Describing a Quantum Channel by State Tomography of a Single Probe State

A general law is presented for (composite) quantum systems which directly describes the time evolution of quantum states (with one or both components) through an arbitrary noisy quantum channel. It is shown that the time evolution of all quantum states through a quantum channel can be completely captured by the evolution of a single 'probe state'. Thus in order to grasp the information of the final output states subject to a quantum channel, especially an unknown one, it only requires quantum state tomography of a single probe state, which dramatically simplifies the practical operations in experiment.