Quantum Weak Coin Flipping

We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias ε if neither player can force the outcome towards their preferred value with probability more than frac12+ε. While it is known that all classical protocols have ε=frac12, Mochon showed in 2007 [arXiv:0711.4114] that quantumly weak coin flipping can be achieved with arbitrarily small bias (near perfect) but the best known explicit protocol has bias 1/6 (also due to Mochon, 2005 [Phys. Rev. A 72, 022341]). We propose a framework to construct new explicit protocols achieving biases below 1/6. In particular, we construct explicit unitaries for protocols with bias approaching 1/10. To go below, we introduce what we call the Elliptic Monotone Align (EMA) algorithm which, together with the framework, allows us to numerically construct protocols with arbitrarily small biases.