Equation 69 · The Bit Comes Back Before the Bearing
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the run for about. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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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 , collecting only a handful more qubits of radiation than were thrown in — not a number that grows with the hole’s size — suppresses 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,
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