Bounding quantum capacities via partial orders and complementarity

Abstract

Quantum capacities are fundamental quantities that are notoriously hard to compute and can exhibit surprising properties such as superadditivity. Thus, a vast amount of literature is devoted to finding tight and computable bounds on these capacities. We add a new viewpoint by giving operationally motivated bounds on several capacities, including the quantum capacity and private capacity of a quantum channel and the one-way distillable entanglement and private key of a quantum state. These bounds are generally phrased in terms of capacity quantities involving the complementary channel or state. As a tool to obtain these bounds, we discuss partial orders on quantum channels and states, such as the less noisy and the more capable order. Our bounds help to further understand the interplay between different capacities, as they give operational limitations on superadditivity and the difference between capacities in terms of the information-theoretic properties of the complementary channel or state. They can also be used as a new approach towards numerically bounding capacities, as discussed with some examples.

Resonant Multilevel Amplitude Damping Channels

“…In this sense a corpus of literature is being built with the aim to produce efficiently computable bounds and approximations of these capacities, see e.g. [38,39,40,41,42,43,44,45]. From this perspective, this paper approaches the evaluation of the quantum capacity Q and the private classical capacity C p of a new class of qudit channels, providing their exact expression for a wide range of noise parameters.…”

Bi-PPT channels are entanglement breaking

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In [HL22], Hirche and Leditzky introduced the notion of bi-PPT channels which are quantum channels that stay completely positive under composition with a transposition and such that the same property holds for one of their complementary channels. They asked whether there are examples of such channels that are not antidegradable.

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“…Hirche and Leditzky [HL22] argue that such examples of channels could be used to exhibit superactivation of the private capacity of quantum channels, since they would have vanishing private capacity. Unfortunately, we can prove that no such example exists.…”