Maximal Privacy without Coherence

Abstract

Privacy is a fundamental feature of quantum mechanics. A coherently transmitted quantum state is inherently private. Remarkably, coherent quantum communication is not a prerequisite for privacy: there are quantum channels that are too noisy to transmit any quantum information reliably that can nevertheless send private classical information. Here, we ask how much private classical information a channel can transmit if it has little quantum capacity. We present a class of channels N[subscript d] with input dimension d[superscript 2], quantum capacity Q(N[subscript d]) ≤ 1, and private capacity P(N[subscript d])= log d. These channels asymptotically saturate an interesting inequality P(N) ≤ (1/2)[log d[subscript A] + Q(N)] for any channel N with input dimension d[subscript A] and capture the essence of privacy stripped of the confounding influence of coherence.