Remarks on Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels

Masanori Ohya, Noboru Watanabe

Shun Watanabe

Two classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.

Christoph Hirche, Felix Leditzky

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 close 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 channel and the one-way distillable entanglement and private key of a bipartite state. Our bounds themselves are generally given by certain capacities of the complementary channel or state. As a tool to obtain these bounds we discuss partial orders on quantum channels, such as the less noisy and the more capable order. Our bounds help to further understand the interplay between different capacities and give operational limitations on superadditivity properties and the difference between capacities. They can also be used as a new approach towards numerically bounding capacities, as discussed with some examples.

Masanori Ohya, Dėnes Petz, Noboru Watanabe

Christoph Hirche, Felix Leditzky

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.

Michael D. Westmoreland, Benjamin W. Schumacher