Constraints on the mixing of states on bipartite quantum systems

We give necessary conditions for the mixing of states on bipartite quantum systems, which are independent of the eigenvalues of these mixed states and based on the algebraic-geometric invariants introduced in [1]. These are further constraints except the previously known constraints based on majorization of eigenvalues ([2,3,4]). One implication of our results is that for some special mixed states, only mixed states in a measure zero set can be used to mix to get them. As indicated in [1] and [5] for many physical problems in composite quantum systems the majorization of some numerical invariants such as eigenvalues is not sufficient. The general problem of quantum operations was considerd in [6],[7],[8], essentially new aspects from the view of quantum information processing was added recently in [9],[10],[11],[12],[13]. All possible physical operations on quantum systems can be divided into (1)state-to-ensemble operations and (2)ensemble-to-state operations(mixing), where ensemble is a set of mixed states {ρi} with ascribed possibilities {pi}. The class (2) can be described by taking convex combination