Partial ordering of weak mutually unbiased bases in finite quantum systems

Abstract

There has being an enormous work on finite quantum systems with variables in Zd, especially on mutually unbiased bases. The reason for this is due to its wide areas of application. We focus on partial ordering of weak mutually un-biased bases. In it, we studied a partial ordered relation which exists between a subsystem ^(q) and a larger system ^(d) and also, between a subgeometry Gq and larger geometry Gd. Furthermore, we show an isomorphism between: (i) the set {Gd} of subgeometries of a finite geometry Gd and subsets of the set {D(d)} of divisors of d. (ii) the set {hd} of subspaces of a finite Hilbert space Hd and subsets of the set {D(d)} of divisors of d. (iii) the set {Y(d)} of subsystems of a finite quantum system ^(d) and subsets of the set {D(d)} of divisors of d. We conclude this work by showing a duality between lines in finite geometry Gd and weak mutually unbiased bases in finite dimensional Hilbert space Hd.