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Higher dimensional bipartite composite systems with the same density matrix: separable, free entangled, or PPT entangled?

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Abstract

Given the density matrix of a bipartite quantum state, could we decide whether it is separable, free entangled, or PPT entangled? Here, we give a negative answer to this question by providing a lot of concrete examples of \(16 \times 16\) density matrices, some of which are well known. We find that both separability and distillability are dependent on the decomposition of the density matrix. To be more specific, we show that if a given matrix is considered as the density operators of different composite systems, their entanglement properties might be different. In the case of \(16 \times 16\) density matrices, we can look them as both \(2 \otimes 8\) and \(4 \otimes 4\) bipartite quantum states and show that their entanglement properties (i.e., separable, free entangled, or PPT entangled) are completely irrelevant to each other.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 60803028 and 61202045), the Fundamental Research Funds for the Central Universities (No. ZYGX2010X014), and the Science and Technology Foundation of Southwest Petroleum University (No. 2012XJZ034). We are very grateful to the anonymous reviewers for their valuable comments and suggestions which help us to improve the quality of our paper.

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Authors and Affiliations

  1. School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu , 611731, People’s Republic of China

    Wei Cheng

  2. School of Sciences, Southwest Petroleum University, Chengdu , 610500, People’s Republic of China

    Fang Xu

  3. School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu , 611731, People’s Republic of China

    Hua Li

  4. School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu , 611731, People’s Republic of China

    Gang Wang

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  1. Wei Cheng
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  2. Fang Xu
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  3. Hua Li
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  4. Gang Wang
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Correspondence to Wei Cheng.

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Cheng, W., Xu, F., Li, H. et al. Higher dimensional bipartite composite systems with the same density matrix: separable, free entangled, or PPT entangled?. Quantum Inf Process 13, 849–862 (2014). https://doi.org/10.1007/s11128-013-0696-4

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  • DOI: https://doi.org/10.1007/s11128-013-0696-4

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