Private Capacity of Quantum Channels Induced by Non-stabilizer Environmental States

This paper investigates the conditions under which quantum convolutional channels can reliably transmit private information. Specifically, it examines how the properties of the environmental state—specifically whether it is a stabilizer state or a 'magic' (non-stabilizer) state—influence the private capacity of these channels in discrete-variable quantum systems.

The authors utilize the recently developed quantum convolution theory to analyze channels derived from global unitary operations, such as discrete beam splitters and amplifiers, coupled with fixed environmental states. They apply the stabilizer formalism to categorize environmental states and use the relative entropy of magic (MRM) to quantify the non-stabilizer resources. By calculating the Holevo information and coherent information, they evaluate the private capacity of these channels under different environmental conditions.

The study establishes that for a broad class of convolutional channels, the private capacity vanishes if the environmental state is a stabilizer state or a mixture of stabilizer states. This demonstrates that magic resources are a necessary condition for achieving non-zero private capacity in these systems. Conversely, the authors provide examples showing that non-zero private capacity can be achieved when using specific magic environmental states. Additionally, for discrete beam splitter unitaries, the authors prove that the private capacity is strictly upper-bounded by the magic resource of the environmental state, reinforcing the critical role of non-stabilizer resources in quantum communication.

This work bridges the gap between magic resource theory and quantum communication capacity. By identifying that magic is a necessary resource for private information transmission in these channels, the paper provides a deeper understanding of the operational requirements for secure quantum communication. It also extends the known limitations of stabilizer-based quantum systems, highlighting the necessity of non-stabilizer resources for achieving quantum advantages in information-theoretic tasks.