Zero Error Correctibility and Phase Retrievability for Twirling Channels

A twirling channel is a quantum channel induced by a continuous unitary representation $\pi ={\sum }_i^{\oplus }{m}_i{\pi }_i$, where ${\pi }_i$ are inequivalent irreducible representations. Motivated by a recent work \cite{Twirling} on minimal mixed unitary rank of ${\mathrm{\Phi }}_{\pi }$, we explore the connections of the independence number, zero error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel ${\mathrm{\Phi }}_{\pi }$ with the irreducible representation multiplicities $m_i$, the irreducible representation dimensions $\dim H_{{\pi }_i}$. In particular we show that the independence number of ${\mathrm{\Phi }}_{\pi }$ is the sum of the multiplicities, the orthogonal index of ${\mathrm{\Phi }}_{\pi }$ is exactly the sum of those representation dimensions, and the zero-error capacity is equal to $\log ({\sum }_{i=1}^d{m}_i)$. We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for $C^n$.