  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.
