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Equation 73 · Part 11 · The Bit Comes Back Before the Bearing

Starting index or lower bound: k=n+1

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

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

k=n+1 appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the surrounding passage

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