Phase retrievability of frames and quantum channels

A phase retrievable quantum channel refers to a quantum channel $\Phi :B\left(H_A\right)\to B\left(H_B\right)$ such that there is a positive operator valued measure (POVM) $\left\{F_j\right\}$ in $B\left(H_B\right)$ and $\left\{{\Phi }^{⁎}\right(F_j\left)\right\}$ 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 Φ of Choi 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 $\left\{F_j\right\}$ such that $\left\{{\Phi }^{⁎}\right(F_j\left)\right\}$ does phase retrieval for $H_A$. Conversely, for a given set of rank-one observables ${\left\{F_j\right\}}_{j=1}^N$, we present a sufficient condition under which, for every given $1\le r\le N$, a phase retrievable quantum channel Φ of Choi rank-r can be explicitly constructed.