  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.
