Large family of quantum weak coin-flipping protocols

Each classical public-coin protocol for coin flipping is naturally associated with a quantum protocol for weak coin flipping. The quantum protocol is obtained by replacing classical randomness with quantum entanglement and by adding a cheat detection test in the last round that verifies the integrity of this entanglement. The set of such protocols defines a family which contains the protocol with bias 0.192 previously found by the author, as well as protocols with bias as low as $1∕6$ described herein. The family is analyzed by identifying a set of optimal protocols for every number of messages. In the end, tight lower bounds for the bias are obtained which prove that $1∕6$ is optimal for all protocols within the family.

Figure 1

A depth 3 binary tree.

Figure 2

The curve $(t^2,\sqrt{1-t})$ for $t∊[0,1]$. The convex hull of the curve is the region ${\Lambda }_2$, with the dashed line serving as lower boundary.

Figure 3

A truncated tree equivalent to ${\mathcal{G}}^{\left(4\right)}$.