The distinguishability of quantum states is important in quantum information theory and has been considered by authors. However, there were no general results considering whether a set of indistinguishable states become distinguishable by viewing them in a larger system without employing extra resources. In this paper, we consider this question for LOCC$_{1}$, PPT and SEP distinguishabilities of states. We use mathematical methods to show 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 LOCC$_{1}$, PPT and SEP distinguishabilities of states are properties of states themselves and independent of the dimension of quantum system. With these results, we can give the maximal number of states which can be distinguished via LOCC$_{1}$ and construct a LOCC indistinguishable basis of product states in a general system. Note that our results are also suitable for unambiguous discriminations. Also, we give a conjecture for other distinguishabilities and a framework by defining the Local-global indistinguishable property. Instead of considering such problems for special sets or special systems, we consider the problems for general states in general systems, which have not been considered yet, for our knowledge.
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