Quantum flags, and new bounds on the quantum capacity of the depolarizing channel

Abstract

A new bound for the quantum capacity of the $d$-dimensional depolarizing channels is presented. Our derivation makes use of a flagged extension of the map where the receiver obtains a copy of a state ${\sigma }_0$ whenever the messages are transmitted without errors, and a copy of a state ${\sigma }_1$ when instead the original state gets fully depolarized. By varying the overlap between the flags states, the resulting transformation nicely interpolates between the depolarizing map (when ${\sigma }_0={\sigma }_1$), and the $d$-dimensional erasure channel (when ${\sigma }_0$ and ${\sigma }_1$ have orthogonal support). In our analysis we compute the product-state classical capacity, the entanglement assisted capacity and, under degradability conditions, the quantum capacity of the flagged channel. From this last result we get the upper bound for the depolarizing channel, which by a direct comparison appears to be tighter than previous available results for $d>2$, and for $d=2$ it is tighter in an intermediate regime of noise. In particular, in the limit of large $d$ values, our findings presents a previously unnoticed $\mathcal{O}(1)$ correction.

arXiv comment8 pages, 2 figures, changing the template with respect to the previous version

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• Entropic singularities give rise to quantum transmission
• Pursuing the fundamental limits for quantum communication
• Aggregating Quantum Networks
• Quantum-enabled communication without a phase reference
• Resonant Multilevel Amplitude Damping Channels
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• Bounding the quantum capacity with flagged extensions
• Superclassical non-Markovian open quantum dynamics
• Additivity of quantum capacities in simple non-degradable quantum channels
• Testing identity of collections of quantum states: sample complexity analysis
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