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Equation 73 · Part 3 · The Bit Comes Back Before the Bearing

Symbol n

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

What this part means

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

Its job in the formula

n 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

…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 S2,16S_{2,16}⟩\rangle = 0.6465 nats, and ⟨\langle S2,64S_{2,64}⟩\rangle = 0.6814 nats, closing in on the ceiling ln⁡\ln 2 = 0.6931 nats. The residual gap to that ceiling shrinks by…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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