Abstract
Motivated by the isomorphic correspondence between quantum channels and their Choi states, we define an entropy function of a quantum channel by the entropy of its Choi state. We show that it satisfies all axioms of the entropy function. We also define the relative entropy of a quantum channel and give the relation between the entropy and the relative entropy of a quantum channel. Moreover, comparing with the entropy of a quantum channel which is defined by Gour and Wilde, we find that the two definitions of the channel entropy are equal for covariant channels. As examples, we compute the entropies of the erasure channel, the d-dimensional depolarizing channel, and a particular kind of Werner–Holevo channels, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The datasets analyzed during the current study are available from the corresponding author on reasonable request.
References
Gyongyosi, L., Imre, S.: Properties of the quantum channel. arXiv:1208.1270 [quant-ph] (2012)
Khatri, S., Wilde, M.M.: Principles of quantum communication theory: a modern approach, arXiv:2011.04672 [quant-ph](2021)
Liu, Y., Yuan, X.: Operational resource theory of quantum channels. Phys. Rev. Res. 2, 012035(R) (2020)
Gour, G., Scandolo, C.M.: Entanglement of a bipartite channel. Phys. Rev. Lett. 103, 062422 (2021)
von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)
Schumacher, B.: Quantum coding. Phys. Rev. A 51, 2738 (1995)
Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concertrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)
Gour, G., Wilde, M.M.: Entropy of a quantum channel. Phys. Rev. Research 3, 023096 (2021)
Cooney, T., Mosonyi, M., Wilde, M.M.: Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication. Commun. Math. Phys. 344, 797 (2016)
Leditzky, F., Kaur, E., Datta, N., Wilde, M.M.: Approaches for approximate additivity of the Holevo information of quantum channels. Phys. Rev. A 97, 012332 (2018)
Gour, G.: Comparison of quantum channels with superchannels. IEEE Trans. Inf. Theory 65, 5880 (2019)
Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40, 147 (1975)
Helevo, A.S.: Remarks on the classical capacity of quantum channel. arxiv:quantum-ph/0212025 (2002)
King, C.: The capacity of the quantum depolarizing channel. IEEE Trans. Inf. Theory 49(1), 221 (2003)
Werner, R.F., Holevo, A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353 (2002)
Acknowledgements
This work was supported by Key Research and Development Project of Guang Dong Province under Grant No. 2020B0303300001 and the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B151 5310016.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chu, Y., Huang, F., Li, MX. et al. An entropy function of a quantum channel. Quantum Inf Process 22, 27 (2023). https://doi.org/10.1007/s11128-022-03778-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-022-03778-1