The Independence of Distinguishability and the Dimension of the System

The are substantial studies on distinguishabilities, especially local distinguishability, of quantum states. It is shown that a necessary condition of a local distinguishable state set is the total Schmidt rank not larger than the system dimension. However, if we view states in a larger system, the restriction will be invalid. Hence, a nature problem is that can indistinguishable states become distinguishable by viewing them in a larger system without employing extra resources. In this paper, we consider this problem for (perfect or unambiguous) LOCC$_{1}$, PPT and SEP distinguishabilities. We demonstrate that if a set of states is indistinguishable in $\otimes _{k=1}^{K} C^{d _{k}}$, then it is indistinguishable even being viewed in $\otimes _{k=1}^{K} C^{d _{k}+h _{k}}$, where $K, d _{k}\geqslant2, h _{k}\geqslant0$ are integers. This shows that such distinguishabilities are properties of states themselves and independent of the dimension of quantum system. Our result gives the maximal numbers of LOCC$_{1}$ distinguishable states and can be employed to construct a LOCC indistinguishable product basis in general systems. Our result is suitable for general states in general systems. For further discussions, we define the local-global indistinguishable property and present a conjecture.