Quantum Anonymous Secret Sharing with Permutation Invariant Codes

Quantum secret sharing schemes are a family of quantum cryptographic protocols which provide secure quantum encodings, mapping one secret to multiple shares of information such that the original secret cannot be accessed without an authorized set of shares present for decoding. In this work, we describe a protocol that enables sender-anonymity during the secret decoding process. By using permutation-invariant QEC codes along with a set of anonymous quantum transmission algorithms, we construct a quantum anonymous secret sharing scheme that achieves sender-anonymity. We quantify information leakage in ramp quantum secret sharing schemes via the quantum conditional min-entropy, justifying it as a valid measure of leaked information by relating it to the Knill-Laflamme quantum error correction conditions. Finally, we evaluate several permutation-invariant codes using this measure to make observations on the information leakage of intermediate shares for each quantum anonymous secret sharing scheme.