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Equation 73 · Part 4 · The Bit Comes Back Before the Bearing

Symbol m

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

What this part means

m occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Its job in the formula

m occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

…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 ] . Fixing m=2 (a single qubit) and letting n — the effective size of everything else the qubit could be entangled with — grow gives ⟨\langle S2,4S_{2,4}⟩\rangle = 0.5095 nats, ⟨\langle…

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Sources cited in the surrounding passage

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