What does it really cost to prepare exotic quantum states?

The Reeh-Schlieder theorem guarantees you can approximate any quantum state by applying an operator in a tiny region of spacetime, but says nothing about the price tag. This work derives a concrete lower bound: states with deeply negative modular energy demand either large operators or probabilistic postselection to prepare. The bound translates to testable predictions in quantum field theory geometries and conformal field theories, revealing a fundamental tension between locality and state preparation cost.