Quantum Systems as an Algebraic-geometric Invariant

In [1] and [2] we defined new invariants of both bipartite and multipartite quantum systems under local unitary transformations via algebraic-geometry of determinantal varieties, and gave a new separability criterion based on our new invariants. The purpose of this note is to show how Schmidt number of pure states in bipartite systems, a classical invariant, can be read out from algebraic-geometric invariants. In [1] and [2], we introduced algebraic sets (ie., the zero locus of several multi-variable homogeneous polynomials, see [4]) in the complex projective spaces and the products of complex projective spaces for mixed states in a