Comments on holographic complexity

Journal of High Energy Physics, Mar 2017

We study two recent conjectures for holographic complexity: the complexity=action conjecture and the complexity=volume conjecture. In particular, we examine the structure of the UV divergences appearing in these quantities, and show that the coefficients can be written as local integrals of geometric quantities in the boundary. We also consider extending these conjectures to evaluate the complexity of the mixed state produced by reducing the pure global state to a specific subregion of the boundary time slice. The UV divergences in this subregion complexity have a similar geometric structure, but there are also new divergences associated with the geometry of the surface enclosing the boundary region of interest. We discuss possible implications arising from the geometric nature of these UV divergences.

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Comments on holographic complexity

Received: February Comments on holographic complexity Dean Carmi 0 1 3 4 6 Robert C. Myers 0 1 3 6 Pratik Rath 0 1 2 3 5 6 Open Access 0 1 3 c The Authors. 0 1 3 0 Berkeley , CA 94720 , U.S.A 1 Tel-Aviv University , Ramat-Aviv 69978 , Israel 2 Department of Physics, University of California 3 31 Caroline Street North , Waterloo, ON N2L 2Y5 , Canada 4 Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy 5 Center for Theoretical Physics, University of California 6 Perimeter Institute for Theoretical Physics We study two recent conjectures for holographic complexity: the complexity=action conjecture and the complexity=volume conjecture. In particular, we examine the structure of the UV divergences appearing in these quantities, and show that the coefcients can be written as local integrals of geometric quantities in the boundary. We also consider extending these conjectures to evaluate the complexity of the mixed state produced by reducing the pure global state to a speci c subregion of the boundary time slice. The UV divergences in this subregion complexity have a similar geometric structure, but there are also new divergences associated with the geometry of the surface enclosing the boundary region of interest. We discuss possible implications arising from the geometric nature of these UV divergences. AdS-CFT Correspondence; Classical Theories of Gravity; Gauge-gravity cor- 1 Introduction 2 3 4 Complexity equals volume conjecture Complexity equals action conjecture Subregion complexity: CV duality Subregion complexity: CA duality A Action user's manual B Example: extremal volume for a spherical boundary C Example: Wheeler-DeWitt action for global AdS D Geometric details for CA duality calculation Introduction Concepts and perspectives from quantum information science are having a rapidly growing in uence in investigations of quantum eld theory and quantum gravity. Quantum complexity is one such concept which has recently begun to be discussed. Loosely speaking, the complexity of a particular state corresponds to the minimum number of simple (universal) gates needed to build a quantum circuit which prepares this state from a particular reference state, e.g., see [1{3]. In the context of the AdS/CFT correspondence, discussions have focused on understanding the growth of the Einstein-Rosen bridge for AdS black holes in terms of quantum complexity in the dual boundary CFT [4{10]. There are two independent proposals to evaluate the complexity of a holographic the complexity of the boundary state is dual to the volume of the extremal codimensionone bulk hypersurface which meets the asymptotic boundary on the desired time slice.1 that similar extremal volumes in the interior of asymptotically at black holes were studied in [12, 13]. More precisely, the CV duality states that the complexity of the state on a time slice CV( ) = max V(B) =@B GN ` where B is the corresponding bulk surface and ` is some length scale associated with the bulk geometry, e.g., the AdS curvature scale or the horizon radius of a black hole. The ambiguity in choosing the latter scale is an unappealing feature of CV duality and provided some motivation for developing CA duality [9, 10]. This second conjecture equates the complexity with the gravitational action evaluated on a particular bulk region, now known as the Wheeler-DeWitt (WDW) patch: CA( ) = The WDW patch can be de ned as the domain of dependence of any Cauchy surface in the bulk which asymptotically approaches the time slice on the boundary. The complexity evaluated with either the CV or CA duality satis es a number of expected properties, e.g., they continue to grow (linearly with time) after the boundary theory reaches thermal equilibrium. However, the second conjecture has certain advantages. In particular, as noted above, CV duality requires choosing an additional length scale, while there are no free parameters in eq. (1.2) for the CA duality. However, the latter faced the obstacle that when the conjecture was originally proposed, there was no rigorous method for evaluating the gravitational action on spacetime regions with null boundaries. This problem was recently overcome with a careful analysis of the boundary terms which must be added to the gravitational action for null boundary surfaces and for joints where such null boundaries intersect with other boundary surfaces [14]. On the gravity side, either of these dualities deals with a geometric entity which extends to the asymptotic AdS boundary and as a result, the holographic complexity is divergent. To understand these divergences, it is natural to draw upon lessons from holographic entanglement entropy [15, 16]. In particular, for both the CV duality and holographic entanglement entropy, the bulk calculations evaluate the volume of an extremal surface extending to the asymptotic boundary. Now UV divergences are found in calculating holographic entanglement entropy, e.g., [ (...truncated)


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Dean Carmi, Robert C. Myers, Pratik Rath. Comments on holographic complexity, Journal of High Energy Physics, 2017, pp. 118, Volume 2017, Issue 3, DOI: 10.1007/JHEP03(2017)118