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Gram Matrices of Mixed-State Ensembles

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Abstract

Gram matrices arise naturally in the consideration of pair-wise overlap of a family of vectors or quantum pure states, and play an important role in synthesizing information of a family of pure states. Since a quantum ensemble in general consists of mixed states, it is desirable to extend the concept of Gram matrix to the case of mixed states. By employing two prominent notions of overlap between mixed states, i.e., quantum affinity and quantum fidelity, one can readily extend the concept of Gram matrices of pure-state ensembles to that of mixed-state ensembles. We study these two extended versions of Gram matrices and reveal their fundamental properties. It is remarkable that while Gram matrix based on quantum affinity is non-negative definite (and thus can be regarded as a quantum state in a fictious system), the one based on quantum fidelity may fail to be non-negative definite. As applications of Gram matrices of mixed-state ensembles, we introduce two quantifiers of quantumness of ensembles via coherence of the corresponding Gram matrices, investigate their properties, and illustrate them in the qubit case.

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Acknowledgements

This work was supported by the National Key R&D Program of China, Grant No. 2020YFA0712700, the National Natural Science Foundation of China, Grant Nos. 11875317, 12005104, and 61833010, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China, Grant No. 20KJB140028.

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Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Yuan Sun

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shunlong Luo & Xiangyun Lei

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Shunlong Luo & Xiangyun Lei

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  1. Yuan Sun
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  2. Shunlong Luo
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  3. Xiangyun Lei
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Correspondence to Shunlong Luo.

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Sun, Y., Luo, S. & Lei, X. Gram Matrices of Mixed-State Ensembles. Int J Theor Phys 60, 3211–3224 (2021). https://doi.org/10.1007/s10773-021-04908-8

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  • DOI: https://doi.org/10.1007/s10773-021-04908-8

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