Diagonal Unitary Covariant Superchannels
Dariusz Chruściński, Vivek Pandey, Sohail
TL;DR
This work presents a complete characterization of diagonal unitary covariant (DU-covariant) quantum superchannels, deriving necessary and sufficient CP/TP conditions and a canonical four-component decomposition that separates diagonal and off-diagonal action in the Choi representation. By exploiting covariance under diagonal unitary actions, the authors show that DU-covariant superchannels act independently on diagonal and coherence sectors, effectively combining classical stochastic processing with constrained coherence transformations. The framework is applied to dephasing, amplitude-damping, bit-flip, and Pauli channels, and yields explicit mappings in terms of classical-quantum block structures and bistochastic matrices, including Pauli superchannels that transfer parameters via a matrix $M$. The results provide a practical toolbox for symmetry-restricted higher-order quantum processes, with implications for channel manipulation, resource theories of processes, and open problems such as the PPT$^2$ conjecture.
Abstract
We present a complete characterization of diagonal unitary covariant (DU-covariant) superchannels, i.e. higher-order transformations transforming quantum channels into themselves. Necessary and sufficient conditions for complete positivity and trace preservation are derived and the canonical decomposition describing DU-covariant superchannels is provided. The presented framework unifies and extends known families of covariant quantum channels and enables explicit analysis of their action on physically relevant examples, including amplitude-damping, bit-flip, and Pauli channels. Our results provide a practical toolbox for symmetry-restricted higher-order quantum processes and offer a setting for exploring open problems such as the PPT$^2$ conjecture.