Quantum state reconstruction is a fundamental task in quantum information science. The standard approach employs many projective measurements on a series of identically prepared systems in order to collect sufficient statistics of an informationally complete set of observables. An alternative procedure is to reconstruct quantum state by performing weak continuous measurement collectively on an ensemble, while simultaneously applying time varying controls . For known dynamics, the measurement history determines the initial state. In current implementations [2,3] the shot noise of the probe dominates over projection noise so that measurement-induced backaction is negligible. We generalize this to the regime where quantum backaction is significant, even for a small number of particles. Using the framework of quantum filtering theory, we model the reconstruction of the state of a qubit through collective spin measurement via the Faraday interaction and magnetic field controls, and develop a maximum-likelihood estimate. We present numerical results indicating that our estimates have an average fidelity of reconstruction that approaches an optimum bound.
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