This slice generally differs from the maximal-volume slice used in holographic complexity, as the maximization is performed over different codimension objects.

To use the theorem, we need to assume $\kappa$ to be $C^1$ for each variable and restrict $\kappa$ to $0<\kappa <{\kappa }_0<\pi /d$.

Here we suppress auxiliary qubits for interior Hilbert space in order to align degrees of freedom for input and output. These qubits are not important in the present context.

The hierarchy was originally motivated in the entropy of evaporating black hole: if radiation is collected for a time interval $T>t_{\text{Page}}$, the radiation entropy grows linearly in time, $S_R\propto T$, and can greatly exceed $S_{\mathrm{BH}}=\log \dim {\mathcal{H}}_B$. In our framework, the interior entanglement entropy $S_A\propto T$ reproduces this growth, suggesting existence of the null states inside the black hole.