Convex rigid cover method in studies of quantum pure-states of many continuous variables

Abstract

In this paper we prove that every pure-state $\Psi ^{(N)}$ of N (N$\geqslant 3)$ continuous variables corresponds to a pair of convex rigid covers (CRCs) structures in the continuous-dimensional Hilbert-Schmidt space. Next we strictly define what are the partial separability and ordinary separability, and discuss how to use CRCs to describe various separability. We discuss the problem of the classification of $\Psi ^{(N)}$ and give a kinematical explanation of the local unitary operations acting upon $\Psi ^{(N)}$. Thirdly, we discuss the invariants of classes and give a possible physical explanation.