Quantum tomography with continuous measurement and a resource limitation*

We propose and analyze quantum state estimation (tomography) for a single qubit and two interacting qubits, where only a single-qubit observable is continuously measured. By utilizing a continuous weak probe, we can turn on sets of qubit oscillations while measuring the single-qubit observable continuously; and the information of qubit coordinates will gradually be transferred to the measured quantity via the oscillations. In the single qubit case, a combination of the weak-continuous σ_z probe and a Rabi oscillation at an angle in the x-z plane is sufficient to extract all three qubit coordinates. For the two interacting qubits, where only σ_z of the first qubit is measured, the information of two-qubit matrix elements can be transferred to the measured observable, via the qubit-qubit interaction and Rabi controls applied locally on each qubit. We simulate tomographic results numerically and analyze the estimation using the Fisher information in the limit of infinitesimally weak measurement.