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Equation 73 · Part 14 · The Bit Comes Back Before the Bearing

Denominator: 2n

⟨Sm,n⟩=∑k=n+1mn1k  −  m−12n\langle S_{m,n}\rangle = \sum_{k=n+1}^{mn}\frac1k \;-\; \frac{m-1}{2n}
2n2n

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

2n occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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∗t_* , collecting only a handful more qubits of radiation than were thrown in — not a number that grows with the hole’s size — suppresses ϵL\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, ⟨Sm,n⟩=∑k=n+1mn1k  −  m−12n\langle S_{m,n}\rangle = \sum_{k=n+1}^{mn}\frac1k \;-\; \frac{m-1}{2n}. is exact [ 4 ] .…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.