Polymatroids and polyquantoids
This work addresses foundational mathematical structures for quantum information theory, providing a framework for analyzing quantum entropy and secret sharing, but it appears incremental as it extends known classical concepts to the quantum domain.
The paper tackles the problem of extending combinatorial structures from classical entropy functions to quantum entropy functions, introducing polyquantoids and quantoids as quantum analogs of polymatroids and matroids, and linking them via linear mappings and duality, with applications to ideal quantum secret sharing schemes described by selfdual matroids.
When studying entropy functions of multivariate probability distributions, polymatroids and matroids emerge. Entropy functions of pure multiparty quantum states give rise to analogous notions, called here polyquantoids and quantoids. Polymatroids and polyquantoids are related via linear mappings and duality. Quantum secret sharing schemes that are ideal are described by selfdual matroids. Expansions of integer polyquantoids to quantoids are studied and linked to that of polymatroids.