, following an application of the min flow-max cut principle from network theory
Summary
Introduction and summary1.1 The big pictureSpacetime physics and quantum information are fundamentally intertwined. The sharpest realization of the interplay between information and gravity is perhaps best captured by the Ryu-Takayanagi (RT) formula [2],1 relating the area of minimal surfaces in a (d + 1)dimensional (bulk) curved spacetime to the entanglement entropy of a state of a quantum field theory living on the d-dimensional (boundary) spacetime, S(A) =. The RT formula can be seen as a generalization of the Bekenstein-Hawking entropy-area relation [8, 9].2 It satisfies all known properties of the von Neumann entropy [14], and can be used to construct other important information theoretic quantities, including (holographic) mutual information and Renyí entropy [15], each of which have dual geometric descriptions akin to (1.1). The RT formula can be seen as a generalization of the Bekenstein-Hawking entropy-area relation [8, 9].2 It satisfies all known properties of the von Neumann entropy [14], and can be used to construct other important information theoretic quantities, including (holographic) mutual information and Renyí entropy [15], each of which have dual geometric descriptions akin to (1.1).
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