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Separability for Positive Operators on Tensor Product of Hilbert Spaces

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Abstract

The separability and the entanglement (that is, inseparability) of the composite quantum states play important roles in quantum information theory. Mathematically, a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space. In this paper, in more general frame, the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces. However, not like the quantum state case, there are different kinds of separability for positive operators with different operator topologies. Four types of such separability are discussed; several criteria such as the finite rank entanglement witness criterion, the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established; some methods to construct separable positive operators by operator matrices are provided. These may also make us to understand the separability and entanglement of quantum states better, and may be applied to find new separable quantum states.

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Acknowledgements

The authors wish to thank the referees who kindly offered many nice suggestions to improve the original manuscript.

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Authors and Affiliations

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, P. R. China

    Jin Chuan Hou & Jin Fei Chai

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  1. Jin Chuan Hou
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  2. Jin Fei Chai
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Correspondence to Jin Chuan Hou.

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Supported by National Natural Science Foundation of China (Grant No. 11171249)

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Hou, J.C., Chai, J.F. Separability for Positive Operators on Tensor Product of Hilbert Spaces. Acta. Math. Sin.-English Ser. 37, 893–910 (2021). https://doi.org/10.1007/s10114-021-0427-1

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  • DOI: https://doi.org/10.1007/s10114-021-0427-1

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