Quantum and private capacities of low-noise channels

Abstract

We determine both the quantum and the private capacities of low-noise quantum channels to leading orders in the channel's distance to the perfect channel. It has been an open problem for more than 20 years to determine the capacities of some of these low-noise channels such as the depolarizing channel. We also show that both capacities are equal to the single-letter coherent information of the channel, again to leading orders. We thus find that, in the low noise regime, super-additivity and degenerate codes have negligible benefit for the quantum capacity, and shielding does not improve the private capacity beyond the quantum capacity, in stark contrast to the situation when noisier channels are considered.

23 pages, 4 figures, comments welcome! v2: improved bounds on degradability parameters and capacities of depolarizing channel and XZ channel (see also ancillary files 'depol-deg-bound.nb' and 'XZ-deg-bound.nb'), extension of results to generalized low-noise channels. v3: strengthened version of Lemma 1

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• Generic nonadditivity of quantum capacity in simple channels
• Entropic singularities give rise to quantum transmission
• Pursuing the fundamental limits for quantum communication
• Optimal uniform continuity bound for conditional entropy of classical--quantum states
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• Bounding quantum capacities via partial orders and complementarity
• Capacities of the covariant Pauli channel
• Entanglement catalysis for quantum states and noisy channels
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• Continuity of entropies via integral representations
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• Information-theoretic aspects of the generalized amplitude damping channel
• Bounding the quantum capacity with flagged extensions
• The superadditivity effects of quantum capacity decrease with the dimension for qudit depolarizing channels
• Exact Quantum Capacity of Decohering Channels in Arbitrary Dimensions