A quantum secret sharing (QSS) scheme is a quantum cryptographic protocol in which a secret is securely distributed among multiple parties. Despite extensive research on these schemes, there has been very little research on optimizing the quantum communication cost during secret recovery. Here we address this problem by initiating the study on communication efficient quantum secret sharing schemes.

In a ((k,n)) quantum threshold secret sharing (QTS) scheme, a secret is encoded and distributed among n parties such that any k of them can collaborate to recover the secret and any set of k or less parties has no information about the secret. In QTS schemes, the communication required during secret recovery from k parties is at least km qudits for a secret having m qudits. We introduce the ((k,n,d)) communication efficient QTS (CE-QTS) schemes which reduce the communication cost while accessing any d (>k) parties. We give two optimal constructions for CE-QTS schemes, one based on Staircase codes and another using concatenation of ramp schemes. A quantum information theoretic model is also provided to study these schemes. We then propose a more flexible class of CE-QTS schemes called the universal CE-QTS schemes in which the communication cost reduces progressively as the number of accessed parties increases. The bounds and constructions for universal CE-QTS schemes are also studied.
Later we propose the communication efficient quantum secret sharing (CE-QSS) schemes which include threshold as well as non-threshold QSS schemes. We generalize our results on CE-QTS schemes for obtaining bounds and constructions in CE-QSS schemes. Particularly, we generalize the Staircase codes based construction for CE-QTS schemes to provide a framework for constructing CE-QSS schemes using extended Calderbank-Shor-Steane codes.