Approximate reconstructability of quantum states and noisy quantum secret sharing schemes

We introduce and analyse approximate quantum secret sharing in a formal cryptographic setting, wherein dealer encodes distributes to players such that authorized structures (sets of subsets players) can approximately reconstruct the omnipotent adversarial agents controlling non-authorized are denied secret. In particular, viewing map encoding shares for an structure as channel, we show reconstructability by these is possible if only information leakage, given terms certain entanglement-assisted capacity complementary channel outside environment, small.