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The Independence of Distinguishability and the Dimension of the System

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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) LOCC1, 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 \(K, d_{k}\geqslant 2, h_{k}\geqslant 0\) 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 LOCC1 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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Acknowledgements

We wish to acknowledge professor Zhu-Jun Zheng who gave advices and helped to check the paper.

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  1. College of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China

    Hao Shu

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  1. Hao Shu
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Shu, H. The Independence of Distinguishability and the Dimension of the System. Int J Theor Phys 61, 146 (2022). https://doi.org/10.1007/s10773-022-05127-5

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