Optimal Coin Flipping

This paper studies the problem of simulating a coin of arbitrary real bias q with a coin of arbitrary real bias p with minimum loss of entropy. We establish a lower bound that is strictly greater than the information-theoretic bound. We show that as a function of q, it is an everywhere-discontinuous self-similar fractal. We provide efficient protocols that achieve the lower bound to within any desired accuracy for \((3-\sqrt 5)/2 < p < 1/2\) and achieve it exactly for p = 1/2.

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