Lattice Theory for Finite Dimensional Hilbert Space with Variables in Z_d

Abstract

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry d2 were expressed as products of lines in near-linear finite geometry 2 p  (where p is a prime). An existence of lattice between any pair of near-linear finite geometry 2 ,( = 1,, ) pj  j k of d2 is confirmed. For q | d , a one-to-one correspondence between the set of subgeometry 2 qj  of d2 and finite geometry 2 d from the subsets of the set {(d)} of divisors of d (where each divisor represents a finite geometry) and set of subsystems {Π(q)} (with variables in q  ) of a finite quantum system Π(d) with variables in d and a finite system from the subsets of the set of divisors of d is established.