Work from the last couple years has made it to the arXiv in the last month or so. Check out the following articles:
Quantum Secure Non-Interactive Reductions (QSNIR) - with Maxwell Gold, Daniel Alabi, Eric Chitambar
I have been thinking for a while now of what a quantum version of classical cryptographic reductions might look like. Our QSNIR framework starts with one of the simplest settings: Reducing classical correlations (specific randomness) to shared bipartite quantum states. We consider a resource from which the correlation can always be correctly derived called a coherent embedding, and then determine the security of these reductions. That is, we quantify how much more information a party can obtain by deviating from the protocol and trying to learn more information from the resource state than if they had been given a black box version of the correlation.
Quantum secret sharing allows parties to reconstruct information that has been shared between them; depending on the scheme, this can be done even without some participants. Which exact combinations of parties can perfectly recover information is defined by the access structure of a quantum secret sharing scheme. In this work we do two things: First, we introduce a type of secret sharing scheme in which any pair of parties that includes a special party called the helper can perfectly recover the information. This allows us to construct examples where just 2 out of n parties can always access the information. Secondly, we use these types of codes to construct a protocol in which a programmer has no information about a secret but remotely decides between various access structures at any point in time. We call this a programmable access structure.
We motivated helper quantum secret sharing in the above companion paper through both the new programmable access structure scheme, as well as information recovery in heterogeneous quantum infrastructure. That is, we think helper codes may be useful in efficiently (a large number of total shareholders, but small number of recovering shareholders) encoding information that protects against the high rate of loss of some type of quantum systems (like photonic qubits) by leveraging the low rates of loss of other systems (like atomic qubits). In this paper, we more closely consider what types of schemes can be constructed. A large number of the examples are derived within the stabilizer formalism of error correcting codes. Our main theorem is a complete characterization of the form of all multi-qubit helper codes.