Modular Lower Bounds on Reeh-Schlieder State Preparation

TL;DR

This paper establishes quantitative lower bounds on the cost of preparing target states in quantum field theory using local operators, linking modular energy to preparation complexity.

Contribution

It isolates a model-independent estimate from the Tomita-Takesaki theory as a lower bound on state preparation costs, applicable in various geometries.

Findings

Deeply negative modular energy targets require large local operators.

Rescaling operators yields a lower bound on postselection overhead.

Explicit bounds are derived in geometries with known modular Hamiltonians.

Abstract

The Reeh-Schlieder theorem says that every target vector can be approximated from the vacuum by an operator localized in an arbitrarily small spacetime region, but it gives no quantitative cost for doing so. This note isolates a standard Tomita-Takesaki estimate as a model-independent preparation bound. Targets with deeply negative modular energy require large local operators. After rescaling such an operator to a physical contraction, the same estimate becomes a lower bound on postselection overhead. In geometries where the modular Hamiltonian is known, the bound becomes explicit. Bisognano-Wichmann turns it into a boost energy statement for wedges, and the Casini-Huerta-Myers formula gives a stress-tensor version for bounded regions of conformal field theories. Local unitaries can only reach states of nonnegative modular energy. Negative modular sectors require nonunitary or postselected outcomes, giving a preparation cost bound that complements vacuum embezzlement in type III local algebras.