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Systematic errors in direct state measurements with quantum controlled measurements

Von Neumann measurement framework describes a dynamic interaction between a target system and a probe. In contrast, a quantum controlled measurement framework uses a qubit probe to control the actions of different operators on the target system, and convenient for establishing universal quantum computation. In this work, we use a quantum controlled measurement framework for measuring quantum states directly. We introduce two types of the quantum controlled measurement framework and investigate the systematic error (the bias between the true value and the estimated values) that caused by these types. We numerically investigate the systematic errors, evaluate the confidence region, and investigate the effect of experimental noise that arises from the imperfect detection. Our analysis has important applications in direct quantum state tomography.