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 Holographic complexity bounds Hai-Shan Liu1,2, H. Lü1, Liang Ma1 & Wen-Di Tan1 1 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin, 300350, China 2 Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou, 310023, China ABSTRACT We study the action growth rate in the Wheeler-DeWitt (WDW) patch for a variety of D ≥ 4 black holes in Einstein gravity that are asymptotic to the anti-de Sitter spacetime, with spherical, toric and hyperbolic horizons, corresponding to the topological parameter k = 1, 0, −1 respectively. We find a lower bound inequality  1T∂I⋅WDW∂S∣∣∣Q,Pth>C for k = 0, 1, where C is some order-one numerical constant. The lowest number in our examples is C = (D − 3)/(D − 2). We also find that the quantity  (I⋅WDW−2PthΔVth)  is greater than, equal to, or less than zero, for k = 1, 0, −1 respectively. For black holes with two horizons, ∆Vth =  V+th  −  V−th , i.e. the difference between the thermodynamical volumes of the outer and inner horizons. For black holes with only one horizon, we introduce a new concept of the volume  V0th  of the black hole singularity, and define  ΔVth=V+th−V0th . The volume  V0th  vanishes for the Schwarzschild black hole, but in general it can be positive, negative or even divergent. For black holes with single horizon, we find a relation between  I⋅WDW  and  V0th , which implies that the holographic complexity preserves the Lloyd’s bound for positive or vanishing  V0th , but the bound is violated when  V0th  becomes negative. We also find explicit black hole examples where  V0th  and hence  I⋅WDW  are divergent. 【关闭窗口】
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