Mocking up a Dephasing Channel with a Minimal-Sized Environment

In order to model the most general quantum operation on a d-dimensional system, a d²-dimensional environment is usually needed. We focus on the quantum dephasing process and find that this channel can be modeled by an environment with the size of at most d dimensions. An experimentally accessible matrix D is defined to characterize the dephasing channel and the minimal number of Kraus operators of the channel from the matrix D for mocking up the dephasing channel is presented in the minimal-sized environment. An experimental simulation of dephasing channels by means of nuclear magnetic resonance techniques is carried out to justify the idea.