Quantum state tomography from observable time traces in closed quantum systems

The task to estimate all the parameters of an unknown quantum state, also called quantum state tomography, is essential for characterizing and controlling quantum systems. In this paper, we utilize observable time traces to identify the initial quantum state of a closed quantum system, based on the state space approach in the control theory. In the informationally complete scenario, we show that with a linear regression estimation (LRE), the mean squared error (MSE) scales as \( O\left( 1/N\right) \) $O\left(1,/,N\right)$ , where N is the resource number. In the informationally incomplete scenario, we introduce regularization LRE to perform the state tomography task. We employ PBH test to demonstrate that closed quantum systems with only one observable are informationally incomplete and propose using \( d-1 \) $d-1$ observables, where d is the dimension of the quantum state, for informational completeness. Numerical examples demonstrate the effectiveness of our method.