Toric Geometry and String Theory Descriptions of Qudit Systems

In this paper, we propose a new way to approach qudit systems using toric geometry and related topics including the local mirror symmetry used in the string theory compactification. We refer to such systems as (n,d) quantum systems where $n$ and $d$ denote the number of the qudits and the basis states respectively. Concretely, we first relate the (n,d) quantum systems to the holomorphic sections of line bundles on n dimensional projective spaces CP^{n} with degree n(d-1). These sections are in one-to-one correspondence with d^n integral points on a n-dimensional simplex. Then, we explore the local mirror map in the toric geometry language to establish a linkage between the (n,d) quantum systems and type II D-branes placed at singularities of local Calabi-Yau manifolds. (1,d) and (2,d) are analyzed in some details and are found to be related to the mirror of the ALE space with the A_{d-1} singularity and a generalized conifold respectively.