# 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

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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.

### ► 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
http:/​/​arxiv.org/​abs/​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

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