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Total, classical and quantum uncertainties generated by channels

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Abstract

States and channels are fundamental and instrumental ingredients of quantum mechanics. Their interplay not only encodes information about states but also reflects uncertainties of channels. In order to quantify intrinsic uncertainties generated by channels, we exploit the action of a channel on an orthonormal basis in the space of observables from three different perspectives. The first concerns the uncertainty generated by a channel via noncommutativity between the Kraus operators of the channel and an orthonormal basis of observables, which can be interpreted as a kind of quantifier of the total uncertainty generated by a channel. The second concerns the uncertainty in terms of the Tsallis-\(2\) entropy of the Jamiołkowski–Choi state associated with the channel via the channel–state duality, which can be interpreted as a quantifier of the classical uncertainty generated by a channel. The third concerns the uncertainty of a channel as the deviation from the identity channel in terms of the Hilbert–Schmidt distance, which can be interpreted as a kind of quantifier of the quantum uncertainty generated by a channel. We reveal basic properties of these quantifiers of uncertainties and establish a relation between them. We identify channels producing the minimal/maximal uncertainties for these three quantifiers. Finally, we explicitly evaluate these uncertainty quantifiers for various important channels, use them to gain insights into the channels from an information-theoretic perspective, and comparatively study the quantifiers.

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Funding

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 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. Department of Engineering Mechanics, Tsinghua University, Beijing, China

    Yizhou Liu

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

    Shunlong Luo

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

    Shunlong Luo

  4. School of Mathematical Sciences, Nanjing Normal University, Nanjing, China

    Yuan Sun

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  1. Yizhou Liu
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  2. Shunlong Luo
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  3. Yuan Sun
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Correspondence to Shunlong Luo.

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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 347–369 https://doi.org/10.4213/tmf10310.

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Liu, Y., Luo, S. & Sun, Y. Total, classical and quantum uncertainties generated by channels. Theor Math Phys 213, 1613–1631 (2022). https://doi.org/10.1134/S0040577922110071

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