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Restricted Quantum Focusing (rQFC)

Establish that, at the semiclassical level, if the quantum expansion Theta vanishes on a spacelike section of a null hypersurface, then Theta cannot become positive away from that section, i.e., prove that Theta=0 implies partial_Theta/partial_lambda <= 0 locally along the null generators.

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Background

The restricted version is a weaker statement than the full QFC but suffices for all currently known applications such as entanglement wedge nesting and the generalized second law. The thesis proves rQFC in JT gravity and presents violations of the unrestricted QFC, arguing rQFC may be the fundamental constraint.

References

Conjecture [Restricted Quantum Focusing] At the semiclassical level, if the quantum expansion vanishes on some spacelike Section of a null hypersurface $\mathcal{N}$, it cannot increase away from this section: \begin{equation} \Theta[V_{\lambda},\vec{y}]=0 \quad \Rightarrow \quad \frac{\partial\Theta[V_{\lambda},\vec{y}]}{\partial\lambda}\leq 0. \end{equation}

— Information-theoretic constraints in quantum gravity and cosmology (2510.15787 - Franken, 17 Oct 2025) in Chapter: Entropy and energy bounds in semiclassical gravity, Section: Quantum Focusing