Communication Efficient Quantum Secret Sharing

Building upon the recent work in communication efficient classical secret sharing schemes of Bitar and El Rouayheb, we study communication efficient quantum secret sharing schemes. In the standard model of quantum secret sharing, typically one is interested in minimal authorized sets for the reconstruction of the secret. In such a setting, reconstruction requires the communication of all the shares of the corresponding authorized set. If we allow for non-minimal authorized sets, then we can trade off the size of the authorized sets with the amount of communication required for reconstruction. Based on the staircase codes proposed by Bitar and El Rouayheb, we propose a class of quantum threshold secret sharing schemes that are also communication efficient. In the standard model of quantum secret sharing, sharing a qudit using a ((k, 2k − 1)) threshold scheme requires k qudits to be communicated to reconstruct the secret. In the proposed schemes, we can recover the secret by communicating d d−k+1 qudits for every qudit of the secret, where d ≥ k. Further, we show that these schemes are optimal.