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Comparing coherence measures for X states: Can quantum states be ordered based on quantum coherence?

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Quantum coherence is an essential resource for quantum information processing and various quantitative measures of it have been introduced. However, the interconnections between these measures are not yet understood properly. Here, using a large set of randomly prepared two-qubit X states (as well as general two-qubit states) and analytically obtained expressions of various measures of coherence (e.g., relative entropy of coherence, l1 norm of coherence, coherence via skew information, and first-order coherence), it is established that these measures of quantum coherence cannot be used to perform ordering of a set of quantum states based on the amount of coherence present in a state. Further, it is shown that for a given value of quantum coherence measured by the relative entropy of coherence, maximally nonlocal mixed states of X type [which are characterized by the maximal violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality] have maximum quantum coherence as measured by l1 norm of coherence. In addition, the amount of coherence measured by l1 norm of coherence for a Werner state is found to be always less than that for a maximally nonlocal mixed state even when they possess an equal amount of coherence measured by the relative entropy of coherence. In our study, we have also explored the connection between the above resource theory-based measures of coherence and the recently introduced first-order coherence and maximum first-order coherence measures for two-qubit systems. We find that while there seems to be no obvious connection between the resource theory-based measures and first-order and maximum first-order coherence, a correlation seems to exist between the maximum first-order coherence and concurrence, both of which are basis independent quantities. These observations could be of use in obtaining a deeper understanding of the interconnections between various measures of quantum coherence.

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Acknowledgements

AP thanks Department of Science and Technology (DST), India for the support provided through the project number EMR/2015/000393. SM thanks Guru Gobind Singh Indraprastha University for STRF. KT thanks the project LO1305 of the Ministry of Education, Youth and Sports of the Czech Republic for support. AP and KT also thank A. Miranowicz, P. Panigrahi and J. Perina Jr. for some fruitful discussion and their interest in this work.

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  1. University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector 16 C, Dwarka, New Delhi, 110078, India

    Sandeep Mishra & Anu Venugopalan

  2. Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP, 201309, India

    Kishore Thapliyal & Anirban Pathak

  3. RCPTM, Joint Laboratory of Optics of Palacky University and Institute of Physics of Academy of Science of the Czech Republic, Faculty of Science, Palacky University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Kishore Thapliyal

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  1. Sandeep Mishra
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Mishra, S., Thapliyal, K., Pathak, A. et al. Comparing coherence measures for X states: Can quantum states be ordered based on quantum coherence? . Quantum Inf Process 18, 295 (2019). https://doi.org/10.1007/s11128-019-2403-6

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