What does this equation mean?

is exact [ 4 ] . Fixing m=2 (a single qubit) and letting n — the effective size of everything else the qubit could be entangled with — grow gives $\langle$ $S_{2,4}$$\rangle$ = 0.5095 nats, $\langle$ $S_{2,16}$$\rangle$ = 0.6465 nats, and $\langle$ $S_{2,64}$$\rangle$ = 0.6814 nats, closing in on the ceiling $\ln$ 2 = 0.6931 nats. The residual gap to that ceiling shrinks by a factor of 3.94 when n quadruples from 4 to 16 , and by 3.98 when it quadruples again from 16 to 64 — converging, as more radiation-sized dimension is added two qubits at a time, on a clean factor of four per doubling of collected qubits, or two per single qubit. This is an exact, hand-computable corroboration of the folklore “each extra collected qubit roughly halves what is left to recover,” not a re-derivation of Hayden and Preskill’s original bound, and it is why $t_L(\epsilon)$ depends on $\epsilon$ only logarithmically: driving $\epsilon$ from 10^{-1} to 10^{-9} costs on the order of thirty extra qubits of radiation, a negligible addition to $t_*$ for any astrophysically sized hole. None of this requires the hole’s dynamics to be literally Haar-random for an unbounded time; Harrow and Low showed that circuits of only polynomial depth already approximate the Haar distribution’s first two moments closely enough for arguments like this one to apply [ 14 ] .