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Equation 78 · Part 2 · The Bit Comes Back Before the Bearing

=

⟨S2,64⟩=0.6814\langle S_{2,64}\rangle = 0.6814
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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 a factor of 3.94 when n quadruples from 4 to 16 , and by 3.98 when it quadruples again from 16 to 64 — converging, as more radiation-sized dimension is added two qubits at a time, on a clean factor of four per doubling of collected qubits, or two per single qubit. This is an exact, hand-computable corroboration of the folklore “each extra…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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