Phase retrievability of frames and quantum channels

A phase retrievable quantum channel refers to a quantum channel $\Phi: B(H_A)\to B(H_B)$ such that there is a positive operator valued measure (POVM) $\{F_{j}\}$ in $B(H_{B})$ and $\{\Phi^*(F_j)\}$ is a phase retrievable operator valued frame. In this paper we examine the phase retrievable quantum channels in terms of their Kraus representations. For quantum channels $\Phi$ of Choi's rank-$2$, we obtain a necessary and sufficient condition under which it is phase retrievable. For the general case, we present several necessary and/or sufficient conditions. In particular, a necessary and sufficient condition is obtained in terms of the relevant matrix-valued joint spectrum of the Kraus operators. Additionally, we also examine, by examples, the problem of constructing quantum channels such that there exists a minimal number of rank-one observables $\{F_{j}\}$ such that $\{\Phi^*(F_j)\}$ does phase retrieval for $H_A$. Conversely, for a given set of rank-one observables $\{F_{j}\}_{j=1}^{N}$, we present a sufficient condition under which, for every $1\leq r\leq N$ given, a phase retrievable quantum channel $\Phi$ of Choi's rank-$r$ can be explicitly constructed.