**Optimal semi-quantum secret sharing schemes via stabilizer codes and twirling of symplectic structures**

As recently shown in [quant-ph/1108.5541], any quantum error-correcting code can be converted into a perfect "hybrid" quantum secret sharing scheme by allowing the sharing of extra classical bits between the dealer and the players. An advantage of this scheme is that it allows the players' quantum shares to be of smaller dimension than the dimension of the encoded secret, which is impossible for regular perfect quantum secret sharing protocols. Whenever the underlying quantum error correcting code is a stabilizer code (this being the case for the vast majority of known quantum error-correcting codes), I provide a general scheme of reducing the amount of classical communication required, then prove that my scheme is optimal for the stabilizer code being used. The optimality proof is based on the fact that the correlations between the dealer and the players can be fully described by an "information group" [Phys. Rev. A 81, 032326 (2010)]; the symplectic structure of the information group effectively gives the minimum number of classical bits required. Finally I provide an explicit protocol that achieves this minimum by employing the notion of "twirling" (or scrambling) the information group. The results are general and valid for any stabilizer code. I will illustrate the results by simple examples.