The Independence of Distinguishability and the Dimension of the System

• URL: http://arxiv.org/abs/2010.03120v7
• Date: Fri, 17 Jun 2022 14:09:07 GMT
• Title: The Independence of Distinguishability and the Dimension of the System
• Authors: Hao Shu
• Abstract summary: We show that if a set of states is indistinguishable in $otimes _k=1K Cd _k$, then it is indistinguishable even being viewed in $otimes _k=1K Cd _k+h _k$. Our result is suitable for general states in general systems.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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