ZERO-ERROR CORRECTIBILITY AND PHASE RETRIEVABILITY FOR TWIRLING CHANNELS

A twirling channel is a quantum channel induced by a continuous unitary representation pi Sigma(circle plus)(i)m(i)pi(i) on a compact group G, where pi(i) are inequivalent irreducible representations. Motivated by a recent work [8] on minimal mixed unitary rank of phi(pi), we explore the connections of the independence number, zero-error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel phi(pi) 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 phi(pi) is the sum of the multiplicities, the orthogonal index of phi(pi) is exactly the sum of those representation dimensions, and the zero-error capacity is equal to log ( n-ary Sigma(d)(i=1)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