Dynamical quantum state tomography with time-dependent channels

In this paper, we establish a dynamical quantum state tomography framework. Under this framework, it is feasible to obtain complete knowledge of any unknown state of a d-level system via only one operator of a special type of positive operator-valued measure (POVM) in dimension d. We define a new channel, referred to as the time-dependent average channel. Utilizing this channel, we show that we can acquire a collection of projective operators that is informationally complete (IC) and thus obtain the corresponding informationally complete POVMs (IC-POVMs). Zauner conjectured that for any dimension d there exists a fiducial vector, such that all remaining d2-1 elements of the desired symmetric informationally complete POVM (SIC-POVM) can be obtained by acting on said vector with unitary matrices representing elements of the Weyl-Heisenberg group. We show that under certain condition, it is possible to obtain infinite families of projective operators that are IC, and obtain infinite families of corresponding IC-POVMs; otherwise, Zauner's conjecture is incorrect. We also show how to simulate a SIC-POVM on any unknown quantum state by using the time-dependent average channel.