Holographic subregion complexity for singular surfaces

Eur. Phys. J. C (2017) 77:665 DOI 10.1140/epjc/s10052-017-5247-1 Regular Article - Theoretical Physics Holographic subregion complexity for singular surfaces Elaheh Bakhshaei1, Ali Mollabashi2,a , Ahmad Shirzad1,3 1 Department of Physics, Isfahan University of Technology, P.O.Box 84156-83111, Isfahan, Iran School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran 3 School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran 2 Received: 24 March 2017 / Accepted: 21 September 2017 © The Author(s) 2017. This article is an open access publication Abstract Recently holographic prescriptions were proposed to compute the quantum complexity of a given state in the boundary theory. A specific proposal known as ‘holographic subregion complexity’ is supposed to calculate the complexity of a reduced density matrix corresponding to a static subregion. We study different families of singular subregions in the dual field theory and find the divergence structure and universal terms of holographic subregion complexity for these singular surfaces. We find that there are new universal terms, logarithmic in the UV cut-off, due to the singularities of a family of surfaces including a kink in (2 + 1) dimensions and cones in even dimensional field theories. We also find examples of new divergent terms such as squared logarithm and negative powers times the logarithm of the UV cut-off parameter. Contents 1 Introduction . . . . . . . . . . . . . . . . . 2 Singular subregions and summary of results Summary of results . . . . . . . . . . . . . . 3 Flat locus singular surfaces . . . . . . . . . . 3.1 Kink k . . . . . . . . . . . . . . . . . . 3.2 Cone cn . . . . . . . . . . . . . . . . . 3.3 Crease k × R m . . . . . . . . . . . . . 3.4 Conical crease cn × R m . . . . . . . . . 4 Curved locus singular surfaces . . . . . . . . 4.1 Crease k ×  . . . . . . . . . . . . . . 4.2 Conical crease cn ×  . . . . . . . . . 5 Discussions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . a e-mail: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Quantum entanglement has been widely studied in the context of holographic field theories after the pioneering proposal of Ryu–Takayanagi (RT) [1,2]. Quantum complexity is another notion in quantum information theory which has been recently included in the context of holographic field theories. Roughly speaking, the quantum complexity of a state is the minimum number of information gates needed to prepare a state from a given reference state. There have been made some efforts to develop a holographic dual for quantities related to this notion in the context of AdS/CFT correspondence [3–11]. From a more geometrical point of view, it is well established that the von Neumann entropy of a subregion in a given state corresponds to the area of a co-dimension two surface in the gravity solution dual of the state. One has also tried to find geometrical duals for other quantities in the context of information theory; such as Renyi entropies [12,13], information metric (fidelity susceptibility) [14,18,19],1 Fisher information [20], etc. Some of these geometrical objects are still co-dimension two objects in the dual theory but some are not. There are two distinct proposals to compute complexity of a state in the dual gravity theory. The first one, which is sometimes called the ‘complexity = volume’ proposal, states that the complexity of a given state at a given time in the boundary theory is given by the volume of an extremal co-dimension one surface in the bulk which meets the corresponding time slice. To be more concrete, one can state this proposal as   V , (1.1) CV = max GN where the maximum is chosen among those co-dimension one surfaces which end on the corresponding time slice on the conformal boundary. In this proposal  is some length scale 1 For related work also see [15–17]. 123 665 Page 2 of 17 Eur. Phys. J. C (2017) 77:665 which should be identified case by case, e.g. the radius of the asymptotically AdS solution or the radius of the horizon in the case of AdS black-hole geometries. This non-recognized length scale seems to be a disadvantage of this proposal. The other proposal, which is sometimes called ‘complexity = action’, states that the complexity of a given state at a given time is equal to the on-shell action of the dual (Einstein) gravity theory computed in the domain of dependence of any Cauchy surface in the bulk which ends on the given time slice at the conformal boundary.2 This region is known as the Wheeler–DeWitt patch, corresponding to the given boundary time slice. Although this proposal (in contrast with the previous one) does not need any length scale by definition, it has its own challenges due to surface terms and corner contributions of the Wheeler–DeWitt patch (see [11,22]). We will come back to this point in the next section. A natural generalization of the ‘complexity = volume’ proposal concerns with generic mixed states. A specific way of constructing a mixed state out of the entire state of a system is to trace out a part of the space-like manifold of the dual field theory. The mixed state constructed in this way is described by what is known as the reduced density matrix. Then the complexity of such a (static) state is proposed to be given by the volume enclosed by the Ryu–Takayanagi surface and the corresponding subregion in the boundary theory.3 To be more concrete the subregion complexity is defined as [18] Csubregion = V (γ ) , 8π G N (1.2) where γ is the RT surface of the corresponding subregion and  is a length scale of the dual geometry. This proposal (up to a numerical factor) reduces to ‘complexity = volume’ given in (1.1) if the subregion is chosen to be the whole time slice of the dual theory. Different proposals for complexity all lead to UV divergent results since they all contain a volume of a surface which reaches the conformal boundary of an asymptotically AdS geometry. This is the same as what happened in the case of holographic entanglement entropy.4 Natural questions as regards such quantities are: What is the divergent structure of this quantity? How it can be regularized? What kind of universal information can be extracted from it? Is it possible to find any monotonic function out of this quantity under the RG flow of the dual theory?5 Specifically for the case of subregion complexity one may also question the (subregion) shape dependence of the divergence structure. 2 Recently some progress has been made for complexity in higher derivative theories in [21]. 3 Recently a covariant generalization of this proposal is given in [11]. 4 A concrete example of a dual CFT calculation for Fischer information metric in the case of marginal deformat (...truncated)