  We propose and analyze quantum state estimation (tomography) using continuous
quantum measurements with resource limitations, allowing the global state of
many qubits to be constructed from only measuring a few. We give a
proof-of-principle investigation demonstrating successful tomographic
reconstruction of an arbitrary initial quantum state for three different
situations: single qubit, remote qubit, and two interacting qubits. The
tomographic reconstruction utilizes only a continuous weak probe of a single
qubit observable, a fixed coupling Hamiltonian, together with single-qubit
controls. In the single qubit case, a combination of the continuous measurement
of an observable and a Rabi oscillation is sufficient to find all three unknown
qubit state components. For two interacting qubits, where only one observable
of the first qubit is measured, the control Hamiltonian can be implemented to
transfer all quantum information to the measured observable, via the
qubit-qubit interaction and Rabi oscillation controls applied locally on each
qubit. We discuss different sets of controls by analyzing the unitary dynamics
and the Fisher information matrix of the estimation in the limit of weak
measurement, and simulate tomographic results numerically. As a result, we
obtained reconstructed state fidelities in excess of 0.98 with a few thousand
measurement runs.
