On the complementary quantum capacity of the depolarizing channel

Debbie Leung; John Watrous

doi:10.22331/q-2017-09-19-28

On the complementary quantum capacity of the depolarizing channel

¹Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo
²Institute for Quantum Computing and School of Computer Science, University of Waterloo

Published:	2017-09-19, volume 1, page 28
Eprint:	arXiv:1510.01366v3
Doi:	https://doi.org/10.22331/q-2017-09-19-28
Citation:	Quantum 1, 28 (2017).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:

v4 corrects errors in equation (38)

Abstract

The qubit depolarizing channel with noise parameter $\eta$ transmits an input qubit perfectly with probability $1-\eta$, and outputs the completely mixed state with probability $\eta$. We show that its complementary channel has positive quantum capacity for all $\eta\gt 0$. Thus, we find that there exists a single parameter family of channels having the peculiar property of having positive quantum capacity even when the outputs of these channels approach a fixed state independent of the input. Comparisons with other related channels, and implications on the difficulty of studying the quantum capacity of the depolarizing channel are discussed.

Featured image: The epolarizing channel where the original depolarizing channel has an X, Y, or Z error each with probability $\epsilon/3$.

► BibTeX data

@article{Leung2017complementary,
  doi = {10.22331/q-2017-09-19-28},
  url = {https://doi.org/10.22331/q-2017-09-19-28},
  title = {On the complementary quantum capacity of the depolarizing channel},
  author = {Leung, Debbie and Watrous, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {1},
  pages = {28},
  month = sep,
  year = {2017}
}

► References

[1] Charles Bennett, David DiVincenzo, and John Smolin. Capacities of quantum erasure channels. Physical Review Letters, 78 (16): 3217, 1997. 10.1103/PhysRevLett.78.3217.
https://doi.org/10.1103/PhysRevLett.78.3217

[2] Dagmar Bruß, David DiVincenzo, Artur Ekert, Christopher Fuchs, Chiara Macchiavello, and John Smolin. Optimal universal and state-dependent quantum cloning. Physical Review A, 57 (4): 2368, 1998. 10.1103/PhysRevA.57.2368.
https://doi.org/10.1103/PhysRevA.57.2368

[3] Toby Cubitt, Mary Beth Ruskai, and Graeme Smith. The structure of degradable quantum channels. Journal of Mathematical Physics, 49 (10): 102104, 2008. 10.1063/1.2953685.
https://doi.org/10.1063/1.2953685

[4] Toby Cubitt, David Elkouss, William Matthews, Maris Ozols, David Pérez-García, and Sergii Strelchuk. Unbounded number of channel uses may be required to detect quantum capacity. Nature Communications, 6, 2015. 10.1038/ncomms7739.
https://doi.org/10.1038/ncomms7739

[5] Igor Devetak. The private classical capacity and quantum capacity of a quantum channel. IEEE Transactions on Information Theory, 51 (1): 44–55, 2005. 10.1109/TIT.2004.839515.
https://doi.org/10.1109/TIT.2004.839515

[6] David DiVincenzo, Peter Shor, and John Smolin. Quantum-channel capacity of very noisy channels. Physical Review A, 57 (2): 830–839, 1998. 10.1103/PhysRevA.57.830.
https://doi.org/10.1103/PhysRevA.57.830

[7] Paweł Horodecki, Michał Horodecki, and Ryszard Horodecki. Binding entanglement channels. Journal of Modern Optics, 47 (2-3): 347–354, 2000. 10.1080/09500340008244047.
https://doi.org/10.1080/09500340008244047

[8] Felix Leditzky, Debbie Leung, and Graeme Smith. Quantum and private capacities of low-noise channels. Unpublished manuscript, available as arXiv:1705.04335. URL http://arxiv.org/abs/1705.04335.
arXiv:1705.04335

[9] Seth Lloyd. Capacity of the noisy quantum channel. Physical Review A, 55: 1613–1622, 1997. 10.1103/PhysRevA.55.1613.
https://doi.org/10.1103/PhysRevA.55.1613

[10] Michael Nielsen and Isaac Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. 10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667

[11] Asher Peres. Separability criterion for density matrices. Physical Review Letters, 77 (8): 1413–1415, 1996. 10.1103/PhysRevLett.77.1413.
https://doi.org/10.1103/PhysRevLett.77.1413

[12] Claude Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27: 379–423, 1948. 10.1002/j.1538-7305.1948.tb01338.x.
https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

[13] Peter Shor. The quantum channel capacity and coherent information. Lecture notes, MSRI Workshop on Quantum Computation, 2002. Available online at http://www.msri.org/realvideo/ln/msri/2002/quantumcrypto/shor/1/.
http://www.msri.org/realvideo/ln/msri/2002/quantumcrypto/shor/1/

[14] Graeme Smith and John Smolin. Detecting incapacity of a quantum channel. Physical Review Letters, 108 (23): 230507, 2012. 10.1103/PhysRevLett.108.230507.
https://doi.org/10.1103/PhysRevLett.108.230507

[15] Shun Watanabe. Private and quantum capacities of more capable and less noisy quantum channels. Physical Review A, 85: 012326, 2012. 10.1103/PhysRevA.85.012326.
https://doi.org/10.1103/PhysRevA.85.012326

Cited by

[1] Mark M. Wilde, "Entanglement cost and quantum channel simulation", Physical Review A 98 4, 042338 (2018).

[2] Satvik Singh and Nilanjana Datta, "Detecting positive quantum capacities of quantum channels", npj Quantum Information 8 1, 50 (2022).

[3] Alireza Tehrani and Rajesh Pereira, "The coherent information on the manifold of positive definite density matrices", Journal of Mathematical Physics 62 4, 042201 (2021).

[4] Felix Leditzky, Debbie Leung, and Graeme Smith, "Quantum and Private Capacities of Low-Noise Channels", Physical Review Letters 120 16, 160503 (2018).

[5] Felix Leditzky, Debbie Leung, and Graeme Smith, "Dephrasure Channel and Superadditivity of Coherent Information", Physical Review Letters 121 16, 160501 (2018).

[6] Vikesh Siddhu, "Entropic singularities give rise to quantum transmission", Nature Communications 12 1, 5750 (2021).

[7] Bong-Hyun Kim, S. Madhavi, and Dalin Zhang, "Method for Quantum Denoisers Using Convolutional Neural Network", Computational Intelligence and Neuroscience 2022, 1 (2022).

[8] Xin Wang, Kun Fang, and Runyao Duan, "Semidefinite Programming Converse Bounds for Quantum Communication", IEEE Transactions on Information Theory 65 4, 2583 (2019).

[9] You-neng Guo, Cheng Yang, Qing-long Tian, Guo-you Wang, and Ke Zeng, "Local quantum uncertainty and interferometric power for a two-qubit system under decoherence channels with memory", Quantum Information Processing 18 12, 375 (2019).

[10] Ivan Sergeev, "Generalizations of 2-Dimensional Diagonal Quantum Channels with Constant Frobenius Norm", Reports on Mathematical Physics 83 3, 349 (2019).

[11] Quntao Zhuang, Elton Yechao Zhu, and Peter W. Shor, "Additive Classical Capacity of Quantum Channels Assisted by Noisy Entanglement", Physical Review Letters 118 20, 200503 (2017).

The above citations are from Crossref's cited-by service (last updated successfully 2023-09-28 08:46:50) and SAO/NASA ADS (last updated successfully 2023-09-28 08:46:51). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.