Quantum-secret-sharing scheme based on local distinguishability of orthogonal multiqudit entangled states

In this study, we propose the concept of judgment space to investigate the quantum-secret-sharing scheme based on local distinguishability (called LOCC-QSS). Because of the proposing of this conception, the property of orthogonal mutiqudit entangled states under restricted local operation and classical communication (LOCC) can be described more clearly. According to these properties, we reveal that, in the previous ($k,n$)-threshold LOCC-QSS scheme, there are two required conditions for the selected quantum states to resist the unambiguous attack: (i) their $k$-level judgment spaces are orthogonal, and (ii) their ($k\ensuremath{-}1$)-level judgment spaces are equal. Practically, if $k<n$ (this is almost always true in the LOCC-QSS scheme), it is very difficult to satisfy the two conditions simultaneously. In the current study, we further establish a simple encoding method, which can concurrently ease the selection of quantum states and ensure the scheme's security, i.e., even if the ($k\ensuremath{-}1$)-level judgment spaces of the selected quantum states are not equal, these states can still be used for defeating the unambiguous attack. With this encoding method, we propose a more secure ($k,n$)-threshold LOCC-QSS scheme, and give two specific examples for illustration.