Comments on holographic spread complexity

Abstract

We revisit the holographic proposal relating the growth rate of spread complexity to the radial momentum of a bulk probe, aiming to identify its underlying assumptions and clarify how classical probe dynamics emerges from quantum dynamics. By quantizing AdS probes directly and using the extrapolation dictionary, we provide a more general derivation of the proposal. In particular, we interpret it as a concrete realization of a classical complexity observable and its quantization in the ``complexity=anything'' framework. Since probe dynamics is not intrinsically tied to the AdS/CFT correspondence, we also examine the proposal in flat and de Sitter spacetimes. In flat spacetime, the physical Hamiltonian does not preserve the coherent-state structure, and spread complexity measures the dispersive broadening of the wave packet rather than classical momentum. In de Sitter space, generic semiclassical wave packets in the static-patch energy representation show the same dispersive behavior. By contrast, coherent states adapted to the principal-series representation can exhibit exponential growth of spread complexity, with a growth rate that has the same time dependence as the classical radial momentum. These examples indicate that a semiclassical limit alone is not sufficient for a momentum--complexity relation.