# 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

### 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.

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### Cited by

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

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

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

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

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

[6] 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).

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

[8] 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).

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