Zero-error correctibility and phase retrievability for twirling channels

A twirling channel is a quantum channel induced by a continuous unitary representation $\pi ={\sum }_i^{\oplus }{m}_i{\pi }_i$ on a compact group G, where π_i are inequivalent irreducible representations. Motivated by a recent work [8] on minimal mixed unitary rank of Φ_π, we explore the connections of the independence number, zero-error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel Φ_π with the irreducible representation multiplicities m_i and the irreducible representation dimensions dim $H_{{\pi }_i}$. In particular, we show that the independence number of Φ_π is the sum of the multiplicities, the orthogonal index of Φ_π is exactly the sum of those representation dimensions, and the zero-error capacity is equal to log $\left({\sum }_{i=1}^d{m}_i\right)$. We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for ℂⁿ