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Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.
We consider the problem of quantifying the information shared by a pair of random variables $X_{1},X_{2}$ about another variable $S$. We propose a new measure of shared information, called extractable shared information, that is left monotonic; that
We develop a connection between tripartite information $I_3$, secret sharing protocols and multi-unitaries. This leads to explicit ((2,3)) threshold schemes in arbitrary dimension minimizing tripartite information $I_3$. As an application we show tha
The interactions between three or more random variables are often nontrivial, poorly understood, and yet, are paramount for future advances in fields such as network information theory, neuroscience, genetics and many others. In this work, we propose
This paper considers a distributed storage system, where multiple storage nodes can be reconstructed simultaneously at a centralized location. This centralized multi-node repair (CMR) model is a generalization of regenerating codes that allow for ban
We explore the duality between the simulation and extraction of secret correlations in light of a similar well-known operational duality between the two notions of common information due to Wyner, and Gacs and Korner. For the inverse problem of simul