Quantum Secret Sharing (QSS) is a cryptographic protocol to share a quantum secret among multiple parties. In ((k,n)) threshold QSS schemes with n parties, sets of k or more parties are allowed to recover the secret (through a combiner downloading their shares). Sets of k-1 or less parties should have no information on the secret. We have earlier introduced a class of threshold QSS schemes called communication efficient threshold QSS schemes (CE-QTS) where the communication cost for secret recovery is reduced when the combiner contacts d>k shares for some fixed d. We also provided an optimal construction for CE-QTS schemes where the communication cost drops from k qudits to d/(d-k+1) qudits.

In our recent work, we extend this class of schemes to introduce the universal CE-QTS schemes. In these schemes, any increase in the number of parties accessed will give a reduction in communication cost i.e. these schemes are communication efficient or all possible d>k. We propose a quantum information theoretic model to study CE-QTS schemes and use this model to prove the bound on communication cost. We provide an optimal construction based on Staircase codes for universal CE-QTS schemes. We also provide a framework based on ramp QSS schemes to design more constructions for fixed d and universal CE-QTS schemes.