Polymatroids and polyquantoids

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.

[1]  James G. Oxley,et al.  Matroid theory , 1992 .

[2]  V. Buzek,et al.  Quantum secret sharing , 1998, quant-ph/9806063.

[3]  Frantisek Matús,et al.  Infinitely Many Information Inequalities , 2007, 2007 IEEE International Symposium on Information Theory.

[4]  Geoff Whittle.  Duality in Polymatroids and Set Functions , 1992, Comb. Probab. Comput..

[5]  T. Helgason.  Aspects of the theory of hypermatroids , 1974 .

[6]  C. McDiarmid.  Rado's theorem for polymatroids , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  F. Mat.  Two Constructions on Limits of Entropy Functions , 2007, IEEE Trans. Inf. Theory.

[8]  H. Narayanan.  Submodular functions and electrical networks , 1997 .

[9]  Zhen Zhang,et al.  On Characterization of Entropy Function via Information Inequalities , 1998, IEEE Trans. Inf. Theory.

[10]  László Lovász,et al.  Submodular functions and convexity , 1982, ISMP.

[11]  A. Winter,et al.  A New Inequality for the von Neumann Entropy , 2004, quant-ph/0406162.

[12]  D. Gottesman.  Theory of quantum secret sharing , 1999, quant-ph/9910067.

[13]  H. Q. Nguyen.  Semimodular functions and combinatorial geometries , 1978 .

[14]  Pradeep Kiran Sarvepalli,et al.  Matroids and Quantum Secret Sharing Schemes , 2009, ArXiv.

[15]  Nicholas Pippenger,et al.  The inequalities of quantum information theory , 2003, IEEE Trans. Inf. Theory.

[16]  Satoru Fujishige,et al.  Submodular functions and optimization , 1991 .

[17]  R. Cleve,et al.  HOW TO SHARE A QUANTUM SECRET , 1999, quant-ph/9901025.

[18]  Pradeep Kiran Sarvepalli.  Quantum codes and symplectic matroids , 2014, 2014 IEEE International Symposium on Information Theory.

[19]  Ernest F. Brickell,et al.  On the classification of ideal secret sharing schemes , 1989, Journal of Cryptology.

[20]  Andreas J. Winter,et al.  Infinitely Many Constrained Inequalities for the von Neumann Entropy , 2011, IEEE Transactions on Information Theory.