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Register a new userLorentzian threads as gatelines and holographic complexity
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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 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 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.
Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be interpreted in terms of a set of Planck-thickness bit threads, which can be either classical or quantum. This interpretation uncovers an aspect of the generalized entropy that admits a neat information-theoretic description, namely, the fact that the quantum corrections can be cast in terms of entanglement distillation of the bulk state. We also prove some general properties of our prescription, including nesting and a quantum version of the max multiflow theorem. These properties are used to verify that our proposal respects known inequalities that a von Neumann entropy must satisfy, including subadditivity and strong subadditivity, as well as to investigate the fate of the holographic monogamy. Finally, using the Iyer-Wald formalism we show that for cases with a local modular Hamiltonian there is always a canonical solution to the program that exploits the property of bulk locality. Combining with previous results by Swingle and Van Raamsdonk, we show that the consistency of this special solution requires the semi-classical Einsteins equations to hold for any consistent perturbative bulk quantum state.
We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions.
We study holographic subregion complexity, and its possible connection to purification complexity suggested recently by Agon et al. In particular, we study the conjecture that subregion complexity is the purification complexity by considering holographic purifications of a holographic mixed state. We argue that these include states with any amount of coarse-graining consistent with being a purification of the mixed state in question, corresponding holographically to different choices of the cutoff surface. We find that within the complexity = volume and complexity = spacetime volume conjectures, the subregion complexity is equal to the holographic purification complexity. For complexity = action, the subregion complexity seems to provide an upper bound on the holographic purification complexity, though we show cases where this bound is not saturated. One such example is provided by black holes with a large genus behind the horizon, which were studied by Fu et al. As such, one must conclude that these offending geometries are not holographic, that CA must be modified, or else that holographic subregion complexity in CA is not dual to the purification complexity of the corresponding reduced state.
We study the holographic complexity conjectures for rotating black holes, uncovering a relationship between the complexity of formation and the thermodynamic volume of the black hole. We suggest that it is the thermodynamic volume and not the entropy that controls the complexity of formation of large black holes in both the Complexity Equals Action and Complexity Equals Volume proposals in general. Our proposal reduces to known results involving the entropy in settings where the thermodynamic volume and entropy are not independent, but has broader scope. Assuming a conjectured inequality is obeyed by the thermodynamic volume, we establish that the complexity of formation is bounded from below by the entropy for large black holes.
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