Explicit quantum weak coin flipping protocols with arbitrarily small bias

          @misc{ scivideos_PIRSA:20030088,
            doi = {10.48660/20030088},
            url = {https://pirsa.org/20030088},
            author = {Arora, Atul},
            keywords = {Quantum Information},
            language = {en},
            title = {Explicit quantum weak coin flipping protocols with arbitrarily small bias},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2020},
            month = {mar},
            note = {PIRSA:20030088 see, \url{https://scivideos.org/index.php/pirsa/20030088}}
          }
          

Abstract

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. A weak coin-flipping protocol has a bias ϵ if neither player can force the outcome towards their preferred value with probability more than 1/2+ϵ. While it is known that classically ϵ=1/2, Mochon showed in 2007 [arXiv:0711.4114] that quantumly weak coin flipping can be achieved with arbitrarily small bias, i.e. ϵ(k)=1/(4k+2) for arbitrarily large k, and he proposed an explicit protocol approaching bias 1/6. So far, the best known explicit protocol is the one by Arora, Roland and Weis, with ϵ(2)=1/10 (corresponding to k=2) [STOC'19, p. 205-216]. In the current work, we present the construction of protocols approaching arbitrarily close to zero bias, i.e. ϵ(k) for arbitrarily large k. We connect the algebraic properties of Mochon's assignments---at the heart of his proof of existence---with the geometric properties of the unitaries whose existence he proved. It is this connection that allows us to find these unitaries analytically.