  In this paper, we study extended linear regression approaches for quantum
state tomography based on regularization techniques. For unknown quantum states
represented by density matrices, performing measurements under certain basis
yields random outcomes, from which a classical linear regression model can be
established. First of all, for complete or over-complete measurement bases, we
show that the empirical data can be utilized for the construction of a weighted
least squares estimate (LSE) for quantum tomography. Taking into consideration
the trace-one condition, a constrained weighted LSE can be explicitly computed,
being the optimal unbiased estimation among all linear estimators. Next, for
general measurement bases, we show that $\ell_2$-regularization with proper
regularization gain provides even lower mean-square error under a cost in bias.
The regularization parameter is tuned by two estimators in terms of a risk
characterization. Finally, a concise and unified formula is established for the
regularization parameter with complete measurement basis under an equivalent
regression model, which proves that the proposed tuning estimators are
asymptotically optimal as the number of samples grows to infinity under the
risk metric. Additionally, numerical examples are provided to validate the
established results.
