addition

The passage around this formula

Hayden and Preskill’s own result, sharpened into an explicit, efficient decoding circuit by Yoshida and Kitaev, is that once scrambling of this kind has run for about $t_*$ , collecting only a handful more qubits of radiation than were thrown in — not a number that grows with the hole’s size — suppresses $\epsilon_L$ exponentially in that small handful [ 15 ] . The mechanism behind that exponential suppression is worth seeing exactly rather than taking on faith, and Page’s own formula for the average entanglement entropy of a subsystem of a random pure state supplies it. For an m -dimensional subsystem embedded in an mn -dimensional Haar-random pure state, $\langle S_{m,n}\rangle = \sum_{k=n+1}^{mn}\frac1k \;-\; \frac{m-1}{2n}$. is exact [ 4 ] .…

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