The Application Of Crofton Formula In Holographic Entanglement Entropy And Holographic Complexity

In this paper,we investigate the holographic complexity,applying the Crofton's formula on Lorentzian Anti-de Sitter spacetime,to enrich the holographic dictionary and interpret that the complexity is bounded from below.In the first place,we review the basics of the entanglement entropy and the complexity in quantum mechanics and quantum field theory separately.The Ryu-Takayanagi conjecture,as the most powerful tool,reveals the relation between the entanglement entropy of a region on the boundary and the area of extremal surface anchored on the entangling surface.Surface/state duality demonstrates the correspondence between a certain codimension-2 surface?and a spe-cific quantum state|?_?.It suggests a new approach to define the complexity in field theories in the form of path-integral complexity of the optimized surface,which is treated as the minimal value of normalizing factor of path integral on the codimension-1 surface M_?anchored on?.Considering a set of half-planes,this paper compares the path-integral complexity and volume,showing good agreement.In this sense,a certain area can be regarded as a rational definition of holographic complexity.The Crofton's formula can be applied to the case of Euclidean Ad S₂.It describes an arbitrary curve by the extremal surface area,or geodesics length in two dimensions,providing the entropic explanation for the general curve in the bulk.Along this line,We explore another quantum information-theoretic quantity,complexity.Without isochronal limitation,different relations between the boundary theory and the three kinds of geodesics including spacelike,null and timelike geodesics make the condition more complicated.To deal with the problem,we deduce Crofton's formula on Lorentzian Ad S₃that links the general surface area to a set of geodesics intersecting with it.As the bridge of geodesics and the complexity,the Crofton's formula links the entropy to the complexity naturally.In particular,a pre-factor is regarded as the volume ratio of U?1?space and Ad S₃spacetime.Finally,the results above allow us to apply the conclusion of tensor network to the com-plexity.It interprets that the lower bound of the complexity depends on the size of the Hilbert space at the lower energy scale.