Quantum secret sharing and random hopping: Using single states instead of entanglement

Quantum secret sharing (QSS) protocols between $N$ players, for sharing classical secrets, either use multipartite entangled states or use sequential manipulation of single $d$-level states only when $d$ is prime (A. Tavakoli et al., arXiv:1501.05582). We propose a sequential scheme which is valid for any value of $d$. In contrast to A. Tavakoli et al. whose efficiency (number of valid rounds) is $\frac{1}{d}$, the efficiency of our scheme is $\frac{1}{2}$ for any $d$. This, together with the fact that in the limit $d\ensuremath{\longrightarrow}\ensuremath{\infty}$ the scheme can be implemented by continuous variable optical states, brings the scheme into the domain of present day technology.