From bulk flows to boundary generators and costs

Abstract

Holographic complexity admits a broad family of bulk observables, yet their relation to boundary state-preparation costs remains unclear. We propose a local prescription that associates suitable bulk observables with quadratic cost geometries on families of states prepared by Euclidean sources. Building on the holographic transport of bulk Hamiltonian flows to Euclidean preparation sources, we use the Hessian of the transported generator at a stationary reference state to define a candidate cost metric. When this Hessian is positive definite, the minimised preparation energy reproduces the change in the bulk observable to quadratic order. We characterise the resulting preparation penalties relative to the Fubini-Study metric and compute them for the maximal volume in scalar perturbations of AdS$_3$. At large-conformal dimension, we recover the known proportionality between cost and Fubini-Study metrics. At finite dimension, we obtain positive but generally unequal penalties for source variations that displace the bulk field and its conjugate momentum, breaking this proportionality even for a single scalar operator. We further study a deformation of volume by quadratic scalar and gradient terms in the large-dimension regime, determining how the couplings modify the penalties and constrain positivity. These additions selectively change field-displacement costs, with the gradient term introducing mode dependence. Our results provide an explicit local connection between the choice of bulk observables and the assignment of boundary preparation penalties.